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SNR or observation time is provided. Therefore, in any task where high accuracy is wanted, the ML rule is optimal not only by the error-probability criterion but also by the estimate accuracy criterion. Certainly, in the real world, high measurement precision is a typical demand and this is why the ML estimates are extensively used.
For the case of non-energy parameter the Cramer–Rao bound acquires on especially simple form and provides a practical tool to calculate the ML estimate variance:
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ð2:59Þ |
varf g ¼ varf"g |
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The presence of SNR q2 ¼ 2E/N0 in the denominator of the right-hand side of (2.59) is no surprise: naturally, for any reasonable estimation rule the greater is SNR, the smaller is the error and the higher is the measurement precision. At the same time, the dependence on the second derivative of a correlation coefficient deserves a more extensive comment. As is well known from mathematical analysis, the second derivative describes the curvature or sharpness of a function at the examined point and for a convex curve is negative. The sharpness of ( ) at zero point, in its turn, shows the sensitivity of a signal towards mismatch in : the sharper it is, the faster a signal copy mismatched in loses its resemblance to the initial copy. Recollect now that estimation is a particular case of signal distinguishing and is the more reliable the smaller the signal similarity is. This gives a complete explanation as to why 00(0) may affect the precision of measuring : when copies of s(t; ) have low resemblance even with close values of , they are more easily distinguished in comparison with the case of their stronger similarity.
The latter fact points at the general trend of signal design in problems of non-energy parameter estimation. To achieve the desired result not at the cost of just ‘brute force’, i.e. energy increase, one may try to find signals with a steep dependence of correlation coefficient ( ) on .
In the following sections we turn to concrete estimation problems, among which examples of measuring both energy and non-energy parameters are considered. The main idea remains as before: to find out where estimation problems may call for the use of spread spectrum.
2.9 Amplitude estimation
The problem of measuring signal intensity (level, power) may be encountered in numerous applications, from TV broadcasting to digital PAM or QAM data transmission and mobile radio. Let us set it as the problem of measuring unknown amplitude A remaining constant during the observation interval [0, T]. In this statement the following signal model can be assumed:
sðt; AÞ ¼ AsðtÞ
where s(t) is some deterministic reference signal whose amplitude is equal to one by convention. Then signal s(t; A) is the result of scaling the reference signal by an
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unknown factor A. Let E be the energy of the reference signal. Then the energy E(A) of the signal with amplitude A and its correlation z(A) with observation y(t) are:
ZT ZT ZT
EðAÞ ¼ s2ðt; AÞ dt ¼ A2 s2ðtÞ dt ¼ A2E zðAÞ ¼ yðtÞsðt; AÞ dt ¼ Az
0 0 0
where
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ð2:60Þ |
z ¼ yðtÞsðtÞ dt |
0
is the correlation of the observation with the reference signal.
Turning to the correlation version of minimum distance rule (2.8) and noticing that the roles of Ej and zj are now played by E(A) and z(A), respectively, the ML estimation procedure may be treated as maximization of the difference z(A) E(A)/2 in A. Using the equations above, this difference takes the form of a quadratic binomial Az A2E/2 in A with the known coefficients. Its maximum is easily found, producing the ML estimate of amplitude:
^ ¼ z A
E
Thus, calculating a correlation of the observed waveform with the reference signal and scaling the result by the constant 1/E is exactly the desired optimal amplitude estimate. After finding the expectation of z from equation (2.60):
ZT ZT ZT
z ¼ yðtÞsðtÞ dt ¼ sðt; AÞsðtÞ dt ¼ A s2ðtÞ dt ¼ AE
0 0 0
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it is readily seen that on average A strictly coincides with the true value of the amplitude, |
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¼ z/E ¼ A, meaning that the ML |
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asymptotically) unbiased. No more difficult is the evaluation of variance of A: |
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where result (2.15) is used and q2 is SNR for the reference signal (i.e. for a signal having a unit amplitude). It may be shown that (2.61) strictly reproduces the Cramer–Rao bound, proving rigorous (not only asymptotic) efficiency of the ML estimate of signal amplitude. This rare case of an estimate’s rigorous optimality is associated with the energy nature of amplitude and will not be met later when non-energy parameters are considered.
Now, what sort of demand does amplitude measuring impose on signal design? As (2.61) demonstrates, nothing but sufficient energy exhaustively determining estimation precision. No complications of the signal modulation law are able to improve amplitude measurement accuracy if they do not govern the signal energy. Consequently, no momentum to the involvement of spread spectrum appears in connection with this classical reception problem.
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2.10 Phase estimation
We now address the situation where the parameter carrying useful information is the initial phase of the signal. This case is typical of coherent radar and navigation receivers, carrier reference recovery loops of PSK/QAM data transmission links, demodulators of 2G and 3G mobile radio receiver, chrominance channels of TV and many more applications.
