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

20 Will-be-set-by-IN-TECH

Fig. 18. Decimating allpass-based minimum-phase IIR COHBF, *n* = 5: (a) optimum SFG (b)

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

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

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

the 1st (2nd) order allpass section, *i* = 0, 1

minimum computational burden.

computation to some extent.

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 allocated as shown in Fig. 20.

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 to [Groth (2003)] in Section 4 .

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

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

Signal Processing 23

The Potential of Halfband Filters in Digital Signal Processing

<sup>259</sup> Most Efficient Digital Filter Structures:

Fig. 21. DF operations: (a) Real HBF prototype centrosymmetric about *H*0(*e*j*π*/2) = <sup>1</sup>

by (38), *o* ∈ {0, 1, 2, 3}, with the RHBF impulse response *h*(*k*) defined by (9). According to (39), highest efficiency is obtained by additionally introducing a suitable complex scaling factor of

j *π*

<sup>4</sup> [*k*(2*o*+1)+*a*] = *h*(*k*)j

0 *k* = 2*l l* = 1, . . . ,(*N* − 3)/4

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

*π*

<sup>2</sup> and *o* ∈ {0, 1, 2, 3}. By directly equating (39), and relating the result

*k*(*o*+ <sup>1</sup> <sup>2</sup> )+ *<sup>a</sup>*

*hk*,*<sup>o</sup>* ∀*k* > 0, *o* ∈ {0, 1, 2, 3}. (41)

<sup>4</sup> ]}, *<sup>o</sup>* <sup>∈</sup> {0, 1, 2, 3}, (42)

Two selected DF filter functions, (c,d) Spectra of decimated DF output signals

*<sup>o</sup>* = *h*(*k*)*e*

to (9) with a suitable choice of the constant *a* = 2*o* + 1 compliant with (29), we get :

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

unity magnitude:

where <sup>−</sup> *<sup>N</sup>*−<sup>1</sup>

*hk*,*o*,*<sup>a</sup>* = *e*

*hk*,*<sup>o</sup>* =

The respective COHBF centre coefficient

*<sup>h</sup>*0,*<sup>o</sup>* <sup>=</sup> <sup>1</sup> 2

<sup>2</sup> <sup>≤</sup> *<sup>k</sup>* <sup>≤</sup> *<sup>N</sup>*−<sup>1</sup>

with the symmetry property:

j*a <sup>π</sup>* <sup>4</sup> *h*(*k*)*z<sup>k</sup>*

 

1 2 j *o*+ <sup>1</sup>

(2*o*+1)*k*

*π*

<sup>4</sup> ] + j sin[(2*<sup>o</sup>* <sup>+</sup> <sup>1</sup>)

j (*k*+1)(*o*+<sup>1</sup>

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

{cos[(2*o* + 1)

<sup>2</sup> , (b)

<sup>2</sup> , (39)

(40)

Fig. 20. FDM input spectrum for selection and separation by two-channel directional filter (DF)

and two output signals Göckler (1996a). For convenience, we map the original odd indices *c* ∈ {1, 3, 5, 7} of the COHBF centre frequencies to natural numbers as defined by

$$f\_o = (2o + 1) \cdot \frac{f\_{\rm n}}{8}, \quad o \in \{0, 1, 2, 3\} \tag{38}$$

for subsequent use throughout Section 3.

Section 3 is organized as follows: In Subsection 3.1, we detail the statement of the problem, and recall the major properties of COHBF needed for our DF investigations. In the main Subsection 3.2, we present and compare two different approaches to implement the outlined LP DF for signal separation with selectable centre frequencies: *i*) A four-channel uniform complex-modulated FDMUX filter bank undercritically decimating by two, where the respective undesired two output signals are discarded, and *ii*) a synergetic connection of two COHBF that share common multipliers and exploit coefficient symmetry for minimum computation. In Subsection 3.3, we apply the transposition rules of [Göckler & Groth (2004)] to derive the dual DF for signal combination (FDM multiplexing). Finally, we draw some further conclusions in Subsection 3.4.

#### **3.1 Statement of the DF problem**

Given a uniform complex-valued FDM signal composed of up to four independent user signals *so*(*kT*n) ←→ *So*(ejΩ) centred at *fo*, *<sup>o</sup>* <sup>=</sup> {0, 1, 2, 3}, according to (38), as depicted in Fig. 20, the DF shall extract any freely selectable two out of the four user signals of the FDM input spectrum, and provide them at the two DF output ports separately and decimated by two: *so*(2*kT*n) :<sup>=</sup> *so*(*mT*d) ←→ *So*(ejΩ(d) ); *T*<sup>d</sup> = 1/ *f*<sup>d</sup> = 2*T*n. Recall that complex-valued time-domain signals and spectrally transformed versions thereof are indicated by underlining.

