**5. References**

234 Applications of Digital Signal Processing

components blood noise clutter power -6dB -20dB 0dB frequency 0.24\**fs* (white noise) -0.08\**fs*

On the basis of the Doppler IQ-signal of the carotid artery collected with the actual Doppler ultrasound system, an example of anti-aliasing signal processing of the Doppler audio is shown in Fig. 22. We use a string phantom (Mark 4 Doppler Phantom: JJ&A Instrument Company) and the ultrasonic diagnosis equipment (SSA-770A: Toshiba Medical Systems Corporation) for generating and collecting the Doppler signal. We use PLT-604AT (6.0 MHz linear probe) at PRF=4 kHz equivalent to *fs*. We collect the IQ-data in PWD mode. Moreover, we set cut-off frequency at an HPF of 200 Hz for clutter removal. The output waveforms of both sides of the Doppler audio and spectrum image obtained from the IQdata are shown in Fig. 22. In this figure, in the vicinity of 0.9 s, the baseline-shift is switched into -0.4\**fs* from 0. At the zero baseline-shift, we observe aliasing in the spectrum image shown in Fig. 22(a) and a negative-side output in Fig. 22(c). However, we confirm that the positive-side display range of the spectrum image expands after a baseline-shift and is interlocked with the Doppler audio. Although it is not observed in Fig. 22, the characteristic of the band-pass filter changes immediately after a baseline-shift. We will continue to examine the transient response of the Doppler audio under this effect and to consider

(a) spectrum image

Time (s)

Time (s)

Time (s)

(b) forward output

(c) reverse output

We developed the direction separation system of a Doppler audio interlocked with the antialiasing processing of a spectrum image using a complex IIR band-pass filter system.

Table 8. Components of simulation input model

implementation technologies, such as muting.

Fig. 22. Doppler spectrum display and audio output waveform

Frequency (kHz)

Amplitude (V)

Amplitude (V)

**4.6 Conclusion** 

**4.5.2 Implementation** 


Heinz G. Göckler

*Germany*

**12**

<sup>2</sup> ) point

*Digital Signal Processing Group, Ruhr-Universität Bochum*

**The Potential of Halfband Filters** 

**in Digital Signal Processing** 

**Most Efficient Digital Filter Structures:** 

<sup>2</sup> ) and half magnitude ( <sup>1</sup>

<sup>n</sup> ), in tree-structured filter banks for FDM de- and

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

[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

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

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);

an odd symmetry about the quarter sample rate (Ω = *<sup>π</sup>*

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>*

**1. Introduction**

(2001)].

Zhang, Y., Wang, Y. & Wang, W. (2003). Denoising quadrature Doppler signals from bidirectional flow using the Wavelet frame, *IEEE Transactions on UFFC,* Vol.50, No.5, pp561-566.
