**Most Efficient Digital Filter Structures: The Potential of Halfband Filters in Digital Signal Processing**

Heinz G. Göckler *Digital Signal Processing Group, Ruhr-Universität Bochum Germany*

#### **1. Introduction**

236 Applications of Digital Signal Processing

Zhang, Y., Wang, Y. & Wang, W. (2003). Denoising quadrature Doppler signals from bi-

pp561-566.

directional flow using the Wavelet frame, *IEEE Transactions on UFFC,* Vol.50, No.5,

A digital halfband filter (HBF) is, in its basic form with real-valued coefficients, a lowpass filter with one passband and one stopband region of unity or zero desired transfer characteristic, respectively, where both specified bands have the same bandwidth. The zero-phase frequency response of a nonrecursive (FIR) halfband filter with its symmetric impulse response exhibits an odd symmetry about the quarter sample rate (Ω = *<sup>π</sup>* <sup>2</sup> ) and half magnitude ( <sup>1</sup> <sup>2</sup> ) point [Schüssler & Steffen (1998)], where Ω = 2*π f* / *f*<sup>n</sup> represents the normalised (radian) frequency and *f*<sup>n</sup> = 1/*T* the sampling rate. The same symmetry holds true for the squared magnitude frequency response of minimum-phase (MP) recursive (IIR) halfband filters [Lutovac et al. (2001); Schüssler & Steffen (2001)]. As a result of this symmetry property, the implementation of a real HBF requires only a low computational load since, roughly, every other filter coefficient is identical to zero [Bellanger (1989); Mitra & Kaiser (1993); Schüssler & Steffen (2001)].

Due to their high efficiency, digital halfband filters are widely used as versatile building blocks in digital signal processing applications. They are, for instance, encountered in front ends of digital receivers and back ends of digital transmitters (software defined radio, modems, CATV-systems, etc. [Göckler & Groth (2004); Göckler & Grotz (1994); Göckler & Eyssele (1992); Renfors & Kupianen (1998)]), in decimators and interpolators for sample rate alteration by a factor of two [Ansari & Liu (1983); Bellanger (1989); Bellanger et al. (1974); Gazsi (1986); Valenzuela & Constantinides (1983)], in efficient multirate implementations of digital filters [Bellanger et al. (1974); Fliege (1993); Göckler & Groth (2004)] (cf. Fig. 1), where the input/output sampling rate *f*n is decimated by *I* cascaded HBF stages by a factor of 2*<sup>I</sup>* to *<sup>f</sup>*<sup>d</sup> <sup>=</sup> <sup>2</sup>−*<sup>I</sup>* · *<sup>f</sup>*<sup>n</sup> (*z*<sup>d</sup> <sup>=</sup> *<sup>z</sup>*2*<sup>I</sup>* <sup>n</sup> ), in tree-structured filter banks for FDM de- and remultiplexing (e.g. in satellite communications) according to Fig. 2 and [Danesfahani et al. (1994); Göckler & Felbecker (2001); Göckler & Groth (2004); Göckler & Eyssele (1992)], etc. A frequency-shifted (complex) halfband filter (CHBF), generally known as Hilbert-Transformer (HT, cf. Fig. 3), is frequently used to derive an analytical bandpass signal from its real-valued counterpart [Kollar et al. (1990); Kumar et al. (1994); Lutovac et al. (2001); Meerkötter & Ochs (1998); Schüssler & Steffen (1998; 2001); Schüssler & Weith (1987)]. Finally, real IIR HBF or spectral factors of real FIR HBF, respectively, are used in perfectly reconstructing sub-band coder (cf. Fig. 4) and transmultiplexer filter banks [Fliege (1993); Göckler & Groth (2004);

Signal Processing 3

<sup>239</sup> Most Efficient Digital Filter Structures:

In addition, it is shown that the complex HBF defined by (1) require roughly the same amount of computation as their original real HBF prototype (*fc* = *f*<sup>0</sup> = 0). Especially, we present the most efficient elementary SFG for sample rate alteration, their main application. The SFG will be given for LP FIR [Göckler (1996b)] as well as for MP IIR HBF for real- and complex-valued input and/or output signals, respectively. Detailed comparison of expenditure is included. In Section 3 we combine *two* of those linear-phase FIR HBF investigated in Section 2 with different centre frequencies out of the set given by (2), to construct efficient SFG of directional filters (DF) for separation of one input signal into two output signals or for combination of two input signals to one output signal, respectively. These DF are generally referred to as two-channel frequency demultiplexer (FDMUX) or frequency multiplexer (FMUX) filter bank [Göckler & Eyssele (1992); Vaidyanathan & Nguyen (1987);

In Section 4 of this chapter we consider the application of the two-channel DF as a building block of a multiple channel tree-structured FDMUX filter bank according to Fig. 2, typically applied for on-board processing in satellite communications [Danesfahani et al. (1994); Göckler & Felbecker (2001); Göckler & Groth (2004); Göckler & Eyssele (1992)]. In case of a great number of channels and/or challenging bandwidth requirements, implementation of the front-end DF is crucial, which must be operated at (extremely) high sampling rates. To cope with this issue, in Section 4 we present an approach to parallelise at least the front end of

In this Section 2 of this chapter we recall the properties of the well-known HBF with real coefficients (real HBF with centre frequencies *fc* ∈ { *f*0, *f*4} = {0, *f*n/2} according to (1)), and

that require roughly the same amount of computation as their real HBF prototype (*fc* = *f*<sup>0</sup> = 0). In particular, we derive the most efficient elementary SFG for sample rate alteration. These will be given both for LP FIR [Göckler (1996b)] and MP IIR HBF for real- and complex-valued input and/or output signals, respectively. The expenditure of all eight versions of HBF

The organisation of Section 2 is as follows: First, we recall the properties of both classes of the afore-mentioned real HBF, the linear-phase (LP) FIR and the minimum-phase (MP) IIR approaches. The efficient multirate implementations presented are based on the polyphase decomposition of the filter transfer functions [Bellanger (1989); Göckler & Groth (2004); Mitra (1998); Vaidyanathan (1993)]. Next, we present the corresponding results on complex HBF (CHBF), the classical HT, by shifting a real HBF to a centre frequency according to (2) with *c* ∈ {2,6}. Finally, complex offset HBF (COHBF) are derived by applying frequency shifts according to (2) with *c* ∈ {1,3,5,7}, and their properties are investigated. Illustrative design

investigate those of the complex HBF with their passbands (stopbands) centred at

*fc* <sup>=</sup> *<sup>c</sup>* · *<sup>f</sup>*<sup>n</sup>

according to (1) is determined and thoroughly compared with each other.

<sup>8</sup> , *<sup>c</sup>* <sup>=</sup> 0, 1, 2, 3, 4, 5, 6, 7. (1)

<sup>8</sup> , *<sup>c</sup>* <sup>=</sup> 1, 2, 3, 5, 6, 7 (2)

with their passbands (stopbands) centred at one point of an equidistant frequency grid

*fc* <sup>=</sup> *<sup>c</sup>* · *<sup>f</sup>*<sup>n</sup>

The Potential of Halfband Filters in Digital Signal Processing

Valenzuela & Constantinides (1983)].

the FDMUX filter bank according to Fig. 2.

examples and implementations thereof are given.

<sup>1</sup> Underlying original publication: Göckler & Damjanovic (2006b)

**2. Single halfband filters**<sup>1</sup>

Fig. 1. Multirate filtering applying dyadic HBF decimators, a basic filter, and (transposed) HBF interpolators

Fig. 2. FDM demultiplexer filter bank; LP/HP: lowpass/highpass directional filter block based on HBF

$$\underbrace{\otimes}\_{\times(k)}\underbrace{\xleftarrow{\coprod}\_{\text{HT}}\underbrace{\bigsqcup\ldots\bigsqcup\ldots\bigsqcup\ldots\circ}\_{\times(2k)}}\_{\times(2k)}\underbrace{\rightsquigarrow\cdots}\_{\times\_{\mathbb{R}}(2k)}\underbrace{\xleftarrow{\hspace{1cm}}\overbrace{\upharpoonright}^{\times\_{\mathbb{R}}(2k)}\xleftarrow{\hspace{1cm}}\overbrace{\upharpoonright}^{\times(2k)}}\_{\text{j}\times(2k)}\underbrace{\xleftarrow{\hspace{1cm}}\overbrace{\upharpoonright}^{\times(k)}}\_{\mathbf{MT}}$$

Fig. 3. Decimating Hilbert-Transformer (a) and its transpose for interpolation by two (b)


Fig. 4. Two-channel conjugated quadrature mirror filter sub-band coder (SBC) filter bank, where the filters *F*(*z*) are spectral factors of a linear-phase FIR HBF

Mitra & Kaiser (1993); Vaidyanathan (1993)], which may apply the discrete wavelet transform [Damjanovic & Milic (2005); Damjanovic et al. (2005); Fliege (1993); Strang & Nguyen (1996)]. Digital linear-phase (LP) FIR and MP IIR HBF have thoroughly been investigated during the last three decades starting in 1974 [Bellanger et al. (1974)] and 1969 [Gold & Rader (1969)], respectively. An excellent survey of this evolution is presented in [Schüssler & Steffen (1998)]. However, the majority of these investigations deal with the properties and the design of HBF by applying allpass pairs [Regalia et al. (1988); Vaidyananthan et al. (1987)], also comprising IIR HBF with approximately linear-phase response [Schüssler & Steffen (1998; 2001); Schüssler & Weith (1987)]. Hence, only few publications on efficient structures e.g. [Bellanger (1989); Bellanger et al. (1974); Lutovac et al. (2001); Man & Kleine (1988); Milic (2009); Valenzuela & Constantinides (1983)], present elementary signal flow graphs (SFG) with minimum computational load. Moreover, only real-valued HBF and complex Hilbert-Transformers (HT) with a centre frequency of *fc* = *f*n/4 (Ω*<sup>c</sup>* = *<sup>π</sup>* <sup>2</sup> ) have been considered in the past.

The goal of Section 2 of this contribution is to show the existence of a family of real and complex HBF, where the latter are derived from the former ones by frequency translation, 2 Will-be-set-by-IN-TECH

Fig. 1. Multirate filtering applying dyadic HBF decimators, a basic filter, and (transposed)

Fig. 2. FDM demultiplexer filter bank; LP/HP: lowpass/highpass directional filter block

Fig. 3. Decimating Hilbert-Transformer (a) and its transpose for interpolation by two (b)

Fig. 4. Two-channel conjugated quadrature mirror filter sub-band coder (SBC) filter bank,

Hilbert-Transformers (HT) with a centre frequency of *fc* = *f*n/4 (Ω*<sup>c</sup>* = *<sup>π</sup>*

The goal of Section 2 of this contribution is to show the existence of a family of real and complex HBF, where the latter are derived from the former ones by frequency translation,

<sup>2</sup> ) have been

Mitra & Kaiser (1993); Vaidyanathan (1993)], which may apply the discrete wavelet transform [Damjanovic & Milic (2005); Damjanovic et al. (2005); Fliege (1993); Strang & Nguyen (1996)]. Digital linear-phase (LP) FIR and MP IIR HBF have thoroughly been investigated during the last three decades starting in 1974 [Bellanger et al. (1974)] and 1969 [Gold & Rader (1969)], respectively. An excellent survey of this evolution is presented in [Schüssler & Steffen (1998)]. However, the majority of these investigations deal with the properties and the design of HBF by applying allpass pairs [Regalia et al. (1988); Vaidyananthan et al. (1987)], also comprising IIR HBF with approximately linear-phase response [Schüssler & Steffen (1998; 2001); Schüssler & Weith (1987)]. Hence, only few publications on efficient structures e.g. [Bellanger (1989); Bellanger et al. (1974); Lutovac et al. (2001); Man & Kleine (1988); Milic (2009); Valenzuela & Constantinides (1983)], present elementary signal flow graphs (SFG) with minimum computational load. Moreover, only real-valued HBF and complex

where the filters *F*(*z*) are spectral factors of a linear-phase FIR HBF

HBF interpolators

based on HBF

considered in the past.

with their passbands (stopbands) centred at one point of an equidistant frequency grid

$$f\_{\mathcal{C}} = \mathbf{c} \cdot \frac{f\_{\mathcal{R}}}{8}, \qquad \mathbf{c} = \mathbf{0}, 1, 2, 3, 4, 5, 6, 7. \tag{1}$$

In addition, it is shown that the complex HBF defined by (1) require roughly the same amount of computation as their original real HBF prototype (*fc* = *f*<sup>0</sup> = 0). Especially, we present the most efficient elementary SFG for sample rate alteration, their main application. The SFG will be given for LP FIR [Göckler (1996b)] as well as for MP IIR HBF for real- and complex-valued input and/or output signals, respectively. Detailed comparison of expenditure is included.

In Section 3 we combine *two* of those linear-phase FIR HBF investigated in Section 2 with different centre frequencies out of the set given by (2), to construct efficient SFG of directional filters (DF) for separation of one input signal into two output signals or for combination of two input signals to one output signal, respectively. These DF are generally referred to as two-channel frequency demultiplexer (FDMUX) or frequency multiplexer (FMUX) filter bank [Göckler & Eyssele (1992); Vaidyanathan & Nguyen (1987); Valenzuela & Constantinides (1983)].