Efficient signal separation and decimation is conceptually achieved by combining two COHBF with their differing passbands centred according to (38), where *o* ∈ {0, 1, 2, 3}, along with twofold polyphase decomposition of the respective filter impulse responses [Göckler & Damjanovic (2006a); Göckler & Groth (2004)]. All COHBF are frequency-shifted versions of a real zero-phase (ZP) lowpass HBF prototype with symmetric impulse response *<sup>h</sup>*(*k*) = *hk* <sup>=</sup> *<sup>h</sup>*−*<sup>k</sup>* ←→ *<sup>H</sup>*0(*e*jΩ) <sup>∈</sup> **<sup>R</sup>** according to Subsection 2.1.1, as depicted in Fig. 21(a) as ZP HBF frequency response [Milic (2009); Mitra & Kaiser (1993)]. A frequency domain representation of a possible DF setting (choice of COHBF centre frequencies *o* ∈ {0, 2}) is shown in Fig. 21(b), and Figs.21(c,d) present the output spectra at port I (*o* = 0) and port II (*o* = 2), respectively, related to the reduced sampling rate *f*<sup>d</sup> = *f*n/2.

A COHBF is derived from a real HBF (9) by applying the frequency shift operation in the time domain by modulating a complex carrier *z<sup>k</sup> <sup>o</sup>* <sup>=</sup> *<sup>e</sup>*j2*πk fo*/ *<sup>f</sup>*<sup>n</sup> <sup>=</sup> *<sup>e</sup>*j*k*(2*o*+1) *<sup>π</sup>* <sup>4</sup> of a frequency prescribed 22 Will-be-set-by-IN-TECH

Fig. 20. FDM input spectrum for selection and separation by two-channel directional filter

*c* ∈ {1, 3, 5, 7} of the COHBF centre frequencies to natural numbers as defined by

*fo* = (2*<sup>o</sup>* <sup>+</sup> <sup>1</sup>) · *<sup>f</sup>*<sup>n</sup>

for subsequent use throughout Section 3.

further conclusions in Subsection 3.4.

decimated by two: *so*(2*kT*n) :<sup>=</sup> *so*(*mT*d) ←→ *So*(ejΩ(d)

(*o* = 2), respectively, related to the reduced sampling rate *f*<sup>d</sup> = *f*n/2.

domain by modulating a complex carrier *z<sup>k</sup>*

**3.1 Statement of the DF problem**

indicated by underlining.

and two output signals Göckler (1996a). For convenience, we map the original odd indices

Section 3 is organized as follows: In Subsection 3.1, we detail the statement of the problem, and recall the major properties of COHBF needed for our DF investigations. In the main Subsection 3.2, we present and compare two different approaches to implement the outlined LP DF for signal separation with selectable centre frequencies: *i*) A four-channel uniform complex-modulated FDMUX filter bank undercritically decimating by two, where the respective undesired two output signals are discarded, and *ii*) a synergetic connection of two COHBF that share common multipliers and exploit coefficient symmetry for minimum computation. In Subsection 3.3, we apply the transposition rules of [Göckler & Groth (2004)] to derive the dual DF for signal combination (FDM multiplexing). Finally, we draw some

Given a uniform complex-valued FDM signal composed of up to four independent user signals *so*(*kT*n) ←→ *So*(ejΩ) centred at *fo*, *<sup>o</sup>* <sup>=</sup> {0, 1, 2, 3}, according to (38), as depicted in Fig. 20, the DF shall extract any freely selectable two out of the four user signals of the FDM input spectrum, and provide them at the two DF output ports separately and

that complex-valued time-domain signals and spectrally transformed versions thereof are

Efficient signal separation and decimation is conceptually achieved by combining two COHBF with their differing passbands centred according to (38), where *o* ∈ {0, 1, 2, 3}, along with twofold polyphase decomposition of the respective filter impulse responses [Göckler & Damjanovic (2006a); Göckler & Groth (2004)]. All COHBF are frequency-shifted versions of a real zero-phase (ZP) lowpass HBF prototype with symmetric impulse response *<sup>h</sup>*(*k*) = *hk* <sup>=</sup> *<sup>h</sup>*−*<sup>k</sup>* ←→ *<sup>H</sup>*0(*e*jΩ) <sup>∈</sup> **<sup>R</sup>** according to Subsection 2.1.1, as depicted in Fig. 21(a) as ZP HBF frequency response [Milic (2009); Mitra & Kaiser (1993)]. A frequency domain representation of a possible DF setting (choice of COHBF centre frequencies *o* ∈ {0, 2}) is shown in Fig. 21(b), and Figs.21(c,d) present the output spectra at port I (*o* = 0) and port II

A COHBF is derived from a real HBF (9) by applying the frequency shift operation in the time

*<sup>o</sup>* <sup>=</sup> *<sup>e</sup>*j2*πk fo*/ *<sup>f</sup>*<sup>n</sup> <sup>=</sup> *<sup>e</sup>*j*k*(2*o*+1) *<sup>π</sup>*

<sup>8</sup> , *<sup>o</sup>* <sup>∈</sup> {0, 1, 2, 3} (38)

); *T*<sup>d</sup> = 1/ *f*<sup>d</sup> = 2*T*n. Recall

<sup>4</sup> of a frequency prescribed

(DF)

