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This process is illustrated in Fig 3.9 using a typical input waveform (a) and a Gaussian-type impulse response (b). Notice that the impulse response starts at the origin so that there is no propagation delay in the filter and the filter outputs begins to respond immediately to an impulse at the input. For reasons of clarity, the input waveform has been replaced by a series of delta functions at t1, t2, etc., separated by Dt where Dt will tend to zero to give the convolution integral. Each impulse excites an impulse response as show in (c), the scaling of which is proportional to the amplitude of ¦(t) and whose time origin coincides with the impulse. The output g(t) at any point (shown in (d)) is calculated by adding the contributions from all the individual impulse response waveforms.
(iii) "Convolution" Theorem - the Transfer Function
Another useful aspect of convolution is know as the convolution theorem. This states that a Fourier transform converts a convolution in the time domain into a multiplication in the frequency domain and vice versa. For example, if F(w) and G(w) are the Fourier transforms of ¦(t) and g(t) and if g(t) = h(t)* ¦(t), then
G(w) = H(w) . F(w) (3.24)
and a proof of this expression is given in Appendix A3. The potential use of this relationship becomes more clear when it is integrated in terms of the excitation and response of a linear system such as a bandpass filter. If ¦(t) is the input and g(t) the output then it has been shown above that h(t) represents the time domain impulse response of the system. Similarly, the Fourier transform H(w) of h(t) is known as the frequency response or transfer function of the system. Thus the output spectrum G(w) is obtained quite simply by multiplying the input spectrum F(w) by the transfer function H(w). Essentially, the transfer defines how each frequency component will propagate (or transfer) through the system. Multiplication in the frequency domain is much less complicated than an equivalent convolution in the time domain.
A practical example of the convolution theorem in operations is illustrated in Fig. 3.10. In this case a multiplication in the time domain leads to a convolution in the frequency domain. The burst of sine waves C has been generated by multiplying the continuous sine wave A by the rectangular pulse B of duration T. In the frequency domain the power spectrum of A is a single spectral line at the oscillation frequency w0. The power spectrum of the pulse is a "sync" function of the form (sin x/x)2 where x = pwT. Thus, since convolution in the frequency domain demands that each spectral line in A should be replaced by the complete spectrum in B, and there is only one spectral line in A, the power spectrum of the sine wave burst consists of the sync function with its origin centred on w0 . The width of this power spectrum is inversely proportional to the length of the sine wave burst and is independent of the sine wave oscillation frequency.
3.1d The Doppler Frequency Spectrum
As a final prelude to the descriptions of Doppler signal analysers, it is useful to examine the properties of a Doppler shift spectrum in order to assess the performance requirements of the frequency analysis system. Although exact spectral characteristics must obviously depend upon the particular choice of ultrasonic transducer, angle of attack, blood vessel and so on, worthwhile design criteria can be obtained by inserting typical values into the Doppler equation. For example, if 10 MHz ultrasound is used to investigate blood flow in a major vessel such as the femoral artery, then assuming an attack angle of about 60 degrees, the Doppler spectrum would extend up to a frequency of about 6 kHz, corresponding to a peak blood velocity of 1000 mms-1 during the passage of the systolic pulse. Furthermore, the blood might accelerate from rest up to this peak velocity in a time interval of less than 0 @ 1 s. (see for example, Chapter 6). Thus the Doppler signal frequency analyser must be able to accommodate frequencies up to 10 kHz and, if it is to follow the changing velocity components, must also be capable of updating the analysis at a rate of at least one hundred spectra per second. This constitutes a typical design specification for analysing Doppler signals from the arterial circulation. However, if a lower ultrasonic frequency were to be used to investigate more slowly moving venous blood, then it would be better to use a lower range of analysis frequencies spaced closer together to maintain the required frequency (and therefore velocity) resolution. Thus a switchable frequency analyser would be more useful where a multi- purpose instrument is required.
3.2 SPECTRUM ANALYSERS
A spectrum analyser is an instrument which can discriminate between frequencies and measure the power level at each. Although spectrum analysers provide the most comprehensive form of Doppler signal processing, they are not always the automatic choice because they can often prove to be unnecessarily complex and expensive. It is not unusual to find that a frequency analysis system complete with display can cost an order of magnitude more than the Doppler flowmeter it is required to serve. In some cases it is more sensible to make use of the Doppler signal processors described later in this chapter. However, in certain situations, for example, where spurious signals have to be recognized and rejected or possibly where signal-to-noise ratios are especially poor, a complete spectral analysis of the Doppler waveform is essential. The problem then comes in deciding which type of spectrum analyser to use.
The are three basic types of spectrum analyser: multichannel (or parallel processing) analysers, swept filter analysers and transform analysers. This section will examine each analysis method in turn describing such concepts as real-time operation, time compression, sampling frequency and aliasing. In addition the relative advantages of analogue or digital implementation will be discussed.
3.2a Multichannel Spectrum Analysers
The multichannel analysers is based on the bandpass filter which can be of either analogue or digital construction. Whichever method is used, the output power of the bandpass unit is proportional to that part of the input spectral power contained within and weighted by the bandpass characteristics of the filter (see Section 3.1b). Using the convolution theorem (Section 3.1c), the output power spectrum G(w)from a bandpass filter can be estimated in the frequency domain simply by multiplying the input spectrum F(w) by the filter characteristic H(w).