In Section 4 of this chapter we consider the application of the two-channel DF as a building block of a multiple channel tree-structured FDMUX filter bank according to Fig. 2, typically applied for on-board processing in satellite communications [Danesfahani et al. (1994); Göckler & Felbecker (2001); Göckler & Groth (2004); Göckler & Eyssele (1992)]. In case of a great number of channels and/or challenging bandwidth requirements, implementation of the front-end DF is crucial, which must be operated at (extremely) high sampling rates. To cope with this issue, in Section 4 we present an approach to parallelise at least the front end of the FDMUX filter bank according to Fig. 2.

### **2. Single halfband filters**<sup>1</sup>

In this Section 2 of this chapter we recall the properties of the well-known HBF with real coefficients (real HBF with centre frequencies *fc* ∈ { *f*0, *f*4} = {0, *f*n/2} according to (1)), and investigate those of the complex HBF with their passbands (stopbands) centred at

$$f\_{\mathbf{c}} = \mathbf{c} \cdot \frac{f\_{\mathbf{n}}}{8}, \qquad \mathbf{c} = 1, 2, 3, 5, 6, 7 \tag{2}$$

that require roughly the same amount of computation as their real HBF prototype (*fc* = *f*<sup>0</sup> = 0). In particular, we derive the most efficient elementary SFG for sample rate alteration. These will be given both for LP FIR [Göckler (1996b)] and MP IIR HBF for real- and complex-valued input and/or output signals, respectively. The expenditure of all eight versions of HBF according to (1) is determined and thoroughly compared with each other.

The organisation of Section 2 is as follows: First, we recall the properties of both classes of the afore-mentioned real HBF, the linear-phase (LP) FIR and the minimum-phase (MP) IIR approaches. The efficient multirate implementations presented are based on the polyphase decomposition of the filter transfer functions [Bellanger (1989); Göckler & Groth (2004); Mitra (1998); Vaidyanathan (1993)]. Next, we present the corresponding results on complex HBF (CHBF), the classical HT, by shifting a real HBF to a centre frequency according to (2) with *c* ∈ {2,6}. Finally, complex offset HBF (COHBF) are derived by applying frequency shifts according to (2) with *c* ∈ {1,3,5,7}, and their properties are investigated. Illustrative design examples and implementations thereof are given.

<sup>1</sup> Underlying original publication: Göckler & Damjanovic (2006b)

Signal Processing 5

<sup>241</sup> Most Efficient Digital Filter Structures:

Hence, the non-causal impulse response of a real zero-phase FIR HBF is characterized by [Bellanger et al. (1974); Göckler & Groth (2004); Mintzer (1982); Schüssler & Steffen (1998)]:

giving rise to efficient implementations. Note that the name Nyquist(2)filter is justified by the zero coefficients of the impulse response (9). Moreover, if an HBF is used as an anti-imaging filter of an interpolator for upsampling by two, the coefficients (9) are scaled by the upsampling factor of two replacing the central coefficient with *h*<sup>0</sup> = 1 [Fliege (1993); Göckler & Groth (2004); Mitra (1998)]. As a result, independently of the application this

Assuming an ideal lowpass desired function consistent with the specification of Fig. 5 with a cut-off frequency of Ω<sup>t</sup> = (Ω<sup>p</sup> + Ωs)/2 = *π*/2 and zero transition bandwidth, and minimizing the integral squared error, yields the coefficients [Göckler & Groth (2004);

> sin(*k <sup>π</sup>* 2 ) *k π* 2

This least squares design is optimal for multirate HBF in conjunction with spectrally white input signals since, e.g in case of decimation, the overall residual power aliased by downsampling onto the usable signal spectrum is minimum [Göckler & Groth (2004)]. To master the Gibbs' phenomenon connected with (10), a centrosymmetric smoothed desired function can be introduced in the transition region [Parks & Burrus (1987)]. Requiring, for instance, a transition band of width ∆Ω = Ω<sup>s</sup> − Ω<sup>p</sup> > 0 and using spline transition functions for *D*(*ej*Ω), the above coefficients (10) are modified as follows [Göckler & Groth

*β*

Least squares design can also be subjected to constraints that confine the maximum deviation from the desired function: The Constrained Least Squares (CLS) design [Evangelista (2001); Göckler & Groth (2004)]. This approach has also efficiently been applied to the design of

, <sup>|</sup>*k*<sup>|</sup> <sup>=</sup> 1, 2, . . . , *<sup>n</sup>*

2

, <sup>|</sup>*k*<sup>|</sup> <sup>=</sup> 1, 2, . . . , *<sup>n</sup>*

0 *k* = 2*l l* = 1, 2, . . . ,(*n* − 2)/4 *h*(*k*) *k* = 2*l* − 1 *l* = 1, 2, . . . ,(*n* + 2)/4 (9)

<sup>2</sup> . (10)

, *β* ∈ **R**. (11)

Fig. 5. Specification of a zero-phase FIR HBF; Ω<sup>p</sup> + Ω<sup>s</sup> = *π*

The Potential of Halfband Filters in Digital Signal Processing

  1

coefficient does never contribute to the computational burden of the filter.

sin(*k*Ωt) *k*Ωt

<sup>=</sup> <sup>1</sup> 2

<sup>2</sup> *k* = 0

*hk* = *<sup>h</sup>*−*<sup>k</sup>* =

Parks & Burrus (1987)] in compliance with (9):

(2004); Parks & Burrus (1987)]:

*hk* <sup>=</sup> <sup>1</sup> 2 sin(*k <sup>π</sup>* 2 ) *k π* 2

high-order LP FIR filters with quantized coefficients [Evangelista (2002)].

sin(*k* ∆Ω <sup>2</sup>*<sup>β</sup>* ) *k* ∆Ω 2*β*

*hk* <sup>=</sup> <sup>Ω</sup><sup>t</sup> *π*

**Design outline**

#### **2.1 Real halfband filters (RHBF)**

In this subsection we recall the essentials of LP FIR and MP IIR lowpass HBF with real-valued impulse responses *h*(*k*) = *hk* ←→ *H*(*z*), where *H*(*z*) represents the associated *z*-transform transfer function. From such a lowpass (prototype) HBF a corresponding real highpass HBF is readily derived by using the modulation property of the *z*-transform [Oppenheim & Schafer (1989)]

$$z\_c^k h(k) \longleftrightarrow H(\frac{z}{z\_c}) \tag{3}$$

by setting in accordance with (1)

$$z\_{\mathcal{L}} = z\_4 = e^{j2\pi f\_4/f\_h} = e^{j\pi} = -1\tag{4}$$

resulting in a frequency shift by *f*<sup>4</sup> = *f*n/2 (Ω<sup>4</sup> = *π*).

#### **2.1.1 Linear-Phase (LP) FIR filters**

Throughout this Section 2 we describe a real LP FIR (lowpass) filter by its non-causal impulse response with its centre of symmetry located at the time or sample index *k* = 0 according to

$$
\hbar\_{-k} = \hbar\_k \quad \forall k \tag{5}
$$

where the associated frequency response *<sup>H</sup>*(*ej*Ω) <sup>∈</sup> **<sup>R</sup>** is zero-phase [Mitra & Kaiser (1993); Oppenheim & Schafer (1989)].

#### **Specification and properties**

A real zero-phase (LP) lowpass HBF, also called Nyquist(2)filter [Mitra & Kaiser (1993)], is specified in the frequency domain as shown in Fig. 5, for instance, for an equiripple or constrained least squares design, respectively, allowing for a don't care transition band between passband and stopband [Mintzer (1982); Mitra & Kaiser (1993); Schüssler & Steffen (1998)]. Passband and stopband constraints *δ*<sup>p</sup> = *δ*<sup>s</sup> = *δ* are identical, and for the cut-off frequencies we have the relationship:

$$
\Omega\_{\mathbf{p}} + \Omega\_{\mathbf{s}} = \boldsymbol{\pi}.\tag{6}
$$

As a result, the zero-phase desired function *<sup>D</sup>*(*ej*Ω) <sup>∈</sup> **<sup>R</sup>** as well as the frequency response *<sup>H</sup>*(*ej*Ω) <sup>∈</sup> **<sup>R</sup>** are centrosymmetric about *<sup>D</sup>*(*ejπ*/2) = *<sup>H</sup>*(*ejπ*/2) = <sup>1</sup> <sup>2</sup> . From this frequency domain symmetry property immediately follows

$$H(e^{j\Omega}) + H(e^{j(\Omega - \pi)}) = 1,\tag{7}$$

indicating that this type of halfband filter is strictly complementary [Schüssler & Steffen (1998)].

According to (5), a real zero-phase FIR HBF has a symmetric impulse response of *odd* length *N* = *n* + 1 (denoted as type I filter in [Mitra & Kaiser (1993)]), where *n* represents the even filter order. In case of a minimal (canonic) monorate filter implementation, *n* is identical to the minimum number *n*mc of delay elements required for realisation, where *n*mc is known as the McMillan degree [Vaidyanathan (1993)]. Due to the odd symmetry of the HBF zero-phase frequency response about the transition region (don't care band according to Fig. 5), roughly every other coefficient of the impulse response is zero [Mintzer (1982); Schüssler & Steffen (1998)], resulting in the additional filter length constraint:

$$N = n + 1 = 4i - 1, \qquad i \in \mathbb{N}.\tag{8}$$

Fig. 5. Specification of a zero-phase FIR HBF; Ω<sup>p</sup> + Ω<sup>s</sup> = *π*

Hence, the non-causal impulse response of a real zero-phase FIR HBF is characterized by [Bellanger et al. (1974); Göckler & Groth (2004); Mintzer (1982); Schüssler & Steffen (1998)]:

$$h\_k = h\_{-k} = \begin{cases} \frac{1}{2} & k = 0\\ 0 & k = 2l \quad l = 1, 2, \dots, (n-2)/4\\ h(k) \; k = 2l - 1 \; l = 1, 2, \dots, (n+2)/4 \end{cases} \tag{9}$$

giving rise to efficient implementations. Note that the name Nyquist(2)filter is justified by the zero coefficients of the impulse response (9). Moreover, if an HBF is used as an anti-imaging filter of an interpolator for upsampling by two, the coefficients (9) are scaled by the upsampling factor of two replacing the central coefficient with *h*<sup>0</sup> = 1 [Fliege (1993); Göckler & Groth (2004); Mitra (1998)]. As a result, independently of the application this coefficient does never contribute to the computational burden of the filter.

#### **Design outline**

4 Will-be-set-by-IN-TECH

In this subsection we recall the essentials of LP FIR and MP IIR lowpass HBF with real-valued impulse responses *h*(*k*) = *hk* ←→ *H*(*z*), where *H*(*z*) represents the associated *z*-transform transfer function. From such a lowpass (prototype) HBF a corresponding real highpass HBF is readily derived by using the modulation property of the *z*-transform [Oppenheim & Schafer

*ch*(*k*) ←→ *<sup>H</sup>*( *<sup>z</sup>*

Throughout this Section 2 we describe a real LP FIR (lowpass) filter by its non-causal impulse response with its centre of symmetry located at the time or sample index *k* = 0 according to

where the associated frequency response *<sup>H</sup>*(*ej*Ω) <sup>∈</sup> **<sup>R</sup>** is zero-phase [Mitra & Kaiser (1993);

A real zero-phase (LP) lowpass HBF, also called Nyquist(2)filter [Mitra & Kaiser (1993)], is specified in the frequency domain as shown in Fig. 5, for instance, for an equiripple or constrained least squares design, respectively, allowing for a don't care transition band between passband and stopband [Mintzer (1982); Mitra & Kaiser (1993); Schüssler & Steffen (1998)]. Passband and stopband constraints *δ*<sup>p</sup> = *δ*<sup>s</sup> = *δ* are identical, and for the cut-off

As a result, the zero-phase desired function *<sup>D</sup>*(*ej*Ω) <sup>∈</sup> **<sup>R</sup>** as well as the frequency response

indicating that this type of halfband filter is strictly complementary [Schüssler & Steffen

According to (5), a real zero-phase FIR HBF has a symmetric impulse response of *odd* length *N* = *n* + 1 (denoted as type I filter in [Mitra & Kaiser (1993)]), where *n* represents the even filter order. In case of a minimal (canonic) monorate filter implementation, *n* is identical to the minimum number *n*mc of delay elements required for realisation, where *n*mc is known as the McMillan degree [Vaidyanathan (1993)]. Due to the odd symmetry of the HBF zero-phase frequency response about the transition region (don't care band according to Fig. 5), roughly every other coefficient of the impulse response is zero [Mintzer (1982); Schüssler & Steffen

*H*(*ej*Ω) + *H*(*ej*(Ω−*π*)

*<sup>H</sup>*(*ej*Ω) <sup>∈</sup> **<sup>R</sup>** are centrosymmetric about *<sup>D</sup>*(*ejπ*/2) = *<sup>H</sup>*(*ejπ*/2) = <sup>1</sup>

*zc*

*zc* <sup>=</sup> *<sup>z</sup>*<sup>4</sup> <sup>=</sup> *<sup>e</sup>j*2*<sup>π</sup> <sup>f</sup>*4/ *<sup>f</sup>*<sup>n</sup> <sup>=</sup> *<sup>e</sup>j<sup>π</sup>* <sup>=</sup> <sup>−</sup><sup>1</sup> (4)

*<sup>h</sup>*−*<sup>k</sup>* = *hk* ∀*<sup>k</sup>* (5)

Ω<sup>p</sup> + Ω<sup>s</sup> = *π*. (6)

*N* = *n* + 1 = 4*i* − 1, *i* ∈ **N**. (8)

<sup>2</sup> . From this frequency

) = 1, (7)

) (3)

*zk*

**2.1 Real halfband filters (RHBF)**

by setting in accordance with (1)

**2.1.1 Linear-Phase (LP) FIR filters**

Oppenheim & Schafer (1989)].