Fig. 21. DF operations: (a) Real HBF prototype centrosymmetric about *H*0(*e*j*π*/2) = <sup>1</sup> <sup>2</sup> , (b) Two selected DF filter functions, (c,d) Spectra of decimated DF output signals

by (38), *o* ∈ {0, 1, 2, 3}, with the RHBF impulse response *h*(*k*) defined by (9). According to (39), highest efficiency is obtained by additionally introducing a suitable complex scaling factor of unity magnitude:

$$\underline{h}\_{k,o,a} = e^{\mathrm{j}a\frac{\pi}{4}}h(k)z\_o^k = h(k)e^{\mathrm{j}\frac{\pi}{4}[k(2o+1)+a]} = h(k)\mathrm{j}^{k(o+\frac{1}{2})+\frac{a}{2}},\tag{39}$$

where <sup>−</sup> *<sup>N</sup>*−<sup>1</sup> <sup>2</sup> <sup>≤</sup> *<sup>k</sup>* <sup>≤</sup> *<sup>N</sup>*−<sup>1</sup> <sup>2</sup> and *o* ∈ {0, 1, 2, 3}. By directly equating (39), and relating the result to (9) with a suitable choice of the constant *a* = 2*o* + 1 compliant with (29), we get :

$$\mathfrak{h}\_{k,o} = \begin{cases} \frac{1}{2} \mathfrak{l}^{o + \frac{1}{2}} & k = 0\\ 0 & k = 2l \quad l = 1, \ldots, (N - 3)/4\\ \mathfrak{j}^{(k+1)(o + \frac{1}{2})} \mathfrak{l}\_k \ k = 2l - 1 \ l = 1, \ldots, (N + 1)/4 \end{cases} \tag{40}$$

with the symmetry property:

$$
\underline{\mathbf{h}}\_{-k,\rho} = -\mathbf{j}^{(2\rho+1)k} \underline{\mathbf{h}}\_{k,\rho} \quad \forall k > 0, \quad \rho \in \{0, 1, 2, 3\}.\tag{41}
$$

The respective COHBF centre coefficient

$$\underline{h}\_{0,o} = \frac{1}{2} \{ \cos[(2o+1)\frac{\pi}{4}] + \text{j}\sin[(2o+1)\frac{\pi}{4}] \}, \ o \in \{0, 1, 2, 3\},\tag{42}$$

Signal Processing 25

The Potential of Halfband Filters in Digital Signal Processing

<sup>261</sup> Most Efficient Digital Filter Structures:

Fig. 22. SFG of directional filter with allowing for 2-out-of-4 channel selection: FDMUX

encompasses all complex signal processing to be performed by the modified causal HBF

An illustrative example with an underlying HBF prototype filter of length *N* = *n* + 1 = 11 is shown in Fig. 22 [Göckler & Groth (2004)]. Due to polyphase decomposition (45) and (46), sample rate reduction can be performed in front of any signal processing (shimming delays: *z*−1). Always two polyphase components of the real and the imaginary parts of the complex input signal share a delay chain in the direct form implementation of the modified causal HBF, where all coefficients are either real- or imaginary-valued except for the centre coefficient

of complex output samples at the two (i.e. all) DF output ports. Furthermore, for the FDMUX DF implementation a total of (3*N* − 5)/2 delays are needed (not counting shimming delays). The calculation of *vp*(*m*), *p* = 0, 1, 2, 3, is readily understood from the signal flow graph (SFG) Fig. 22, where for any filter length *N always one* of these quantities vanishes as a result of the zero coefficients of (9). Hence, the *I* = 4 point IDFT, depicted in Fig. 23(a,b) in detailed form, requires only 4 real additions to provide a complex output sample at any of the output ports *o* ∈ {0, 1, 2, 3}; Fig. 23(b). Channel selection, for instance as shown in Fig. 21, is simply achieved by selection of the respective two output ports of the SFG of Figs.22 and 23(a), respectively. Moreover, the remaining two unused output ports may be deactivated by

<sup>4</sup> . As a result, only *N* + 3 real multiplications must be performed to calculate a set

approach; *N* = 11

prototype.

*<sup>h</sup>*<sup>0</sup> = <sup>1</sup> <sup>2</sup> <sup>e</sup><sup>j</sup> *<sup>π</sup>*

disconnection from power supply.

is the only truly complex-valued coefficient, where its real and imaginary parts always possess identical moduli. All other coefficients are either purely imaginary or real-valued. Obviously, all frequency domain symmetry properties, including also those related to strict complementarity, are retained in the respective frequency-shifted versions, cf. Subsection 2.3.1 and [Göckler & Damjanovic (2006a)].

#### **3.2 Linear-phase directional separation filter**

We start with the presentation of the FDMUX approach [Göckler & Groth (2004); Göckler & Eyssele (1992)] followed by the investigation of a synergetic combination of two COHBF [Göckler (1996a;c); Göckler & Damjanovic (2006a)].

#### **3.2.1 FDMUX approach**

Using time-domain convolution, the *I* = 4 potentially required complex output signals, decimated by 2 and related to the channel indices *o* ∈ {0, 1, 2, 3}, are obtained as follows:

$$\underline{\boldsymbol{y}}\_{o}(m\boldsymbol{T}\_{\rm d}) := \underline{\boldsymbol{y}}\_{o}(m) = \sum\_{\kappa=0}^{N-1} \underline{\boldsymbol{x}}(2m-\kappa) \underline{\boldsymbol{h}}\_{o}(\kappa - \frac{N-1}{2}),\tag{43}$$

where the complex impulse responses of channels *o* are introduced in causal (realizable) form. Replacing the complex impulse responses with the respective modulation forms (39), and setting the constant to *a* = (2*o* + 1)(*N* − 1)/2, we get:

$$\underline{\mathbf{v}}\_{o}(m) = \sum\_{\kappa=0}^{N-1} \underline{\mathbf{x}}(2m-\kappa)h(\kappa - \frac{N-1}{2})\mathbf{e}^{\mathbf{j}\cdot\overline{\mathbf{x}}\kappa(2o+1)}\,\tag{44}$$

