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which is the familiar axial resolution restriction imposed on any echo ranging system operating over a limited bandwidth. This equation implies that the axial resolution capability of a frequency modulated system is independent of the modulation rate and depends only on the spectral range of transmitted frequencies. The reason for this is that an increased sweep rate is counteracted by a reduction in frequency resolution and vice versa. For example, if the modulating ramp becomes steeper then the differences in the mixer output frequencies from closely-spaced targets will increase. However, the modulating ramp will have to be reset (or reversed) earlier thereby limiting the duration of the mixer output and degrading the frequency resolution capability. Thus, although the difference in frequency returns from adjacent targets will be greater, the inherent resolution of the system is degraded by an equivalent amount. Conversely, a slowly rising
modulation ramp will take much longer to sweep through the ultrasonic bandwidth producing a long continuous mixer output. However, the resulting improvement in frequency resolution is now necessary (and just sufficient) to separate the more closely spaced frequencies returned from the adjacent targets.
A more detailed analysis of FM range discrimination is to be found in an excellent article by Bertram (1979) which shows that the mixer output can only consist of harmonics at the ramp repetition frequency. Echoes from targets at those ranges which do not correspond to pure harmonics are obliged to contribute their energy in adjacent "allowable" spectral lines. In other words, a target range corresponding to an anharmonic beat frequency produces a complete spectrum of allowable harmonics. Furthermore, Bertram shows how the flyback interval pollutes the overall spectrum and degrades the signal-to-noise performance of the system.
(iii) Velocity Resolution
A fundamental feature of FM coding is that the range of the target is indicated by the frequency of the demodulator output. If the target is moving then the output frequency will vary at a rate proportional to target velocity. Thus, in theory it should be possible to resolve velocity using an FM system but in practice multiple targets confuse the analysis. Using a simple sawtooth sweep modulation it is not possible to differentiate a stationary target at one range from a moving target with accompanying Doppler shift, at a different range. lt. might be possible to devise a triangular sweep in which the apparent range of targets during an increasing frequency transmission is compared with that during a decreasing frequency transmission. The range and Doppler shift components will then either add or subtract depending on whether the transmission frequency sweep is increasing or decreasing. This complex type of system has to the authors' knowledge never been used for monitoring blood flow.
2.3c Random Signal Coding
From the previous sections it seems that a simple coding of either the amplitude or frequency of the transmitted ultrasonic wave cannot prevent range-velocity ambiguities. Basically the velocity limitation arises when the target is travelling at such a high velocity that it moves more than one quarter of a wavelength between transmission coding sequences. Increasing the repetition rate solves this particular problem but eventually limits the range of the system. The range ambiguity arises because successive transmission sequences are indistinguishable and it is not permissible for more than one sequence to be propagating at any instant in time. However if successive transmissions could be made to retain their individuality in some way, then additional pulses could be transmitted to interrogate target movement before previous ones had returned. This is the principle of the random signal flow measurement system first applied to ultrasonics by Newhouse et al. (1976) and also the coded pulse devices described by Waag et al. (1972). These two systems are similar in concept but slightly different in detail. The random signal device transmits white noise whereas the coded pulse system transmits a pseudo-random binary sequence (or PRBS). Only the random signal, or noise, Doppler will be described in detail here since the PRBS analysis would follow similar lines.
(i) Basic Principles
The basic components of the random noise device are shown in Fig. 2.16. The source transducer driven by a wideband noise generator transmits a signal which consists of random noise limited only by the transducer bandwidth. (This transmission can be thought of as being a series of short pulses. all slightly different, and occurring at a repetition rate which is so high that successive pulses overlap to produce a continuous wave of quasi- random amplitude and phase.) For the simple case illustrated, this random noise signal is reflected from the single point target and returns to the receiving element. The random noise signal is also routed through a variable delay line and then correlated with the received signal.
(ii) Correlation
Correlation is a process for comparing two waveforms. The result of a correlation indicates to what extent the two waveforms are similar. One correlation method is to multiply the two waveforms together and then average the result in a low-pass filer. To illustrate this process, two uncorrelated random noise waveforms A and B are crudely represented in Fig. 2.17a by a random sequence of equally spaced impulses with amplitude of either + 1 or - 1. If A and B are multiplied together to give waveform C then the result is equally likely to be + 1 as it is to be - 1 . When averaged in a low-pass filter with a sufficiently long time constant, the contributions will tend to cancel giving zero output D corresponding to zero correlation. However, if the two waveforms show some degree of correlation, that is a + 1 in waveform A is more likely to coincide with + 1 in waveform B as shown in Fig. 2.17b, then the filtered output D will be a value whose magnitude indicates the degree of correlation. In particular, if a waveform is correlated with an exact replica of itself (Fig. 2.17c) then the result of each multiplication will be +l, shown in C, since (+1 x +1)=+1 and ( -1 x -1 ) = +1 . The averaged output D will be at a maximum level and proportional to the power of the input waveform.
