Constructing an input signal whose response is determined by the impulse response

Question

I'm self-studying Signals and Systems by Alan Oppenheim, and I have some trouble understanding the solution to a homework problem.

P5.13 tries to find the sufficient and necessary condition on the impulse response $h[n]$ such that for any input $x[n]$,

$$max\{|x[n]|\} \geq max\{|y[n]|\}$$

In the 2nd part of the proof, when it tries to prove $\sum_{k=-\infty}^{+\infty} |h[k]|$ is a necessary condition of the inequation above, it claims that it's always possible to construct $x[n]$ as a series of +1 and -1 so that $y[n] = \sum_{k=-\infty}^{+\infty} |h[k]|$.

I'm not convinced by this claim. I think because of the shifting during convolution, it not always possible to construct x[n] satisfying the inequation.

Say h[n] has 3 non-zero components: $h[0]$, $h[1]$ positive, h[2] negative. We construct $x[-\tau]$ at time (0, 1, 2) as (1, 1, -1), and $y[0] = h[0]+h[1]-h[2] = \sum_{k=-\infty}^{+\infty} |h[k]|$.

When we shift $x[-\tau]$ to the right by 1 to compute y[1], setting $x[-\tau+1]$ at time (0, 1, 2) as (1(new), 1, 1), the last two components having been determined by $y[0]$, then $y[1] = h[0]+h[1]+h[2] \neq \sum_{k=-\infty}^{+\infty} |h[k]|$.

Can anyone confirm my analysis or point out of the flaw of it? Any insight is appreciated. Thanks.

Link to homework problem and solution

Answer 1

You're right that you generally can't find a sequence $x[n]$ such that

$$y[n]=\sum_{k=-\infty}^{\infty}\big|h[k]\big|\tag{1}$$

is satisfied for all values of $n$, but that's also not necessary. It is enough to see that there's a sequence $x[n]$ with $|x[n]|=1$ such that $(1)$ is satisfied for one specific value of $n$. If that's the case, then you've shown that the condition

$$\sum_{k=-\infty}^{\infty}\big|h[k]\big|\le 1\tag{2}$$

is not only sufficient but also necessary for

$$\max_n \big|x[n]\big|\ge\max_n \big|y[n]\big|\tag{3}$$

to hold.

Answer 2

A more detailed proof based on Matt's answer.

Let $n_0$ be the argument to $\max\{|y[n]|\}$, i.e. $\max\{|y[n]|\} = |y[n_0]|$.

We can always find $x[n]$ as a series of +1's and -1's, s.t. $$|y[n_0]| = \sum_{k=-\infty}^{+\infty} |h[k]|.$$

Given $$\max\{|y[n]|\} \leq \max\{|x[n]|\}$$,

we can derive $$\max\{|y[n]|\} = y[n_0] = \sum_{k=-\infty}^{+\infty} |h[k]| \leq \max\{|x[n]|\} = 1$$,