# Linear Time Invariant System Response

1. Jan 29, 2008

### Bibble

1. The problem statement, all variables and given/known data
The Relationship between the input x(n) and the output y(n) for the discrete System A is described by the expression:
$$\frac{x(n) - 2x(n-1) + x(n-2)}{2}$$
What is:
(i) The impulse resopnse function h(n)?
(ii) The frequency response function H(f)?
(iii) The aplitude response function A(f)?
(iv) The phase response function?
2. Relevant equations

3. The attempt at a solution
For (i), I got h(0)=$$\frac{1}{2}$$, h(1)=-1, h(2)=$$\frac{1}{2}$$.
What I am looking for is a layman's explanation of what the responses are/how they are used. E.g. I have managed to decipher the fact that the impulse response function provides the values given by a system with an instantaneos impulse 1 is applied at n=0, and that it can be used to make up the system equation (if h(0)=1, h(1)=4 and h(2)=12, y(n)=x(n)+4x(n-1)+12x(n-2) ). Unless I have misunderstood completely.

Now, as I see it, the frequency response function are the actions applied to any given input frequency (so, if the inputs were 1Hz, 3Hz, and 4Hz, and the outputs were 1Hz, 9Hz, and 16Hz respectively, then H(f) = (x(f))^2).

The amplitude response function is the ratio of input amplitude to output amlitude, plotted against the frequency, and the phase response function is the lag of the output wave, in comparison to it's input wave, plotted against frequency.

These have been niggling at me, because I am not sure whether the entire premise that I am working from is flawed, or whether I am just not seeing what I am doing.