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It will be remembered from Section 2.4e(ii) that heterodyning is a process whereby a signal is multiplied by a sinusoid in the time domain such that the effect in the frequency domain is to shift the origin of the signal to the frequency of the sinusoid. In mathematical terms, if the sinusoid is represented by exp (iw_ht) and the signal is ¦(t), then the heterodyned signal h₁(t) is given by

h₁(t) = ¦(t) ^. e^iwht

The Fourier transform H₁(w) of h(t) is

Since Eqn (3.27) has the form of a Fourier integral with (w-w_h) replacing w it can be rewritten

H₁(w) = F(w-w_h) (3.28)

where F represents the Fourier transform of ¦(t). Thus the spectrum of a function multiplied by a sinusoid at a frequency w_h is the spectrum of the function with its origin shifted to w_d. Figures 3.l7a-d show the heterodyned Doppler spectrum as the heterodyne frequency w_d is swept downwards infrequency. It can be seen that this causes the fixed-frequency bandpass filter to effectively sweep upwards in frequency through the Doppler spectrum. If w_h is swept sufficiently slowly then the detector output draws out a replica (e) of the frequency input spectrum.

In the practical instrument, the magnetic tape loop on which the Doppler signal is recorded forms the outer surface of a revolving disc. This disc is mounted coaxially with a display drum as shown diagrammatically in Fig. 3.18. This display drum is metallic and is covered with a charge sensitive paper which darkens when current flows through it. A marker pen is fixed to a screwed vertical drive and supplied with a current which is proportional to the detected output from the bandpass filter. This pen touches the current-sensitive paper and marks a spot on the paper whenever there is an output from the detector. Over a limited dynamic range the intensity of the mark is related to the detector output level, Also, the vertical position of the pen is monitored by a potentiometer and the tapped output is used to define the heterodyne frequency generated in the voltage controlled oscillator (or VCO). The potentiometer is connected so that when the pen is at its lowest position, marking the bottom of the paper, the heterodyne frequency is set so that the filter passes Doppler shifts near DC. When the pen is at its highest position, marking the top of the paper, the filter passes Doppler signals at, for example, 10kHz. Thus, as the pen traverses the paper, the filter effectively scans a frequency range from 0-10 kHz through the Doppler spectrum.

To analyse the signal the Doppler record is replayed at a faster speed through the heterodyne filter by rotating the disc and drum. (The bandpass filter and heterodyne frequency are suitably scaled to take account of the replay/record ratio.) The vertical screw drives the marker pen slowly up the display drum and the potentiometer causes the heterodyne frequency to follow this traverse. Pen movement is synchronized to the drum rotation rate so that the bandpass filter increments by approximately its own bandwidth for each complete replay of the Doppler record. This compromise ensures maximum frequency resolution without redundant replays, Thus for each rotation of the drum, the Doppler record is effectively replayed through a bandpass filter at a frequency determined by the vertical position of the marker pen. Furthermore, the detected output level is continuously indicated by the intensity of the mark drawn on the current sensitive paper. A typical display produced from a Doppler record of the femoral artery is shown in Fig. 6.8. The horizontal axis represents time and the vertical axis frequency. The degree of blackening indicates the detected output level at the particular frequency and time. Thus, this intensity modulation can be interpreted as representing the amount of blood flowing at any particular velocity and time. This type of display is generally known as a "sonogram" and is referred to extensively in the clinical sections of this book.

(iii) Time Compression Analysers

In general, the time taken to analyse a signal into a bandwidth B is approximately 1/B. The only way of decreasing the analysis time is to increase the bandwidth (see Section 3.1b). In order to maintain the original frequency resolution the signal spectrum must be similarly expanded by recording the signal at one speed and then playing it back at a faster rate. The time-compression analyser is essentially a special case of the swept filter analyser where the recorded signal is replayed at such a fast rate that real-time operation can be maintained. As was mentioned in the previous section, if the time required to analyse the recorded data is to be less than the time required to acquire that data, then the speed-up ratio n must be equal to or greater than the number of analysis channels, N. Time compression analysers with real-time capabilities in excess of 10 kHz are commercially available using digital memory as the high-speed recording medium.

Figure 3.l9 illustrates a typical layout of a time-compression analyser based on a recirculating memory. The incoming waveform is low-pass filtered to remove unwanted components and so prevent aliasing effects (see Section 3.2b (ii)). The signal is then sampled and digitized in the analogue to digital converter (ADC). The sampling rate is chosen to be greater than about three times the anti-aliasing cut-off frequency so that, for example, Doppler signals extending to l0 kHz would be sampled at not less than 30kHz. Each new digitized word of data is stored in memory in such a way that it replaces the most ancient data. In this way the memory is continuously updated and always contains the most recent data epoch ready for analysis. The memory is then replayed repeatedly at a faster rate through a digital-to-analogue converter (DAC) and then fed to a swept filter analyser with a suitably increased bandwidth NB, where B is the required frequency resolution. Because the bandwidth is wider, the filter can be swept more rapidly through the repeating waveform. (The operations required to write data into the correct storage location while recirculating the memory to produce the time-compressed output requires complex logic control. In the schematic shown in Fig. 3.19, the necessary write-read commands are generated by the memory input/output controllers.)