where *h*[*k* − (*N* − 1)/2] represents the real HBF prototype (9) in causal form. Next, in order to introduce an *I*-component polyphase decomposition for efficient decimation, we split the convolution index *κ* into two indices:

$$
\kappa = rI + p = 4r + p\_\prime \tag{45}
$$

where *p* = 0, 1, 2, *I* − 1 = 3 and *r* = 0, 1, . . . , (*N* − 1)/*I* = (*N* − 1)/4. As a result, it follows from (44):

$$\underline{\mathbf{y}}\_o(m) = \sum\_{p=0}^3 \sum\_{r=0}^{\left\lfloor \frac{N-1}{4} \right\rfloor} \underline{\mathbf{x}}(2m - 4r - p) h(4r + p - \frac{N-1}{2}) \cdot \mathbf{e}^{\mathbf{i}\frac{\pi}{4}(4r + p)(2o + 1)}.\tag{46}$$

Rearranging the exponent of the exponential term according to *<sup>π</sup>* <sup>4</sup> (4*r* + *p*)(2*o* + 1) = 2*πro* + *πr* + *p <sup>π</sup>* <sup>4</sup> <sup>+</sup> <sup>2</sup>*<sup>π</sup>* <sup>4</sup> *op*, (46) can compactly be rewritten as [Oppenheim & Schafer (1989)]:

$$\underline{\underline{y}}\_{o}(m) = \sum\_{p=0}^{3} \underline{\underline{v}}\_{p}(m) \cdot \mathrm{e}^{\mathrm{j}\frac{2\pi}{4}op} = 4 \cdot \mathrm{IDFT}\_{4}\{\underline{\underline{v}}\_{p}(m)\},\tag{47}$$

where the quantity

$$\underline{\mathbf{v}}\_{p}(m) = \sum\_{r=0}^{\left\lfloor \frac{N-1}{4} \right\rfloor} \underline{\mathbf{x}}(2m - 4r - p)h(4r + p - \frac{N-1}{2})(-1)^{r}\mathbf{e}^{\mathbf{j}p\frac{\pi}{4}}\tag{48}$$

24 Will-be-set-by-IN-TECH

is the only truly complex-valued coefficient, where its real and imaginary parts always possess identical moduli. All other coefficients are either purely imaginary or real-valued. Obviously, all frequency domain symmetry properties, including also those related to strict complementarity, are retained in the respective frequency-shifted versions, cf. Subsection 2.3.1

We start with the presentation of the FDMUX approach [Göckler & Groth (2004); Göckler & Eyssele (1992)] followed by the investigation of a synergetic combination of two

Using time-domain convolution, the *I* = 4 potentially required complex output signals, decimated by 2 and related to the channel indices *o* ∈ {0, 1, 2, 3}, are obtained as follows:

> *N*−1 ∑ *κ*=0

where the complex impulse responses of channels *o* are introduced in causal (realizable) form. Replacing the complex impulse responses with the respective modulation forms (39), and

*<sup>x</sup>*(2*<sup>m</sup>* <sup>−</sup> *<sup>κ</sup>*)*h*(*<sup>κ</sup>* <sup>−</sup> *<sup>N</sup>*−<sup>1</sup>

where *h*[*k* − (*N* − 1)/2] represents the real HBF prototype (9) in causal form. Next, in order to introduce an *I*-component polyphase decomposition for efficient decimation, we split the

where *p* = 0, 1, 2, *I* − 1 = 3 and *r* = 0, 1, . . . , (*N* − 1)/*I* = (*N* − 1)/4. As a result, it

*<sup>x</sup>*(2*<sup>m</sup>* <sup>−</sup> <sup>4</sup>*<sup>r</sup>* <sup>−</sup> *<sup>p</sup>*)*h*(4*<sup>r</sup>* <sup>+</sup> *<sup>p</sup>* <sup>−</sup> *<sup>N</sup>*−<sup>1</sup>

<sup>4</sup> *op*, (46) can compactly be rewritten as [Oppenheim & Schafer (1989)]:

*<sup>x</sup>*(2*<sup>m</sup>* <sup>−</sup> <sup>4</sup>*<sup>r</sup>* <sup>−</sup> *<sup>p</sup>*)*h*(4*<sup>r</sup>* <sup>+</sup> *<sup>p</sup>* <sup>−</sup> *<sup>N</sup>*−<sup>1</sup>

*vp*(*m*) · <sup>e</sup><sup>j</sup> <sup>2</sup>*<sup>π</sup>*

*<sup>x</sup>*(2*<sup>m</sup>* <sup>−</sup> *<sup>κ</sup>*)*ho*(*<sup>κ</sup>* <sup>−</sup> *<sup>N</sup>*−<sup>1</sup>

<sup>2</sup> )e<sup>j</sup> *<sup>π</sup>*

<sup>4</sup> *κ*(2*o*+1)

*κ* = *r I* + *p* = 4*r* + *p*, (45)

<sup>2</sup> ) · <sup>e</sup><sup>j</sup> *<sup>π</sup>*

<sup>4</sup> (4*r*+*p*)(2*o*+1)

<sup>4</sup> *o p* <sup>=</sup> <sup>4</sup> · IDFT4{*vp*(*m*)}, (47)

<sup>2</sup> )(−1)*r*ej*<sup>p</sup> <sup>π</sup>*

<sup>4</sup> (4*r* + *p*)(2*o* + 1) = 2*πro* +

<sup>2</sup> ), (43)

, (44)

. (46)

<sup>4</sup> (48)

(*m*) =

and [Göckler & Damjanovic (2006a)].

