Discrete Time Multiplication

Hello there,

Please refer to the image below.



Thanks!

x[-2] -- this expression does not depend on "n".

Can you please elaborate a bit? here n = -2? What do you mean that it does NOT depend on n?

The expression "-2" does not include any variables.

x[-2] is independent of the value of n.

for example, if n = 10000, x[-2] = 1. (just guessing, there is no vertical scale). if n =-10000, x[-2] = 1. if n = 0, x[-2] = 1. if n = -2, x[-2] = 1.

Umm.. I think this is exactly where my confusion is.

Let me elaborate my question.

Q1: In the image below, we are multiplying the values in the green and the blue box, right?



Q2: If so, then that means it is going to be "n by n" i.e.
(a) x[-2] times h[-2], which will be a non-negative value
(b) x[-2] times h[-1], is this correct?
(c) x[-2] times h[0], is this correct?

If, (a), (b) and (c) above are correct, then

Q3: What is the difference between the blue box and the yellow box? Just the magnitude at n = -2?

Thanks much

x[-2] evaluates to an value that is not dependent on n, or any other variables.

Specifically:
y_{-2}[-2] = x[-2] h[n+2] = x[-2] h[-2 + 2] = x[-2] h[0]
y_{-2}[-1] = x[-2] h[n+2] = x[-2] h[-1 + 2] = x[-2] h[1]
y_{-2}[0] = x[-2] h[n+2] = x[-2] h[0 + 2] = x[-2] h[2]

So for Q1, in the green and yellow this is a multiplication. the yellow box is described as above.
Q2: a,b,c: yes. They will also be non-zero.

Q3: There is no difference. There are probably more slides that provide more insight into convolution. y_{-2}[n] appears to be "the value of the output based only on an impulse at x[-2]".

For some reason people always give confusing example problems for convolution.

Specifically:
y_{-2}[-2] = x[-2] h[n+2] = x[-2] h[-2 + 2] = x[-2] h[0]
y_{-2}[-1] = x[-2] h[n+2] = x[-2] h[-1 + 2] = x[-2] h[1]
y_{-2}[0] = x[-2] h[n+2] = x[-2] h[0 + 2] = x[-2] h[2]

Click to expand...

Why is it x[-2] h[0], x[-2] h[1] and x[-2] h[2]? Isn't h[n] shifted by 2 to the left giving us the signal in the blue box (in my previous comment)? So shouldn't it be
x[-2] h[-2]
x[-2] h[-1]
x[-2] h[0] instead?

Thanks

when n = -2, n+2 = -2 + 2 = 0.

In the last picture, they are showing the convolution of h with a version of "x" that is not shown on other diagrams. This version has a value of x[-2] at n=0 and is 0 everywhere else.

It can be confusing if you think the inputs are shown on other charts. I'm assuming there is text that explains what the lower 2 charts represent. It isn't hard to explain convolution. These charts seem to try to simplify things in a way that only leads to more confusion.

I'm going to try to explain step by step.
According to post #1 in the first graph we have x having the following values (roughly, because the y-axis has no units):

x(-2)=1
x(0) = 1.2
x(3) = -0.8
0 for all other "n".

We know that delta(k) is 1 is k=0, and 0 for all others value of k.

This means delta(n+2)=1 if and only if n=-2, then x*delta(n+2)=x(-2)*delta(n+2)=1 if n=-2 and 0 for any other value of n. This is represented in the second graph. The dependency from "n" is given by the function delta(n+2), since x(-2)=1 and doesn't depend from n.

Let consider now h as:

h(0) = 1
h(1) = 0.8
h(2) = 0.5

that is h(n+2) will be:

n=0 ==> h(2)=0.5
n=-1 ==> h(1)=0.8
n=-2 ==> h(0)=1

if we now multiply x(-2), that is 1, we obtain:

y-2(0)=1*0.5
y-2(-1)=1*0.8
y-2(-2)=1*1

as represented in the last graph.