DOPPLER3 (Раздаточные материалы), страница 6

P_i = (a_i + jb_i)² + (a_i - jb_i)² (3.30)

or

P_i = 2(a_i² + b_i²) (3.31)

This value is averaged or processed as required for each of the N/2 frequency points to form the FFT power output spectrum of Fig. 3.21b.

(ii) The Instrument

Figure 3.22 is a block diagram of an FFT analyser and typifies the construction of most commercially available instruments. The processing sequence is under strict control. of the

internal timing unit (usually a micro-processor) which generates appropriate commands for sampling the input performing the transform and displaying the output. The input waveform enters through the anti-aliasing filter which prevents sampling artefacts. The synchronous ADC then samples the data at a suitable rate, converts it into digital form and stores the result in the input buffer memory. When full, the contents of this memory are transferred to the processor memory. (The time taken to fill a given length of input buffer memory depends on the selected sampling rate.) The processor then computes the FFT of this data and stores the results in the output buffer memory. An output of power P(¦_i) at each frequency point ¦_i can then be generated by repetitively scanning the output buffer, usually at a rate fast enough to provide a flicker-free trace on an oscilloscope screen. By utilizing the buffer memories in this way, the analyser can continue recording new data and displaying the most recent transform while current signals are being processed so that real-time analysis is maintained. Notice that FFT analysers operate on blocks of data which must be gathered beforehand. This is in marked contrast to time compression analysers which continuously update the memory with the most recent data available.

(iii) Performance Characteristics

The computing speed of the processor determines the rate at which transforms can be computed and this eventually limits the real-time bandwidth of the instrument. Once again a numerical example best illustrates the principles of performance. Suppose it takes the processor 30 milliseconds to compute a 512 point non-redundant transform from the 1024 real words stored in the processor memory. From the definition of real-time analysis, for maximum analysis bandwidth the 1024 words must be sampled at equally spaced intervals in precisely 30 milliseconds, otherwise a part of the input data would either have to be missed or would be inadequately sampled producing aliasing artefact. The sampling rate ¦_s must therefore be 1024/(30 x 10^-3) Hz which is approximately 34^.1 kHz giving a Nyquist frequency of around 17 kHz. After transformation, the 512 frequency points are equally spaced from DC to 17 kHz and thus the line spacing is 17kHz/512 or 33^.3 Hz. Notice that this is also the reciprocal of the input data period (30 milliseconds). Thus each line in the FFT corresponds to a harmonic of that fundamental frequency which is defined by the reciprocal of the time period it takes to gather the input data. Thus, this particular FFT analyser is capable of a real-time spectrum analysis producing one spectrum every 30 milliseconds comprising 512 channels spaced at 33-3 Hz and ranging from DC to about 17 kHz.

More generally, if it takes an FFT processor DT seconds to compute an N point transform, then the maximum sampling frequency ¦_s^max which can maintain real-time operation is given by

and since the maximum allowed input frequency ¦_max is equal to ¦_s^max/2,

The channel spacing D¦ of the N/2 positive frequency points computed from this data is

The frequency resolution is therefore the reciprocal of the input sample length.

Equations (3.32) through (3.34) indicate how the characteristics of the transform can be manipulated to suit the Doppler spectrum under analysis. Reducing the sampling rate will decrease the maximum analysis frequency but improve the frequency resolution. Decreasing the number of points in the transform will maintain the maximum analysis frequency but degrade the frequency resolution allowing more spectra per second. Table 3.1 shows some possible combinations of processing parameters and transform characteristics. Notice that because of the almost linear relationship between input sample length and processing time, Eqns (3.33) and (3.34) hold for all cases. In effect the transformation time per input data word remains constant.