**3.2.1 FDMUX approach**

**3.2 Linear-phase directional separation filter**

*yo*

COHBF [Göckler (1996a;c); Göckler & Damjanovic (2006a)].

(*mT*d) := *yo*

*N*−1 ∑ *κ*=0

setting the constant to *a* = (2*o* + 1)(*N* − 1)/2, we get:

*yo* (*m*) =

3 ∑ *p*=0

> *yo* (*m*) =

*vp*(*m*) =

 *<sup>N</sup>*−<sup>1</sup> <sup>4</sup> ∑ *r*=0

Rearranging the exponent of the exponential term according to *<sup>π</sup>*

3 ∑ *p*=0

 *<sup>N</sup>*−<sup>1</sup> <sup>4</sup> ∑ *r*=0

convolution index *κ* into two indices:

follows from (44):

*πr* + *p <sup>π</sup>*

*yo* (*m*) =

<sup>4</sup> <sup>+</sup> <sup>2</sup>*<sup>π</sup>*

where the quantity

Fig. 22. SFG of directional filter with allowing for 2-out-of-4 channel selection: FDMUX approach; *N* = 11

encompasses all complex signal processing to be performed by the modified causal HBF prototype.

An illustrative example with an underlying HBF prototype filter of length *N* = *n* + 1 = 11 is shown in Fig. 22 [Göckler & Groth (2004)]. Due to polyphase decomposition (45) and (46), sample rate reduction can be performed in front of any signal processing (shimming delays: *z*−1). Always two polyphase components of the real and the imaginary parts of the complex input signal share a delay chain in the direct form implementation of the modified causal HBF, where all coefficients are either real- or imaginary-valued except for the centre coefficient *<sup>h</sup>*<sup>0</sup> = <sup>1</sup> <sup>2</sup> <sup>e</sup><sup>j</sup> *<sup>π</sup>* <sup>4</sup> . As a result, only *N* + 3 real multiplications must be performed to calculate a set of complex output samples at the two (i.e. all) DF output ports. Furthermore, for the FDMUX DF implementation a total of (3*N* − 5)/2 delays are needed (not counting shimming delays). The calculation of *vp*(*m*), *p* = 0, 1, 2, 3, is readily understood from the signal flow graph (SFG) Fig. 22, where for any filter length *N always one* of these quantities vanishes as a result of the zero coefficients of (9). Hence, the *I* = 4 point IDFT, depicted in Fig. 23(a,b) in detailed form, requires only 4 real additions to provide a complex output sample at any of the output ports *o* ∈ {0, 1, 2, 3}; Fig. 23(b). Channel selection, for instance as shown in Fig. 21, is simply achieved by selection of the respective two output ports of the SFG of Figs.22 and 23(a), respectively. Moreover, the remaining two unused output ports may be deactivated by disconnection from power supply.

Signal Processing 27

<sup>263</sup> Most Efficient Digital Filter Structures:

SFG, the coefficient multipliers can obviously be shared with all transfer functions *Ho*(*z*), *o* ∈ {0, 1, 2, 3}; however, the respective outbound delay chains must essentially be duplicated. Merging all of the above considerations, a signal separating DF requiring minimum computation that, in addition, allows for simple channel selection or switching, respectively,

1. Multiply the incoming decimated polyphase signal samples concurrently and consecutively by the complete set of all real coefficients (9) to allow for the exploitation of

2. Form a real and imaginary (R/I) sub-sequence of DF output signals being *independent* of the selected channel transfer functions, i.e. *o*I, *o*II ∈ {0, 1, 2, 3}, by using all **R**-set coefficients of

3. Form an R and I sub-sequence of DF output signals being likewise *independent* of the selected channels *<sup>o</sup>*I, *<sup>o</sup>*II by using all **<sup>I</sup>**-set coefficients of Table 8 multiplied by (−1)*<sup>o</sup>* to

4. Form R/I sub-sequences of DF output signals being *dependent* of the selected channels

5. Combine all of the above R/I sub-sequences considering the sign rules of Table 8 to select

Based on the outlined DF implementation strategy, an illustrative example is presented in Fig. 24 with an underlying RHBF of length *N* = 11. The front end for polyphase decomposition and sample rate reduction by 2 is identical to that of the FDMUX approach of Fig. 22. Contrary to the former approach, the delay chains for the odd-numbered coefficients are outbound *and* duplicated (rather than interlaced) to allow for simple channel selection. As a result, channel selection is performed by combining the respective sub-sequences that have passed the **R**-set coefficients (cf. Table 8) with those having passed the corresponding **I**-set coefficients, where

Multipliers and delays for the centre coefficient *h*0,*oi* signal processing are implemented similarly to Fig. 22 without need for duplication of delays. However, the post-delay inner lattice must be realized for each transfer function individually; its channel dependency follows

where *oi* ∈ {0, 1, 2, 3}, *i* ∈ {I, II} and *h*<sup>0</sup> = 1/2 according to (9). Rearranging (49) yields with