**Specification and properties**

frequencies we have the relationship:

domain symmetry property immediately follows

(1998)], resulting in the additional filter length constraint:

resulting in a frequency shift by *f*<sup>4</sup> = *f*n/2 (Ω<sup>4</sup> = *π*).

(1989)]

(1998)].

Assuming an ideal lowpass desired function consistent with the specification of Fig. 5 with a cut-off frequency of Ω<sup>t</sup> = (Ω<sup>p</sup> + Ωs)/2 = *π*/2 and zero transition bandwidth, and minimizing the integral squared error, yields the coefficients [Göckler & Groth (2004); Parks & Burrus (1987)] in compliance with (9):

$$h\_k = \frac{\Omega\_\text{lt}}{\pi} \frac{\sin(k\Omega\_\text{lt})}{k\Omega\_\text{lt}} = \frac{1}{2} \frac{\sin(k\frac{\pi}{2})}{k\frac{\pi}{2}}, \quad |k| = 1, 2, \dots, \frac{n}{2}. \tag{10}$$

This least squares design is optimal for multirate HBF in conjunction with spectrally white input signals since, e.g in case of decimation, the overall residual power aliased by downsampling onto the usable signal spectrum is minimum [Göckler & Groth (2004)]. To master the Gibbs' phenomenon connected with (10), a centrosymmetric smoothed desired function can be introduced in the transition region [Parks & Burrus (1987)]. Requiring, for instance, a transition band of width ∆Ω = Ω<sup>s</sup> − Ω<sup>p</sup> > 0 and using spline transition functions for *D*(*ej*Ω), the above coefficients (10) are modified as follows [Göckler & Groth (2004); Parks & Burrus (1987)]:

$$h\_k = \frac{1}{2} \frac{\sin(k\frac{\pi}{2})}{k\frac{\pi}{2}} \left[\frac{\sin(k\frac{\Lambda\Omega}{2\beta})}{k\frac{\Lambda\Omega}{2\beta}}\right]^\beta, |k| = 1, 2, \dots, \frac{n}{2}, \beta \in \mathbb{R}.\tag{11}$$

Least squares design can also be subjected to constraints that confine the maximum deviation from the desired function: The Constrained Least Squares (CLS) design [Evangelista (2001); Göckler & Groth (2004)]. This approach has also efficiently been applied to the design of high-order LP FIR filters with quantized coefficients [Evangelista (2002)].

Signal Processing 7

<sup>243</sup> Most Efficient Digital Filter Structures:

Fig. 6. Polyphase representation of a decimator (a,b) and an interpolator (c) for sample rate

<sup>d</sup> :<sup>=</sup> *<sup>z</sup>*−<sup>1</sup>

Fig. 7. Optimum SFG of LP FIR HBF decimator (a) and interpolator (b) of order *n* = 10

*N*<sup>A</sup> *n*/2 + 1 *n*/2 *N*Op 3*n*/4 + 3/2 3*n*/4 + 1/2 Table 1. Expenditure of real linear-phase FIR HBF; *n*: order, *n*mc: McMillan degree, *N*M(*N*A):

*concurrently* exploited. (Note that this concurrent exploitation of coefficient symmetry *and* minimum memory property is not possible for Nyquist(*M*)filters with *M* > 2. As shown in [Göckler & Groth (2004)], for Nyquist(*M*)filters with *M* > 2 only *either* coefficient symmetry

The application of the multirate transposition rules on the optimum decimator according to Fig. 7(a), as detailed in Section 3 and [Göckler & Groth (2004)], yields the optimum LP FIR HBF interpolator, as depicted in Fig. 6(c) and Fig. 7(b), respectively. Table 1 shows that the interpolator obtained by transposition requires less memory than that published in [Bellanger

*n*mc *n n*/2 + 1 *N*<sup>M</sup> (*n* + 2)/4

number of multipliers (adders), *f*Op: operational clock frequency

*or* the minimum memory property can be exploited.)

(1989); Bellanger et al. (1974)].

MoR: *f*Op = *f*<sup>n</sup> Dec: *f*Op = *f*n/2 Int: *f*Op = *f*n/2

alteration by two; shimming delay: *z*−1/2

The Potential of Halfband Filters in Digital Signal Processing

Subsequently, all comparisons are based on equiripple designs obtained by minimization of the maximum deviation max *<sup>H</sup>*(*ej*Ω) <sup>−</sup> *<sup>D</sup>*(*ej*Ω) ∀Ω on the region of support according to [McClellan et al. (1973)]. To this end, we briefly recall the clever use of this minimax design procedure in order to obtain the exact values of the predefined (centre and zero) coefficients of (9), as proposed in [Vaidyanathan & Nguyen (1987)]: To design a two-band HBF of even order *n* = *N* − 1 = 4*m* − 2, as specified in Fig. 5, start with designing *i*) a single-band zero-phase FIR filter *g*(*k*) ←→ *G*(*z*) of odd order *n*/2 = 2*m* − 1 for a passband cut-off frequency of 2Ωp which, as a type II filter [Mitra & Kaiser (1993)], has a centrosymmetric zero-phase frequency response about *G*(*ejπ*) = 0, *ii*) upsample the impulse response *g*(*k*) by two by inserting between any pair of coefficients an additional zero coefficient (without actually changing the sample rate), which yields an interim filter impulse response *h* (*k*) ←→ *H* (*z*2) of the desired odd length *N* with a centrosymmetric frequency response about *H* (*ejπ*/2) = 0 [Göckler & Groth (2004); Vaidyanathan (1993)], *iii*) lift the passband (stopband) of *H* (*ej*Ω) to 2 (0) by replacing the zero centre coefficient with 2*h*(0) = 1, and *iv*) scale the coefficients of the final impulse response *<sup>h</sup>*(*k*) ←→ *<sup>H</sup>*(*z*) with <sup>1</sup> 2 .

#### **Efficient implementations**

Monorate FIR filters are commonly realized by using one of the direct forms [Mitra (1998)]. In our case of an LP HBF, minimum expenditure is obtained by exploiting coefficient symmetry, as it is well known [Mitra & Kaiser (1993); Oppenheim & Schafer (1989)]. The count of operations or hardware required, respectively, is included below in Table 1 (column MoR). Note that the "multiplication" by the central coefficient *h*<sup>0</sup> does not contribute to the overall expenditure.

The minimal implementation of an LP HBF decimator (interpolator) for twofold down(up)sampling is based on the decomposition of the HBF transfer function into two (type 1) polyphase components [Bellanger (1989); Göckler & Groth (2004); Vaidyanathan (1993)]:

$$H(z) = E\_0(z^2) + z^{-1}E\_1(z^2). \tag{12}$$

In the case of decimation, downsampling of the output signal (cf. upper branch of Fig. 1) is shifted from filter output to system input by exploiting the noble identities [Göckler & Groth (2004); Vaidyanathan (1993)], as shown in Fig. 6(a). As a result, all operations (including delay and its control) can be performed at the reduced (decimated) output sample rate *f*<sup>d</sup> = *f*n/2: *Ei*(*z*2) := *Ei*(*z*d), *i* = 0, 1. In Fig. 6(b), the input demultiplexer of Fig. 6(a) is replaced with a commutator where, for consistency, the shimming delay *z*−1/2 <sup>d</sup> :<sup>=</sup> *<sup>z</sup>*−<sup>1</sup> must be introduced [Göckler & Groth (2004)].

As an example, in Fig. 7(a) an optimum, causal real LP FIR HBF decimator of *n* = 10th order and for twofold downsampling is recalled [Bellanger et al. (1974)]. Here, the odd-numbered coefficients of (9) are assigned to the zeroth polyphase component *E*0(*z*d) of Fig. 6(b), whereas the only non-zero even-numbered coefficient *h*<sup>0</sup> belongs to *E*1(*z*d).

For implementation we assume a digital signal processor as a hardware platform. Hence, the overall computational load of its arithmetic unit is given by the total number of operations *N*Op = *N*<sup>M</sup> + *N*A, comprising multiplication (M) and addition (A), times the operational clock frequency *f*Op [Göckler & Groth (2004)]. All contributions to the expenditure are listed in Table 1 as a function of the filter order *n*, where the McMillan degree includes the shimming delays. Obviously, both coefficient symmetry (*N*<sup>M</sup> < *n*/2) and the minimum memory property (*n*mc < *n* [Bellanger (1989); Fliege (1993); Göckler & Groth (2004)]) are 6 Will-be-set-by-IN-TECH

Subsequently, all comparisons are based on equiripple designs obtained by minimization of

[McClellan et al. (1973)]. To this end, we briefly recall the clever use of this minimax design procedure in order to obtain the exact values of the predefined (centre and zero) coefficients of (9), as proposed in [Vaidyanathan & Nguyen (1987)]: To design a two-band HBF of even order *n* = *N* − 1 = 4*m* − 2, as specified in Fig. 5, start with designing *i*) a single-band zero-phase FIR filter *g*(*k*) ←→ *G*(*z*) of odd order *n*/2 = 2*m* − 1 for a passband cut-off frequency of 2Ωp which, as a type II filter [Mitra & Kaiser (1993)], has a centrosymmetric zero-phase frequency response about *G*(*ejπ*) = 0, *ii*) upsample the impulse response *g*(*k*) by two by inserting between any pair of coefficients an additional zero coefficient (without actually

 

∀Ω on the region of support according to

(*k*) ←→ *H*

(*z*2)

(*ejπ*/2) = 0

(*ej*Ω) to

*<sup>H</sup>*(*ej*Ω) <sup>−</sup> *<sup>D</sup>*(*ej*Ω)

changing the sample rate), which yields an interim filter impulse response *h*

of the desired odd length *N* with a centrosymmetric frequency response about *H*

[Göckler & Groth (2004); Vaidyanathan (1993)], *iii*) lift the passband (stopband) of *H*

2 (0) by replacing the zero centre coefficient with 2*h*(0) = 1, and *iv*) scale the coefficients of

Monorate FIR filters are commonly realized by using one of the direct forms [Mitra (1998)]. In our case of an LP HBF, minimum expenditure is obtained by exploiting coefficient symmetry, as it is well known [Mitra & Kaiser (1993); Oppenheim & Schafer (1989)]. The count of operations or hardware required, respectively, is included below in Table 1 (column MoR). Note that the "multiplication" by the central coefficient *h*<sup>0</sup> does not contribute to the overall

The minimal implementation of an LP HBF decimator (interpolator) for twofold down(up)sampling is based on the decomposition of the HBF transfer function into two (type 1) polyphase components [Bellanger (1989); Göckler & Groth (2004); Vaidyanathan (1993)]:

In the case of decimation, downsampling of the output signal (cf. upper branch of Fig. 1) is shifted from filter output to system input by exploiting the noble identities [Göckler & Groth (2004); Vaidyanathan (1993)], as shown in Fig. 6(a). As a result, all operations (including delay and its control) can be performed at the reduced (decimated) output sample rate *f*<sup>d</sup> = *f*n/2: *Ei*(*z*2) := *Ei*(*z*d), *i* = 0, 1. In Fig. 6(b), the input demultiplexer of Fig. 6(a) is replaced with

As an example, in Fig. 7(a) an optimum, causal real LP FIR HBF decimator of *n* = 10th order and for twofold downsampling is recalled [Bellanger et al. (1974)]. Here, the odd-numbered coefficients of (9) are assigned to the zeroth polyphase component *E*0(*z*d) of Fig. 6(b), whereas

For implementation we assume a digital signal processor as a hardware platform. Hence, the overall computational load of its arithmetic unit is given by the total number of operations *N*Op = *N*<sup>M</sup> + *N*A, comprising multiplication (M) and addition (A), times the operational clock frequency *f*Op [Göckler & Groth (2004)]. All contributions to the expenditure are listed in Table 1 as a function of the filter order *n*, where the McMillan degree includes the shimming delays. Obviously, both coefficient symmetry (*N*<sup>M</sup> < *n*/2) and the minimum memory property (*n*mc < *n* [Bellanger (1989); Fliege (1993); Göckler & Groth (2004)]) are

a commutator where, for consistency, the shimming delay *z*−1/2

the only non-zero even-numbered coefficient *h*<sup>0</sup> belongs to *E*1(*z*d).

*H*(*z*) = *E*0(*z*2) + *z*−1*E*1(*z*2). (12)

<sup>d</sup> :<sup>=</sup> *<sup>z</sup>*−<sup>1</sup> must be introduced

2 .

the maximum deviation max

**Efficient implementations**

[Göckler & Groth (2004)].

expenditure.