(i) Practical Aspects
Figure 3.11a illustrates how a multichannel spectrum analyser can be constructed from a bank of bandpass filters centred on a staggered range of consecutive frequencies from F1 to Fn. The filters themselves can be based on either analogue or digital networks and the relative merits of each will be discussed later. The components of the detector which follows the bandpass network in each channel and the waveforms at each stage are shown in more detail below the main diagram in Fig.3.11b. Each filter output is fed to a full-wave rectifier which inverts the negative half-cycles producing a unipolar waveform. This is then smoothed in an integrator to give a voltage A(Fi) which, so long as the filter band width is kept constant and narrow, is a good estimate spectral density of the input waveform at the filter centre frequency Fi. The smoothed outputs from each channel are then sampled consecutively by a rotating switch (which in more modern form is usually a bank of semiconductor analogue gates opened sequentially using logic pulses generated by a central control unit). These gates are scanned sufficiently rapidly to produce a flicker-free display on an oscilloscope screen, forming a histogram of A(Fi) against Fi as shown in Fig. 3.11a which represents the amplitude frequency spectrum of the input signal. Alternatively the output voltages can be squared to give the linear power spectrum and then logarithmically compressed to give the log power spectrum. Another useful feature usually included on commercially available analysers is a "hold" facility which allows the averaged histogram to be frozen for more leisurely viewing or perhaps permanent recording.
(ii) Performance
The bandwidth of the filter and the smoothing-time constant in the integrator determine the general performance characteristics of the spectrum analyser. The bandwidth of the filter not only defines the frequency (and hence velocity) resolution capability of the analyser but also determines the maximum rate at which the output spectrum can follow input frequency changes. This follows from the convolution theorem which shows (Fig. 3.12a) that the filter output G(w) in the frequency domain can be calculated by multiplying the input spectrum ¦(w) by the bandpass characteristic of the filter H(w). Because a multiplication in the frequency domain is equivalent to a convolution in the time domain, the filter output waveform g(t) can also be obtained by convolving the input signal ¦(t) with the impulse response h(t) of the filter as shown in Fig. 3.12b. The impulse response waveform has by definition a frequency spectrum which is identical to the filter characteristic H(¦) and, as was mentioned in Section 3.1c, can be generated by exciting the filter with a delta function. Furthermore, the impulse response of Gaussian-shaped filter is in the form of an exponentially damped cosine wave oscillating at the filter centre frequency and with a damping constant which is inversely proportional to the filter bandwidth. This means that as the bandwidth is reduced, the frequency resolution improves but the response and recovery time of the filter is increased. Slow response followed by extended filter "ring down" can cause problems if the spectral content of the Doppler signal rapidly with time due, for example, to acceleration of the blood during systole. The filter output lags behind the changing input thereby distorting not only the analysed Doppler spectrum but also any velocity waveforms which might subsequently be computed from it.
The integrator time constant provides a controllable averaging facility which is available on most commercial analysers and which can conveniently be used to improve signal-to-noise ratios and also to reduce the effects of fluctuations in Doppler signal strength caused by the variations in the effective backscattering coefficient of blood described in Chapter 1. However, spectrum averaging must be with caution when analysing rapidly changing Doppler spectra. In fact, in some cases, the band pass filter itself might not be wide enough to follow the rapidly changing signal from an accelerating target. For example, during passage of the systolic pulse through the femoral artery, the blood can accelerate to its peak velocity in less than 0 .1 s. The band width of the analysis channels required to follow this signal must be greater than 40 Hz, which at 10 MHz automatically imposes a lower limit to velocity resolution of about 6 mms-1. In fact this figure is independent of the ultrasonic carrier since the Doppler shift excursions which the filter if required to follow also decrease with ultrasonic frequency. Thus, in order to follow rapidly changing signals, the resolution capability of the analyser must be degraded.
(iii) Effect of Filter Characteristic
Multichannel analysers are generally constructed from banks of constant relative percentage filters (see Section 3.1b (ii)). Octave or third-octave characteristics are usually the preferred choice but whichever type is used the constant Q factor that the lower frequency filters respond slower and are capable of more precise absolute resolution than the higher frequency filters. The analysed spectrum is therefore non-uniform in both its resolution and time response. If required, the spectrum can be normalized by adjusting the gain and averaging characteristics of the integrator stages in the detectors.
3.2b Digital Filters
The multichannel spectrum analyser is perhaps the most straightforward method of processing Doppler signals. However the analogue bandpass filter-bank suffers from two main disadvantages. Firstly, it is inherently inflexible because each filter is turned to a preselected centre frequency so that changing the frequency range of the analyser requires a complete re-design of the filter components. Secondly, the instrument is usually complex and bulky because each processing network including filter, rectifier and integrator has to be duplicated for each frequency analysis channel. The inherent inflexibility of analogue networks can be overcome by using digital filters. This section will describe the basic concepts of digital filtering and will also examine some of the problems encountered when dealing with sampled data. The latter topic will provide a useful basis for the description of time-compression and transform analysers dealt with later.
(i) Basic Principles
A digital filter is really a digital processor which takes in a sequence of input data values and performs a digital operation on them so that the output becomes a filtered version of the input. It is not the intention here to go into the details of digital processing theory which have been dealt with comprehensively elsewhere (see for example, Oppenheim and Schafer (1975), Rabiner and Gold (1975)). Instead the basic principles will be explained by investigating the construction and operation of a simple digital filter network. In addition the technique will be contrasted with other forms of digital signal processors such as transform analysers which are described later in Section 3.2d.
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