<sup>√</sup><sup>2</sup> [(−1)*oi* <sup>+</sup> <sup>j</sup>](−1)*oi*/2 <sup>=</sup> *<sup>h</sup>*<sup>0</sup>

It is easily recognized that the inner lattices of Fig. 24 implement the operations within the brackets of (50) with their results displayed at the respective inner nodes A, B, C, D. In compliance with (50), these inner node sequences must be multiplied by the respective signs *di* = (−1)*oi*/2; *oi* <sup>∈</sup> {0, 1, 2, 3}, *<sup>i</sup>* <sup>∈</sup> {I, II}, prior to their combination with the above R/I

To calculate a set of complex output samples at the two DF output ports, obviously the minimum number of (*N* + 5)/2 real multiplications must be carried out. Furthermore, for

the latter sub-sequences are pre-multiplied by *bi* = (−1)*oi* ; *oi* <sup>∈</sup> {0, 1, 2, 3}, *<sup>i</sup>* <sup>∈</sup> {I, II}.

*oi* <sup>=</sup> *<sup>h</sup>*<sup>0</sup> √2 

(*z*), *oi* ∈ {0, 1, 2, 3}, *i* ∈ {I, II}.

(−1)*oi*/2 <sup>+</sup> <sup>j</sup>(−1)*oi*/2

<sup>√</sup><sup>2</sup> [*bi* <sup>+</sup> <sup>j</sup>] *di*. (50)

, (49)

is readily developed as follows:

eliminate channel dependency.

the desired DF transfer functions *Hoi*

*<sup>h</sup>*0,*oi* <sup>=</sup> *<sup>h</sup>*<sup>0</sup> √2

*<sup>h</sup>*0,*oi* <sup>=</sup> *<sup>h</sup>*<sup>0</sup>

(1 + j)j

Table 8.

from Table 8 and (40):

obvious abbreviations:

sub-sequences.

coefficient symmetry (41) in compliance with Table 8.

The Potential of Halfband Filters in Digital Signal Processing

*o*I, *o*II that are derived from centre coefficients *h*0,*o*.

Fig. 23. *I* = 4 point IDFT of FDMUX approach; *N* = 11: (a) general (b) pruned for channels *o* = 0, 1


Table 8. Properties of COHBF coefficients in dependence of channel index *o* ∈ {0, 1, 2, 3}; **I**: **C** with *Re*{•} = 0

#### **3.2.2 COHBF approach**

For this novel approach, we combine two decimating COHBF of different centre frequencies *fo*, *o* ∈ {0, 1, 2, 3}, according to (38) in a synergetic manner to construct a DF for signal separation that requires *minimum* computation. To this end, we first study the commonalities of the impulse responses (40) of the four transfer functions *Ho*(*z*), *o* ∈ {0, 1, 2, 3} (underlying constant in (39) subsequently: *a* = 2*o* + 1). These impulse responses are presented in Table 8 as a function of the channel number *o* ∈ {0, 1, 2, 3} for the non-zero coefficients of (40), related to the respective real RHBF coefficients.

Except for the centre coefficient exhibiting identical real and imaginary parts, one half of the coefficients is real (**R**) and *independent of the desired centre frequency* represented by the channel indices *o* ∈ {0, 1, 2, 3}. Hence, these coefficients are common to all four transfer functions. The other half of the coefficients is purely imaginary (**I**: i.e., their real parts are zero) and dependent of the selected centre frequency. However, this dependency on the channel number is identical for all these coefficients and just requires a simple sign operation. Finally, the repetitive pattern of the coefficients, as a result of coefficient symmetry (41), is reflected in Table 8.

A COHBF implementation of a demultiplexing DF aiming at *minimum computational load* must exploit the inherent coefficient symmetry (41), cf. Table 8. To this end, we consider the COHBF as depicted in Fig. 17 of Subsection 2.3.1, applying input commutators for sample rate reduction. In contrast to the FDMUX approach of Fig. 22, the SFG of Fig. 17 is based on the transposed FIR direct form Bellanger (1989); Mitra (1998), where the incoming signal samples are concurrently multiplied by the complete set of all coefficients, and the delay chains are directly connected to the output ports. When combining two of these COHBF

26 Will-be-set-by-IN-TECH

Fig. 23. *I* = 4 point IDFT of FDMUX approach; *N* = 11: (a) general (b) pruned for channels

*k* -5 -3 -1 0 1 3 5

type **R I R C I R I**

Table 8. Properties of COHBF coefficients in dependence of channel index *o* ∈ {0, 1, 2, 3}; **I**: **C**

For this novel approach, we combine two decimating COHBF of different centre frequencies *fo*, *o* ∈ {0, 1, 2, 3}, according to (38) in a synergetic manner to construct a DF for signal separation that requires *minimum* computation. To this end, we first study the commonalities of the impulse responses (40) of the four transfer functions *Ho*(*z*), *o* ∈ {0, 1, 2, 3} (underlying constant in (39) subsequently: *a* = 2*o* + 1). These impulse responses are presented in Table 8 as a function of the channel number *o* ∈ {0, 1, 2, 3} for the non-zero coefficients of (40), related