 

the final impulse response *<sup>h</sup>*(*k*) ←→ *<sup>H</sup>*(*z*) with <sup>1</sup>

Fig. 6. Polyphase representation of a decimator (a,b) and an interpolator (c) for sample rate alteration by two; shimming delay: *z*−1/2 <sup>d</sup> :<sup>=</sup> *<sup>z</sup>*−<sup>1</sup>

Fig. 7. Optimum SFG of LP FIR HBF decimator (a) and interpolator (b) of order *n* = 10


Table 1. Expenditure of real linear-phase FIR HBF; *n*: order, *n*mc: McMillan degree, *N*M(*N*A): number of multipliers (adders), *f*Op: operational clock frequency

*concurrently* exploited. (Note that this concurrent exploitation of coefficient symmetry *and* minimum memory property is not possible for Nyquist(*M*)filters with *M* > 2. As shown in [Göckler & Groth (2004)], for Nyquist(*M*)filters with *M* > 2 only *either* coefficient symmetry *or* the minimum memory property can be exploited.)

The application of the multirate transposition rules on the optimum decimator according to Fig. 7(a), as detailed in Section 3 and [Göckler & Groth (2004)], yields the optimum LP FIR HBF interpolator, as depicted in Fig. 6(c) and Fig. 7(b), respectively. Table 1 shows that the interpolator obtained by transposition requires less memory than that published in [Bellanger (1989); Bellanger et al. (1974)].

Signal Processing 9

<sup>245</sup> Most Efficient Digital Filter Structures:

In order to compare MP IIR and LP FIR HBF, we subsequently consider elliptic filter designs. Since an elliptic (minimax) HBF transfer function satisfies the conditions (6) and (13), the design result is uniquely determined by specifying the passband Ωp (stopband Ωs) cut-off frequency and one of the three remaining parameters: the odd filter order *n*, allowed minimal stopband attenuation *A*<sup>s</sup> = −20log(*δ*<sup>s</sup> ) or allowed maximum passband attenuation *A*<sup>p</sup> =

There are two most common approaches to elliptic HBF design. The first group of methods is performed in the analogue frequency domain and is based on classical analogue

transfer function *H*(*z*) to be designed is mapped onto an analogue frequency domain by applying the bilinear transformation [Mitra (1998); Oppenheim & Schafer (1989)]. The magnitude response of the analogue elliptic filter is approximated by appropriate iterative procedures to satisfy the design requirements [Ansari (1985); Schüssler & Steffen (1998; 2001); Valenzuela & Constantinides (1983)]. Finally, the analogue filter transfer function is remapped

The other group of algorithms starts from an elliptic HBF transfer function, as given by (17).

techniques minimizing the peak stopband deviation. For a given transition bandwidth, the maximum deviation is minimized e.g. by the Remez exchange algorithm or by Gauss-Newton

For the particular class of elliptic HBF with *minimal Q-factor*, closed-form equations for calculating the exact values of stopband and passband attenuation are known allowing for straightforward designs, if the cut-off frequencies and the filter order are given [Lutovac et al.

In case of a monorate filter implementation, the McMillan degree *n*mc is equal to the filter order *n*. Having the same hardware prerequisites as in the previous subsection on FIR HBF, the computational load of hardware operations per output sample is given in Table 2 (column MoR). Note that multiplication by a factor of 0.5 does not contribute to the overall expenditure. In the general decimating structure, as shown in Fig. 9(a), decimation is performed by an input commutator in conjunction with a shimming delay according to Fig. 6(b). By the underlying exploitation of the noble identities [Göckler & Groth (2004); Vaidyanathan (1993)], the cascaded second order allpass sections of the transfer function (17) are transformed to

methods [Valenzuela & Constantinides (1983); Zhang & Yoshikawa (1999)].

<sup>1</sup>+*aiz*−<sup>2</sup> :<sup>=</sup> *ai*+*z*−<sup>1</sup>

d 1+*aiz*−<sup>1</sup> d

, *i* = 0, 1, ..., *<sup>n</sup>*−<sup>1</sup>

 *D*(*ej*Ω) 

<sup>2</sup> − 1) are obtained by iterative nonlinear optimization

of the elliptic HBF

<sup>2</sup> − 1, as illustrated in Fig. 9(b).

Fig. 8. Magnitude specification of minimum-phase IIR lowpass HBF;

filter design techniques: The desired magnitude response

<sup>s</sup> = 1, Ω<sup>p</sup> + Ω<sup>s</sup> = *π*

The Potential of Halfband Filters in Digital Signal Processing

to the *z*-domain by the bilinear transformation.

The filter coefficients *ai*, *i* = 0, 1, ...,( *<sup>n</sup>*−<sup>1</sup>

(<sup>1</sup> <sup>−</sup> *<sup>δ</sup>*p)<sup>2</sup> <sup>+</sup> *<sup>δ</sup>*<sup>2</sup>

**Design outline**

−20log(1 − *δ*p).

(2001)].

**Efficient implementation**

first order allpass sections: *ai*+*z*−<sup>2</sup>

#### **2.1.2 Minimum-Phase (MP) IIR filters**

In contrast to FIR HBF, we describe an MP IIR HBF always by its transfer function *H*(*z*) in the *z*-domain.

#### **Specification and properties**

The magnitude response of an MP IIR lowpass HBF is specified in the frequency domain by *D*(*ej*Ω) , as shown in Fig. 8, again for a minimax or equiripple design. The constraints of the designed magnitude response *H*(*ej*Ω) are characterized by the passband and stopband deviations, *δ*p and *δ*s, according to [Lutovac et al. (2001); Schüssler & Steffen (1998)] related by

$$(1 - \delta\_{\mathbf{p}})^2 + \delta\_{\mathbf{s}}^2 = 1. \tag{13}$$

The cut-off frequencies of the IIR HBF satisfy the symmetry condition (6), and the squared magnitude response *H*(*ej*Ω) 2 is centrosymmetric about *D*(*ejπ*/2) 2 = *H*(*ejπ*/2) 2 = <sup>1</sup> 2 . We consider real MP IIR lowpass HBF of odd order *n*. The family of the MP IIR HBF comprises Butterworth, Chebyshev, elliptic (Cauer-lowpass) and intermediate designs [Vaidyananthan et al. (1987); Zhang & Yoshikawa (1999)]. The MP IIR HBF is doubly-complementary [Mitra & Kaiser (1993); Regalia et al. (1988); Vaidyananthan et al. (1987)], and satisfies the power-complementarity

$$\left| H(e^{j\Omega}) \right|^2 + \left| H(e^{j(\Omega - \pi)}) \right|^2 = 1 \tag{14}$$

and the allpass-complementarity conditions

$$\left| H(e^{j\Omega}) + H(e^{j(\Omega - \pi)}) \right| = 1. \tag{15}$$

*H*(*z*) has a single pole at the origin of the z-plane, and (*n* − 1)/2 complex-conjugated pole pairs on the imaginary axis within the unit circle, and all zeros on the unit circle [Schüssler & Steffen (2001)]. Hence, the odd order MP IIR HBF is suitably realized by a parallel connection of two allpass polyphase sections as expressed by

$$H(z) = \frac{1}{2} \left[ A\_0(z^2) + z^{-1} A\_1(z^2) \right],\tag{16}$$

where the allpass polyphase components can be derived by alternating assignment of adjacent complex-conjugated pole pairs of the IIR HBF to the polyphase components. The polyphase components *Al*(*z*2), *l* = 0, 1 consist of cascade connections of second order allpass sections:

$$H(z) = \frac{1}{2} \left( \underbrace{\prod\_{i=0,2,\dots}^{\frac{\mu-1}{2}-1} \frac{a\_i + z^{-2}}{1 + a\_i z^{-2}}}\_{A\_0(z^2)} + z^{-1} \underbrace{\prod\_{i=1,3,\dots}^{\frac{\mu-1}{2}-1} \frac{a\_i + z^{-2}}{1 + a\_i z^{-2}}}\_{A\_1(z^2)} \right),\tag{17}$$

where the coefficients *ai*, *i* = 0, 1, ...,( *<sup>n</sup>*−<sup>1</sup> <sup>2</sup> − 1), with *ai* < *ai*+1, denote the squared moduli of the HBF complex-conjugated pole pairs in ascending order; the complete set of *n* poles is given by 0, ±*j* <sup>√</sup>*a*0, <sup>±</sup>*<sup>j</sup>* <sup>√</sup>*a*1, ..., <sup>±</sup>*<sup>j</sup> a* <sup>n</sup>−<sup>1</sup> <sup>2</sup> −1 [Mitra (1998)].

Fig. 8. Magnitude specification of minimum-phase IIR lowpass HBF; (<sup>1</sup> <sup>−</sup> *<sup>δ</sup>*p)<sup>2</sup> <sup>+</sup> *<sup>δ</sup>*<sup>2</sup> <sup>s</sup> = 1, Ω<sup>p</sup> + Ω<sup>s</sup> = *π*

#### **Design outline**

8 Will-be-set-by-IN-TECH

In contrast to FIR HBF, we describe an MP IIR HBF always by its transfer function *H*(*z*) in the

The magnitude response of an MP IIR lowpass HBF is specified in the frequency domain by

deviations, *δ*p and *δ*s, according to [Lutovac et al. (2001); Schüssler & Steffen (1998)] related by

The cut-off frequencies of the IIR HBF satisfy the symmetry condition (6), and the squared

We consider real MP IIR lowpass HBF of odd order *n*. The family of the MP IIR HBF comprises Butterworth, Chebyshev, elliptic (Cauer-lowpass) and intermediate designs [Vaidyananthan et al. (1987); Zhang & Yoshikawa (1999)]. The MP IIR HBF is doubly-complementary [Mitra & Kaiser (1993); Regalia et al. (1988); Vaidyananthan et al.

*H*(*ej*(Ω−*π*)

) 2

) 

*A*0(*z*2) + *z*−1*A*1(*z*2)

+*z*−<sup>1</sup>

[Mitra (1998)].

*n*−1 <sup>2</sup> −1 ∏ *i*=1,3,... *ai* + *z*−<sup>2</sup> 1 + *aiz*−<sup>2</sup>

<sup>2</sup> − 1), with *ai* < *ai*+1, denote the squared moduli

 *A*1(*z*<sup>2</sup>)

(<sup>1</sup> <sup>−</sup> *<sup>δ</sup>*p)<sup>2</sup> <sup>+</sup> *<sup>δ</sup>*<sup>2</sup>

is centrosymmetric about

*H*(*ej*Ω) + *H*(*ej*(Ω−*π*)

*H*(*z*) has a single pole at the origin of the z-plane, and (*n* − 1)/2 complex-conjugated pole pairs on the imaginary axis within the unit circle, and all zeros on the unit circle [Schüssler & Steffen (2001)]. Hence, the odd order MP IIR HBF is suitably realized by a

where the allpass polyphase components can be derived by alternating assignment of adjacent complex-conjugated pole pairs of the IIR HBF to the polyphase components. The polyphase components *Al*(*z*2), *l* = 0, 1 consist of cascade connections of second order allpass sections:

of the HBF complex-conjugated pole pairs in ascending order; the complete set of *n* poles is

*ai* + *z*−<sup>2</sup> 1 + *aiz*−<sup>2</sup>

 *A*0(*z*<sup>2</sup>)

> *a* <sup>n</sup>−<sup>1</sup> <sup>2</sup> −1

 *H*(*ej*Ω) 

, as shown in Fig. 8, again for a minimax or equiripple design. The constraints of

 

*D*(*ejπ*/2)

are characterized by the passband and stopband

 2 = 

<sup>s</sup> = 1. (13)

*H*(*ejπ*/2)

= 1 (14)

<sup>=</sup> 1. (15)

, (16)

, (17)

 2 = <sup>1</sup> 2 .

**2.1.2 Minimum-Phase (MP) IIR filters**

**Specification and properties**

magnitude response

the designed magnitude response

 *H*(*ej*Ω) 2

(1987)], and satisfies the power-complementarity

*<sup>H</sup>*(*z*) = <sup>1</sup> 2

where the coefficients *ai*, *i* = 0, 1, ...,( *<sup>n</sup>*−<sup>1</sup>

<sup>√</sup>*a*0, <sup>±</sup>*<sup>j</sup>*

given by

 0, ±*j*

and the allpass-complementarity conditions

 *H*(*ej*Ω) 2 + 

> 

parallel connection of two allpass polyphase sections as expressed by

*<sup>H</sup>*(*z*) = <sup>1</sup> 2 

*n*−1 <sup>2</sup> −1 ∏ *i*=0,2,...

<sup>√</sup>*a*1, ..., <sup>±</sup>*<sup>j</sup>*

*z*-domain.

 *D*(*ej*Ω) 

> In order to compare MP IIR and LP FIR HBF, we subsequently consider elliptic filter designs. Since an elliptic (minimax) HBF transfer function satisfies the conditions (6) and (13), the design result is uniquely determined by specifying the passband Ωp (stopband Ωs) cut-off frequency and one of the three remaining parameters: the odd filter order *n*, allowed minimal stopband attenuation *A*<sup>s</sup> = −20log(*δ*<sup>s</sup> ) or allowed maximum passband attenuation *A*<sup>p</sup> = −20log(1 − *δ*p).

> There are two most common approaches to elliptic HBF design. The first group of methods is performed in the analogue frequency domain and is based on classical analogue filter design techniques: The desired magnitude response *D*(*ej*Ω) of the elliptic HBF transfer function *H*(*z*) to be designed is mapped onto an analogue frequency domain by applying the bilinear transformation [Mitra (1998); Oppenheim & Schafer (1989)]. The magnitude response of the analogue elliptic filter is approximated by appropriate iterative procedures to satisfy the design requirements [Ansari (1985); Schüssler & Steffen (1998; 2001); Valenzuela & Constantinides (1983)]. Finally, the analogue filter transfer function is remapped to the *z*-domain by the bilinear transformation.

