# 0.11 Timing recovery  (Page 10/11)

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Consider a resampler with input $x\left(k{T}_{s}\right)$ and output $x\left(k{T}_{s}+\tau \right)$ . Use the approximation in [link] (b) to derive an approximate gradient descent algorithm that minimizesthe fourth power performance function

${J}_{FP}\left(\tau \right)=\frac{1}{N}\sum _{k={k}_{0}+1}^{{k}_{0}+N}{x}^{4}\left(k{T}_{s}+\tau \right)$

for a suitably large $N$ .

Implement the algorithm from [link] using clockrecOP.m as a basis and compare the behavior with the output power maximization. Consider

1. Convergence speed
2. Resilience to noise
3. Misadjustment when ISI is present

Now answer the same questions for the fourth power algorithm of [link] .

This problem explores a receiver that requires carrier recovery, timing recovery, and automatic gain correctionin order to function properly.

1. Write a carrier-recovery routine for the signal generated by BigEx1.m . You may want to use the receiver in BigIdeal.m for inspiration. Pay particular attention to the given parameters (Hint: the sampling rate isbelow the IF frequency, so the receiver front-end effectively employs subsampling). Describe the carrier recovery method used and plot the tracking of $\Phi$ . If the receiver employs preprocessing, take care to design the BPF appropriately.
2. Add a timing recovery algorithm to the receiver so that the receiver samples at the appropriate time. State the timing recovery method usedand plot the tracking of $\tau$ .
3. Add an AGC to the code and decode the message. Does the code recover the message appropriately?Calculate the error rate of the receiver.

## Two examples

This section presents two examples in which timing recovery plays a significant role.The first looks at the behavior of the algorithms in the nonideal setting. When there is channel ISI, theanswer to which the algorithms converge is not the same as in the ISI-free setting. This happens because the ISIof the channel causes an effective delay in the energy that the algorithm measures.The second example shows how the timing recovery algorithms can be used to estimate (slow) changes in theoptimal sampling time. When these changes occur linearly, they are effectively a change in the underlying period,and the timing recovery algorithms can be used to estimate the offset of the period in the same way that thephase estimates of the PLL in [link] can be used to find a (small) frequency offset in the carrier.

Modify the simulation in clockrecDD.m by changing the channel:

% T/m channel chan=[1 0.7 0 0 .5];

With an oversampling of m=2 , 2-PAM constellation, and beta=0.5 , the output of the output power maximization algorithm clockrecOP.m is shown in [link] . With these parameters, the iteration begins in a closed eye situation. Because of the channel, nosingle timing parameter can hope to achieve a perfect $±1$ outcome. Nonetheless, by finding a good compromise position (in this case, converging to anoffset of about 0.6), the hard decisions are correct once the eye has opened (which first occurs around iteration500).

Example  [link] shows that the presence of ISI changes the convergent valueof the timing recovery algorithm. Why is this?

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