Except for the centre coefficient exhibiting identical real and imaginary parts, one half of the coefficients is real (**R**) and *independent of the desired centre frequency* represented by the channel indices *o* ∈ {0, 1, 2, 3}. Hence, these coefficients are common to all four transfer functions. The other half of the coefficients is purely imaginary (**I**: i.e., their real parts are zero) and dependent of the selected centre frequency. However, this dependency on the channel number is identical for all these coefficients and just requires a simple sign operation. Finally, the repetitive pattern of the coefficients, as a result of coefficient symmetry (41), is reflected in

A COHBF implementation of a demultiplexing DF aiming at *minimum computational load* must exploit the inherent coefficient symmetry (41), cf. Table 8. To this end, we consider the COHBF as depicted in Fig. 17 of Subsection 2.3.1, applying input commutators for sample rate reduction. In contrast to the FDMUX approach of Fig. 22, the SFG of Fig. 17 is based on the transposed FIR direct form Bellanger (1989); Mitra (1998), where the incoming signal samples are concurrently multiplied by the complete set of all coefficients, and the delay chains are directly connected to the output ports. When combining two of these COHBF

√2 j *o* j

(−1)*<sup>o</sup>* <sup>−</sup><sup>1</sup> <sup>−</sup> <sup>j</sup>

(−1)*<sup>o</sup>*

(−1)*<sup>o</sup>* <sup>1</sup> <sup>1</sup>+<sup>j</sup>

*hk*,*<sup>o</sup>*

*hk* <sup>−</sup><sup>1</sup> <sup>−</sup> <sup>j</sup>

*o* = 0, 1

Table 8.

with *Re*{•} = 0

**3.2.2 COHBF approach**

to the respective real RHBF coefficients.

SFG, the coefficient multipliers can obviously be shared with all transfer functions *Ho*(*z*), *o* ∈ {0, 1, 2, 3}; however, the respective outbound delay chains must essentially be duplicated. Merging all of the above considerations, a signal separating DF requiring minimum computation that, in addition, allows for simple channel selection or switching, respectively, is readily developed as follows:


Based on the outlined DF implementation strategy, an illustrative example is presented in Fig. 24 with an underlying RHBF of length *N* = 11. The front end for polyphase decomposition and sample rate reduction by 2 is identical to that of the FDMUX approach of Fig. 22. Contrary to the former approach, the delay chains for the odd-numbered coefficients are outbound *and* duplicated (rather than interlaced) to allow for simple channel selection. As a result, channel selection is performed by combining the respective sub-sequences that have passed the **R**-set coefficients (cf. Table 8) with those having passed the corresponding **I**-set coefficients, where the latter sub-sequences are pre-multiplied by *bi* = (−1)*oi* ; *oi* <sup>∈</sup> {0, 1, 2, 3}, *<sup>i</sup>* <sup>∈</sup> {I, II}.

Multipliers and delays for the centre coefficient *h*0,*oi* signal processing are implemented similarly to Fig. 22 without need for duplication of delays. However, the post-delay inner lattice must be realized for each transfer function individually; its channel dependency follows from Table 8 and (40):

$$\underline{h}\_{0,o\_l} = \frac{h\_0}{\sqrt{2}} (1+\mathbf{j}) \mathbf{j}^{o\_l} = \frac{h\_0}{\sqrt{2}} \left[ (-1)^{\lceil o\_l/2 \rceil} + \mathbf{j} (-1)^{\lfloor o\_l/2 \rfloor} \right],\tag{49}$$

where *oi* ∈ {0, 1, 2, 3}, *i* ∈ {I, II} and *h*<sup>0</sup> = 1/2 according to (9). Rearranging (49) yields with obvious abbreviations:

$$
\underline{h}\_{0,o\_i} = \frac{h\_0}{\sqrt{2}} \left[ (-1)^{o\_i} + \mathrm{j} \right] (-1)^{\lfloor o\_i/2 \rfloor} = \frac{h\_0}{\sqrt{2}} \left[ b\_i + \mathrm{j} \right] d\_i. \tag{50}
$$

It is easily recognized that the inner lattices of Fig. 24 implement the operations within the brackets of (50) with their results displayed at the respective inner nodes A, B, C, D. In compliance with (50), these inner node sequences must be multiplied by the respective signs *di* = (−1)*oi*/2; *oi* <sup>∈</sup> {0, 1, 2, 3}, *<sup>i</sup>* <sup>∈</sup> {I, II}, prior to their combination with the above R/I sub-sequences.

To calculate a set of complex output samples at the two DF output ports, obviously the minimum number of (*N* + 5)/2 real multiplications must be carried out. Furthermore, for

Signal Processing 29

<sup>265</sup> Most Efficient Digital Filter Structures:

APPROACH multiplications/sample delays FDMUX *N* + 3 (3*N* − 5)/2

COHBF (*N* + 5)/2 (5*N* − 11)/2

FDMUX ex.: *N* = 11 14 14

COHBF ex.: *N* = 11 8 22

at the expense of a higher count of delay elements. Finally, it should be noticed that the DF

Using transposition techniques, we subsequently derive DF being complementary (dual) to those presented in Subsection 3.2: They combine two complex-valued signals of identical sampling rate *f*<sup>d</sup> that are likewise oversampled by at least 2 to an FDM signal, where different

An example can be deduced from Fig. 21 by considering the signals *so*(*mT*d) ←→

sampling rates of both signals to *f*<sup>n</sup> = 2 *f*d, and provides the filtering operations shown in Fig. 21(b), *ho*(*kT*n) ←→ *Ho*(ejΩ), *<sup>c</sup>* <sup>=</sup> 0, 2, to form the FDM output spectrum being exclusively

