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2.12 Estimation of the bandpass signal time delay
2.12.1 Estimation algorithm
The problem we address in this section is among the most frequently encountered. It is typical of TV broadcasting (synchronization channels), digital mobile radio (pilot channels, timing recovery loops), radar (target distance measurement), space-based and groundbased navigation (beacon distance measurement) and so forth. To operate adequately, practically any modern information processing system needs to retrieve timing data from the received waveform, and this is exactly what is meant by time delay estimation.
Suppose that a bandpass signal (2.37) s(t) ¼ Re[S_(t) exp (j2 f0t)] passing through the channel acquires unknown time delay and initial phase ’0, i.e. takes the form:
sðt; ; ’0Þ ¼ sðt ; ’0Þ ¼ RefS_ðt Þ exp½j2 f0ðt Þ þ j’0&g
In many situations phase ’0 is random and uniformly distributed over the interval [ , ], i.e. it has no bearing on the only object of interest—the time delay . Let us incorporate the phase component caused by delay into the integral initial phase ’ ¼ 2 f0 þ ’0. The latter, remaining random and uniformly distributed over [ , ], is again independent of , i.e. contains no information on it due to the destructive contribution of ’0. Then the received signal may be represented by the following:
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where time delay is an unknown useful parameter to be measured and ’ is a useless initial phase, whose uncertainty may only complicate the measurement of .
As was stated in Section 2.8, any estimation procedure is a particular case of signal distinguishing. In the case in question, we need to distinguish between multiple copies of signal (2.69) that differ from each other by value of the time shift , a nuisance param- eter—initial phase ’—being an extra care. Fortunately, there is a straightforward way to overcome the signal indeterminacy related to the randomness of ’: in Section 2.5 it was shown that optimal choice between noncoherent signals is performed by distinguishing between their deterministic modulation laws, i.e. complex envelopes. In the delay estimation, consequently, time-shifted copies of the signal complex envelope S_(t) should be compared and one of them declared received. It is the time shift of the latter which is given out as the ML estimate ^ of time delay. Certainly, preference for this copy over the rest is based on its minimum distance from the received complex envelope Y_ (t) or, taking into account that time delay is a non-energy parameter, on maximal correlation with Y_ (t). This correlation is evaluated by correlation modulus (2.47), which can be rewritten taking into account that the role of the signal number now belongs to the value of :
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Based on this entity estimation rule Z(^) ¼ max Z( ) is absolutely transparent physic-
ally: the ML estimate ^ is just the time delay under which the signal modulation law has maximal resemblance with the observed one.
Classical reception problems and signal design |
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Figure 2.18 Correlator-bank implementation of the ML estimate of time delay
One possible solution to implement this estimation rule is a bank of correlators, shown in Figure 2.18. The observed complex envelope is processed in parallel by M correlators, whose reference signals are time-shifted copies of the signal complex envelope. At the correlator outputs values of Z( i), i ¼ 1, 2, . . . , M are present, and the rightmost block compares them to select the largest. The ML estimate is the delay of the reference in the correlator whose output is maximal.
Of course, this structure treats the time delay as though it takes only discrete values. When this is not the case it simply quantizes continuous and the number of correlators (or, which is the same, discrete values i) should be chosen sufficient to make the quantization error tolerably low.
The matched filter offers an alternative version of delay estimator. Let the observation y(t) be applied to the filter matched to the signal s(t). Find the output waveform r(t) using the convolution integral and filter pulse response h(t) ¼ s(T t):
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Comparing this result with the general bandpass signal model (2.37), one can see that the square brackets single out nothing but the complex envelope at the filter output.
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Figure 2.19 Matched filter implementation of the ML estimate of time delay
Therefore, the real output envelope (amplitude modulation) defined as the modulus of
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i.e. replicates in real time (with immaterial scaling by a factor 1/2) a copy of the correlation modulus (2.70) time-shifted by the known signal duration. This points directly to the possible structure of ML estimator of delay shown in Figure 2.19a. The observation y(t) is first filtered by a matched filter and then passes through the envelope detector. The last unit in the structure registers the time moment tm when the detector output waveform rd (t) takes a maximum value and the desired ML estimate ^ is obtained after subtraction of the known constant T from tm (Figure 2.19b).