> The other group of algorithms starts from an elliptic HBF transfer function, as given by (17). The filter coefficients *ai*, *i* = 0, 1, ...,( *<sup>n</sup>*−<sup>1</sup> <sup>2</sup> − 1) are obtained by iterative nonlinear optimization techniques minimizing the peak stopband deviation. For a given transition bandwidth, the maximum deviation is minimized e.g. by the Remez exchange algorithm or by Gauss-Newton methods [Valenzuela & Constantinides (1983); Zhang & Yoshikawa (1999)].

> For the particular class of elliptic HBF with *minimal Q-factor*, closed-form equations for calculating the exact values of stopband and passband attenuation are known allowing for straightforward designs, if the cut-off frequencies and the filter order are given [Lutovac et al. (2001)].

#### **Efficient implementation**

In case of a monorate filter implementation, the McMillan degree *n*mc is equal to the filter order *n*. Having the same hardware prerequisites as in the previous subsection on FIR HBF, the computational load of hardware operations per output sample is given in Table 2 (column MoR). Note that multiplication by a factor of 0.5 does not contribute to the overall expenditure. In the general decimating structure, as shown in Fig. 9(a), decimation is performed by an input commutator in conjunction with a shimming delay according to Fig. 6(b). By the underlying exploitation of the noble identities [Göckler & Groth (2004); Vaidyanathan (1993)], the cascaded second order allpass sections of the transfer function (17) are transformed to first order allpass sections: *ai*+*z*−<sup>2</sup> <sup>1</sup>+*aiz*−<sup>2</sup> :<sup>=</sup> *ai*+*z*−<sup>1</sup> d 1+*aiz*−<sup>1</sup> d , *i* = 0, 1, ..., *<sup>n</sup>*−<sup>1</sup> <sup>2</sup> − 1, as illustrated in Fig. 9(b).

Signal Processing 11

<sup>247</sup> Most Efficient Digital Filter Structures:

(NOp,n) overview

(21,26) (24,30) (27,34) (30,38) (18,22)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Fig. 10. Expenditure curves of real linear-phase FIR and minimum-phase IIR HBF decimators

A complex HBF, a classical Hilbert-Transformer [Lutovac et al. (2001); Mitra & Kaiser (1993); Schüssler & Steffen (1998; 2001); Schüssler & Weith (1987)], is readily derived from a real HBF according to Subsection 2.1 by applying the *z*-transform modulation theorem (3) by setting in

thus shifting the real prototype HBF to a passband centre frequency of *f*±<sup>2</sup> = ± *f*n/4 (Ω±<sup>2</sup> =

In the FIR CHBF case the frequency shift operation (3) is immediately applied to the impulse response *h*(*k*) in the time domain according to (3). As a result of the modulation of the impulse response (9) of any real LP HBF on a carrier of frequency *f*<sup>2</sup> according to (18), the

<sup>2</sup> <sup>−</sup> *<sup>n</sup>*

is obtained. (Underlining indicates complex quantities in time domain.) By directly equating

<sup>2</sup> <sup>≤</sup> *<sup>k</sup>* <sup>≤</sup> *<sup>n</sup>*

*zc* <sup>=</sup> *<sup>z</sup>*±<sup>2</sup> <sup>=</sup> *<sup>z</sup>*∓<sup>6</sup> <sup>=</sup> *<sup>e</sup>j*2*<sup>π</sup> <sup>f</sup>*±2/ *<sup>f</sup>*<sup>n</sup> <sup>=</sup> *<sup>e</sup>*±*<sup>j</sup> <sup>π</sup>*

±*π*/2). For convenience, subsequently we restrict ourselves to the case *fc* = *f*2.

*hk* <sup>=</sup> *<sup>h</sup>*(*k*)*ejk <sup>π</sup>*

FIR IIR

(12,14)

(15,18)

NOp: Number of operations

based on equiripple minimax designs [Parks & Burrus (1987)]

2f p /fn

n: The filter order (5,3)

(9,5)

<sup>2</sup> = ±*j*, (18)

<sup>2</sup> (19)

(13,7)

(17,9)

(21,11)

0

compliance with (2)

**2.2 Complex Halfband Filters (CHBF)**

**2.2.1 Linear-Phase (LP) FIR filters**

complex-valued CHBF impulse response

(19) and relating the result to (9), we get:

10

20

30

40

50

As [dB]

60

70

80

(6,6)

(9,10)

The Potential of Halfband Filters in Digital Signal Processing

90

100

Fig. 9. Optimum minimum-phase IIR HBF decimator block structure (a) and SFG of the 1st (2nd) order allpass sections (b)


Table 2. Expenditure of real minimum-phase IIR HBF; *n*: order, *n*mc: McMillan degree, *NM* (*NA*): number of multipliers (adders), *f*Op: operational clock frequency

Hence, the polyphase components *Al*(*z*2) := *Al*(*z*d), *l* = 0, 1 of Fig. 9(a) operate at the reduced output sampling rate *f*<sup>d</sup> = *f*n/2, and the McMillan degree *n*mc is almost halved. The optimum interpolating structure is readily derived from the decimator by applying the multirate transposition rules (cf. Section 3 and [Göckler & Groth (2004)]). Computational complexity is presented in Table 2, also indicating the respective operational rates *f*Op for the *N*Op arithmetical operations.

Elliptic filters also allow for multiplierless implementations with small quantization error, or implementations with a reduced number of shift-and-add operations in multipliers [Lutovac & Milic (1997; 2000); Milic (2009)].

#### **2.1.3 Comparison of real FIR and IIR HBF**

The comparison of the Tables 1 and 2 shows that *NFIR* Op <sup>&</sup>lt; *<sup>N</sup>IIR* Op for the same filter order *n*, where all operations are performed at the operational rate *f*Op, as given in these Tables. Since, however, the filter order *n*IIR < *n*FIR or even *n*IIR *n*FIR for any type of approximation, the computational load of an MP IIR HBF is generally smaller than that of an LP FIR HBF, as it is well known [Lutovac et al. (2001); Schüssler & Steffen (1998)].

The relative computational advantage of equiripple minimax designs of monorate IIR halfband filters and polyphase decimators [Parks & Burrus (1987)], respectively, is depicted in Fig. 10 where, in extension to [Lutovac et al. (2001)], the expenditure *N*Op is indicated as a parameter along with the filter order *n*. Note that the IIR and FIR curves of the lowest order filters differ by just one operation despite the LP property of the FIR HBF.

A specification of a design example is deduced from Fig. 10: *n*IIR = 5 and *n*FIR = 14, respectively, with a passband cut-off frequency of *f*<sup>p</sup> = 0.1769 *f*<sup>n</sup> at the intersection point of the associated expenditure curves: Fig. 11. As a result, the stopband attenuations of both filters are the same (cf. Fig. 10). In addition, for both designs the typical pole-zero plots are shown [Schüssler & Steffen (1998; 2001)]. From the point of view of expenditure, the MP IIR HBF decimator (*N*Op = 9, *n*mc = 3) outperforms its LP FIR counterpart (*N*Op = 12, *n*mc = 8).

10 Will-be-set-by-IN-TECH

Fig. 9. Optimum minimum-phase IIR HBF decimator block structure (a) and SFG of the 1st

*N*<sup>A</sup> 3(*n* − 1)/2 + 1 3(*n* − 1)/2 *N*Op 2*n* − 1 2*n* − 2 Table 2. Expenditure of real minimum-phase IIR HBF; *n*: order, *n*mc: McMillan degree,

Hence, the polyphase components *Al*(*z*2) := *Al*(*z*d), *l* = 0, 1 of Fig. 9(a) operate at the reduced output sampling rate *f*<sup>d</sup> = *f*n/2, and the McMillan degree *n*mc is almost halved. The optimum interpolating structure is readily derived from the decimator by applying the multirate transposition rules (cf. Section 3 and [Göckler & Groth (2004)]). Computational complexity is presented in Table 2, also indicating the respective operational rates *f*Op for the

Elliptic filters also allow for multiplierless implementations with small quantization error, or implementations with a reduced number of shift-and-add operations in multipliers

where all operations are performed at the operational rate *f*Op, as given in these Tables. Since, however, the filter order *n*IIR < *n*FIR or even *n*IIR *n*FIR for any type of approximation, the computational load of an MP IIR HBF is generally smaller than that of an LP FIR HBF, as it is

The relative computational advantage of equiripple minimax designs of monorate IIR halfband filters and polyphase decimators [Parks & Burrus (1987)], respectively, is depicted in Fig. 10 where, in extension to [Lutovac et al. (2001)], the expenditure *N*Op is indicated as a parameter along with the filter order *n*. Note that the IIR and FIR curves of the lowest order

A specification of a design example is deduced from Fig. 10: *n*IIR = 5 and *n*FIR = 14, respectively, with a passband cut-off frequency of *f*<sup>p</sup> = 0.1769 *f*<sup>n</sup> at the intersection point of the associated expenditure curves: Fig. 11. As a result, the stopband attenuations of both filters are the same (cf. Fig. 10). In addition, for both designs the typical pole-zero plots are shown [Schüssler & Steffen (1998; 2001)]. From the point of view of expenditure, the MP IIR HBF decimator (*N*Op = 9, *n*mc = 3) outperforms its LP FIR counterpart (*N*Op = 12, *n*mc = 8).

Op <sup>&</sup>lt; *<sup>N</sup>IIR*

Op for the same filter order *n*,

*n*mc *n* (*n* + 1)/2

*N*<sup>M</sup> (*n* − 1)/2

*NM* (*NA*): number of multipliers (adders), *f*Op: operational clock frequency

MoR: *f*Op = *f*<sup>n</sup> Dec: *f*Op = *f*n/2 Int: *f*Op = *f*n/2

(2nd) order allpass sections (b)

*N*Op arithmetical operations.

[Lutovac & Milic (1997; 2000); Milic (2009)].

**2.1.3 Comparison of real FIR and IIR HBF**

The comparison of the Tables 1 and 2 shows that *NFIR*

well known [Lutovac et al. (2001); Schüssler & Steffen (1998)].

filters differ by just one operation despite the LP property of the FIR HBF.

Fig. 10. Expenditure curves of real linear-phase FIR and minimum-phase IIR HBF decimators based on equiripple minimax designs [Parks & Burrus (1987)]

#### **2.2 Complex Halfband Filters (CHBF)**

A complex HBF, a classical Hilbert-Transformer [Lutovac et al. (2001); Mitra & Kaiser (1993); Schüssler & Steffen (1998; 2001); Schüssler & Weith (1987)], is readily derived from a real HBF according to Subsection 2.1 by applying the *z*-transform modulation theorem (3) by setting in compliance with (2)

$$z\_{\mathfrak{c}} = z\_{\pm 2} = z\_{\mp 6} = e^{j2\pi f\_{\pm 2}/f\_n} = e^{\pm j\frac{\pi}{2}} = \pm j,\tag{18}$$

thus shifting the real prototype HBF to a passband centre frequency of *f*±<sup>2</sup> = ± *f*n/4 (Ω±<sup>2</sup> = ±*π*/2). For convenience, subsequently we restrict ourselves to the case *fc* = *f*2.

#### **2.2.1 Linear-Phase (LP) FIR filters**

In the FIR CHBF case the frequency shift operation (3) is immediately applied to the impulse response *h*(*k*) in the time domain according to (3). As a result of the modulation of the impulse response (9) of any real LP HBF on a carrier of frequency *f*<sup>2</sup> according to (18), the complex-valued CHBF impulse response

$$
\underline{h}\_k = h(k)e^{jk\frac{n}{2}} \qquad -\frac{n}{2} \le k \le \frac{n}{2} \tag{19}
$$

is obtained. (Underlining indicates complex quantities in time domain.) By directly equating (19) and relating the result to (9), we get:

Signal Processing 13

<sup>249</sup> Most Efficient Digital Filter Structures:

Fig. 12. Optimum SFG of decimating LP FIR HT (a) and its interpolating multirate transpose

where Ωp<sup>+</sup> represents the upper passband cut-off frequency and Ωs<sup>−</sup> the associated stopband

The optimum implementation of an *n* = 10th order LP FIR CHBF for twofold downsampling is again based on the polyphase decomposition of (20) according to (12). Its SFG is depicted in Fig. 12(a) that exploits the odd symmetry of the HT part of the system. Note that all imaginary units are included deliberately. Hence, the optimal FIR CHBF interpolator according to Fig. 12(b), which is derived from the original decimator of Fig. 12(a) by applying the multirate transposition rules [Göckler & Groth (2004)], performs the dual operation with respect to the underlying decimator. Since, however, an LP FIR CHBF is strictly rather than power complementary (cf. (23)), the inverse functionality of the decimator is only approximated

In addition, Fig. 13 shows the optimum SFG of an LP FIR CHBF for decimation of a complex signal by a factor of two. In essence, it represents a doubling of the SFG of Fig. 12(a). Again, the dual interpolator is readily derived by transposition of multirate systems, as outlined in

The expenditure of the half- (**R C**) and the full-complex (**C** → **C**) CHBF decimators and their transposes is listed in Table 3. A comparison of Tables 1 and 3 shows that the overall

are almost the same as those of the real FIR HBF systems depicted in Fig. 7. Only the number

Op of the half-complex CHBF sample rate converters (cf. Fig. 12)

) + *H*(*ej*(Ω<sup>±</sup> *<sup>π</sup>*

2 )

) = 1, (23)

*n*mc 3*n*/4 + 1/2 *n* + 2 *N*<sup>M</sup> (*n* + 2)/4 (*n* + 2)/2 *N*<sup>A</sup> *n*/2 *n* + 2 *n N*Op 3*n*/4 + 1/2 3*n*/2 + 3 3*n*/2 + 1 Table 3. Expenditure of linear-phase FIR CHBF; *n*: order, *n*mc: McMillan degree, *N*M(*N*A):

number of multipliers (adders), operational clock frequency: *f*Op = *f*n/2

*H*(*ej*(Ω<sup>∓</sup> *<sup>π</sup>*

where (3) is applied in the frequency domain.

