# Find the expression and the system impulse response

I've started to learn signal fundamentals and I have to do one exercise and I can't understand something.

It is said that $$x[n]=1.5\cos(0.025 \Pi n)(u[n+40]-u[n-40]))$$ and that the signal $u[n-m]$ is a unit step with the value $0$ for $n<m$ and $1$ for $n\geq m$ It is also said that $$y_{1}[n]=0.4*x[n-1] + 0*x[n-2] + 0.3*x[n-3] - 0.4*x[n-4]$$

I have to find the expression and plot the impulse response of the system $y_{1}[n]$

In the example that I have, I have this Matlab code:

n = -40:40;
ind = find(n);
x = zeros(size(n));
x(ind) = 1.5*cos(0.025*pi*n(ind));
h = [0.4 0 0.6 -0.4];
y = conv(x, h);
plot(n, x,'r', n, y(1:end-3),'b');


I understand what goes on from line 1 to 4 but after that I'm lost.

I can see that $h$ is a vector with the multiplying indexes of $x[n]$ on $y_{1}[n]$ but I cant understand what does y = conv(x, h); do. I've came across with the following conv example (MatLab code) but I don't understand how does that values come up.

>> u = [1 2 3 4];
>> v = [10 20 30];
>> conv(u,v)
ans =
10    40   100   160   170   120
>>


Finally, in the plot, I understand that y(1:end-3) is to make the vectors x and y the same size but why that indexes?

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I'm not sure I can explain it as well as video would, so you should probably checkout youtube.

Basically,

$$\left(\text{conv}(a,b)\right)_n=\sum_{i=-\infty}^{\infty}a_i b_{n-i}$$

So that what happens here is both vectors are padded with zeros on both sides, that is:

u = ... 0, 0, 1, 2, 3, 4, 0, 0 ...

v = ... 0, 0, 10, 20, 30, 0, 0 ...

Then one of them is reversed:

v' = ... 0, 0, 30, 20, 10, 0, 0 ...

Then for every location $n$ you're interested in, you perform a shift left on the second vector by $n$ and perform a dot product.

Eventually you get a vector that's

0

(30,20,10,0,0,0)(0,0,1,2,3,4) = 30*0+20*0+10*1 = 10

(...30,20,10,0,0...)(...0,1,2,3,4...) = 30*0+20*1+10*2 = 40

(...30,20,10,0...)(...1,2,3,4...) = 30*1+20*2+10*3 = 100

(...0,30,20,10...)(...1,2,3,4...) = 30*2+20*3+10*4 = 160

(...0,0,30,20,10...)(...1,2,3,4,0...) = 30*3+20*4+10*0 = 170

(...0,0,0,30,20,10...)(...1,2,3,4,0,0...) = 30*4+20*0+10*0 = 120

0

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Hi.Sorry for the late feedback. Many thanks for your explanation and video –  Favolas Mar 3 '13 at 22:50