The scheme of Figure 2.19 seems more transparent for explaining the idea, but many practical software-based ML estimators may appear not so directly identifiable with either of the two structures just discussed.
2.12.2 Estimation accuracy
According to equation (2.59), the variance of estimation of depends on the steepness with which the time-shifted signal copy loses its similarity with the initial one. But in the case of random-phase signals only the deterministic complex envelopes are being compared to carry out the ML estimate. The similarity of time-mismatched copies of the complex envelope is characterized by the envelope of the signal ACF (2.65):
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Figure 2.20 Illustration to the time measuring accuracy
and therefore its steepness should affect the error variance of measuring time delay, which can be strongly argued on physical grounds. Indeed, as is seen from equations (2.70) and (2.71), ACF modulus (2.72) is the noise-free envelope at the matched filter output (neglecting constant factor) taking a maximum value at some ‘true’ time-position tm0 (Figure 2.20a, solid line). When noise is added the time-position tm of this maximum fluctuates with respect to the true point (Figure 2.20a, dashed line) spanning the range depending on the sharpness of the signal envelope at the filter output, i.e. of the ACF modulus (2.72).
To make the latter more evident, note that fixing the moment of a maximum of the detector output rd (t) is tantamount to registering the time-point where its derivative r0d (t) crosses the zero level (provided it is selected properly against spurious ‘zero-points’ caused by possible side maximums). This is illustrated by Figure 2.20b. When SNR is high enough, deviation " ¼ tm tm0 is small and we may assume that the noisy (dashed) curve r0d (t) is linear within the range [tm0, tm], having the same slope as the noiseless (solid) curve has at the point tm0. Hence, solving the right triangle seen in Figure 2.20b, " can be found as a result of division of its dashed-dotted leg by the steepness of the noiseless curve r0d (t) at the point tm0, i.e. by the second derivative r00d (tm0). The latter, in its turn, is exactly 000(0), so that " r0d (tm0)/ 000(0). On the other hand, variance of scattering of r0d (tm0) around the noiseless zero value is greater the smaller is SNR and the higher is the rate of random change of the noisy detector output rd (t) (dashed line in Figure 2.20a). The sharpness of ACF of a random process tells us about the rate of its change, and ACF of the detector output random process under high SNR repeats the envelope of ACF at the detector input. The latter envelope, when the filter is matched to the signal, is nothing but the envelope 0( ) of the signal ACF. Since the sharpness of any ACF is measured by its second derivative at zero point with a minus sign, the variance of r0d (tm0) is proportional to 000(0). Then:
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We thereby came to quite an important conclusion: time-measurement accuracy is critically governed by the signal ACF sharpness, and the sharper is ACF, the smaller is the variance of the ML estimate of time delay .
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Spread Spectrum and CDMA |
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In parallel with 000(0), another indicator of signal ACF sharpness can be introduced, which we will call the correlation spread and denote c. This parameter characterizes signal ACF width (see Figure 2.20a) and—like duration or bandwidth—should be defined by a convention, since ACF can be of rather complicated shape and fall to zero only asymptotically. In the light of the meaning of ACF, we believe that signal copies (or copies of the complex envelope) with mutual time-shift < c have significant resemblance while with > c their resemblance is negligible. It is clear that the conclusion above may be reformulated in terms of this new entity: signals with narrow ACF, i.e. small correlation spread, are generally preferable for the high precision of time-delay estimate.
Continuing, we may recall one of the basic facts of spectrum analysis following directly from result (2.72) after applying the Parseval theorem to it: signal ACF and energy spectrum are related to each other by the Fourier transform. In terms of the complex envelope:
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It may appear difficult to understand why a measure like this reports about the spectrum width. In this case analogy with the more customary probabilistic scattering parameter is advisable. Variance of the random variable x with zero mean and PDF W(x) is by definition varfxg ¼ 11 x2W(x)dx, characterizing a scattering range of x around its which is equivalent, width of PDF W(x). But the normalized energy
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Thus, roughly speaking, the criterion for good signals in time-measurement tasks is: signals with short ACF (small correlation spread c) or, equivalently, wide bandwidths W, are of the primary interest. Compared with the previously considered two estimation problems the situation looks pretty new: there exists a sustainable resource to improve estimation fidelity beyond a brute-force course, i.e. just extra energy expenditure.