Thus, when the delayed version of the random noise transmitted signal is correlated with the received echo, the correlator output will be averaged to zero unless the particular delay which is selected in the delay line exactly equals the time of flight to and from the target. In this case the two input waveforms will be identical and so the correlator output will reach a maximum value indicating that the range of the target corresponds to the selected delay time.
(iii) Analysis-Point Target
A mathematical description of the random signal technique has been described by Bendick and Newhouse (1974) and is outlined below. It will be seen that this analysis shows quite well how the range resolution capabilities of a random noise flowmeter are identical to those of a pulse-Doppler or FM flowmeter working under the same transmitted-frequency bandwidth restrictions.
Suppose the transmitted signal can be described by the function T(t). The echo R(t) received from a single point scatterer moving at a velocity v towards the transducer will be of the form
where A is the backscattering cross-section of the target and ts, is the time of flight of the signal at t = 0. This equation says that the received signal will be a replica of the transmitted signal but delayed by the flight time of the echo to and from the target. This flight interval is changing at a rate 2v/c as the target moves at velocity v towards the transducer.
The transmitted signal T(t) is also delayed artificially in the delay line by a time td, giving a delay line output L(t) where
When R(t) and L(t) are fed to the correlator input the time averaged correlator output C(t) will be
where E{...} denotes a time average which can more usefully be expressed in terms of the autocorrelation function Z(t) of the transmitted function giving
This equation follows from the definition of the correlation function of any waveform g(t)
together with the fact that the correlator output C(t) is the time average of the transmitted function T(t) multiplied by a time-delayed version of itself. This time delay is defined by the difference between (2vt/c - ts), the time-of-flight and td, the time delay interval. Furthermore it can also be shown in general that the Fourier transform of the autocorrelation function is identical to the power spectrum G(¦) of the original signal. In this case the transmitted power spectrum is limited and defined by the frequency response of the ultrasonic transducer and is usually a Gaussian-shaped function similar to that shown in Fig. 2.18a where
where B is the bandwidth and ¦c the ultrasonic centre frequency. The inverse Fourier transform of G(¦) is the autocorrelation function Z(t) and is of the general form
Z(t) = e-pBt cos 2p¦ct (2.41)
Substituting this expression for Z(t) into Eqn (2.38) gives for the correlator output
This function is sketched in Fig. 2.18b.
In order to interpret Eqn (2.42) first suppose that the target is stationary so that v = 0. The output of the correlator will then be a constant amplitude level determined by the value of both the envelope term and the phase term. Notice that although both these terms depend on the value of (ts - td), the phase term oscillates indefinitely and changes rapidly with time whereas the envelope term varies more slowly and also confines the output C(t) to the region where ts is approximately equal to td This is equivalent to saying that the correlator output is non-zero only when the ultrasonic and line delays are approximately the same so that the two random noise signals superimpose.
Consider now the situation illustrated in Fig. 2.19 where the target is moving through the region where ts = td so that the function Z(t) is swept out in time. (Note that for the sake of clarity it has artificially been assumed in Fig. 2.19 that the random noise epoch shown in (a) is transmitted identically at five different and separate times to interrogate the target and produce echoes (b)-(f). This simplification makes it possible to use waveform (a) as the reference for all five echoes. It would otherwise have been necessary to show both the delay line signal and the received echo for each transmission-making the illustration even more complicated! This simplification is valid because the autocorrelation functions of all random noise epochs are identical.) Returning to the analysis, the phase term in Eqn (2.42) predicts that the output will oscillate at the Doppler shift frequency 2v¦0/c although the reason for this may not be immediately apparent. In fact this effect is exactly similar to the way in which the coherently-demodulated samples of a pulse-Doppler system (see Section 2.3a (v)) vary with time at the Doppler difference frequency and can be understood as follows. Suppose that the target is situated at a range where ts = td The returning echo shown in Fig, 2.19b will then be superimposed precisely on the signal (a) emerging from the delay line and the two waveforms correlate maximally to give a positive peak at the correlator output (g). As the target moves, all the returning echoes will gradually shift in phase relative to the "reference" provided by the delay line signal and, after the target moves a distance l0/4, a situation of negative correlation (shown in (c)) will be reached where each positive excursion of the echo corresponds to a negative excursion of the reference and vice versa. The correlator output (g) at this point will be a peak negative signal. Further target movement will begin to increase the correlator output to another positive peak (d). This second peak exists because the quasi-random signal is bandwidth limited rather than pure white noise. This means that successive parts of the transmitted waveform cannot be completely independent since the rate of change of the signal envelope is limited by the transducer bandwidth. However the second peak will not be as large as the one produced by exact superimposition and this is accounted for by the slowly decaying envelope term in Eqn (2.42) The output completes one cycle in the time it takes for the target to move a distance l0/2 since this movement increases the go and return ultrasonic path by one wavelength. Thus the frequency of oscillation ¦d is given by
which is the familiar Doppler relation.
(iv) Blood Target-Range-Velocity Resolution
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