# Impulse response of system with $\max\left\{\lvert x[n] \rvert\right\}\ge\max\left\{\lvert y[n]\rvert \right\}$

Find a relation/condition for the impulse response, so for any $$x[n]$$ (input) with $$y[n]$$ (output) this relation is true: $$\max\left\{\lvert x[n] \rvert\right\} \ge \max\left\{\lvert y[n]\rvert \right\}$$ The system is discrete, linear and time invariant.

I tried to replace the $$y[n]$$ with the convolution formula,but it doesn't work. I tried to use Fourier transformation, but it doesn't work either. Any suggestions?

• Hi! I added a few line breaks and the > citation marker to make your question easier to read. Welcome here! – Marcus Müller Aug 26 at 8:04

Not sure what exactly the question is: find some LTI system that meets this constraint (which is trivial) or create a testable condition for any LTI system. (Which is not trivial).

For the latter: Discrete LTI systems are guaranteed to not increase the maximum amplitude if the absolute sum of the impulse response is smaller than $$1$$, i.e.

$$\sum |h(n)| \leq 1$$

The proof is a bit tedious, so I'll skip it unless there is need for it.

• The question is to create a testable condition for any LTI system.I think that the sum you wrote above is too close to the solution,can you give me a link with the proof? – JIm Aranikout Aug 26 at 9:43

To simplify notations (and, honestly, put convergence troubles aside), let us consider a finite impulse response system. Every output then becomes, in a sliding way, an weighted average of samples $$x[k]$$: $$y[\cdot] =\sum_k a_kx[k]$$. Easily, you get:

$$|y[\cdot]| =\left|\sum_k a_kx[k]\right|\le \sum_k |a_k||x[k]|\le \max |x[k]|\sum_k |a_k|$$

So $$\max |y[k]|$$ is smaller than $$\max |x[k]|$$ when $$\sum_k |a_k|\le 1$$. If $$\sum_k |a_k|= 1$$, choose $$x[k] = \textrm{sign}\, a_k$$, to see that the bound is tight. This could be a testable starting point.

In a nutshell: as long as $$\sum_k |a_k|\le 1$$, the condition is satisfied. And you can design an input signal with a subsequence of values, defined by the signs of the flipped impulse response, to attain the maximum.

You're overthinking it. The literally simplest possible linear system (that's not output=input) can do this for you, if you're using the right parameter.

Another solution (which is a special case of the recommended system above) is also

$$y[n] = 0\text.$$

• Yes,but i think that i must find a solution,that has impulse response in it.ex h[n]>2x[n] Is this relation true for every system?(max{|x[n]|}≥max{|y[n]|}) – JIm Aranikout Aug 26 at 8:17
• $h[n] = 2x[n]$ is not an impulse response of a time-invariant system. Get your technical terms right! I think what you mean is right, but the way you say it isn't. Also, you can easily test your hypothesis yourself. Not doing your homework! – Marcus Müller Aug 26 at 8:33
• Its August,i don't have any homework,Im trying to prepare myself for some exams and im stuck with this problem.The ex h[n]>2x[n] was an example so people can understand my question.It was not a possible solution.So dont blame me that i want you to solve my "homework".Thank you for your time. – JIm Aranikout Aug 26 at 8:57
• but that's the problem: your $h[n]$ is not an example of an impulse response of a linear time invariant system. And with "doing your homework" I meant: doing the checking that you can do you yourself very well. – Marcus Müller Aug 26 at 9:28
• Now i understand your statement,yep h[n]>2x[n] was technically wrong,maybe ∑|h(n)|≤1 is a more appropriate example for the kind of solution im looking for. – JIm Aranikout Aug 26 at 9:36

this is i believe what you were looking for

https://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/assignments/MITRES_6_007S11_hw05_sol.pdf