0.5 Sampling with automatic gain control

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As foreshadowed in [link] , transmission systems cannot be fully digital because themedium through which the signal propagates is analog. Hence, whether the signal begins as analog (such as voiceor music) or whether it begins as digital (such as mpeg, jpeg or wav files), it will be converted to a highfrequency analog signal when it is transmitted. In a digital receiver, the received signal must betransformed into a discrete-time signal in order to allow subsequent digital processing.

This chapter begins by considering the sampling process in both the time domainand in the frequency domain. Then "Exploring Sampling in MATLAB" discusses how M atlab can be used to simulate the sampling process. This is not completely obvious becauseanalog signals cannot be represented exactly in the computer. Two simple tricks are suggested. The first expressesthe analog signal in functional form and takes samples at the desired times. The second oversamples the analog signal so that it is represented at a high datarate; the “sampling” can then be done on the oversampled signal.

Sampling and quantization are important because they translate the signal from analog to digital. It is equally important to be ableto translate from digital back into analog, and the celebrated Nyquist sampling theorem shows that this is possible for any bandlimited signal, assuming the sampling rate is fast enough.When the goal of this translation is to rebuild a copy of the transmitted signal, this is called reconstruction . When the goal is to determine the value of the signalat some particular point, it is called interpolation . Techniques (and M atlab code) for both reconstruction and interpolation appear in "Interpolation and Reconstruction" .

[link] shows the received signal passing through a BPF (which removes out-of-band interferenceand isolates the desired frequency range) followed by a fixed demodulation to the intermediate frequency (IF) wheresampling takes place. The automatic gain control (AGC) accounts for changes in the strength of thereceived signal. When the received signal is powerful, the gain $a$ is small; when the signal strength is low, the gain $a$ is high. The goal is to guarantee that the analog to digital converterdoes not saturate (the signal does not routinely surpass the highest level that can be represented), and that it doesnot lose dynamic range (the digitized signal does not always remain in a small number of the possible levels).The key in the AGC is that the gain must automatically adjust to account for the signal strength, which may vary slowlyover time.

The AGC provides the simplest example of a system element that must adapt to changes in its environment (recall the “fifthelement” of Chapter  [link] ). How can such elements be designed? Software Receiver Design suggests a general method based on gradient-directed optimization. First, a “goal” and an associated “objective function” arechosen. Since it is desired to maintain the output of the AGC at a roughly constantpower, the associated objective function is defined to be the average deviation of the power from that constant; the goal is to minimizethe objective function. The gain parameter is then adjusted according to a“steepest descent” method that moves the estimate “downhill” towards the optimal value that minimizes theobjective. In this case the adaptive gain parameter is increased(when the average power is too small) or decreased (when the average power is too large), thus maintaininga steady power. While it would undoubtedly be possible to design asuccessful AGC without recourse to such a general optimization method, the framework developed in Sections "Iteration and Optimization" through "Automatic Gain Control" will also be useful in designing other adaptive elementssuch as the phase tracking loops of Chapter  [link] , the clock recovery algorithms of Chapter  [link] , and the equalization schemes of Chapter  [link] .