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With beta=0 , the SRRC is exactly the sinc. Redo the above exercises tryingvarious values of beta between 0 and 1.

The function srrc.m is available on the website. Its help file is

 s=srrc(syms, beta, P, t_off);  Generate a Square-Root Raised Cosine Pulse 'syms' is 1/2 the length of srrc pulse         in symbol durations 'beta' is the rolloff factor:         beta=0 gives the sinc function 'P' is the oversampling factor  t_off is the phase (or timing) offset

M atlab also has a built-in function called resample , which has the following help file:

 Change the sampling rate of a signal  Y = resample(X,P,Q) resamples the sequence in vector X at P/Q times the original sample  rate using a polyphase implementation. Y is P/Q times the length of X.  P and Q must be positive integers.

This technique is different from that used in [link] . It is more efficient numerically at reconstructing entire waveforms, but it only workswhen the desired resampling rate is rationally related to the original. The method of [link] is far more efficient when isolated (not necessarilyevenly spaced) interpolating points are required, which is crucial for synchronization tasks in Chapter  [link] .

Iteration and optimization

An important practical part of the sampling procedure is that the dynamic range of the signal at the input tothe sampler must remain within bounds. This can be accomplished using an automatic gain control, which isdepicted in [link] as multiplication by a scalar a , along with a “quality assessment” block that adjusts a in response to the power at the output of the sampler.This section discusses the background needed to understand how the quality assessment works.The essential idea is to state the goal of the assessment mechanism as an optimization problem.

Many problems in communications (and throughout engineering) can be framed in terms of an optimization problem. Solving suchproblems requires three basic steps:

  1. Setting a goal—choosing a “performance” or “objective” function.
  2. Choosing a method of achieving the goal—minimizing ormaximizing the objective function.
  3. Testing to make sure the method works as anticipated.

“Setting the goal” usually consists of finding a function that can be minimized (or maximized), and for whichlocating the minimum (or maximum) value provides useful information about theproblem at hand. Moreover, the function must be chosen carefully so that it(and its derivative) can be calculated based on quantities that are known, or which can be derived from signalsthat are easily obtainable. Sometimes the goal is obvious, and sometimes it is not.

There are many ways of carrying out the minimization or maximization procedure. Some of these are direct. For instance, if the problem is tofind the point at which a polynomial function achieves its minimum value, this can be solved directly by finding the derivative and setting itequal to zero. Often, however, such direct solutions are impossible, and even when they are possible, recursive(or adaptive) approaches often have better properties when the signals are noisy. This chapter focuses on arecursive method called steepest descent , which is the basis of many adaptive elements used in communications systems(and of all the elements used in Software Receiver Design ).

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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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