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A block diagram of a phase-domain processor is shown in Fig. 2.28b along with the notation describing the waveform after each stage of the processing. A mathematical interpretation of this system is most readily derived by starting with the pair of "cosine only" waveforms given in Eqn (2.60). The corresponding waveforms emerging from the p/2 phase shifters are

Combining the channels as shown in Fig. 2.28b gives

which is the forward flow channel, and

which is the reverse Flow component.

(v) Phase Requirements

The unwanted Doppler sideband will cancel only if each Doppler component is shifted by precisely p/2 radians at the particular Doppler frequency. Thus the p/2 phase shifter is required to delay each component of the Doppler difference waveform by a time period which is inversely proportional to the Doppler difference frequency. (The higher the frequency the shorter the time delay required to produce a p/2 phase shift.) Furthermore, separation of the sidebands requires precise equalization of gain in the two channels otherwise the unwanted sideband will once again not cancel completely. In fact, it can be shown that a one per cent matching in terms of phase and gain will reduce the crosstalk to approximately 40 dB. The practical details of a phase-domain processor incorporating a passive p/2 shifting network are described in Appendix A1.d (ii).

(vi) Frequency Domain Processing

Phase domain processing separates the Doppler signal into distinct forward and reverse Flow channels. This feature is essential if it is required to frequency process the Doppler signals using devices such as zero-crossing counters or mean frequency followers (see Section 3.3). However, if the signal is to be frequency analysed into discrete frequency bands then it is sometimes advantageous to combine the forward and reverse components so that only one spectrum analyser is required. Frequency domain processing of quadrature-phase demodulation signals is a method which essentially shifts the Doppler signals to a pilot frequency whilst maintaining directional information. Figure 2.28c shows a block diagram of a frequency domain processor. The direct and quadrature demodulated channels enter twin multiplying networks fed with the sine and cosine waves at the pilot frequency w_p. When the two multiplier outputs are added together in the summing amplifier the effect is to offset the Doppler difference frequency so that the Doppler sidebands lie on either side of the pilot frequency.

In mathematical terms, if as shown in Fig. 2.28c the direct channel waveform is multiplied by sin w_pt generating D_s(t) while the quadrature channel waveform Q(t) is multiplied by cos w_pt generating Q_c(t), then using Eqn (2.59) the multiplier outputs are given by

Summing the outputs

or, using the trigonometric expressions for sin(A+B) and sin(A-B)

Thus the stationary clutter A is offset to the pilot frequency and the forward and reverse Flow signals have become sidebands lying on either side of w_p. Frequency analysis in a single channel is now possible, producing the type of sonogram shown in Fig. 6.8. As in phase processing, if care is taken to equalize the phase and gain response of each channel, it is usually possible to reduce crosstalk to around 40 dB. Practical details of a frequency domain processor are included in Appendix A1.b.

2.4g Comparison of Directional Systems

By way of a summary, Table 2.1 compares the properties of the various methods of directional demodulation that have been described in this chapter. In the authors' opinion it would seem that single sideband filtering is difficult to implement since the crystal oscillator and filters have to be carefully chosen and matched to the ultrasonic transducer. Furthermore, the output requires some additional processing before it is suitable for display on a single channel spectrum analyser. Conversely, heterodyne detection provides a fully directional display but the output, due to its frequency offset, is unsuitable for processing in zero-crossing counters or peak frequency followers (see Chapter 3). However, quadrature demodulation followed by both phase and frequency domain processing produces good channel separation and offers the added advantage of a fully directional display on a single channel spectrum analyser. Furthermore, this method uses relatively straightforward electronic circuits (see Appendix Al.d).

Time domain processing of quadrature signals is a very basic method which suffers severe limitations and can produce spurious results. It should be used with caution.

Apart from time domain processing, all the remaining techniques when properly tuned ought to be capable of reducing crosstalk between channels to around 40 dB. Once aligned, crystal filters ought to remain stable for a long period of time. Heterodyne and quadrature systems could need more frequent attention to adjust filters and equalize phase and gain performance although practical experience with the quadrature systems described in Appendix AI has indicated no detectable deterioration in performance over periods of several years).

2.4h Summary

This summary brings to an end a chapter in which most aspects of ultrasonic Doppler flowmeters have been mentioned. It has been the intention here to provide the reader with an introduction to the variety of Doppler methods available. As well as describing the elements of each system every attempt has been made to show that although superficially the Doppler principle is quite straightforward, the analysis of any practical application rapidly becomes more complicated. Useful modifications, such as range discrimination, inevitably introduce undesirable effects such as range velocity ambiguities and transit-time fluctuations which eventually limit Doppler performance. The Doppler ultrasonologist ought to be fully aware of such phenomena which could sometimes cause spurious outputs and affect the interpretation of results.

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