DOPPLER3 (1040789), страница 4
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Figure 3.13a shows how a digital low-pass filter of a type commonly used for exponential averaging can be constructed from the basic units comprising multiplier delay line and summing amplifier. In fact these active elements can be either analogue or digital in form but, with the increased availability of micro-electronics, digital implementation is becoming increasingly convenient. The input data is a sequence of samples, generated by taking the value of the input waveform at regular time intervals defined by the sampling frequency ¦s. For each sample period the new output value is generated by adding the current input data value to A times the previous output which has been stored in the delay line. In order to understand how this procedure results in a filtering action consider how this circuit responds to a unit amplitude impulse when A equals say 0.5. (It is assumed that one of the sample clock pulses is timed so that it captures the impulse producing the input data train shown in Fig. 3.13b). Before the impulse arrives both input and output equal zero. When the impulse enters the filter the previous output (=0) is multiplied by 0.5 (giving zero) and added to the new input (=1) to give the new output (=1). At the next clock pulse, the previous output (=1) is multiplied by 0.5 (giving 0.5 ) and added to the input (=0) to give new output (= 0.5). A similar set events occur at each clock pulse producing the output data sequence (1, 0.5, 0.25, 0.125,...) as shown in Fig. 3.13c: this is the digital equivalent of a decaying exponential waveform. Thus the relatively straightforward digital filter configuration gives a useful low-pass or exponentially averaging filter characteristic.
There are two important points to notice: firstly, digital filtering is a continuous operation whereby every value produces a processed output value. This is in marked contrast to transform analysers which input and then analysers complete blocks of data at a time. Secondly, if the multiplier coefficient A if held constant then the filter response time is defined solely by the sampling frequency ¦s. Increasing the sampling frequency shortens the exponential decay rate while decreasing ¦s lengthens the decay. In the frequency domain this is equivalent to increasing and decreasing the cut-off frequency of the low-pass averaging filter which this digital network represents. However, the shape of the filter remains fixed since this is defined by the filter configuration and multiplication coefficients.
More complicated bandpass filters can be constructed using the basic units of delay line, multiplier and adder. Figure 3.14 shows a block diagram of a general two-pole digital filter. The multiplier coefficients H0, A0, A1, A2, B1, B2 define the shape of the response and by suitable selection it is possible to generate digital equivalents of all the well known bandpass filter types such as Butterworth, Chebyshev, Bessel and so on. In fact it is even possible to design digital filters which are not physically realizable in analogue form, but these generally have peculiar gain and phase properties which are not particularly useful for signal analysis. Like analogue filters multiple pole digital filters with steep roll-offs can be formed by cascading a series of two-pole networks with appropriate choice of multiplier coefficients.
In conclusion, the main advantage of the digital filter is that the same basic units can be construct almost any required filter shape simply by changing the filter coefficients. For example a high-pass filter has the same construction as a low-pass unit but with different coefficients. Furthermore, the centre or cut-off frequency is determined solely by the data sampling rate. Once constructed, the digital nature of the device means that the filter characteristics will remain stable and the filter should never need trimming.
(ii) Aliasing
It has been assumed in the above description of digital filters that the sampled signal fed to the filter is an accurate representation of the analogue waveform. This will only be the case if the input waveform is sampled at a sufficiently high rate. Consider the two cases of waveform sampling shown in Fig. 3.15. In each case the signal frequency is ¦ and the sampling frequency is ¦s. If, as shown in (a) the sampling frequency is much greater than the waveform frequency then the data output can be reconstructed to form an accurate representation of the input waveform. However, if the sampling rate is less than one-half the waveform frequency, that is¦s < ¦/2, then the sampled data will be a sine wave at some distorted frequency as shown by the dashed line in Fig. 3.15b. This misinterpretation of high frequencies which are above one-half the sampling frequency is known as aliasing. It is commonly encountered as the stroboscopic effect which causes cartwheels in Western films to appear to rotate backwards, or slowly forwards, due to the sampling effects induced by the filming. In general, the maximum frequency which can be accommodated in a sampled system is defined by Shannon's Theorem which states that a sampled time signal must not contain components at frequencies above half the sampling rate. In other words, the maximum allowed analogue frequency ¦max is given by
and is generally known as the Nyquist frequency. Components above the Nyquist limit must be removed by analogue anti-aliasing filters before the signal is sampled. Thus, the user must be aware of the spectral content of the input signal before choosing the sampling frequency. Otherwise aliasing effects will shift non-allowable frequencies above ¦s/2 so that they appear as components below the Nyquist limit.