(iv) Time Windows and Weighting Functions

The data input to time compression or transform analysers is artificially restricted in length by the input memory of the instrument. Restricting the length of a time waveform is the same as multiplying it by a rectangular time window and this is illustrated in Fig. 3.10. In the frequency domain this multiplication corresponds to a convolution of the input spectrum with the window or weighting function spectrum (see Section 3.1c (iii)). Thus, because the analyser sees a waveform which seems to start and finish abruptly, the output spectrum becomes distorted by the weighting function. For the most simple case of a rectangular weighting the distortion takes the form of particularly undesirable sidebands arising from the (sin x/x)² form of the windowing spectrum. In order to reduce these sidebands, smoother weighting functions such as Gaussian, Hanning or Hamming weighting functions can be applied to the input waveform. To apply the weighting, the waveform in the input memory is multiplied at each time point by the corresponding value of the weighting function. The resulting signal is then processed as normal.

Figure 3.23 shows four types of weighting functions in the time domain and their effect in the frequency domain. The relative performance of these windows is compared in Table 3.2. Each function is assumed to have the same total length T. The Gaussian (which is effectively infinite in length) is terminated at about seven standard deviations and this has little or no effect over a 65 dB dynamic range. It should be pointed out that the frequency domain plots in Fig.3.23 represent the bandpass characteristics that each analysis channel will possess after the weighting function has been applied. Notice that to show more detail, the frequency plots are on a log-log scale with the logarithmic base of the frequency axis equal to l/T. The zero frequency point in each plot (which is impossible to show on a logarithmic axis) corresponds to the centre frequency of each particular analysis channel.

Looking at each time window in more detail, the Gaussian function (see Fig. 3 . 23d) has the peculiar and unique property that its Fourier Transform is another Gaussian function. Thus Gaussian weighting produces a relatively wide bandpass filter with no sidelobes. A Hanning function (b) is a (cosine)² and Hamming weighting (c) is a Hanning function mounted on a small rectangular pedestal. The height of the Hamming pedestal is chosen so that the second sidelobe of the rectangular function tends to cancel the first sidelobe of the Hanning function with which it combines. Unfortunately the remaining sidelobes are dominated by the rectangular function and fall off more slowly than for the Hanning window. In general, slowly rising and falling time windows decrease sidelobes but, because they reduce the effective time duration of the signal, they also tend to increase the width of the main lobe and therefore degrade primary resolution. Choice of time window summarized in Table 3.2 is a compromise between a narrow main lobe with accompanying sidelobes or a wider main lobe with sidelobe suppression. The latter is usually more useful in Doppler signal analysis. Most commercial transform analysers and some time compression instruments incorporate a Hanning or Hamming weighting option. This operates by multiplying the digitally stored input data by the chosen weighting function before the analysis is performed. The effect on the output spectrum can be dramatic if the input is a pure sine wave at a single frequency. The effects of weighting on a complex Doppler spectrum are usually more subtle and, quite frankly, in most cases almost unobservable. Frequency analysis is the most reliable method of processing the Doppler signal since only then can the spectral information be displayed completely, allowing artefacts and spurious effects to be recognized by the operator. However in some clinical situations, comprehensive frequency analysis is an expensive luxury which generates a mass of data, most of which is eventually discarded. This is especially so in cases where the Doppler diagnosis is based on the shape of the peak or mean frequency versus time waveform. Although the required information can be obtained from a sonogram, a more simple analysis can sometimes be equally useful. This section will therefore explore both the established and new techniques which have been devised to extract velocity-related waveforms directly from the time domain signal.

3.3a Zero Crossing Counters

The essential problem in time domain analysis is to process the Doppler signal in some simple way such that particular parameters related to its frequency content are revealed. The first step is to examine the time domain signal to find clues to its frequency content. For the simple case of a pure sinusoid the problem is elementary: the frequency of the sine wave is equal to one half the rate at which the waveform crosses its own mean level. However, for the case of a Doppler shift signal from a blood vessel the problem is much less straightforward. For instance, low frequency signals originating from slow moving blood near the wall combine with higher frequency signals from the centre of the vessel to produce a complicated waveform shape. Also noise on the signal can produce unwanted zero crossings. Relating zero crossings to frequency in the practical situation is quite difficult but nevertheless this principle forms the basis for the most popular type of Doppler shift processor currently in use. However, it is perhaps true that this widespread popularity has been gained more from the simplicity of the device rather than from its reliable and accurate performance characteristics.

