DOPPLER3 (Раздаточные материалы)

3. DOPPLER SHIFT SIGNALS

.3. PROCESSING AND ANALYSIS

OF DOPPLER SHIFT SIGNALS

The previous chapter has show how the echoes reflected from a moving target are shifted by a frequency which is related to target velocity. Up until this point it has been assumed that these difference frequencies could be analysed in some way in order to compute target velocity using the basic Doppler equations. This chapter will be devoted to the subject of Doppler signal analysis. By describing and comparing the various processing methods which are available, it is hoped to show and emphasize that proper implementation of the most suitable processing technique plays a vital part in the successful use of ultrasonic flowmeters in the clinical situation.

The chapter begins by reviewing some general concepts of frequency analysis, including a short section on Fourier theory and how it can be used. These principles are then applied to the particular case of Doppler signal processing. Various methods of spectral analysis, including both analogue and digital techniques, are examined and their relative advantages discussed. In addition the concept of real-time analysis is introduced and devices for displaying the vast amounts of data generated by real-time analysers are described/ Finally, and with an eye to the future, the increasingly popular Doppler processing methods which do not require analysis are investigated. These include established techniques such as

zero crossing counters and also more sophisticated devices such as peak velocity followers, mean velocity computers and mean flow computers.

3.1 SIGNAL PROCESSING

From now on "the Doppler signal" will be taken to mean that Doppler difference waveform output from an ultrasonic flowmeter which contains the demodulated velocity information from the interrogated targets. The Doppler signal output is a time varying voltage waveform and usually looks something like the trace show in Fig.3.1a. Because the velocity information is related to the frequency content of this trace, it is obviously of little value to view the Doppler signal directly by displaying it in the conventional way on an oscilloscope screen. Instead, the waveform requires some form of analysis to extract the frequency components and so reveal their velocity content.

It is fortuitous that using ultrasonic frequencies in the 1-10 MHz range, human blood flow velocity produce Doppler shifts which lie in the audio frequency band. Thus a subjective impression of frequency content can be gained simply by listening to the Doppler signal which can be amplified and broadcast over a loudspeaker. In fact, the human ear-brain combination possesses an extremely powerful spectral analysis capability which with practice can process and recognize the most subtle characteristics in blood flow signals. As anyone who has listened to a Doppler signal and then looked at its corresponding frequency spectrum will know, for qualitative work the human ear represents a frequency analysis system with which no man-made instrument can compete. However, more quantitative investigation requires frequency analysis of the Doppler signal so that its spectral characteristics can be displayed in that is commonly know as the "frequency domain".

3.1a Frequency Domain Analysis

Waveforms are usually displayed graphically by plotting the way in which their amplitude varies with time: this is the conventional time domain representation of a signal show in Fig. 3.1a. Similarly, the frequency domain is simply a graphical display of signal amplitude (or possibly power or phase) as a function of frequency. Figure 3.1b shows the Doppler waveform of Fig. 3.1a displayed in the frequency domain. More generally Fig. 3.2 is an illustration showing the relationship showing the relationship between the more conventional time domain and the perhaps less familiar frequency domain. The sinusoidal waveforms at frequencies ¦₁ and ¦₂ which are immediately recognizable in the time domain can equally well be represented by the single spectral lines in the frequency domain. The height and location of the spectral line corresponds to the amplitude and frequency of the wave respectively. (Incidentally, it is interesting to note that if a pure sine wave were to be broadcast over a loudspeaker, then the ear-brain combination of the listener would automatically perceive the signal in the frequency domain and recognize it as a single tone corresponding to the single spectral line).

The advantages of frequency domain representation become more apparent when sinusoids at different frequencies are combined as show in Fig. 3.3. Even when only three sine waves A, B and C are combined, the resultant time domain waveform D in Fig. 3.3a is complex and its constituents almost unrecognizable. However, the domain representation in Fig. 3.3b still shows clearly the components of the composite signal.

More practically, suppose these three sine waves had been generated by Doppler shifts from the three targets moving in the beam of a CW flowmeter. The amplitude (or intensity) of each component would then be determined by both the size (or backscattering cross-section) of the target and its position within the (non-uniform) ultrasonic beam. The frequency on the other hand would be defined by the target velocity. Thus the frequency domain provides a convenient and easily interpretable display which not only indicates the presence of the three targets, but also shows their relative echo strengths and meters their velocity in the direction of the transducer. Fourier methods provide a link for converting waveforms between the time and frequency domains and therefor form the basis of Doppler signal processing.

