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

Consider performing an iterative maximization of

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

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

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

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: $\frac{d\left|f\right(b\left)\right|}{db}=\mathrm{sign}\left(f\left(b\right)\right)\frac{df\left(b\right)}{db}$ 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.