Equation (2.28) is quite similar to the Woodward formula for potentially achievable accuracy of time measurement met in Section 2.12.2:

stressing that a tracking loop is an adequate means of time measurement.

There are three parameters affecting the steady-state accuracy of DLL: signal level with respect to noise A2P/N0, loop noise bandwidth BN and early–late separation . The first is a brute force resource and needs no special comment. Optimization of the second is not a straightforward task, since a mechanical reduction of noise bandwidth without careful design of a loop filter may dramatically deteriorate the dynamic properties of the system like pull-in duration and ability of tracking a varying-delay signal. Choosing the separation is a matter of compromise, too: reduction of versus chip duration D provides higher estimation accuracy, thanks to positive correlation of noises at the outputs of early and late branches. At the same time the smaller is separation , the narrower is the discriminator characteristic itself (see Figure 8.6). This imposes more rigid demands on the acquisition precision, since the latter should guarantee falling time mismatch of the local reference and received signal inside an active (non-zero) zone of discriminator characteristic. Another factor to be kept in mind is the risk of loss of synchronism increasing with narrowing discriminator characteristic. A popular way to reconcile the conflicting requirements for the parameters of a tracking loop is adaptation: at the initial stage of pulling-in wider noise bandwidth and larger separation may be used, which after finishing the transient processes are reduced to come to a higher steady-state precision.

Problems

8.1. A serial search should be organized with a constant dwell time Td ¼ 2 ms. The discrete signal to be searched occupies bandwidth 1 MHz and has code length L ¼ 1000. No prior information on code phase is known and the initial frequency bias of the local clock versus the signal carrier frequency lies in the range 10 kHz. Estimate roughly the minimal number of cells to be tested.

8.2.Find asymptotic approximations of overall average probability and average number of steps of a serial search if the false alarm probability becomes very small (Mpf 1). Try to explain the results physically.

8.3.A serial search is used with fixed dwell time and threshold, which are optimized for some signal power P. What happens to the overall average probability of acquisition and average number of steps in two limiting cases: P ! 0 and P ! 1 if no readjustment of dwell time and threshold is done? Give physical reasoning for the results.

8.4.Find expressions for probabilities of false alarm and detection and then for dwell time per cell necessary to secure given pf , pd , if signal amplitude fluctuates according to the Rayleigh law (3.12) and the initial phase is a random constant uniformly distributed over the interval [ , ]. (Hint: the easiest way to do this is by treating a signal as a Gaussian process independent of noise.)

8.5.Find expressions for probability of correct acquisition and acquisition time for Stiffler’s rapid acquisition sequence of length L ¼ 2n.

8.6.Build a discriminator curve of a coherent DLL for the cases ¼ D, 0 < < D, and D < < 2D. Find an extension of the maximal-slope zone. Why is > 2D irrelevant?

8.7.Build a discriminator curve of a noncoherent DLL with separation ¼ D. Why is¼ 2D irrelevant?

8.8.Prove that a voltage controlled oscillator of DLL operates as an integrator of the error signal. Suppose no noise is present on the input and signal delay is constant. Prove that steady-state error at the DLL output is zero. Is the same true if the signal delay changes linearly and the DLL contains no more integrators?

8.9.Consider the DLL where no additional loop filter is used. Suppose the discriminator slope is 0:5 V/ms and VCO changes its frequency by 100 kHz per one volt. The initial difference of frequencies of VCO and received signal is 10 kHz. Find a steady-state noise-free error of DLL.

8.10.Find noise bandwidth and variance of the output error in terms of discriminator slope and VCO gain of a DLL having no loop filter. Give physical reasoning for the dependences of these quantities on system parameters.

Matlab-based problems

8.11.Write a program evaluating the average acquisition time (8.19) and plotting dependences similar to those of Figure 8.4 for an arbitrary search region, given the overall probability of a correct acquisition. Running the program for various values of M, Pc, observe and comment on the behaviour of optimal probability of detection per cell.

8.12.Modify this program for the signal model of Problem 8.4. On running the program, observe and try to explain the difference in optimal detection probability and acquisition time against the previous case.

8.13.Write a program illustrating the dynamics of serial search of a baseband m-sequence. Recommended steps:

(a)Generate a f 1g m-sequence of memory 7–10 and oversample it two times in order to have two search steps per chip.

(b)Repeat this sequence as a reference with a random cyclic shift.

(c)Form several Gaussian noise realizations with standard deviation about p

(2n 1)/8 higher than the signal amplitude, add them to the original sequence and plot with superposition the observation realizations obtained.

(d)Plot a reference sequence.

(e)Repeat the iteration steps, calculating every time the correlation between reference and an observation, each time updating the noise realization; if a current correlation normalized to the length exceeds the threshold 0.5 break the cycle and declare the search finished, otherwise shift the reference by one position (half a chip) and continue to the next step;

(f)At every step plot the correlation versus a current cell number and current reference to see its motion against the signal; use the operator ‘pause’ for a better visualization.

(g)Run the program, varying the threshold, and observe events like false alarm, signal miss and repeated cycles.

8.14.Leaning upon the results of Problem 8.5, write a program calculating search time for Stiffler’s rapid acquisition sequences, given the probability of correct acquisition. Running the program for various code lengths, estimate the gain in acquisition time versus a serial search of an ordinary binary sequence of the same length. Plot the dependences of both acquisition times on n for different probabilities of correct acquisition.

8.15.Write a program calculating and plotting the discriminator curves of coherent and noncoherent DLL for various chip forms and early–late separations. Run the program for rectangular and half-cosine chips and comment on the results.

8.16.Write a program for simulating and exploring coherent DLL with no loop filter. Recommended steps:

(a)Generate a f 1g m-sequence of length L ¼ 63 and oversample it 100 times.

(b)Set early–late separation within the range 0 < 2D and form early and late references as shifted copies of the m-sequence.

(c)Add noise with a standard deviation exceeding signal amplitude by 15 times to the m-sequence of item (a) to obtain observations.

(d)Calculate correlations (normalized to reference energy) with early and late references and error signal as their difference.

(e)Multiply the error signal by gain G and round the result.

(f)Shift references according to the scaled error signal of the previous item.

(g)Plot ‘pure’ signal, observation and references.

(h)Repeat items (c)–(g) 1000 times and observe the behaviour of the DLL. Use the operator ‘pause’ inside the cycle to get a proper visualization.

(i)Changing gain in the range 10–40 and varying separation (e.g. ¼ D/2, D, 2D) observe and compare with the theoretical prediction the dependence of output variance on these parameters.

(j)Find an experimental estimation of error variance and compare its value with the one found in Problem 8.10.