The theory of operation of a zero crossing detector is based on a classic theoretical analysis by Rice (1944, 1945) who analyses the problem by predicting the expected number of zero crossings from the spectral content of the signal. The zero crossing rate is predicted from the probability density of finding a zero crossing event in a given time interval. Flax et al. (l973) have adapted this theory to determine the relationship between the Doppler spectrum and the zero crossing frequency and show that for a Doppler power spectrum P(w) the zero crossing rate N is given by

provided there is no constant phase relation between components at different frequencies. The zero crossing rate is proportional to the root mean square (or RMS) frequency of the input signal. Kato et al. (l965) applied this type of theory to Doppler spectra and concluded that N could then be related to the RMS value of the flow velocity, but precautions must be taken to reduce artefacts.

Firstly, as mentioned above. vessel wall movement during systole causes large deviations of the Doppler signal commonly heard as "wall thump". These low frequency clutter signals must be removed by high pass filtering otherwise blood zero crossings would be missed. Also noise triggering can be reduced by using the SET-RESET system illustrated in Fig. 3.24. After a positive-going zero crossing the signal must first go below a negative going threshold before another positive going crossing can be counted. In effect the signal must reach a minimum peak-to-peak threshold before zero crossings register. Most commercial zero crossing detectors use this SET-RESET system with an operator-controllable threshold level.

Further problems result from the assumption implicit in Eqn (3.7) that backscattered power relates directly to blood volume. If the vessel is not insonated uniformly then this assumption need not be true. More will be said about these problems in Section 3.3c which describes mean flow computers. In addition Eqn (3.35) predicts the RMS frequency and this need not be uniquely related to the mean or peak flow velocity. Lunt (1975) reviews the effects of velocity profile on the ratio ^-v^-:v_RMS. With a flat profile this ratio is unity, increasing to 1^.16 for parabolic flow.

Directional systems based on zero crossing counters have been developed by McLeod (1967) but suffer from a fundamental limitation when there is simultaneous forward and reverse flow in two vessels in the ultrasonic beam. Although the McLeod system will indicate both forward and reverse flow, it cannot be used quantitatively under these conditions. Lunt (l975) explains that this problem is caused by the directional system distributing the zero crossing count between forward and reverse channels. Although the forward-reverse ratio is correct, the actual zero crossing frequency for each channel is reduced. This prevents quantitative analysis.

In conclusion, the zero crossing detector is available in most commercial Doppler devices. However it has certain limitations which must be understood and respected if it is to be used effectively. However, in spite of these shortcomings, the instrument should be capable of indicating component blood velocity to about 20% and can detect changes in velocity of a few per cent.

3.3b Peak-frequency Followers

Most clinical applications of Doppler data are based on analysis of the peak-frequency versus time waveform. Historically this was found to be the simplest parameter to extract from a sonographic display since it merely involved drawing an envelope round the sonogram (see Fig. 6.8). In addition the peak frequency waveform seemed to be a parameter which was closely related to the physiologically significant flow patterns in the vessel. Thus, although it may now be better to use the less noisy mean frequency waveforms to characterize the flow, the peak frequency still retains its significance as a simple to measure and well proven flow parameter.

Drawing an outline by hand around sonograms rapidly becomes a time-consuming and laborious pastime for the Doppler ultrasonographer. Furthermore, human intervention prevents on-line input of the peak frequency data into a computer or other data processing device. It was for these two reasons that automatic peak frequency followers have been combined with spectrum analysers to provide time-saving methods of processing Doppler signals,