A simple example should help illustrate the operating principles of the device. Suppose the sampling frequency is 30 kHz and also that the memory is 1200 words long representing 400 (= 1200/3) cycles at the highest frequency (10 kHz) passed by the anti-aliasing filter. At any one time, the memory contains the most recent 40 milliseconds (= 1200/30 kHz) of signal.

The frequency resolution is limited to 25 Hz ( = 1/40 milliseconds) and so the spectrum comprises 400 equally-spaced lines extending from 25 Hz to 10 kHz. If real-time operation is to be maintained then the waveform in the memory must be analysed into 400 lines in one memory period, that is, in 40 milliseconds. This can be done by speeding-up the replay by 400 times giving a word rate of 12 MHz (= 400 x 30 kHz) and an effective memory length of 100 µs. Each of the 400 lines now takes 100 µs to analyse in a l0 kHz bandwidth swept filter which sweeps from 10 kHz (25Hz x 400) to 4MHz (10 kHz x 400) to cover the input signal frequency range. The total analysis time is therefore 40 milliseconds ( = 400 x 100µs) which, because it equals the time taken to acquire the data, means that real-time operation can be maintained.

To change the frequency range it is merely necessary to alter the input sampling rate and the anti-aliasing cut-off frequency. The output spectrum will always comprise 400 equally-spaced lines extending from DC to one-third the sampling frequency. The upper limit to the frequency range is ultimately limited by the maximum word recirculation rate. which is usually in the region of 12 MHz.

3.2d Fast-Fourier Transform Analysers

Fourier analysis provides the most direct link between the time and frequency domains and is therefore a useful toot in Doppler signal processing. The mathematical basis for all frequency analysis procedures is the Continuous Fourier Transform (CFT) which has long been used to investigate the spectral content of waveforms. However, when a waveform is sampled so that the spectrum can be analysed on a digital computer, the finite discrete version of the Fourier Transform, the Discrete Fourier Transform or DFT must be used. The Fast Fourier Transform (FFT) is simply an efficient method for calculating the DFT. Since it was first published by Cooley and Tukey (1965), the FFT algorithm has revolutionized the field of signal analysis. At first the calculation could only be implemented on large computers in a high level language. Nowadays the algorithm is performed rapidly in dedicated micro-processors making possible high-resolution real-time spectral analysis.

A comprehensive description of the FFT is beyond the purpose of this text and the interested reader is referred to an excellent primer on the topic by Bergland (1969). However, the' FFT analyser represents such an important part of Doppler signal analysis that some basic aspects of the transform and its properties will be examined here. It is also worthwhile describing the operation of spectrum analysis instrumentation based on the FFT principle.

(i) Mathematical Basis

In its most general form the CFT is described by the Fourier integral pair of equations (3.1). The discrete equivalents of these integrals can be written

and these two equations are known as the Discrete Fourier Transform or DFT. Equation (3.29a) is the forward translation and Eqn (3.29b) the inverse transform. By examination it can be seen that each of the continuous functions F(w) and ¦(t) which extend from -µ to +µ in the CFT formulation given in Eqns (3.l) has been replaced by a finite number N of discrete samples F(k) and ¦(n) respectively. If D¦ is the frequency increment between the N samples and Dt the time increment then k D¦ is equivalent to w and n Dt corresponds to t. The advantage of the DFT is that even though the unmanageable infinite integrals of the CFT have been replaced by finite summations, the DFT still retains many useful properties of the CFT.

Because the DFT is formulated as at summation over a finite series of points it is ideally suited for processing on a digital computer. However, direct calculation is a lengthy and inefficient procedure requiring some N² complex multiplications and additions to complete an N point transform. The FFT reduces the number of multiplications required to approximately Nlog₂N and the drastic relative reduction in computation as N increases is illustrated in Fig. 3.20.

The first formulation of the FFT devised by Cooley and Tukey comprises a set of recursive equations operating on the sampled input waveform which must be an integer power of two (i.e., 2ⁿ) samples long. In the generalized forward transform N complex time domain points (Fig. 3.21a) transform into N complex frequency domain points which are evenly spaced from zero up to the sampling frequency. However, it can be shown that the transform is periodic with a repeat cycle equal to the sampling frequency l/T as illustrated in Fig.3.21b. Furthermore, if the time domain points are real valued (which is always the case in Doppler signal analysis) then the transform will be conjugate-even which means that the positive frequency components can be generated from the negative ones (and vice versa) as shown in the figure. Thus it can be seen that only the N/2 positive complex frequency points from 0 to N/Z are needed to define the complete transform of the N input data points. Each of the frequency points will have the complex form (not shown in Fig. 3.21b) F(i) = a_i+jb_i and for each positive components there will be a corresponding negative component
F(i) = a_i-jb_i. The power P_i contained in the i-th frequency component (frequency i D¦) is the sum of the powers of the positive and negative frequency components so that

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