3.2c Swept Filter Analysis
Digital filtering removes the limitations of fixed frequency ranges imposed by analogue filter networks. However, it does nothing to solve the problem of a large component count: each analysis channel still requires its own bandpass filter following by a detector circuit. This section will investigate the technique of swept filter analysis, which as the name suggests uses a single filter channel which is swept over the frequency range of interest.
(i) Basic Principles
Suppose for the moment that the Doppler shift spectrum does not change with time and that the Doppler signal always contains the same frequency components. Suppose also that it is possible to design a bandpass filter channel whose centre frequency can be varied in a controlled manner. The Doppler spectrum could then be analysed by slowly sweeping the bandpass region of the filter through the frequency content of the Doppler spectrum. This process is illustrated in Fig. 3.16. The output of the ideal rectangular bandpass filter is detected (rectified and smoothed) in exactly the same way as for the multichannel analyser described in Section 3.2a. The detector output is proportional to the area (shown shaded) in Fig. 3.16 which is common to both filter and spectrum. This represents the power if the input spectrum in the region of the filter. Thus, as the filter is swept in frequency (a-d) the detector output sweeps out a profile (e) of the input Doppler spectrum. The rate at which the bandpass filter can be swept depends upon the filter settling time: the faster the output settles, the sooner the analysis can move on to the frequency point.
The advantage of the swept filter concept is that unlike multichannel analysis, only one signal processing channel is required. Unfortunately, direct application of the swept filter principle to Doppler analysis simply would not work for the basic reason that the Doppler frequencies are not constant but change rapidly with time during the heart cycle. In signal processing jargon the Doppler signal is termed "non-stationary" and so the swept filter does not have the opportunity to scan the complete Doppler spectrum before its content changes. The solution to this problem is to record the Doppler signal on, for example, a magnetic tape loop and then play back the signal repeatedly through the filter which is incremented in frequency after each complete replay. This procedure is time consuming but can be speeded up by replaying the tape at a rate faster than the recording speed. For instance, if it were required to analyse the Doppler signal into N channels, then (if the replay rate were the same as the record rate) it would take NT seconds to process a waveform which lasted for T seconds. However, if the replay were to be speeded-up n times, then it would take only NT/n seconds to analyse the data. For the particular case where n = N, the analysis time equals the recording time and it is possible, in theory at least, to analyse completely one frame of data while the next one is being collected. An analyser working under these conditions is said to be operating in "real-time",
Real-time analysers can always be recognized because of their ability to process completely the input signal without creating a backlog of data awaiting analysis. The multichannel analyser described in Section 3.2a is an example of a real-time instrument. Notice, however, that "real-time" does not mean the same as "instantaneous". In the real-time swept filter device (and also in transform analysers described later) the analysed output lag: one frame behind the signal input. Similarly, the bandpass filters in the multichannel analyser take time to respond depending on their bandwidth.
For the case of the analogue swept filter, it is usually not possible to speed up the replay sufficiently to achieve real-time operation. This is mainly because of practical limitations imposed by the frequency response of the recording medium. For example, if it were required to analyse a Doppler spectrum extending to 10 kHz into fifty channels then the recording medium would have to replay signals at frequencies up to 500 kHz. This bandwidth is beyond the range of all but the most expensive analogue storage devices. (It is, however, possible to digitize the signal and circulate it using a fast digital memory and this type of time compression analyser will be described later.) In spite of not being able to maintain real-time operation, analogue swept filter analysers are regularly used for Doppler signal analysis and therefore merit a more detailed examination.
(ii) Analogue Systems
The term swept filter analyser is something of a misnomer when applied to the practical instrument. It is very difficult to design and control a constant bandwidth bandpass filter which is able to sweep over a range of centre frequencies. Thus instead of sweeping the filter across the spectrum, it is more usual to sweep the complete Doppler spectrum past a fixed-frequency bandpass filter. The equivalence of these two methods is illustrated in Figs 3.16 and 3.17 which shows the similarity of the detector outputs in both cases.
The electro-mechanical layout of an analogue swept "filter" analyser is shown in Fig. 3.18. This system uses a heterodyne unit to sweep the frequencies in the Doppler spectrum past the fixed bandpass filter.
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