## Impulse Response of discrete system

Impulse signal can be represented as: d[n] = 1, if n=0 d[n] = 0, otherwise it can also be written like d=[1,0,0,0,â€¦] Impulse Response The impulse response h(n) is the response of filter L() at time n to unit impulse occurring at time 0. h(n)=L(d(n)) Lets see how discrete system can be described when impulse response is known We know that: In the linear system this can be written as follows: Because h(n-k)=L(d(n-k)) Then: What do we get? There is obvious, that linear system can be described by its impulse response. The last expression is called convolution. This is the heart of DSP Filtering. To write this sum in more convenient matter is assumed that: Matlab example Matlab example: % Plot an unit impulse signal n = -7:7; x = [0 0 0 0 0 0 0 1 2 3 0 0 0 0 0]; subplot(4,2,1); stem(n, x); limit=[min(n), max(n), 0, 5]; axis(limit); title(‘Input x[n]’); subplot(4,2,3); x0=0*x; x0(8)=x(8); stem(n, x0); axis(limit); h=text(0, x0(8), ‘x[0]’); set(h, ‘horiz’, ‘center’, ‘vertical’, ‘bottom’); subplot(4,2,4); y0=0*x; index=find(x0); for i=index:length(n) y0(i)=x0(index)*exp(-(i-index)/2); end stem(n, y0); axis(limit); h=text(0, x0(8), ‘x[0]*h[n-0]’); set(h, ‘vertical’, ‘bottom’); subplot(4,2,5); x1=0*x; x1(9)=x(9); stem(n, x1); axis(limit); h=text(1, x1(9), ‘x[1]’); set(h, ‘horiz’, ‘center’, ‘vertical’, ‘bottom’); subplot(4,2,6);…