It should be stressed now that ‘wide band’ and ‘spread spectrum’ are not synonyms. As a matter of fact, if conventions adopted on the correlation spread c and the signal duration T definitions are sound enough, signal copies shifted by more than T are practically non-overlapping, i.e. have negligible inner product or resemblance. Therefore, c T, showing that shortening signal ACF and, consequently, widening its bandwidth can be achieved trivially by shortening the signal itself.
However, going this way one should not forget that SNR depends on signal energy E ¼ kpPT where P is the signal peak power and kp is a coefficient determined by the signal shape. It has to be clear that preserving SNR when the signal is being shortened requires a proportional increase of peak power. Consequently, in the pursuit of higher and higher precision, it is possible to arrive at the situation where the necessary peak power becomes prohibitively large. Generally, very high transmitted power entails large mass and dimension of transmitter equipment and an energy source. In addition to that, short high-power pulses may appear substantially damaging for neighbouring systems and the surrounding ecology.
A more elegant way of improving the time-measurement accuracy is prompted by the fact that signal shortening is not the only method of widening the spectrum or, which is the same, reducing the correlation spread. Consider a signal whose duration T is large enough to provide necessary energy, i.e. SNR, in combination with acceptably low peak power P. Suppose that the internal angle modulation law is found, making a signal correlation spread much smaller than signal duration: c T. Then, signal ACF is sharp, providing high-accuracy estimation of time delay, despite the long duration of the signal itself. But in the light of the dependence between the correlation spread and the bandwidthc 1/W inequality c T means that the signal has a large time–frequency product WT 1, i.e. it is a spread spectrum one. Putting it another way, involvement of the spread spectrum allows the contradiction between peak power and estimation precision to be removed: necessary energy is put into a signal at the cost of duration, not power, while high measurement accuracy is achieved by designing an appropriate modulation law.
Physically, fulfilment of the condition c T means that a long signal becomes short after processing in the matched filter of the estimator shown in Figure 2.19a. It should be clear that this matched filter time-compression phenomenon is achievable only with spread spectrum signals. In principle, any signal may be time-shortened in some purposely designed (generally mismatched) filter, e.g. an equalizer, but for plain signals
52 |
Spread Spectrum and CDMA |
the price of this is loss in SNR, and only spread spectrum signals promise best ‘noiseclearance’ simultaneously with the time compression. At the same time it has to be understood that the spread spectrum condition WT 1 is just the necessary one, and finding signals combining long proper duration with a sharp ACF is a rather sophisticated task. There is further discussion of this in Chapter 5.
Consider Figure 2.21 where waveforms are shown simulated by Matlab. The plots in column (a) present, respectively, a plain bandpass bell-shaped pulse, matched filter response to it (delayed ACF) and 10 superimposed noisy realizations at the detector output. The plots of column (b) show similar waveforms but for the spread spectrum signal, which is a linear-frequency-modulated pulse having the same shape, duration and energy as that of column (a). The time-compression effect for case (b) manifests itself clearly and results in a noticeably narrower range of fluctuations of the time position of a maximum at the detector output in comparison with case (a). This demonstrates convincingly how spread spectrum can improve time-measurement accuracy without compromising peak power.
We now formulate the following conclusion. When no peak power limitation is imposed, the classical problem of time-delay estimation does not appeal strongly to the spread spectrum. However, spread spectrum is an imperative demand when tough peak power constraints need to be obeyed. Note in passing that the latter situation is quite typical of the pulse radar, which explains why this application area has for decades been stimulating the development of spread spectrum technology.
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Figure 2.21 Illustration of time estimation and matched filter time-compression effect