Classical reception problems and signal design |
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2.8 Signal parameter estimation
2.8.1 Problem statement and estimation rule
Everywhere in radio systems we encounter the problem of signal parameter measurement or estimation. It describes any situation when information which is interesting to an observer is carried by a current value of some signal parameter (e.g. amplitude, frequency, initial phase, time delay etc.). Therefore, to extract necessary information the observer needs to measure or estimate the corresponding parameters.
Let us turn to some transparent examples. In conventional AM (FM) broadcasting, dependence of amplitude (frequency) on time carries audio information: the volume and pitch of a tone. To restore the audio message and give it out to the listener, instant values of the amplitude (frequency) should first be measured and reproduced as a continuous-time waveform. Another similar example is conventional analog TV, where both amplitude and frequency are involved in information transmission. To restore colour moving images amplitude measurement should be performed, since the luminance and chrominance components are broadcast via AM, while audio transmission is accomplished via FM and, hence, in any TV receiver frequency measurement is present.
Another parameter estimation task is found in the synchronization or timing problem, where time–frequency mismatch between the received signal and a local reference clock should be measured to synchronize the second with the first. This procedure is characteristic of a great number of systems, ranging from TV horizontal and vertical synchronization channels to pilot channels of 2G and 3G mobile radio.
Numerous estimation problems are typical of radar and navigation: measurement of time delay and signal arrival direction provides knowledge of the mutual distance and angle coordinates between a receiver and a target; if knowledge of a target velocity and manoeuvre is necessary Doppler frequency shift should be measured etc.
The list of examples could easily be continued, since parameter measurement is an integral part of practically any system in which information is transmitted, recovered and processed.
In terms convenient for our context, the parameter estimation problem may be stated in the following manner. The observation y(t) along with noise contains the signal s(t; ), which is deterministic except for the unknown constant value of the parameter. The latter may be a vector or a scalar, depending on the specific situation. The observer, based on the analysis of y(t), should produce a decision on what value, within the range of possible ones, is taken by the signal parameter in question. This decision, in
association with the problem itself also called estimate, is denoted by ^. Since noise is
always present in y(t), in any separate session of reception ^ differs from the unknown
true value of the parameter , and the question is how to make the optimal decision, which guarantees the smallest harm caused by this discrepancy.
The simplest clue to this issue may be found in understanding that in principle the estimation problem is not anything radically new with respect to the problem of distinguishing M signals studied in Section 2.3. In fact, suppose at first that a parameterto be measured is discrete and takes one of M competitive values 1, 2, . . . , M . Then the decision about which of these possible values is assumed by a signal parameter in
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this specific observation is nothing more than just determining between M hypotheses on which of M competitive signals s1(t), s2(t), . . . , sM (t) is being received, where the signals are just copies of s(t; ), differing from each other only in value of parameter: sk(t) ¼ s(t; k). To cover the case of a continuous parameter with this reasoning too, one can just imagine an infinite (up to uncountable) number M of the parameter values and, consequently, of the signals to be recognized.
The conclusion from these arguments is that the already well-known optimal decision strategy, the ML rule, remains applicable to the parameter estimation. This means that
among all competitive values of the one should be picked as the estimate ^ which
maximizes the probability of transforming the sent signal s(t; ) into the observed waveform y(t) at the channel output. For the AWGN channel this rule is equivalent to the minimum distance one, which, rewritten in the current designations, looks as follows:
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where s is vector notation of signal s(t; ). This rule produces the maximum likelihood
estimate ^ by finding the value of under which signal s(t; ) is closest to the observa-
tion y(t) by the Euclidean distance. Figure 2.14 gives an illustration of it. The signal s(t; ) may be thought of as a vector s , which moves, tracing the changes of the parameter . Its extreme point travels along some trajectory, points of which map one-to-one to specific values of (Figure 2.14a). The point of the trajectory closest to the observation vector y is found according to rule (2.54) and the corresponding value ofis announced as the estimate, which is also seen in Figure 2.14b, showing the dependence of the distance between the observation and the signal copy on the value
of . The ML estimate ^ is the parameter value minimizing this distance.