The goal of transposition is to derive a system that is complementary or dual to the original one: The various filter transfer functions must be retained, demultiplexing and decimating operations must be replaced with the dual operations of multiplexing and interpolation,

The types of systems we want to transpose, Figs.22 and 24, represent complex-valued 4 × 2 multiple-input multiple-output (MIMO) multirate systems. Obviously, these systems are composed of *complex monorate* sub-systems (complex filtering of polyphase components) and

While the transposition of real MIMO monorate systems is well-known and unique [Göckler & Groth (2004); Mitra (1998)], in the context of *complex* MIMO monorate systems the *Invariant* (ITr) and the *Hermitian* (HTr) transposition must be distinguished, where the former

detailed in [Göckler & Groth (2004)], the ITr is performed by applying the transposition rules known for real MIMO monorate systems *provided that* all imaginary units "j", both of the complex input and output signals *and* of the complex coefficients, are conceptually considered and treated as multipliers within the SFG<sup>3</sup> (denoted as truly complex implementation), as to

The transposition of an *M*-downsampler, representing a real single-input single-output (SISO) multirate system, uniquely leads to the corresponding *M*-upsampler, the complementary

<sup>3</sup> The imaginary units of the input signals and the coefficients *must not* be eliminated by simple multiplication and consideration of the correct signs in subsequent adders; this approach would transform the original complex MIMO SFG to a corresponding real SFG, where the direct transposition

*<sup>o</sup>* (*z*) = *Ho*(*z*) ∀*o*, as desired in our application. As

*real multirate* sub-systems (down- and upsampler), cf. [Göckler & Groth (2004)].

(dual) multirate system, and vice versa [Göckler & Groth (2004)].

of the latter would perform the HTr [Göckler & Groth (2004)].

), *o* = 0, 2, of Figs.21(c,d) as input signals. The multiplexing DF increases the

Table 9. Comparison of expenditure of FDMUX and COHBF DF approaches

group delay is independent of its (FDMUX or COHBF) implementation.

**3.3 Linear-phase directional combination filter**

*So*(ejΩ(d)

composed of *So*(ejΩ), *o* = 0, 2.

respectively [Göckler & Groth (2004)].

retains the original transfer functions, *H*<sup>T</sup>

be seen from Figs.22 and 24.

oversampling factors allow for different bandwidths.

The Potential of Halfband Filters in Digital Signal Processing

**3.3.1 Transposition of complex multirate systems**

Fig. 24. COHBF approach to *demultiplexing* DF implementation with selectable transfer functions; *<sup>N</sup>* <sup>=</sup> 11, *bi* = (−1)*oi* , *di* = (−1)*oi*/2; *oi* <sup>∈</sup> {0, 1, 2, 3}, *<sup>i</sup>* <sup>∈</sup> {I, II}

Fig. 25. DF separator: Sign-setting for selection of desired channel transfer functions

the COHBF approach to DF implementation a total of (5*N* − 11)/2 delays are needed (not counting shimming delays, *z*−1, and the two superfluous delays at the input nodes of the outer delay chains, indicated in grey).

Finally, we want to show and emphasise the simplicity of the channel selection procedure. There is a total of 8 summation points, the inner 4 lattice output nodes A, B, C, and D, and the 4 system output port nodes, where the signs of some input sequences of the output port nodes must be set compliant to the desired channel transfer functions: *oi* ∈ {0, 1, 2, 3}, *i* ∈ {I, II}. The sign selection is most easily performed as shown in Fig. 25.

A concise survey of the required expenditure of the two approaches to the implementation of a demultiplexing DF is given in Table 9, not counting sign manipulations for channel selection. Obviously, the COHBF approach requires the minimum number of multiplications 28 Will-be-set-by-IN-TECH

Fig. 24. COHBF approach to *demultiplexing* DF implementation with selectable transfer

Fig. 25. DF separator: Sign-setting for selection of desired channel transfer functions

outer delay chains, indicated in grey).

sign selection is most easily performed as shown in Fig. 25.

the COHBF approach to DF implementation a total of (5*N* − 11)/2 delays are needed (not counting shimming delays, *z*−1, and the two superfluous delays at the input nodes of the

Finally, we want to show and emphasise the simplicity of the channel selection procedure. There is a total of 8 summation points, the inner 4 lattice output nodes A, B, C, and D, and the 4 system output port nodes, where the signs of some input sequences of the output port nodes must be set compliant to the desired channel transfer functions: *oi* ∈ {0, 1, 2, 3}, *i* ∈ {I, II}. The

A concise survey of the required expenditure of the two approaches to the implementation of a demultiplexing DF is given in Table 9, not counting sign manipulations for channel selection. Obviously, the COHBF approach requires the minimum number of multiplications

functions; *<sup>N</sup>* <sup>=</sup> 11, *bi* = (−1)*oi* , *di* = (−1)*oi*/2; *oi* <sup>∈</sup> {0, 1, 2, 3}, *<sup>i</sup>* <sup>∈</sup> {I, II}


Table 9. Comparison of expenditure of FDMUX and COHBF DF approaches

at the expense of a higher count of delay elements. Finally, it should be noticed that the DF group delay is independent of its (FDMUX or COHBF) implementation.
