DOPPLER3 (Раздаточные материалы), страница 7

(i) Spectrum Samplers

The most straightforward way to detect peak frequency is automatically to sample the output channels of a spectrum analyser and decide which is the highest frequency channel registering a signal. Although it is then still necessary to perform complete frequency analysis, this method allows direct input of a flow-characterizing waveform into a computer or other signal processing device. Figure 3.25 illustrates how a peak frequency follower might operate. The analysis channels are arranged in order of ascending frequency F₁ to F_n and each detected output is fed to a voltage comparator which decides if the signal is above a preset noise threshold. The comparator output is a bistable signal which indicates when the channel voltage exceeds the preset level. A priority encoder rapidly scans the comparator outputs. This network (which is available in integrated circuit form) produces a voltage proportional to the ranking of the highest channel which contains a signal. As the spectrum changes, the output voltage from the priority encoder follows the peak frequency content of the spectrum.

This method can usually provide the most accurate and reliable method of peak frequency following (for more details of practical implementation, see, for example, Skidmore (1979)). Its main disadvantage is that not only is a complete frequency analysis required, but the complexity of the instrument is also increased still further because each analysed frequency channel now requires its own comparator circuitry. In addition, as is the case for all peak following methods, the rigidly defined logic finds it difficult to discard artefacts which might be present in the spectrum and which could be recognized by a human operator. For example, because the scattering from blood is a statistical process, the Doppler power fluctuates in time and periodically approaches zero (see Chapter 1). This means that the amplitude of the peak frequency will periodically dip below inherent noise levels and the peak follower will encode a lower frequency maximum. Under similar circumstances the human operator would probably recognize this temporary drop-out as a "fading" artefact and ignore it. Although this effect can be reduced by low pass filtering of the priority encoded output it can never be completely eliminated. This illustrates the fundamental problem in the concept of peak frequency following; by definition, the signal-to-noise ratio of the true peak frequency component must be close to unity since if it were not (assuming no discontinuities in the Doppler spectrum) it would not be the maximum frequency present. Inherent noise and signal fading make it difficult to track the true peak frequency.

(ii) Frequency Tracking Filters

lt. is inefficient to perform complete spectral analysis simply to detect peak frequency and so a number of attempts have been made to derive the peak frequency waveform directly from the time domain signal. Sainz et al. (l976) have described one method based on the ubiquitous phase-locked-loop.

A phase locked loop (PLL) consists of a coherent demodulator (see Section 2.4b) and a voltage controlled oscillator (VCO) interconnected as shown in Fig. 3.26a. The coherent demodulator output is a low-pass filtered signal proportional to the phase difference between the input and reference signals. The reference signal is generated by the VCO at a frequency controlled by the output voltage from the coherent demodulator. The loop is arranged so that the control voltage always changes the VCO frequency in a direction which reduces the phase difference between the input and reference signals. Thus the reference becomes "locked" in phase with the input signal. The control voltage is then proportional to the frequency of the input signal.

To use the PLL as a peak frequency follower. the Doppler signal is first quadrature demodulated and frequency domain shifted (see Section 2.4f(vi)) to a pilot frequency of around 80 kHz, well above Doppler frequencies. (This process separates the Doppler sidebands and locates them on either side of the pilot frequency). The heterodyned signal is then input to a PLL designed to "lock in" to signals around 80 kHz. Because of the complex and rapidly changing frequency content of the Doppler signal the PLL is often out of lock, searching for a consistent component on which to fix. If the low pass filter in the coherent demodulator has a sufficiently short time constant then this searching operation is rapid and the PLL spends brief periods transiently locked onto various dominant Doppler components. Part of the time will be spent locked onto the "maximum" frequency signal and the output will look similar to the noisy trace shown in Fig. 3.26b from which an estimate of peak and, possibly, mean frequency (and thus velocity) can be made.

Apart from this rather sketchy, intuitive analysis it is difficult to end or derive a precise analysis of the operation of this device. Sainz (op. cit.) states: "Empirically, it has been found that the transient locking onto peak frequencies will occur reliably only if these frequencies have an amplitude which is at least half of the strongest coincident component." Thus the output can at best only be proportional to the true peak frequency component which, by definition must have an amplitude only slightly greater than noise.

Another way of tracking the peak frequency described by Skidmore (1978) is to use a high pass filter in a feedback loop. Figure 3.27a is a block diagram of such a system and Fig. 3.27b illustrates its operation which is perhaps easier to understand in the frequency domain. The central element in the network is the voltage controlled high pass filter. The cut-off frequency of this filter can be controlled by the voltage V_c and the network is arranged so that the cut-off frequency increases with control voltage. The Doppler signal is fed to the filter and the output is detected (rectified and smoothed) and applied to the control voltage input. With no input signal the offset is adjusted so that the filter frequency rests at slightly above zero frequency. When a Doppler signal is fed into the system, the filter passes the signal and the control voltage increases thereby raising the filter frequency and cutting off the Doppler input. The feedback loop settles to an equilibrium position shown in Fig. 3.27b where the signal power passed by the filter is just sufficient to hold the filter in position. The control voltage is roughly proportional to the area shaded in the figure which is common to both the Doppler spectrum and the filter pass band. As the maximum Doppler frequency increases the filter "rides" on the upper edge of the spectrum maintaining the slight overlap necessary to produce the required controlling voltage. The feedback characteristics are controlled by the loop filter which is designed to reduce overshoot. A low pass, one pole filter with a suitable time constant is usually sufficient. Further practical details on the design and construction of a peak frequency follower based on a tracking filter are described in Appendix A3.

