**2.2 Conventional analog signal-processing**

An analog phase-shift processing system that consists of all-pass filters has been used in the direction separation processing. The outline of it is shown in Fig. 3. This is a processing system that shifts the phase between the IQ-signals of 90 degree, and adds them or subtracts them. Since an all-pass filter has the characteristic that the phase reverses on cut-off frequency, this system shifts the phase in a target frequency range combining all-pass filter arrays. If it assumes that the input IQ-signal *x(t)* has a frequency component of Z*d* .

$$\mathbf{x}(t) = \exp(j \cdot o\_d \cdot t) \tag{4}$$

Complex Digital Filter Designs for Audio Processing in Doppler Ultrasound System 215

*PQ t x t j* ( ) Im ( ) exp( ( ( ))) sin( ( )) > @ *<sup>d</sup>* I Z

( ) ( ) ( ) ( ( ) 1) sin( ( )) *Forward t PI t PQ t sign d d*

Re ( ) ( ) ( ) ( ( ) 1) sin( ( )) *d d verse t PI t PQ t sign*

IQ-signals are separable into positive-component and negative-component. Comparison of direction separation performance is shown in Fig. 5. The frequency-characteristic in the velocity range 4kHz (-2kHz to +2kHz) that is well used in diagnosis of the cardiac or abdomen is shown. A solid line shows the positive-component (*Forward*) and a dashed line shows the negative-component (*Reverse*). The direction separation performance of the phaseshift system (conventional analog system) is shown in Fig. 5(a), and the direction separation performance of the complex IIR filter system (digital system referenced in section 3.2) is

Z

(a) the phase-shift system (analog)

*Reverse*

*Reverse*

(b) the complex IIR system (digital)

In Fig. 5, a filter-order to which hardware size becomes same is set up. In the complex IIR filter system, sufficient separation performance (more than 30 dB) is got except for near a low frequency and near the Nyquist frequency. On the one hand a ringing has occurred by the phase shift system, there is little degradation near the Nyquist frequency. Although the direction separation performance near a low frequency and near the Nyquist frequency can improve if the filter-order is raised in the complex IIR filter system, the processing load becomes large. It is although the ringing will decrease if range of the phase-shift system is

Z

> Z

Z

/ 2 when Doppler frequency

Here, D is S

negative. So ( ) *<sup>d</sup> sign*

zero output. And when

shown in Fig. 5(b).

addition-output *Reverse(t)* are

Z

From the formulas (7) and (8), when

Power (dB)

Power (dB)

Fig. 5. Direction separation performance

divided finely, processing load becomes large similarly.

Z

 Z

*d* is positive, and

means the polarity. The subtraction-output *Forward(t)* and the

*<sup>d</sup>* is negative, only the *Reverse(t)* serves as a non-zero output. Thus,

*Forward*

*Forward*

 IZ

Z

Z

*<sup>d</sup>* is positive, only the *Forward(t)* serves as a non-

Frequency (Hz)

Frequency (Hz)

D is S

> IZ

> > IZ

*t* (6)

/2 when

*t* (7)

*t* (8)

Z*<sup>d</sup>* is

Fig. 3. Outline of analog direction separation system

Fig. 4. Frequency characteristics of all-pass filters

In Fig. 4(b), the phase characteristics of I-channel and Q-channel are delayed as frequency becomes high. Here, the phase characteristics of I-channel and Q-channel are defined to be I( ) Z D and I( ) Z , respectively. The output of I-channel and Q-channel are set to *PI(t)* and *PQ(t)*.

$$PI(t) = \operatorname{Re}\left[\mathbf{x}(t) \cdot \exp(\cdot (\phi(\alpha o) + \alpha))\right] = \operatorname{sign}(\alpha o\_d) \cdot \sin(\alpha o\_d \cdot t + \phi(\alpha))\tag{5}$$

214 Applications of Digital Signal Processing

Phase-shifter 䃥㻔䃨㻕

Phase-shifter 䃥㻔䃨㻗䃐㻕

Frequency (Hz) Frequency (Hz)

(a) Phase characteristics of *PI*, *PQ* and

All-pass filter H4(z)

All-pass filter H3(z)

Phse-shifter 䃥(䃨) (degree)

filter Subtraction

All-pass filter H8(z) *PI(t)*

*PQ(t)*

All-pass filter H7(z)

H3 H5

Frequency (Hz)

 Z

 IZ*sign t* (5)

Frequency (Hz)

䃐

Phse-shifter䃥(䃨+䃐)

All-pass filter H6(z)

All-pass filter H5(z)

H1,H3,H5,H7 (degree)

H1

䃐

Addition

Phse-shifter 䃥(䃨+䃐) (degree)

H7

*Forward(t)*

*Reverse(t)*

Low-pass filter

I-channel signal

Q-channel signal

H2,H4,H6,H8 (degree)

I( ) Z Dand

*PQ(t)*.

