modulation component B_ (t) and unknown initial phase . After multiplying the input signal with the references and extracting the low-frequency component the two resulting bandpass signals of difference carrier frequency f0 f1 will have complex envelopes AB_ (t)S_(t)S_ (t " /2) exp [j( #)]. Suppose that bandpass filters after the multipliers have pulse response with complex envelope H_ (t). Then real envelopes at the filter outputs calculated in terms of the convolution integral (see Section 2.12.1) are:

If the filter pulse response is rectangular of duration T equal to data symbol duration, and data modulation is PSK, then samples of output real envelopes at the moment t ¼ T are:

being proportional to the modulus of the corresponding value of ACF ( ) of the spreading complex envelope. The difference of squared moduli again gives a discriminator curve of form similar to the one of Figure 8.6 (see Problems 8.7 and 8.15).

Implementation of the scheme of Figure 8.8 may run into trouble in the form of parameter imbalance of the early and late branches. In order to get round it various solutions are known [9,18,77], including the tau-dither loop (another name is ‘timeshared’), where only a single branch is involved, switching by turns between the early and late references.

8.4.3 DLL noise performance

DLL is just a particular case of a phase-lock loop and exposes the difficulties as to the analysis of its behaviour, which are generic to nonlinear feedback systems [89,90]. Still, one of the most important characteristics of DLL performance–noise error of a steadystate delay tracking–is easily calculated whenever the linear approximation is applicable.

In practice, rather small noise error is usually wanted, meaning good filtering capability of the loop against noise. The fluctuations at the loop output may be considered small if the error, i.e. the difference between the true current signal delay and its estimation ^ delivered by DLL, is held within the linear zone of the discriminator curve with probability close to one. If this condition is met, one can believe that the discriminator curve is linear in the infinite range of error " ¼ ^ . This allows linearizing the system model as follows.

Let us limit ourselves to a baseband (or equivalently, coherent) discriminator of

¼ N0Er/2, where Er is reference energy over integration time T. With E being, as

entirely linear, and variance varf^g of random fluctuations at its output may be found, based on the superposition principle, independently of signal component, as:

~

where hl(f ) is the transfer function of a closed loop. To find the last quantity it is enough to apply the delta function to the loop input. Then the output spectrum, being exactly

A dummy spectrum of delay fluctuations N (f ) is spread over the bandwidth Wf ¼ 1/T, which is typically much wider as compared to the bandwidth of the closed loop; otherwise the latter could not smooth noise fluctuations effectively. This allows the following version of (8.25):

Returning now to (8.23) and (8.24) and noting that ( ) ¼ 1 /D, 0 < < D we may write (8.27) in the form:

is called SNR in the loop. The reason for such a name is obvious: the numerator of (8.29) contains an actual signal power, since P ¼ E/T is a power of a standard (having amplitude A ¼ 1) signal. At the same time, the denominator presents a noise power within the noise bandwidth of the loop.