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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

### Images

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

- Introduction → Signals, Systems
- Signal Basics → Signal Categorization, Basic Signal Types
- Signal Operations with Examples → Amplitude and Time Operations, Examples
- System Classification with Examples → System Classifications and Properties, Examples
- Sinusoidal and Complex Exponential Signals → Sinusoidal and Exponential Signals in Continuous and Discrete Time

### LTI Systems and Convolution

- LTI System Response and Convolution → Linear System Interconnection (Cascade, Parallel, Feedback), Delayed Impulses, Convolution Sum and Integral
- LTI Convolution Properties → Commutative, Associative and Distributive Properties of LTI Convolution
- System Representation in Discrete-Time using Difference Equations → Linear Constant-Coefficient Difference Equations, Block Diagram Representation (Direct Form I and II)
- System Representation in Continuous-Time using Differential Equations → Linear Constant-Coefficient Differential Equations, Block Diagram Representation (Direct Form I and II)
- Exercises on LTI System Properties → Superposition, Impulse Response and System Classification Examples

### Fourier Series and Transform

- Continuous-Time Periodic Signals & Fourier Series → Input Decomposition, Fourier Series, Analysis and Synthesis
- Continuous-Time Aperiodic Signals & Fourier Transform → Aperiodic Signals, Envelope Representation, Fourier and Inverse Fourier Transforms, Fourier Transform for Periodic Signals
- Continuous-Time Fourier Transform Properties → Linearity, Time-Shifting (Translation), Conjugate Symmetry, Time and Frequency Scaling, Duality, Differentiation and Integration, Parseval's Relation, Convolution and Multiplication Properties
- Discrete-Time Fourier Series & Transform → Getting into Discrete-Time, Fourier Series and Transform, Synthesis and Analysis Equations
- Discrete-Time Fourier Transform Properties → Differences with Continuous-Time, Periodicity, Linearity, Time and Frequency Shifting, Conjugate Summetry, Differencing and Accumulation, Time Reversal and Expansion, Differentation in Frequency, Convolution and Multiplication, Dualities

## 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!