(i) Fourier Analysis

In general terms the Fourier theorem states that a waveform of any shape can be constructed by adding together a set of sine and cosine waves of suitable amplitude, frequency and relative phase. Thus Fig. 3.3 is essentially a simple example of the Fourier principle in operation showing that the complex-looking waveform D can be constructed from (or analysed into) the three component sine waves A + B + C.

The translation between the time and frequency domains of non-periodic signals is described by the Fourier Integral. If ¦(t) is the time t description of a waveform and F(w) its corresponding frequency w representation, then

and

and the coupled equations are know as a Fourier transform pair. The conventional notation adopted here is lower case letters for time functions (¦(t), g(t), etc.) and upper case letters for their corresponding frequency domain representations (F(w), G(w), etc.). More comprehensive derivations and descriptions of Fourier transform theory are beyond purpose of this book and can be found elsewhere.

It will be sufficient for the present to interpret Eqn (3.1a) to mean that any waveform ¦(t) can be constructed by adding together an infinite set of sinusoidal waves of frequency w and complex amplitude F(w) which is given by Eqn (3.1b). Conversely Eqn (3.1a) also implies that ¦(t) can be resolved or analysed into a set of constituent sinusoids. For the simple case shown in Fig. 3.3, F(w) would be zero except at the three frequencies w_A, w_B and w_c

(ii) The Power Spectrum

Because ¦(t) is multiplied by exp(-iwt) in the integral equation (3.1b), F(w) can be a complex function of frequency containing both real and imaginary parts. The real part of F(w) corresponds to the amplitude of the cosine wave at frequency w while the imaginary part corresponds to the amplitude of the sine component.

Varying the relative amplitudes of the real and imaginary parts of F(w) allows the constituent sinusoids to be summed with the correct phase relationships, a crucial factor in producing the required waveform. However, is Doppler signal analysis, the phase of the returning echoes are arbitrarily determined by the range of the target measured in units of the ultrasonic wavelength. The relative amplitudes of the real and imaginary components are therefore not relevant to the problem of determining the size and velocity of the target. It is much more important to define the power contained in each frequency band since in Chapter 1 it has been shown that this relates to the volume of the flowing blood. The so-called power spectrum P(w) can be derived from F(w) by first rewriting Eqn (3.1b) in the form

or, expanding into real and imaginary parts

Separating the real R and imaginary T components gives the relationships

The power spectrum P(w) is defined as

P(w) = a²(w) + b²(w) (3.5)

and its modulus A(w) is given by

For the purposes of this book, the amplitude frequency spectrum will be taken to mean the same as the modulus or square root of the power spectrum. The output from a spectrum analyser is usually in the form of either the power spectrum or amplitude spectrum of the input time waveform.

In order to understand why the power spectrum forms the basis of Doppler signal processing, consider the situation illustrated in Fig. 3.4a. To simplify the analysis it has been assumed that a unit length of the blood vessel is uniformly insonated with continuous-wave ultrasound. Furthermore, the small attenuation of ultrasound by the blood is neglected and it is also assumed for the moment that the blood constitutes a homogeneous backscatterer. (In spite of the conclusions of Chapter 1, this latter approximation can be validated later when the power spectrum is averaged over time periods which are much longer than the statistical fading rate). The Doppler spectrum derived from the Doppler signal backscattered by the flowing blood is shown in Fig. 3.4b. Because of the velocity profile existing across the vessel diameter, the Doppler spectrum extends from low frequencies corresponding to returns from slow moving blood near the vessel wall, to high frequency components generated by blood moving at the peak velocity along the axis of the vessel. If n(v) is defined as the velocity of red cells then the effective volume of target (that is the number of corpuscles) travelling at
velocities between v and (v+dv) is equal to n(v)dv. Similarly, the echo power backscattered into any frequency band between w and w + dw is proportional to P(w)dw Thus, if the backscattering cross-section is proportional simply to the volume of the target, then because of the simplifying assumptions made above

P(w)dw µ n(v)dv (3.7)

where w and v are linked by the Doppler equation

2v

w = ------- w₀ cos (3.8)

c

In other words, the power spectral density is an indicator of the fractional volumes of blood flowing at velocity n(v) corresponding to Doppler shift w. By definition, the mean velocity ^-v^- is given by

Since P(w) is linearly related to n(v)dv and because velocity v can be expressed in terms of Doppler frequency w, Eqn (3.90 can be rewritten

The quotient of the integrals on the right-hand side of this expression is equal to the mean frequency v of the Doppler power spectrum, giving the not unexpected final relationship