The Potential of Halfband Filters in Digital Signal Processing

**Efficient implementations**

[Göckler & Groth (2004)].

numbers of operations *N*CFIR

of delays is, for obvious reasons, higher in the case of CHBF.

Section 3.

cut-off frequency. Obviously, strict complementarity (7) is retained as follows

2 )

Dec: **R** → **C** Int: **C** → **R** Dec: **C** → **C** Int: **C** → **C**

(b)

Fig. 11. RHBF design examples: Magnitude characteristics and pole-zero plots

$$
\underline{\mu}\_{k} = \begin{cases}
\frac{1}{2} & k = 0 \\
0 & k = 2l \\
f^{k}h(k) \; k = 2l - 1 \; l = 1, 2, \ldots, (n-2)/4
\end{cases}
\tag{20}
$$

where, in contrast to (5), the imaginary part of the impulse response

$$
\underline{\mathfrak{h}}\_{-k} = -\underline{\mathfrak{h}}\_{k} \quad \forall k > 0 \tag{21}
$$

is skew-symmetric about zero, as it is expected from a Hilbert-Transformer. Note that the centre coefficient *h*<sup>0</sup> is still real, whilst all other coefficients are purely imaginary rather than generally complex-valued.

#### **Specification and properties**

All properties of the real HBF are basically retained except of those which are subjected to the frequency shift operation of (18). This applies to the filter specification depicted in Fig. 5 and, hence, (6) modifies to

$$
\Omega\_{\rm p} + \frac{\pi}{2} + \Omega\_{\rm s} + \frac{\pi}{2} = \Omega\_{\rm p+} + \Omega\_{\rm s-} = 2\pi,\tag{22}
$$

248 Applications of Digital Signal Processing Most Efficient Digital Filter Structures: The Potential of Halfband Filters in Digital Signal Processing 13 <sup>249</sup> Most Efficient Digital Filter Structures: The Potential of Halfband Filters in Digital Signal Processing

12 Will-be-set-by-IN-TECH

= 0.1769, As

= 43.9 dB

−1 −0.5 0 0.5 1

*<sup>h</sup>*−*<sup>k</sup>* = −*hk* ∀*<sup>k</sup>* > <sup>0</sup> (21)

<sup>2</sup> <sup>=</sup> <sup>Ω</sup>p<sup>+</sup> <sup>+</sup> <sup>Ω</sup>s<sup>−</sup> <sup>=</sup> <sup>2</sup>*π*, (22)

IIR, nIIR = 5

Real Part

(20)

f p /fn

<sup>0</sup> 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 −60

f/fn

Imaginary Part

−1

0 *k* = 2*l l* = 1, 2, . . . ,(*n* − 2)/4

*kh*(*k*) *<sup>k</sup>* <sup>=</sup> <sup>2</sup>*<sup>l</sup>* <sup>−</sup> <sup>1</sup> *<sup>l</sup>* <sup>=</sup> 1, 2, . . . ,(*<sup>n</sup>* <sup>+</sup> <sup>2</sup>)/4

is skew-symmetric about zero, as it is expected from a Hilbert-Transformer. Note that the centre coefficient *h*<sup>0</sup> is still real, whilst all other coefficients are purely imaginary rather than

All properties of the real HBF are basically retained except of those which are subjected to the frequency shift operation of (18). This applies to the filter specification depicted in Fig. 5 and,

*π*

−0.5

0

0.5

1

−50 −40 −30 −20 −10 0

> −1 −0.5 0 0.5 1 1.5

generally complex-valued.

**Specification and properties**

hence, (6) modifies to

Imaginary Part

FIR, nFIR = 14 IIR, nIIR = 5

−1 <sup>0</sup> <sup>1</sup> <sup>2</sup> −1.5

*hk* =

Real Part

  1

where, in contrast to (5), the imaginary part of the impulse response

Ω<sup>p</sup> + *π*

<sup>2</sup> <sup>+</sup> <sup>Ω</sup><sup>s</sup> <sup>+</sup>

*j*

Fig. 11. RHBF design examples: Magnitude characteristics and pole-zero plots

<sup>2</sup> *k* = 0

14

FIR, nFIR = 14

Magnitude [dB]

Fig. 12. Optimum SFG of decimating LP FIR HT (a) and its interpolating multirate transpose (b)


Table 3. Expenditure of linear-phase FIR CHBF; *n*: order, *n*mc: McMillan degree, *N*M(*N*A): number of multipliers (adders), operational clock frequency: *f*Op = *f*n/2

where Ωp<sup>+</sup> represents the upper passband cut-off frequency and Ωs<sup>−</sup> the associated stopband cut-off frequency. Obviously, strict complementarity (7) is retained as follows

$$
\underline{H}(e^{j(\Omega \mp \frac{\pi}{2})}) + \underline{H}(e^{j(\Omega \pm \frac{\pi}{2})}) = 1,\tag{23}
$$

where (3) is applied in the frequency domain.

#### **Efficient implementations**

The optimum implementation of an *n* = 10th order LP FIR CHBF for twofold downsampling is again based on the polyphase decomposition of (20) according to (12). Its SFG is depicted in Fig. 12(a) that exploits the odd symmetry of the HT part of the system. Note that all imaginary units are included deliberately. Hence, the optimal FIR CHBF interpolator according to Fig. 12(b), which is derived from the original decimator of Fig. 12(a) by applying the multirate transposition rules [Göckler & Groth (2004)], performs the dual operation with respect to the underlying decimator. Since, however, an LP FIR CHBF is strictly rather than power complementary (cf. (23)), the inverse functionality of the decimator is only approximated [Göckler & Groth (2004)].

In addition, Fig. 13 shows the optimum SFG of an LP FIR CHBF for decimation of a complex signal by a factor of two. In essence, it represents a doubling of the SFG of Fig. 12(a). Again, the dual interpolator is readily derived by transposition of multirate systems, as outlined in Section 3.

The expenditure of the half- (**R C**) and the full-complex (**C** → **C**) CHBF decimators and their transposes is listed in Table 3. A comparison of Tables 1 and 3 shows that the overall numbers of operations *N*CFIR Op of the half-complex CHBF sample rate converters (cf. Fig. 12) are almost the same as those of the real FIR HBF systems depicted in Fig. 7. Only the number of delays is, for obvious reasons, higher in the case of CHBF.

Signal Processing 15

<sup>251</sup> Most Efficient Digital Filter Structures:

Fig. 14. Decimating allpass-based minimum-phase IIR HT: (a) optimum block structure (b)

Dec: **R** → **C** Int: **C** → **R** Dec: **C** → **C** Int: **C** → **C**

*N*<sup>A</sup> 3(*n* − 1)/2 3(*n* − 1) + 2 3(*n* − 1) *N*Op 2*n* − 2 4*n* − 2 4*n* − 4

doubling this structure, as depicted in Fig. 15, the IIR CHBF for decimating a complex signal by two is obtained. Multirate transposition [Göckler & Groth (2004)] can again be applied to

The expenditure of the half- (**R C**) and the full-complex (**C** → **C**) CHBF decimators and their transposes is listed in Table 4. A comparison of Tables 2 and 4 shows that, basically, the half-complex IIR CHBF sample rate converters (cf. Fig. 14) require almost the same

As it is obvious from the similarity of the corresponding expenditure tables of the previous subsections, the expenditure chart Fig. 10 can likewise be used for the comparison of CHBF

*n*mc (*n* + 1)/2 *n* + 1 *N*<sup>M</sup> (*n* − 1)/2 *n* − 1

Table 4. Expenditure of minimum-phase IIR CHBF; *n*: order, *n*mc: McMillan degree, *N*M(*N*A): number of multipliers (adders), operational clock frequency: *f*Op = *f*n/2

SFG of the 1st (2nd) order allpass sections

The Potential of Halfband Filters in Digital Signal Processing

Fig. 15. Block structure of decimating minimum-phase IIR CHBF

derive the corresponding dual structures for interpolation.

expenditure as the real IIR HBF systems depicted in Fig. 9.

**2.2.3 Comparison of FIR and IIR CHBF**

Fig. 13. Optimum SFG of decimating linear-phase FIR CHBF

#### **2.2.2 Minimum-Phase (MP) IIR filters**

In the IIR CHBF case the frequency shift operation (3) is again applied in the *z*-domain. Using (18), this is achieved by substituting the complex *z*-domain variable in the respective transfer functions *H*(*z*) and all corresponding SFG according to:

$$z := \frac{z}{z\_2} = z e^{-j\frac{n}{2}} = -jz. \tag{24}$$

#### **Specification and properties**

All properties of the real IIR HBF are basically retained except of those subjected to the frequency shift operation of (18). This applies to the filter specification depicted in Fig. 8 and, hence, (6) is replaced with (22). Obviously, power (14) and allpass (15) complementarity are retained as follows

$$|\underline{H}(e^{j(\Omega \mp \frac{\pi}{2})})|^2 + |\underline{H}(e^{j(\Omega \pm \frac{\pi}{2})})|^2 = 1,\tag{25}$$

$$\left| \underline{H}(e^{j(\Omega \mp \frac{\pi}{2})}) + \underline{H}(e^{j(\Omega \pm \frac{\pi}{2})}) \right| = 1,\tag{26}$$

where (3) is applied in the frequency domain.

#### **Efficient implementations**

Introducing (24) into (16) performs a frequency-shift of the transfer function *H*(*z*) by *f*<sup>2</sup> = *f*n/4 (Ω<sup>2</sup> = *π*/2):

$$\underline{H}(z) = \frac{1}{2} \left[ A\_0(-z^2) + jz^{-1}A\_1(-z^2) \right]. \tag{27}$$

The optimum general block structure of a decimating MP IIR HT, being up-scaled by 2, is shown in Fig. 14(a) along with the SFG of the 1st (system theoretic 2nd) order allpass sections (b), where the noble identities [Göckler & Groth (2004); Vaidyanathan (1993)] are exploited. By 14 Will-be-set-by-IN-TECH

In the IIR CHBF case the frequency shift operation (3) is again applied in the *z*-domain. Using (18), this is achieved by substituting the complex *z*-domain variable in the respective transfer

= *ze*−*<sup>j</sup> <sup>π</sup>*

All properties of the real IIR HBF are basically retained except of those subjected to the frequency shift operation of (18). This applies to the filter specification depicted in Fig. 8 and, hence, (6) is replaced with (22). Obviously, power (14) and allpass (15) complementarity

<sup>2</sup> <sup>+</sup> <sup>|</sup>*H*(*ej*(Ω<sup>±</sup> *<sup>π</sup>*

) + *H*(*ej*(Ω<sup>±</sup> *<sup>π</sup>*

Introducing (24) into (16) performs a frequency-shift of the transfer function *H*(*z*) by *f*<sup>2</sup> =

The optimum general block structure of a decimating MP IIR HT, being up-scaled by 2, is shown in Fig. 14(a) along with the SFG of the 1st (system theoretic 2nd) order allpass sections (b), where the noble identities [Göckler & Groth (2004); Vaidyanathan (1993)] are exploited. By

*<sup>A</sup>*0(−*z*2) + *jz*−1*A*1(−*z*2)

2 ) )|

2 ) ) 

<sup>2</sup> = −*jz*. (24)

<sup>2</sup> = 1, (25)

<sup>=</sup> 1, (26)

. (27)

*<sup>z</sup>* :<sup>=</sup> *<sup>z</sup> z*2

> 2 ) )|

> > 2 )

<sup>|</sup>*H*(*ej*(Ω<sup>∓</sup> *<sup>π</sup>*

*H*(*ej*(Ω<sup>∓</sup> *<sup>π</sup>*

 

*<sup>H</sup>*(*z*) = <sup>1</sup> 2 

where (3) is applied in the frequency domain.

Fig. 13. Optimum SFG of decimating linear-phase FIR CHBF

functions *H*(*z*) and all corresponding SFG according to:

**2.2.2 Minimum-Phase (MP) IIR filters**

**Specification and properties**

are retained as follows

**Efficient implementations**

*f*n/4 (Ω<sup>2</sup> = *π*/2):

Fig. 14. Decimating allpass-based minimum-phase IIR HT: (a) optimum block structure (b) SFG of the 1st (2nd) order allpass sections

Fig. 15. Block structure of decimating minimum-phase IIR CHBF


Table 4. Expenditure of minimum-phase IIR CHBF; *n*: order, *n*mc: McMillan degree, *N*M(*N*A): number of multipliers (adders), operational clock frequency: *f*Op = *f*n/2

doubling this structure, as depicted in Fig. 15, the IIR CHBF for decimating a complex signal by two is obtained. Multirate transposition [Göckler & Groth (2004)] can again be applied to derive the corresponding dual structures for interpolation.