All signal parameters can be categorized as energy or non-energy ones, the terms reflecting whether the parameter affects signal energy. If is of the second type, the energy of the signal s(t; ) does not depend on the value of :
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Figure 2.14 Illustration of ML estimation
Classical reception problems and signal design |
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Amplitude and duration, for example, are energy parameters, whereas time delay, frequency and initial phase are non-energy ones. It should be clear now that estimation of a non-energy parameter is the particular case of the problem of distinguishing between competitive signals of equal energies, for which correlation rule (2.8), in the new designations, can be presented as:
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where according to (2.7): |
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is a correlation between the observed waveform y(t) and the signal s(t; ) in dependence on the value of the measured parameter .
In the light of the physical content of the correlation, the estimation rule (2.55) has
quite a transparent interpretation: the ML estimate ^ is the value of under which
signal s(t; ) has maximal resemblance with the observed waveform y(t).
2.8.2 Estimation accuracy
Let us recollect now the fact discussed in depth in Sections 2.2–2.3 that the fidelity of signal distinguishing is critically governed by the correlation coefficient (2.14). In the case of parameter estimation, signals to be distinguished are just copies of s(t; ) with different values of . In many practical situations the correlation of any two such copies s(t; 1), s(t; 2) depends only on their mismatch in , i.e. the difference 2 1, rather than on values 1, 2 separately, so that putting 1 ¼ 0, 2 ¼ leads to the following representation of the correlation coefficient (2.14) for the case of non-energy parameter :
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As is customary for a correlation coefficient, this entity characterizes the resemblance of two signal copies depending on their mismatch in the parameter . It is evident that( ) (0) ¼ 1, which has an instructive implication: signal copies mismatched in can not be more similar to each other than fully identical ones, which have, in their turn, unity correlation. Another property of the quantity (2.57) induced by its dependence on only ¼ 2 1 is evenness: ( ) ¼ ( ).
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ρ (λ)
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Figure 2.15 Typical curves of ( ) in dependence on
Figure 2.15 gives example curves of ( ) in dependence on for two hypothetical signals. The solid curve is flatter than the dashed one, which means that the first signal is less sensitive to changing : the resemblance between its copies mismatched in is higher than that between copies of the second with the same mismatch.
Now it is necessary to stress one detail, which was deliberately omitted above. As a matter of fact, one or another decision rule is always optimal only in some strict sense specified by an optimality criterion. The ML rule referred to here is optimal in the sense of the minimum probability of error criterion, which is quite natural and adequate when discrete signals are distinguished or a discrete parameter is measured. But it does not look that adequate when a continuous parameter is measured. It seems much more reasonable in this case to characterize estimation fidelity by a precision, i.e. magnitude
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of deviation " ¼ |
of estimate from the true value of . First of all, it looks quite |
normal to require that expectation of error " over all possible observations y(t), true
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value fixed, be zero for any , i.e. that estimate be equal to the true on average: |
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An estimate meeting this condition is called unbiased. But fulfilment of (2.58) does not yet allow us to consider the estimate good, since the magnitude of random scattering of the estimate around the true value is of critical importance. The variance of error
var ( ^ )2 is a traditional and very adequate measure of this scattering, and f"g ¼
seeking for the rule providing unbiased estimate with minimal varf"g over all true :
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would be highly justified. Thus, minimization of variance of unbiased estimate is a natural way of pursuing greatest measurement accuracy.
In estimation theory the fundamental Cramer–Rao bound is proved bordering the variance of any unbiased estimate from below. An estimate whose variance lies on this bound is called an efficient one. ‘Purely’ efficient estimates are rather infrequent but this is not a big problem from the application standpoint. The matter is that the ML estimate is asymptotically unbiased and efficient as it is again established in classical estimation theory. Physically the term ‘asymptotically’ means ‘in situations where high measurement accuracy is necessary’, or, to put it even more practically, when sufficiently high