H2

Low-pass filter

(fc=20kHz)

(fc=20kHz)

All-pass

All-pass filter H1(z)

H4 H6 H8

Phse-shifter䃥(䃨)

Phase (degree)

*PI*

Phase (degree)

Fig. 4. Frequency characteristics of all-pass filters

I( ) Z

*QI*

(b) Difference

䃐

䃐

I Z D

between *PI* and *PQ*

, respectively. The output of I-channel and Q-channel are set to *PI(t)* and

 Z

In Fig. 4(b), the phase characteristics of I-channel and Q-channel are delayed as frequency becomes high. Here, the phase characteristics of I-channel and Q-channel are defined to be

*PI t x t j* ( ) Re ( ) exp( ( ( ) )) ( ) sin( ( )) > @ *d d*

Fig. 3. Outline of analog direction separation system

H2(z)

$$PQ(t) = \operatorname{Im} \left[ \mathbf{x}(t) \cdot \exp(\cdot (\phi(\alpha))) \right] = -\sin(\alpha \mathbf{\bar{\jmath}} \cdot t + \phi(\alpha)) \tag{6}$$

Here, D is S / 2 when Doppler frequency Z*d* is positive, and D is S /2 when Z*<sup>d</sup>* is negative. So ( ) *<sup>d</sup> sign* Z means the polarity. The subtraction-output *Forward(t)* and the addition-output *Reverse(t)* are

$$Forward(t) = PI(t) + PQ(t) = (sign(o\_d) + 1) \cdot \sin(o\_d \cdot t + \phi(o))\tag{7}$$

$$\text{Re}\,\text{vers}(t) = PI(t) - PQ(t) = (\text{sign}(o\_d) - 1) \cdot \sin(o\_d \cdot t + \phi(o))\tag{8}$$

From the formulas (7) and (8), when Z*<sup>d</sup>* is positive, only the *Forward(t)* serves as a nonzero output. And when Z*<sup>d</sup>* is negative, only the *Reverse(t)* serves as a non-zero output. Thus, IQ-signals are separable into positive-component and negative-component. Comparison of direction separation performance is shown in Fig. 5. The frequency-characteristic in the velocity range 4kHz (-2kHz to +2kHz) that is well used in diagnosis of the cardiac or abdomen is shown. A solid line shows the positive-component (*Forward*) and a dashed line shows the negative-component (*Reverse*). The direction separation performance of the phaseshift system (conventional analog system) is shown in Fig. 5(a), and the direction separation performance of the complex IIR filter system (digital system referenced in section 3.2) is shown in Fig. 5(b).

Fig. 5. Direction separation performance

In Fig. 5, a filter-order to which hardware size becomes same is set up. In the complex IIR filter system, sufficient separation performance (more than 30 dB) is got except for near a low frequency and near the Nyquist frequency. On the one hand a ringing has occurred by the phase shift system, there is little degradation near the Nyquist frequency. Although the direction separation performance near a low frequency and near the Nyquist frequency can improve if the filter-order is raised in the complex IIR filter system, the processing load becomes large. It is although the ringing will decrease if range of the phase-shift system is divided finely, processing load becomes large similarly.

Complex Digital Filter Designs for Audio Processing in Doppler Ultrasound System 217

Since spectrum image signal processing involves 256-point FFT, an acceptable frequency (velocity) resolution is obtained. However, when the frequency resolution of the Doppler audio is unacceptable, similar to that of a small-pitch Doppler image, we set the target resolution to be *fs*/100. The frequency range is determined from sample frequency. However, the frequency resolution is proportional to the reciprocal of observation time. For example, in FFT, it is equivalent to the main robe width of the sampling function

Although operation load is dependent on the hardware-architecture, such as DSP, ASIC, and FPGA, lighter load is more advantageous to cost, size, and power consumption in

Six kinds of digital signal-processing systems that were pre-existing or newly devised are

(a) the Hilbert transform system (b) the complex FIR system (c) the complex IIR system

(d) the FFT/IFFT system (e) the modulation/demodulation system (f) the phase-shift system

The delay of (filter tap length)/2 is given to I-channel of IQ-signals. It and the Hilbert transform output of Q-signal are subtracted or added. The direction separated signals are calculates by formulas (9) and (10). Here a convolution is indicated . The tap number is

*F n X n nta* 1( ) Re ( / 2) Im ( ) 1( ) *p X n h ntap*  (9)

*R n X n nta* 1( ) Re ( /2) Im ( ) 1( ) *p X n h ntap*  (10)

determined from observation time width and the window function.

**3.2 Six kinds of digital signal-processing ideas** 

Fig. 6. Six kinds of digital signal-processing systems

set to 128 in the estimation of the calculation load shown in Table 3.

The coefficient *h1* of Hilbert transform is given by a formula (11).

**Hilbert transform system:** 

examined. They are shown in Fig. 6.

**Frequency resolution:** 

**Calculation load:** 

common.
