2.13 Estimation of carrier frequency

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which is then selected to derive the ML estimate F as a frequency mismatch of the filter outputting the largest Z(Fk). Clearly, continuous F here is again quantized and the number of filters M should be sufficient to tolerate the quantization error.

Precision of the frequency measurement in accordance with equation (2.59) is governed not only by SNR but also by the sharpness of the correlation between frequencyshifted signal copies in dependence on their mutual frequency mismatch. Due to the phase randomness, we operate with the complex envelope, and the resemblance between its frequency-detuned copies S_(t; 0) and S_(t; F) is characterized by the modulus of the correlation coefficient (2.44):

As is seen, this correlation coefficient as a function of F duplicates in its shape the amplitude spectrum of the signal squared real envelope. Since it is a real envelope, i.e. the amplitude modulation law is always some baseband signal, it follows from the Fourier transform properties that the longer is the signal, the sharper is 0(F). An appropriate name for the extension of 0(F) along the frequency axes is the ‘envelope frequency spread’. Denoting it by Fe we have from the fact just mentioned Fe 1/T. Thus, beyond the ‘brute force’ way of increasing the signal energy, it is possible to improve the accuracy of the frequency measurement by employing a signal having a rather compact spectrum of the envelope (small Fe), i.e. sufficiently long duration T. Formally, this conclusion may be deduced again by evaluation of the second derivative of (F) and substituting it into the Cramer–Rao bound (2.59):

where the rms duration Trms characterizes the extension of a signal in time just as rms bandwidth Wrms characterizes a signal spectrum extension.

Two physical aspects may be referred to at this point. First, increase of precision with signal duration T is easily understood: frequency is the velocity of a signal complete phase angle and, like any velocity, is measured more reliably when the angle increment is observed over a longer time interval. Second, the time–frequency duality shows itself in comparison of the frequency and time-delay estimates. Indeed, while the time-measurement precision is governed by a signal extension in the frequency domain (bandwidth W), frequencymeasurement precision is controlled by a signal extension in the time domain (duration T).

The result above bring us to the conclusion that when the only informative parameter is a signal frequency there is no momentum to resort to the spread spectrum, since, apart from the energy, only signal duration is influential as to the estimate accuracy. This explains why the Doppler speed-monitoring radar very often operates with the plainest unmodulated CW harmonic signal.

2.14 Simultaneous estimation of time delay and frequency

Our next subject is the situation where both time and frequency F shifts of the received signal are unknown and informative, i.e. to be measured. This matches many practical scenarios. In digital communications, e.g. 2G and 3G mobile radio, a receiving session typically starts with synchronizing the local reference with the received signal. This procedure consists of measuring time–frequency misalignment of the local clock against the received signal and subsequent time–frequency adjustment of the former towards its synchronism with the latter. In radar technology to measure simultaneously the distance and the velocity of the target relative to the receiver the signal time delay and Doppler frequency are estimated. In navigation, e.g. GPS, much the same sort of estimation serves to measure the user’s own location and velocity. There are many similar examples.

In distinction to the material of Sections 2.9–2.13 the parameter to be estimated is now the two-dimensional vector ¼ ( , F) rather than a scalar. Accordingly, the model of the received signal combines the two of Sections 2.12 and 2.13:

sðt; ; F; ’Þ ¼ Re S_ðt; ; FÞ expðj2 f0tÞ expðj’Þ

where S_(t; , F) ¼ S_(t ) exp (j2 Ft) is the complex envelope incorporating both the time delay and the frequency shift; and ’ is, as before, a non-informative initial phase. It is obvious that now the correlation rule of estimating , F operates with the correlation modulus:

envelope S(t; , F) resembles the observed complex envelope Y(t). Finding and F,

As previously, one can imagine the correlator-bank structure of the ML estimator where references of different correlators are the copies S_(t; , F) ¼ S_(t ) exp (j2 Ft) with different values , F. More instructive, however, is the matched-filter-based scheme combining those of Figures 2.22 and 2.19a and illustrated by Figure 2.23. It follows directly from the comparison of equations (2.74) and (2.71) showing that (neglecting a constant factor) Z( , F) is duplicated by the real envelope at the output of the matched filter, the filter being frequency-detuned by F. M filter-detector branches are taken and each of them is tuned to its specific frequency, securing thereby estimation of F, while estimation of is obtained by fixing the point of a maximum value at the detector output. The ‘Selection of largest’ unit implements both these operations together, fixing the time moment where the global maximum among the output values of all the branches occurs. Then this moment itself (after subtracting the signal duration) gives^, while the frequency tuning of the branch at whose output the global maximum is

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registered provides F.

The estimation precision depends on the rapidity with which the resemblance between the time–frequency mismatched copies S_(t; 0, 0) and S_(t; , F) of the signal complex