The expenditure of the half- (**R C**) and the full-complex (**C** → **C**) CHBF decimators and their transposes is listed in Table 4. A comparison of Tables 2 and 4 shows that, basically, the half-complex IIR CHBF sample rate converters (cf. Fig. 14) require almost the same expenditure as the real IIR HBF systems depicted in Fig. 9.

#### **2.2.3 Comparison of FIR and IIR CHBF**

As it is obvious from the similarity of the corresponding expenditure tables of the previous subsections, the expenditure chart Fig. 10 can likewise be used for the comparison of CHBF

Signal Processing 17

The Potential of Halfband Filters in Digital Signal Processing

<sup>253</sup> Most Efficient Digital Filter Structures:

Fig. 16. Optimum SFG of decimating LP FIR COHBF (a) and its transpose for interpolation

where Ωp<sup>+</sup> represents the upper passband cut-off frequency and Ωs<sup>−</sup> the associated stopband

The optimum implementation of an *n* = 10th order LP FIR COHBF for twofold downsampling is again based on the polyphase decomposition of (40). Its SFG is depicted

The optimum FIR COHBF interpolator according to Fig. 16(b) is readily derived from the original decimator of Fig. 16(a) by applying the multirate transposition rules, as discussed in Section 3. As a result, the overall expenditure is again retained (c.f. invariant property of

In addition, Fig. 17 shows the optimum SFG of an LP FIR COHBF for decimation of a complex signal by a factor of two. It represents essentially a doubling of the SFG of Fig. 16(a). The dual

The expenditure of the half- (**R C**) and the full-complex (**C** → **C**) LP COHBF decimators and their transposes is listed in Table 5 in terms of the filter order *n*. A comparison of Tables 3 and 5 shows that the implementation of any type of COHBF requires just two or four extra

) + *H*(*ej*(Ω−*π*(1+*c*/4))) = 1. (33)

cut-off frequency. Obviously, strict complementarity (7) reads as follows

4 )

*H*(*ej*(Ω−*<sup>c</sup> <sup>π</sup>*

in Fig. 16(a) that exploits the coefficient symmetry as given by (41).

interpolator can be derived by transposition [Göckler & Groth (2004)].

(b)

**Efficient implementations**

transposition [Göckler & Groth (2004)]).

decimators. Both for FIR and IIR CHBF, the number of operations has to be substituted: *NCHBF* Op :<sup>=</sup> *<sup>N</sup>HBF* Op − 1.

#### **2.3 Complex Offset Halfband Filters (COHBF)**

A complex offset HBF, a Hilbert-Transformer with a frequency offset of ∆*f* = ± *f*n/8 relative to an RHBF, is readily derived from a real HBF according to Subsection 2.1 by applying the zT modulation theorem (3) with *c* ∈ {1, 3, 5, 7}, as introduced in (2):

$$z\_c = e^{j2\pi f\_c/f\_n} = e^{jc\frac{\pi}{4}} = \cos(c\frac{\pi}{4}) + j\sin(c\frac{\pi}{4}) = \pm\frac{1\pm j}{\sqrt{2}}.\tag{28}$$

As a result, the real prototype HBF is shifted to a passband centre frequency of *fc* ∈ ± *f*n <sup>8</sup> , <sup>±</sup><sup>3</sup> *<sup>f</sup>*<sup>n</sup> 8 . In the sequel, we predominantly consider the case *fc* = *f*<sup>1</sup> (Ω<sup>1</sup> = *π*/4).

#### **2.3.1 Linear-Phase (LP) FIR filters**

Again, the frequency shift operation (3) is applied in the time domain. However, in order to get the smallest number of full-complex COHBF coefficients, we introduce an additional complex scaling factor of unity magnitude. As a result, the modulation of a carrier of frequency *fc* according to (28) by the impulse response (9) of any real LP FIR HBF yields the complex-valued COHBF impulse response:

$$
\underline{h}\_k = e^{j\varepsilon \frac{\pi}{4}} h(k) z\_c^k = h(k) e^{j(k+1)c\frac{\pi}{4}} = h(k) j^{c(k+1)/2},\tag{29}
$$

where <sup>−</sup> *<sup>n</sup>* <sup>2</sup> <sup>≤</sup> *<sup>k</sup>* <sup>≤</sup> *<sup>n</sup>* <sup>2</sup> and *c* = 1, 3, 5, 7. By directly equating (39) for *c* = 1, and relating the result to (9), we get:

$$\underline{h}\_{k} = \begin{cases} \frac{1}{2} \frac{1+j}{\sqrt{2}} & k=0\\ 0 & k=2l \quad l=1,2,\ldots,(n-2)/4\\ j^{(k+1)/2} h(k) \; k=2l-1 \; l=1,2,\ldots,(n+2)/4 \end{cases} \tag{30}$$

where, in contrast to (21), the impulse response exhibits the symmetry property:

$$
\underline{\mathfrak{h}}\_{-k} = -\mathfrak{j}^{\rm ck} \underline{\mathfrak{h}}\_{k} \quad \forall k > 0. \tag{31}
$$

Note that the centre coefficient *h*<sup>0</sup> is the only truly complex-valued coefficient where, fortunately, its real and imaginary parts are identical. All other coefficients are again either purely imaginary or real-valued. Hence, the symmetry of the impulse response can still be exploited, and the implementation of an LP FIR COHBF requires just one multiplication more than that of a real or complex HBF [Göckler (1996b)].

#### **Specification and properties**

All properties of the real HBF are basically retained except of those which are subjected to the frequency shift operation according to (28). This applies to the filter specification depicted in Fig. 5 and, hence, (6) modifies to

$$
\Omega\_{\mathbb{P}} + c\frac{\pi}{4} + \Omega\_{\mathbb{S}} + c\frac{\pi}{4} = \Omega\_{\mathbb{P}^+} + \Omega\_{\mathbb{S}^-} = \pi + c\frac{\pi}{2}.\tag{32}
$$

16 Will-be-set-by-IN-TECH

decimators. Both for FIR and IIR CHBF, the number of operations has to be substituted:

A complex offset HBF, a Hilbert-Transformer with a frequency offset of ∆*f* = ± *f*n/8 relative to an RHBF, is readily derived from a real HBF according to Subsection 2.1 by applying the zT

As a result, the real prototype HBF is shifted to a passband centre frequency of *fc* ∈

Again, the frequency shift operation (3) is applied in the time domain. However, in order to get the smallest number of full-complex COHBF coefficients, we introduce an additional complex scaling factor of unity magnitude. As a result, the modulation of a carrier of frequency *fc* according to (28) by the impulse response (9) of any real LP FIR HBF yields

*<sup>c</sup>* <sup>=</sup> *<sup>h</sup>*(*k*)*ej*(*k*+1)*<sup>c</sup> <sup>π</sup>*

*k* = 0

Note that the centre coefficient *h*<sup>0</sup> is the only truly complex-valued coefficient where, fortunately, its real and imaginary parts are identical. All other coefficients are again either purely imaginary or real-valued. Hence, the symmetry of the impulse response can still be exploited, and the implementation of an LP FIR COHBF requires just one multiplication more

All properties of the real HBF are basically retained except of those which are subjected to the frequency shift operation according to (28). This applies to the filter specification depicted in

<sup>4</sup> <sup>=</sup> <sup>Ω</sup>p<sup>+</sup> <sup>+</sup> <sup>Ω</sup>s<sup>−</sup> <sup>=</sup> *<sup>π</sup>* <sup>+</sup> *<sup>c</sup>*

*π*

where, in contrast to (21), the impulse response exhibits the symmetry property:

*<sup>h</sup>*−*<sup>k</sup>* <sup>=</sup> <sup>−</sup>*<sup>j</sup>*

*π*

. In the sequel, we predominantly consider the case *fc* = *f*<sup>1</sup> (Ω<sup>1</sup> = *π*/4).

<sup>4</sup> ) + *<sup>j</sup>* sin(*<sup>c</sup>*

<sup>4</sup> = *h*(*k*)*j*

<sup>2</sup> and *c* = 1, 3, 5, 7. By directly equating (39) for *c* = 1, and relating the result

0 *k* = 2*l l* = 1, 2, . . . ,(*n* − 2)/4

(*k*+1)/2*h*(*k*) *<sup>k</sup>* <sup>=</sup> <sup>2</sup>*<sup>l</sup>* <sup>−</sup> <sup>1</sup> *<sup>l</sup>* <sup>=</sup> 1, 2, . . . ,(*<sup>n</sup>* <sup>+</sup> <sup>2</sup>)/4

*π*

<sup>4</sup> ) = <sup>±</sup><sup>1</sup> <sup>±</sup> *<sup>j</sup>*

<sup>√</sup><sup>2</sup> . (28)

*<sup>c</sup>*(*k*+1)/2, (29)

*ckhk* <sup>∀</sup>*<sup>k</sup>* <sup>&</sup>gt; 0. (31)

*π*

<sup>2</sup> . (32)

(30)

<sup>4</sup> = cos(*c*

*NCHBF*

 ± *f*n <sup>8</sup> , <sup>±</sup><sup>3</sup> *<sup>f</sup>*<sup>n</sup> 8 

where <sup>−</sup> *<sup>n</sup>*

to (9), we get:

Op :<sup>=</sup> *<sup>N</sup>HBF*

Op − 1.

**2.3.1 Linear-Phase (LP) FIR filters**

<sup>2</sup> <sup>≤</sup> *<sup>k</sup>* <sup>≤</sup> *<sup>n</sup>*

**Specification and properties**

Fig. 5 and, hence, (6) modifies to

the complex-valued COHBF impulse response:

*hk* =

*hk* <sup>=</sup> *<sup>e</sup>jc <sup>π</sup>*

 

than that of a real or complex HBF [Göckler (1996b)].

Ω<sup>p</sup> + *c*

*π*

<sup>4</sup> <sup>+</sup> <sup>Ω</sup><sup>s</sup> <sup>+</sup> *<sup>c</sup>*

*j*

<sup>4</sup> *h*(*k*)*z<sup>k</sup>*

1 2 1+*j* √2

**2.3 Complex Offset Halfband Filters (COHBF)**

modulation theorem (3) with *c* ∈ {1, 3, 5, 7}, as introduced in (2):

*zc* = *<sup>e</sup>j*2*<sup>π</sup> fc*/ *<sup>f</sup>*<sup>n</sup> = *<sup>e</sup>jc <sup>π</sup>*

Fig. 16. Optimum SFG of decimating LP FIR COHBF (a) and its transpose for interpolation (b)

where Ωp<sup>+</sup> represents the upper passband cut-off frequency and Ωs<sup>−</sup> the associated stopband cut-off frequency. Obviously, strict complementarity (7) reads as follows

$$
\underline{H}(e^{j(\Omega - c\frac{\pi}{4})}) + \underline{H}(e^{j(\Omega - \pi(1+c/4))}) = 1. \tag{33}
$$

#### **Efficient implementations**

The optimum implementation of an *n* = 10th order LP FIR COHBF for twofold downsampling is again based on the polyphase decomposition of (40). Its SFG is depicted in Fig. 16(a) that exploits the coefficient symmetry as given by (41).

The optimum FIR COHBF interpolator according to Fig. 16(b) is readily derived from the original decimator of Fig. 16(a) by applying the multirate transposition rules, as discussed in Section 3. As a result, the overall expenditure is again retained (c.f. invariant property of transposition [Göckler & Groth (2004)]).

In addition, Fig. 17 shows the optimum SFG of an LP FIR COHBF for decimation of a complex signal by a factor of two. It represents essentially a doubling of the SFG of Fig. 16(a). The dual interpolator can be derived by transposition [Göckler & Groth (2004)].

The expenditure of the half- (**R C**) and the full-complex (**C** → **C**) LP COHBF decimators and their transposes is listed in Table 5 in terms of the filter order *n*. A comparison of Tables 3 and 5 shows that the implementation of any type of COHBF requires just two or four extra

Signal Processing 19

<sup>255</sup> Most Efficient Digital Filter Structures:

*N*<sup>A</sup> 3(*n* − 1) 6(*n* − 1) + 2 6(*n* − 1) *N*Op 4*n* − 3 8*n* − 4 8*n* − 6

All properties of the real IIR HBF are basically retained except of those subjected to the frequency shift operation of (28). This applies to the filter specification depicted in Fig. 8 and, hence, (6) is replaced with (32). Obviously, power (14) and allpass (15) complementarity

<sup>2</sup> <sup>+</sup> <sup>|</sup>*H*(*ej*(Ω−*π*(1+*c*/4)))<sup>|</sup>

) + *H*(*ej*(Ω−*π*(1+*c*/4)))

√2

Introducing (34) in (16), the transfer function is frequency-shifted by *f*<sup>1</sup> = *fn*/8 (Ω = *π*/4):

*<sup>A</sup>*0(−*jz*2) + <sup>1</sup> <sup>+</sup> *<sup>j</sup>*

The optimal structure of an *n* = 5th order MP IIR COHBF decimator for real input signals is shown in Fig. 18(a) along with the elementary SFG of the allpass sections Fig. 18(b). Doubling of the structure according to Fig. 19 allows for full-complex signal processing. Multirate transposition [Göckler & Groth (2004)] is again applied to derive the corresponding

The expenditure of the half- (**R C**) and the full-complex (**C** → **C**) COHBF decimators and their transposes is listed in Table 6. A comparison of Tables 2 and 6 shows that the half-complex IIR COHBF sample rate converter (cf. Fig. 18(a)) requires almost twice, whereas the full-complex IIR COHBF (cf. Fig. 19) requires even four times the expenditure of that of

LP FIR COHBF structures allow for implementations that utilize the coefficient symmetry property. Hence, the required expenditure is just slightly higher than that needed for CHBF. On the other hand, the expenditure of MP IIR COHBF is almost twice as high as that of the corresponding CHBF, since it is not possible to exploit memory and coefficient sharing. Almost the whole structure has to be doubled for a full-complex decimator (cf. Fig. 19).

