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,…]
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);
y1=0*x;
index=find(x1);
for i=index:length(n)
y1(i)=x1(index)*exp(-(i-index)/2);
end
stem(n, y1);
axis(limit);
h=text(1, x1(9), 'x[1]*h[n-1]'); set(h, 'vertical', 'bottom');
subplot(4,2,7);
x2=0*x;
x2(10)=x(10);
stem(n, x2);
axis(limit);
h=text(2, x2(10), 'x[2]'); set(h, 'horiz', 'center', 'vertical', 'bottom');
subplot(4,2,8);
y2=0*x;
index=find(x2);
for i=index:length(n)
y2(i)=x2(index)*exp(-(i-index)/2);
end
stem(n, y2);
axis(limit);
h=text(2, x2(10), 'x[2]*h[n-2]'); set(h, 'vertical', 'bottom');
subplot(4,2,2);
stem(n, y0+y1+y2);
axis(limit);
title('Output y[n] = x[0]*h[n-0] + x[1]*h[n-1] + x[2]*h[n-2]');
As you can see if input queue has N samples, impulse response has M after convolution there will be total N+M-1 samples.