We modify the bandpass signal model (2.24), separating the constant (during observation interval) initial phase ’, which is unknown and to be measured:
sðt; ’Þ ¼ SðtÞ cosð2 f0t þ ðtÞ þ ’Þ
Since ’ is a non-energy parameter, E(’) ¼ E and ML estimation of phase consists in maximizing z(’) ¼ R0T y(t)s(t; ’) dt over all ’ 2 [ , ]. To make use of equation (2.59), note that by definition (’) is the cosine of the angle between two signal copies phaseshifted by ’; that is, between two vectors separated by the angle ’. Therefore(’) ¼ cos ’, 00(0) ¼ 1 and variance of the ML estimate ’^ is:
1 varf’^g q2 ; q 1
Again, in common with amplitude measuring, precision of phase estimation is governed only by SNR. Thus, this classical problem is also indifferent to the signal modulation law, whenever signal energy is maintained constant, and does not stimulate spreading a signal spectrum.
2.11 Autocorrelation function and matched filter response
Spread spectrum theory rests to a very large extent on the notion of a signal autocorrelation function (ACF), which is defined as the inner product of two copies of the same signal time-shifted to each other by seconds:
Rð Þ ¼ ðs0; s Þ ¼ |
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ð2:62Þ |
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Signal time delay is a non-energy parameter (E( ) ¼ E) and scaling (2.62) by E 1 produces normalized ACF, which is simply a correlation coefficient of the time-shifted signal copies:
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Clearly, the latter shows how rapidly the likeness of the signal time-spaced copies dies away with the delay mismatch . According to the general properties of a correlation coefficient ( ) indicated in Section 2.8, ACF is an even function of , attaining its maximum at zero point:
Rð Þ Rð0Þ ¼ E; Rð Þ ¼ Rð Þ , ð Þ ð0Þ ¼ 1; ð Þ ¼ ð Þ |
ð2:64Þ |
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Using equations (2.39) and (2.34) it is not difficult to verify that for any bandpass signal (2.37) ACF:
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is the ACF of the complex envelope S_(t), or, in other normalized version of ACF (2.66):
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words, the modulation law. The
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being the correlation coefficient of two time-shifted copies of the complex envelope S_(t), serves (after taking the modulus) as a measure of the rapidness with which the timeshifted modulation law loses similarity with the initial one when the mismatch grows. As may be seen from (2.65), ACF ( ) of a bandpass signal s(t) is a bandpass signal itself whose modulation law is ACF ( ) of the complex envelope of s(t). In particular, the real envelope 0( ) of ACF of s(t) is the modulus of ( ): 0( ) ¼ j ( )j, which is illustrated by Figure 2.16.
Any ACF can be obtained physically as an output of the correlator, i.e. the device running the straightforward operations set up by definitions (2.62) or (2.66). In this case calculations for a range of are fulfilled point by point, i.e. repeatedly in time or in hardware. An alternative solution is the matched filter, i.e. a linear system with the pulse response reproducing a mirror image of the signal: h(t) ¼ s(T t), where T is, as usual, signal duration and an immaterial proportionality factor is set to be one. This filter emerges every now and again as an integral element of an optimal receiver in the
1
ρ (τ)
ρ 0(τ)
τ
Figure 2.16 Bandpass signal ACF
Classical reception problems and signal design |
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AWGN channel, but its optimality often goes far beyond this specific channel model. In particular, it maximizes output SNR among all linear systems, signal given. In our current context the matched filter is important due to its ability to calculate and reproduce ACF as a real-time output waveform. To examine this, apply the signal s(t) to the input of the filter matched to s(t). The filter response r(t) may then be calculated as the convolution integral:
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Z1 sð Þhðt Þ d ¼ |
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ð2:68Þ |
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and duplicates the ACF in real time with predictable delay equal to the signal duration. To elucidate what was said, consider Figure 2.17. The rectangular baseband pulse s(t) of duration T (Figure 2.17a) has a triangular ACF R( ) of duration 2T with a maximum at the zero point (b, dashed line). In accordance with equation (2.68) the matched filter response reproduces a copy of this ACF delayed by the signal duration T so that maximal voltage at the filter output occurs at the moment when the input signal ends (b, solid line). If the pulse were bandpass with a rectangular envelope s(t), its ACF would be a triangular bandpass pulse (c, dashed bold line) and its T-delayed copy would appear at the bandpass matched filter output (c, solid bold line). The maximum of the filter response to the signal the filter is matched to always occurs at the moment of signal ending (at least no earlier), since this filter processes the whole signal. It is very instructive to note that for a bandpass signal moments of maximal envelope and maximal value of carrier cosine at the matched filter output always coincide, since ACF always assumes its maximum at the zero point (see also
Figure 2.16).
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Figure 2.17 Illustration to the definition of ACF and its forming by the matched filter