We have recalled basic properties and design outlines of linear-phase FIR and minimum-phase IIR halfband filters, predominantly for the purpose of sample rate alteration by a factor of two, which have a passband centre frequency out of the specific set defined by (1). Our

 

*<sup>z</sup>*−1*A*1(−*jz*2)

<sup>2</sup> = 1, (35)

<sup>=</sup> 1, (36)

. (37)

*n*mc *n* 2*n N*<sup>M</sup> *n* 2*n*

Table 6. Expenditure of minimum-phase IIR COHBF; *n*: order, *n*mc: McMillan degree, *N*M(*N*A): number of multipliers (adders), operational clock frequency: *f*Op = *f*n/2

<sup>|</sup>*H*(*ej*(Ω−*<sup>c</sup> <sup>π</sup>*

*H*(*ej*(Ω−*<sup>c</sup> <sup>π</sup>*

 

The Potential of Halfband Filters in Digital Signal Processing

*<sup>H</sup>*(*z*) = <sup>1</sup> 2 

**2.4 Conclusion: Family of single real and complex halfband filters**

where (3) is applied in the frequency domain.

4 ) )|

4 )

**Specification and properties**

are retained as follows

**Efficient implementations**

dual structure for interpolation.

the real IIR HBF system depicted in Fig. 9.

**2.3.3 Comparison of FIR and IIR COHBF**

Dec: **R** → **C** Int: **C** → **R** Dec: **C** → **C** Int: **C** → **C**

Fig. 17. Optimum SFG of linear-phase FIR COHBF decimating by two


Table 5. Expenditure of linear-phase FIR COHBF; *n*: order, *n*mc: McMillan degree, *N*M(*N*A): number of multipliers (adders), operational clock frequency: *f*Op = *f*n/2

operations over that of a classical HT (CHBF), respectively (cf. Figs. 12 and 13). This is due to the fact that, as a result of the transition from CHBF to COHBF, only the centre coefficient changes from trivially real (*h*<sup>0</sup> = <sup>1</sup> <sup>2</sup> ) to simple complex (*h*<sup>0</sup> <sup>=</sup> <sup>1</sup>+*<sup>j</sup>* 2 √2 ) calling for only one extra multiplication. The number *n*mc of delays is, however, of the order of *n*, since a (nearly) full delay line is needed both for the real and imaginary parts of the respective signals. Note that the shimming delays are always included in the delay count. (The number of delays required for a monorate COHBF corresponding to Fig. 17 is 2*n*.)

#### **2.3.2 Minimum-Phase (MP) IIR filters**

In the IIR COHBF case the frequency shift operation (3) is again applied in the *z*-domain. This is achieved by substituting the complex *z*-domain variable in the respective transfer functions *H*(*z*) and all corresponding SFG according to:

$$z \coloneqq \frac{z}{z\_1} = z e^{-j\frac{\pi}{4}} = z \frac{1-j}{\sqrt{2}}.\tag{34}$$


Table 6. Expenditure of minimum-phase IIR COHBF; *n*: order, *n*mc: McMillan degree, *N*M(*N*A): number of multipliers (adders), operational clock frequency: *f*Op = *f*n/2

#### **Specification and properties**

18 Will-be-set-by-IN-TECH

Fig. 17. Optimum SFG of linear-phase FIR COHBF decimating by two

number of multipliers (adders), operational clock frequency: *f*Op = *f*n/2

*<sup>z</sup>* :<sup>=</sup> *<sup>z</sup> z*1

changes from trivially real (*h*<sup>0</sup> = <sup>1</sup>

**2.3.2 Minimum-Phase (MP) IIR filters**

*H*(*z*) and all corresponding SFG according to:

for a monorate COHBF corresponding to Fig. 17 is 2*n*.)

Dec: **R** → **C** Int: **C** → **R** Dec: **C** → **C** Int: **C** → **C**

operations over that of a classical HT (CHBF), respectively (cf. Figs. 12 and 13). This is due to the fact that, as a result of the transition from CHBF to COHBF, only the centre coefficient

multiplication. The number *n*mc of delays is, however, of the order of *n*, since a (nearly) full delay line is needed both for the real and imaginary parts of the respective signals. Note that the shimming delays are always included in the delay count. (The number of delays required

In the IIR COHBF case the frequency shift operation (3) is again applied in the *z*-domain. This is achieved by substituting the complex *z*-domain variable in the respective transfer functions

= *ze*−*<sup>j</sup> <sup>π</sup>*

<sup>4</sup> = *z*

1 − *j*

<sup>2</sup> ) to simple complex (*h*<sup>0</sup> <sup>=</sup> <sup>1</sup>+*<sup>j</sup>*

2 √ 2 ) calling for only one extra

<sup>√</sup><sup>2</sup> . (34)

*n*mc *n n* + 2 *N*<sup>M</sup> (*n* + 6)/4 (*n* + 6)/2 *N*<sup>A</sup> *n*/2 + 1 *n* + 4 *n* + 2 *N*Op 3*n*/4 + 5/2 3*n*/2 + 7 3*n*/2 + 5 Table 5. Expenditure of linear-phase FIR COHBF; *n*: order, *n*mc: McMillan degree, *N*M(*N*A): All properties of the real IIR HBF are basically retained except of those subjected to the frequency shift operation of (28). This applies to the filter specification depicted in Fig. 8 and, hence, (6) is replaced with (32). Obviously, power (14) and allpass (15) complementarity are retained as follows

$$|\underline{H}(e^{j(\Omega-c\frac{\pi}{4})})|^2 + |\underline{H}(e^{j(\Omega-\pi(1+c/4))})|^2 = 1,\tag{35}$$

$$\left| \underline{H}(e^{j(\Omega - c\frac{\pi}{4})}) + \underline{H}(e^{j(\Omega - \pi(1+c/4))}) \right| = 1,\tag{36}$$

where (3) is applied in the frequency domain.

#### **Efficient implementations**

Introducing (34) in (16), the transfer function is frequency-shifted by *f*<sup>1</sup> = *fn*/8 (Ω = *π*/4):

$$\underline{H}(z) = \frac{1}{2} \left[ \underline{A}\_0(-jz^2) + \frac{1+j}{\sqrt{2}} z^{-1} \underline{A}\_1(-jz^2) \right]. \tag{37}$$

The optimal structure of an *n* = 5th order MP IIR COHBF decimator for real input signals is shown in Fig. 18(a) along with the elementary SFG of the allpass sections Fig. 18(b). Doubling of the structure according to Fig. 19 allows for full-complex signal processing. Multirate transposition [Göckler & Groth (2004)] is again applied to derive the corresponding dual structure for interpolation.

The expenditure of the half- (**R C**) and the full-complex (**C** → **C**) COHBF decimators and their transposes is listed in Table 6. A comparison of Tables 2 and 6 shows that the half-complex IIR COHBF sample rate converter (cf. Fig. 18(a)) requires almost twice, whereas the full-complex IIR COHBF (cf. Fig. 19) requires even four times the expenditure of that of the real IIR HBF system depicted in Fig. 9.

#### **2.3.3 Comparison of FIR and IIR COHBF**

LP FIR COHBF structures allow for implementations that utilize the coefficient symmetry property. Hence, the required expenditure is just slightly higher than that needed for CHBF. On the other hand, the expenditure of MP IIR COHBF is almost twice as high as that of the corresponding CHBF, since it is not possible to exploit memory and coefficient sharing. Almost the whole structure has to be doubled for a full-complex decimator (cf. Fig. 19).

#### **2.4 Conclusion: Family of single real and complex halfband filters**

We have recalled basic properties and design outlines of linear-phase FIR and minimum-phase IIR halfband filters, predominantly for the purpose of sample rate alteration by a factor of two, which have a passband centre frequency out of the specific set defined by (1). Our

Signal Processing 21

<sup>257</sup> Most Efficient Digital Filter Structures:

HBF Decimator 12 8 7 9 3 9 CHBF: **R** → **C** 11 11 12(a) 8 3 14 CHBF: **C** → **C** 24 16 13 18 6 15 COHBF: **R** → **C** 13 14 16(a) 17 5 18 COHBF: **C** → **C** 28 16 17 36 10 19 Table 7. Expenditures of real and complex HBF decimators based on the design examples of Fig. 11; *N*Op: number of operations, *n*mc: McMillan degree; operational clock frequency:

contribution. This sectoral computational advantage of LP FIR COHBF is, despite *n*IIR < *n*FIR, due to the fact that these FIR filters still allow for memory sharing in conjunction with the exploitation of coefficient symmetry [Göckler (1996b)]. However, the amount of storage *n*mc

In this Section 3, we address a particular class of efficient directional filters (DF). These DF are composed of two real or complex HBF, respectively, of different centre frequencies out of the set given by (1). To this end, we conceptually introduce and investigate two-channel frequency demultiplexer filter banks (FDMUX) that extract from an incoming complex-valued frequency division multiplex (FDM) signal, being composed of up to *four* uniformly allocated independent user signals of identical bandwidth (cf. Fig. 20), two of its constituents by concurrently reducing the sample rate by *two* Göckler & Groth (2004). Moreover, the DF shall allow to select any pair of user signals out of the four constituents of the incoming FDM signal, where the individual centre frequencies are to be selectable with minimum switching effort. At first glance, there are two optional approaches: The selectable combination of two filter functions out of a pool of *i*) two RBF according to Subsection 2.1 and two CHBF (HT), as described in Subsection 2.2, where the centre frequencies of this filter quadruple are given by (1) with *c* ∈ {0, 2, 4, 6}, or *ii*) four COHBF, as described in Subsection 2.3, where the centre frequencies of this filter quadruple are given by (1) with *c* ∈ {1, 3, 5, 7}. Since centre frequency switching is more crucial in case one (switching between real and/or complex filters), we subsequently restrict our investigations to case two, where the FDM input spectrum must be

These DF with easily selectable centre frequencies are frequently used in receiver front-ends to meet routing requirements [Göckler (1996c)], in tree-structured FDMUX filter banks [Göckler & Felbecker (2001); Göckler & Groth (2004); Göckler & Eyssele (1992)], and, in modified form, for frequency re-allocation to avoid hard-wired frequency-shifting [Abdulazim & Göckler (2007); Eghbali et al. (2009)]. Efficient implementation is crucial, if these DF are operated at high sampling rates at system input or output port. To cope with this high rate challenge, we introduce a systematic approach to system parallelisation according

In continuation of the investigations reported in Section 2, we combine two linear-phase (LP) FIR complex *offset* halfband filters (COHBF) with different centre frequencies, being characterized by (1) with *c* ∈ {1, 3, 5, 7}, to construct efficient directional filters for one input

required for IIR HBF is always below that of their FIR counterparts.

The Potential of Halfband Filters in Digital Signal Processing

*f*Op = *f*n/2

**3. Halfband filter pairs**<sup>2</sup>

allocated as shown in Fig. 20.

to [Groth (2003)] in Section 4 .

<sup>2</sup> Underlying original publication: Göckler & Alfsmann (2010)

LP FIR MP IIR *N*Op *n*mc Fig. *N*Op *n*mc Fig.

Fig. 18. Decimating allpass-based minimum-phase IIR COHBF, *n* = 5: (a) optimum SFG (b) the 1st (2nd) order allpass section, *i* = 0, 1

Fig. 19. Block structure of decimating (a) and interpolating (b) minimum-phase IIR COHBF

main emphasis has been put on the presentation of optimum implementations that call for minimum computational burden.

It has been confirmed that, for the even-numbered centre frequencies *c* ∈ {0, 2, 4, 6}, MP IIR HBF outperform their LP FIR counterparts the more the tighter the filter specifications. However, for phase sensitive applications (e.g. software radio employing quadrature amplitude modulation), the LP property of FIR HBF may justify the higher amount of computation to some extent.

In the case of the odd-numbered HBF centre frequencies of (2), *c* ∈ {1, 3, 5, 7}, there exist specification domains, where the computational loads of complex FIR HBF with frequency offset range below those of their IIR counterparts. This is confirmed by the two bottom rows of Table 7, where this table lists the expenditure of a twofold decimator based on the design examples given in Fig. 11 for all centre frequencies and all applications investigated in this


Table 7. Expenditures of real and complex HBF decimators based on the design examples of Fig. 11; *N*Op: number of operations, *n*mc: McMillan degree; operational clock frequency: *f*Op = *f*n/2

contribution. This sectoral computational advantage of LP FIR COHBF is, despite *n*IIR < *n*FIR, due to the fact that these FIR filters still allow for memory sharing in conjunction with the exploitation of coefficient symmetry [Göckler (1996b)]. However, the amount of storage *n*mc required for IIR HBF is always below that of their FIR counterparts.
