Mathematics - Signals And Systems - Exercise on Convolution

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Introduction

Hey it's a me again @drifter1!

Today we continue with my mathematics series about Signals and Systems in order to cover Exercise on Convolution.

So, without further ado, let's dive straight into it!


Discrete-Time Convolution Example [Based on P4.2 from Ref1]

Let's determine the discrete-time convolution of the following two signals:

Solution

The convolution sum that we want to calculate is defined as:

Its easy to notice that the convolution sum requires an reflected version of h[n]. Reflecting h[n] about the origin and changing the unit from n to k leads to h[-k]:



which is basically h[n - k] for n = 0.

For the values n < 0 time-shifting leads to values of h[n - k] which result to zero contribution to the convolution sum. More specifically, the resulting h[n - k] are not in the closed range [0, 3] in which x[k] is non-zero. For example n = -1 and n = -2:

As such, the sums that need to be calculated start at n = 0, which leads us to the question: "When do we stop?".

Let's start increasing n, which will of course lead to a shifting of h[n - k] to the right by 1, 2, 3 etc.

After n = 5 the graph of h[n - k] again contributes zero to the convolution sum. Therefore, the convolution can be easily calculated by using n in the range [0, 5], which leads to a result, y[n], of length 6.

Generally, the result is equal to the sum of the lengths of the individual discrete-time signal sequences x[n] and h[n] minus 1:

So, how do we get the final result? Well, its simply multiplying x[k] by the various shifts of h[n - k] and summing up those values. The result of this sum is then the value of the convolution y[n] for each specific point in time.

The values n = 0 and n = 1 give us:



which give us the values of the convolution y[n]:

Similarly, for the values n = 2 and n = 3 we have:



which gives us two 8 's.

Lastly, for the values n = 4 and n = 5:



and so the convolution contributions 6 and 4, respectively.

Finally, the convolution between x[n] and h[n], which is y[n], can be now visualized as follows:


RESOURCES:

References

  1. Alan Oppenheim. RES.6-007 Signals and Systems. Spring 2011. Massachusetts Institute of Technology: MIT OpenCourseWare, License: Creative Commons BY-NC-SA.

Images

  1. https://commons.wikimedia.org/wiki/File:From_Continuous_To_Discrete_Fourier_Transform.gif

Mathematical equations used in this article were made using quicklatex.

Block diagrams and other visualizations were made using draw.io


Previous articles of the series

Basics

LTI Systems and Convolution

Fourier Series and Transform


Final words | Next up

And this is actually it for today's post!

Next time we will get into exercises on other topics that we covered!

See Ya!

Keep on drifting!