DOPPLER3 (1040789), страница 2
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Thus the mean blood velocity is directly related by the Doppler equation to the mean frequency which can readily be computed from the power spectrum. It is emphasized that this relation can be precisely true only if the stringent conditions of uniform insonation, negligible attenuation and homogeneous backscattering are satisfied. If any of these assumptions is not satisfied then the basic premise that the backscattered power is linearly related to the volume of the blood is invalid and so the linking relationship in Eqn (3.7) cannot be applied.
In spite of its shortcomings, it is hoped that this brief analysis has helped to show why the power spectrum plays an important part in Doppler signal analysis. Later in this chapter, various methods for generating the power spectrum of a Doppler signal will be investigated. In preparation for these descriptions it is worthwhile pausing for a moment to examine some of the basic principles of signal processing theory.
3.1b Bandwidth and Bandpass Filters
"Bandwidth" is a term which is often used in signal processing jargon and it is important to understand its true meaning and how it relates to frequency resolution. The term originates from the use of bandpass filters which pass only that part of the spectrum which lies within a narrow range of frequencies, know as the passband. The power transmission characteristics of a so-called "ideal" bandpass filter are show in Fig. 3.5. The filter transmits with zero attenuation all components lying within the bandpass region which is of width B centred on the filter frequency. All other frequency components, that is those lying outside the passband (in the so-called "stop band") are totally attenuated.
The characteristics of any practical filter network will differ from those of an ideal filter in three major respects: first, the power transmission will not be completely uniform over the passband and this effect is know as passband ripple; second, signals in the stopband can never be completely eliminated and produce stopband ripple; thirdly, the sides of the filter characteristic will not be vertical but will fall away gradually at the filter roll-off rate. These characteristics are also illustrated in Fig. 3.5.
For the case of a practical filter, the effective noise bandwidth is defined as being the width of an ideal filter which would transmit the same power as the practical filter when both filters are fed with the same white-noise source. Since white noise has a flat frequency spectrum, this is equivalent to saying that the integrated areas under the power transmission characteristics will be the same in both cases. Figure 3.5 shows ideal and practical bandpass filters with the same effective noise bandwidth.
(i) Record Length
It is important to realize that the bandwidth associated with any particular analysis need not necessarily be determined by the filter bandwidth. (This will become more obvious when transform analysers are described later in Section 3.2d since they do not use conventional filtering techniques). The other parameter on which the effective bandwidth depends is the length of the record on which the analysis is made. In fact it can be show that if the record length is limited to T seconds then the maximum bandwidth possible is 1/T Hz. The origins of this very simple relationship will become clearer in the next sections which deal with concepts of convolution. The relevance of the relationship, however, is immediately apparent: at a Doppler signal is restricted in length to time T then the frequency resolution will be restricted to 1/T. Thus the longer a moving target can be interrogated, the more precisely its velocity can be measured. It might remembered that it was this principle which formed the basis of the spatial-velocity resolution limitations discussed in Sections 2.2b and 2.3a (viii).
(ii) Filter Response Time
When a signal is suddenly applied to the input of a bandpass filter, it takes time for the output to respond. If, for example, the input is a sine wave at a frequency within the passband (Fig. 3.6a) then the filter output will be a sine wave which gradually increases to its final amplitude as show in Fig. 3.6b.
If the filter has a passband gain characteristic of unity then the final output amplitude will be equal to the input amplitude. Also the time T¦ required for the output to approach this final value is inversely proportional to the filter bandwidth B. This may be expressed as
B . T¦ ~ 1 (3.12)
which is another way of saying that a measurement with a bandwidth @ requires a measurement time of at least 1/B.
(iii) Filter Characteristics
Before leaving the subject of bandwidth and bandpass filters, it is worthwhile examining the two basic types of bandpass characteristic which are in common use. These are the constant absolute bandwidth filter and the constant relative (or percentage) bandwidth filter where the absolute bandwidth is a fixed percentage of the tuned centre frequency. Figure 3.7 shows the two characteristics for the case of an ideal rectangular filter at three different centre frequencies plotted on a linear frequency scale.
