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

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

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.

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