3.3c Mean Velocity Computers

(i) Basic Principles

There has long been a need for a transcutaneous method of measuring volume flow through a blood vessel, and later chapters explain the clinical interest in this parameter. The volume flow Q of blood through a vessel of cross sectional area A is given by the relation:

Q = ^-v^- . A (3.36)

where ^-v^- is the mean blood velocity. The velocity profile across the vessel determines the relationship between the peak velocity v_p and ^-v^-. For parabolic flow it can be shown that
^-v^- = 2v_p/3 whereas for plug flow ^-v^- = v_p. Since the ratio v_p/^-v^- changes for flow patterns between these two extremes it is not reliable to use v_p as an indicator of ^-v^-. Instead the mean frequency itself has to be computed from the returning Doppler signal. The most direct method is to perform a frequency analysis and then calculate the first moment of the Doppler power spectrum P(w) using the relationship

The mean frequency w can be related to v using the basic Doppler equation so that

This relationship assumes that: (1) the complete cross-section of the vessel is uniformly illuminated and (2) the blood is a homogeneous scatterer of ultrasound. These assumptions have to be satisfied so that the magnitude of the backscattered power at a certain Doppler frequency can be directly and linearly related to the volume of blood flowing at the corresponding velocity to the vessel were not uniformly insonated then the echo from blood flowing in the more intensely insonated regions would artificially weight the integral in Eqn (3.37). The same effect could also be caused if certain volumes of blood exhibited an increased backscattering power. In the clinical situation. assumption (l) can be satisfied by choosing the correct beam geometry. Assumption (2) can be validated by averaging the Doppler power spectrum over a period which is larger than the fluctuation intervals caused by the fading of the signal backscattered by blood (see Chapter 1). Except for a narrow "plasma sheath" next to the vessel wall. blood appears to be a relatively homogeneous scatterer of ultrasound.

This direct method for computing mean velocity suffers from the disadvantage of requiring a complete frequency analysis of the Doppler signal. Nevertheless the mean velocity waveform computed in this way would be significantly less noisy than that generated by inherently noisy peak frequency followers, since the averaging process would tend to eliminate the effects of signal fading. However perhaps because of its complexity, this direct method of calculating mean velocity is not in widespread clinical use.

(ii) An Analogue Mean Velocity Computer

An ingenious and relatively straightforward method of computing w directly from the Doppler time signal has been described by Arts and Roevros (1972). The device is essentially an analogue mean frequency computer which calculates w from Eqn (3.37). The system uses the mathematical relationship that

which shows how differentiating a sine wave has the effect of shifting the phase of the oscillation by p/2 and more importantly, multiplying its amplitude A by its frequency w. When a Doppler signal is similarly differentiated, the effect is to weight the amplitude of each component by its frequency. lt. can be shown (see for example Arts and Roevros) that if the received Doppler signal R(t) is a function of a real variable, then the Fourier transform of the differentiated Doppler signal R'(t) is related to R(t) by

F[R'(t)] = iwF[R(t)] (3.39)

Arts and Roevros develop a general theory showing that the mean frequency of any Doppler spectrum can be computed using analogue techniques. The method can be illustrated by investigating how the system processes a simple combination of two sine waves.

Figure 3.28 is a block diagram of a mean velocity computer showing the state of the signal

after each stage of processing. Suppose the returning ultrasonic echo R(t) contains two Doppler shifted components of amplitude A and B so that

R(t) = Acos[(w₀+w_A)t+f_A] + Bcos[(w₀+w_B)t+f_B], (3.40)

where w_A, w_B are the Doppler shifts and f_A, f_B the phases of the two components. In Channel 1 the ultrasonic echo is multiplied by cosw₀t and in Channel 2 by sinw₀t. The multiplied outputs M(t) are thus

M₁(t) = {Acos[(w₀+w_A)t+f_A] + Bcos[(w₀+w_B)t+f_B]}cosw₀t, (3.41a)

M2(t) = {Acos[(w₀+w_A)t+f_A] + Bcos[(w₀+w_B)t+f_B]}sinw₀t, (3.41b)

The products of Eqn (3.41) are low pass filtered to remove components at the ultrasonic frequency giving the twin coherent-modulated signals D(t) where

D₁(t) = Acos(w_At+f_A) + Bcos(w_Bt+f_B), (3.42a)