# 0.5 Sampling with automatic gain control  (Page 14/19)

 Page 14 / 19

Consider performing an iterative maximization of

$J\left(x\right)=8-6|x|+6cos\left(6x\right)$

via [link] with the sign on the update reversed (so that the algorithm will maximize ratherthan minimize). Suppose the initialization is $x\left[0\right]=0.7$ .

1. Assuming the use of a suitably small stepsize $\mu$ , determine the convergent value of $x$ .
2. Is the convergent value of $x$ in part (a) the global maximum of $J\left(x\right)$ ? Justify your answer by sketching the error surface.

Suppose that a unimodal single-variable performance function has only one point with zeroderivative and that all points have a positive second derivative. TRUE or FALSE: A gradient descent methodwill converge to the global minimum from any initialization.

Consider the modulated signal

$r\left(t\right)=w\left(t\right)cos\left(2\pi {f}_{c}t+\Phi \right)$

where the absolute bandwidth of the baseband message waveform $w\left(t\right)$ is less than ${f}_{c}/2$ . The signals $x$ and $y$ are generated via

$\begin{array}{cc}\hfill x\left(t\right)& =\text{LPF}\left\{r\left(t\right)cos\left(2\pi {f}_{c}t+\theta \right)\right\}\hfill \\ \hfill y\left(t\right)& =\text{LPF}\left\{r\left(t\right)sin\left(2\pi {f}_{c}t+\theta \right)\right\}\hfill \end{array}$

where the LPF cutoff frequency is ${f}_{c}/2$ .

1. Determine $x\left(t\right)$ in terms of $w\left(t\right)$ , ${f}_{c}$ , $\Phi$ , and $\theta$ .
2. Show that
$\frac{\partial }{\partial \theta }\left\{\frac{1}{2}{x}^{2}\left(t\right)\right\}=-x\left(t\right)y\left(t\right)$
using the fact that derivatives and filters commute as in [link] .
3. Determine the values of $\theta$ maximizing ${x}^{2}\left(t\right)$ .

Consider the function

$J\left(x\right)={\left(1,-,|,x,-,2,|\right)}^{2}.$
1. Sketch $J\left(x\right)$ for $-5\le x\le 5$ .
2. Analytically determine all local minima and maxima of $J\left(x\right)$ for $-5\le x\le 5$ . Hint: $\frac{d\phantom{\rule{4pt}{0ex}}|f\left(b\right)|}{db}=\mathrm{sign}\left(f\left(b\right)\right)\frac{d\phantom{\rule{3.33333pt}{0ex}}f\left(b\right)}{db}$ where $\mathrm{sign}\left(a\right)$ is defined in [link] .
3. Is $J\left(x\right)$ unimodal as a function of $x$ ? Explain your answer.
4. Develop an iterative gradient descent algorithm for updating $x$ to minimize $J$ .
5. For an initial estimate of $x=1.2$ , what is the convergent value of $x$ determined by an iterative gradient descent algorithm with a satisfactorily small stepsize.
6. Compute the direction (either increasing $x$ or decreasing $x$ ) of the update from (d) for $x=1.2$ .
7. Does the direction determined in part (f) point from $x=1.2$ toward the convergent value of part (e)? Should it (for a correct answer to (e))? Explain your answer.

## Automatic gain control

Any receiver is designed to handle signals of a certain average magnitude most effectively. The goal of an AGC is toamplify weak signals and to attenuate strong signals so that they remain (as much as possible) within the normaloperating range of the receiver. Typically, the rate at which the gain varies is slow compared with the data rate, though itmay be fast by human standards.

The power in a received signal depends on many things: the strength of the broadcast, the distance from the transmitter to the receiver, thedirection in which the antenna is pointed, and whether there are any geographic features such as mountains (or tall buildings) that block,reflect, or absorb the signal. While more power is generally better from the point of view of trying to decipher the transmitted message,there are always limits to the power handling capabilities of the receiver. Hence if the received signal is too large (on average), itmust be attenuated. Similarly, if the received signal is weak (on average), then it must be amplified.

[link] shows the two extremes that the AGC is designed to avoid. In part (a), the signal is much larger than the levels of thesampling device (indicated by the horizontal lines). The gain must be made smaller. In part (b), the signal ismuch too small to be captured effectively, and the gain must increased.

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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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