The constant bandwidth filter maintains uniform resolution no matter what the centre frequency. Thus, because the bandwidth remains fixed, the
response time of the is independent of its tuned centre frequency. Conversely, the bandwidth of a constant percentage filter increases with tuned frequency. This means that the absolute bandwidth is very narrow at low frequencies (show response) and becomes wider (faster response) at higher frequencies. Constant percentage bandwidth a constant Q ( = B/¦0) and so such filters are relatively easier to design if conventional analogue techniques are used. Two special classes of constant percentage bandwidth filters are widely used, namely octave and third-octave filters. The upper limiting frequency ¦u of an octave filter is a always twice the lower limiting frequency ¦l that the bandwidth B is given by
B = ¦u - ¦l (3.13)
or, since ¦u = 2¦l
B = ¦l (3.14)
If the centre frequency ¦c is defined as the geometric mean of these two limits, then
or
The relative percentage bandwidth, also know as the reciprocal of Q, is given by
or
Thus an octave filter is always characterized by a bandwidth is about 70% of its tuned frequency.
Similarly, third-octave filters are generated by dividing an octave filter into three geometrically-equal subsections. The upper and lower limits of a third-octave filter are related by the expression
¦u = 21/3¦l (3.19)
The nominal centre frequency of a third-octave filter is
and so the relative percentage bandwidth is
Thus a third-octave filter is always characterized by a bandwidth is about 23% of its tuned centre frequency.
3.1c Convolution
Convolution is an extremely useful mathematical procedure which can be used to predict how an analysis system will perform in both the time and frequency domains. However, before its full use can be appreciated, it is necessary to review the related topics of the delta function and impulse response.
(i) The Delta Function
The delta function is another mathematical concept which is of considerable use in signal processing theory. A typical delta function occurring at time t = t0 is conventionally represented as d(t - t0) and has the property that is value is zero everywhere except at time,
t = t0 where its value becomes infinite (see Fig.3.8a), Furthermore, the function is considered to be infinitely narrow so that the result of integrating over any time which includes t0 gives unity. In effect, the delta function can be considered to be the limiting case of any pulse with unit are whose duration is made infinitesimally short while its height is made infinitely large, thus maintaining the enclosed unit area. The function can also be scaled by a factor which might or might not have physical dimensions. Integrating over a time period which includes the delta function then gives the value of this scaling factor. For example, the frequency power spectrum of a continuous sine wave is a delta function located at the oscillation frequency. The scaling factor is chosen so that the power measured in the time domain corresponds to the power measured in the frequency domain.
Figure 3.8 illustrates the properties of a delta function in the time and frequency domains. Notice that the spectrum contains all frequency components from -¥ to +¥ at a constant amplitude. It is this feature which makes the delta function so very useful for analysing systems since it provides an input which can interrogate the response of the system at all frequencies.
(i) Impulse Response
As its name implies, the impulse response of a system is the output waveform excited by a impulse at the input. In theory, the impulse should really be a true Dirac delta function whose spectrum covers all frequencies but in practice it is possible to use a rectangular pulse which is much shorter that the minimum time constant of the system. (Or in other words the spectrum (B = 1/T) of the input must be essentially flat over the frequency response of the system).
Returning to convolution, it is first necessary to define the principal mathematical terms and also to give practical examples of its application so that its physical meaning can be appreciated. The convolution g(t) of two time functions ¦(t) and h(t) is written as
where t is a dummy time (or delay) variable. For convenience Eqn (3.23) is often expressed as
g(t) = ¦(t) * h(t) (3.23)
where the symbol * is the notation for "convolved with". Thus ¦(t) is convolved with h(t) to give g(t).
As a practical example, suppose ¦(t) represents the input to a filter which has an impulse response h(t). The convolution principle can be used to predict that the filter output will be g(t) as given by Eqn (3.22). To understand the reason for this, consider the input waveform ¦(t) expressed as a function of the time variable t (t runs along the same time axis as t and is distinguished from t only for the purpose of convenience in this analysis). Each point t on ¦(t) is considered as being a delta function d(t) weighted by the scaling factor ¦(t). Each impulse excites an impulse response h(t) from the filter with scaling factor ¦(t) and originating at time t. The output signal g(t) at time t is the sum of these scaled impulse response waveform, each delayed by the appropriate time interval (t - t) which corresponds to the difference in time between excitation (t) and measurement (t). (Causality demands that the difference is (t - t) and not (t - t) since the response cannot be measured before it has been excited!) Thus in the limit g(t) is given by the integral of Eqn (3.22) which represents the summation of the scaled impulse responses generated by the input train of delta functions which go to make ¦(t) .
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