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0.11 Timing recovery  (Page 9/11)

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The error surface for the output power maximization algorithm can be drawn using thesame “experimental” method as was used in clockrecDDcost.m . Replacing the line that calculates theperformance function with cost(i)=sum(x.^2)/length(x); calculates the error surface for the output power algorithm [link] . [link] shows this, along with three variants:

  1. the average value of the absolute value of the output of the sampler avg { | x [ k ] | } ,
  2. the average of the fourth power of the output of the sampler avg { x 4 [ k ] } , and
  3. the average of the dispersion avg { ( x 2 [ k ] - 1 ) 2 } .
Output of the program clockrecOP.m shows the estimates of the symbols in the top plot and the trajectory of the offset estimates in the bottom.
Output of the program clockrecOP.m shows the estimatesof the symbols in the top plot and the trajectory of the offset estimates in the bottom.
Four performance functions that can be used for timing recovery, plotted as a function of the timing offset τ. In this figure, the optimal answer is at τ=0. Some of the performance functions must be minimized and some must be maximized.
Four performance functions that can be used for timing recovery, plotted as a function of the timing offset τ . In this figure, the optimal answer is at τ = 0 . Some of the performance functions must be minimized and some must be maximized.

Clearly, some of these require maximization (the output power and the absolute value), while othersrequire minimization (the fourth power and the dispersion). While they all behave more or lessanalogously in this easy setting (the figure shows the 2-PAM case with a SRRC pulse shape with beta=0.5 ), the maxima (or minima) may occur atdifferent values of τ in more extreme settings.

TRUE or FALSE: The optimum settings oftiming recovery via output power maximization with and without intersymbol interference in the analog channelare the same.

Use the code in clockrecOP.m to “play with” the output power clock recovery algorithm.How does mu affect the convergence rate? What range of stepsizes works? How does the signal constellation ofthe input affect the convergent value of tau (try 4-PAM and 8-PAM)?

Implement a rectangular pulse shape. Does this work better or worse than the SRRC?

Add noise to the signal (add a zero mean noise to the received signal using theM atlab randn function). How does this affect the convergence of the timing offset parameter tau . Does it change the final converged value?

Modify clockrecOP.m by setting toffset=-1 . This starts the iteration in a closed eye situation. How many iterations does it take to open the eye?What is the convergent value? Try other values of toffset . Can you predict what the final convergent value will be?Try toffset=-2.3 . Now let the oversampling factor be m = 4 and answer the same questions.

Redo [link] using a sinc pulse shape. What happens to the Output Power performance function?

Redo [link] using a T -wide Hamming pulse shape. Which of the four performance functions need to be minimizedand which need to be maximized?

Consider the sampled version x ( k T + τ ) of the baseband signal x ( t ) recovered by the receiver in a 2-PAM communication system with source alphabet ± 1 . In ideal circumstances, the baud-timing variable τ could be set so x ( k T + τ ) exactly matched the source symbol sequence at the transmitter.Suppose τ is selected as the value that optimizes

J D M ( τ ) = 1 N ∑ k = k 0 + 1 k 0 + N ( 1 - x 2 ( k T s + τ ) ) 2

for a suitably large N .

  1. Should J D M be minimized or maximized?
  2. Use the choice in part (a) and the approximation
    d ( x ( k T s + τ ) ) d τ ≈ x ( k T s + τ ) - x ( k T s + τ - ϵ ) ϵ
    for small ϵ > 0 to derive a gradient algorithm that optimizes J D M .
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Source:  OpenStax, Software receiver design. OpenStax CNX. Aug 13, 2013 Download for free at http://cnx.org/content/col11510/1.3
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