# Mathematics - Signals And Systems - Exercises on Discrete-Time Fourier Series and Transform

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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 Exercises on Discrete-Time Fourier Series and Transform.

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

## Discrete-Time Fourier Series Coefficients Calculation [Based on 10.3 from Ref1]

Let's consider the following periodic sequence:

Find the appropriate expression for the envelope of the Fourier series coefficients and sample it.

### Solution

In discrete-time an orthogonal pulse in the range [-N1, N1] can be represented as follows:

Using complex exponentials, the Fourier series coefficient is thus given as follows:

As such, the samples can be taken from the sampling function, which leads us to the following envelope:

In the case of this example, N1 = 2, and N = 10, and so:

## Discrete-Time Fourier Transform Calculation [Based on 11.1 from Ref1]

Compute the discrete-time Fourier transform of the following signals:

### Solution

The Fourier transform can be easily calculated with the help of Fourier transform pair tables and properties that we discussed throughout the series of articles.

#### a.

The first case is a slighly different representation of one of the known Fourier transform pairs, which is:

For |a| > 1, this can be re-written as:

Thus, for a = 6, which is the first signal, the result is:

#### b.

This case is quite similar to the first one. Let's start re-formulating in order to achieve such a form:

From the previous example we know that:

Due to linearity its possible to multiply both sides by 36, which yields:

From the time-shifting property, we then get the final result:

#### c.

The orthogonal pulse can be easily described using unit step functions, in the following manner:

The Fourier transform of the unit step function is:

And time-shifting by 3 leads to the following result:

So, the final result is:

## Output of LTI System [Based on 11.2 from Ref1]

The following linear constant-coefficient difference equation describes an LTI system initially at rest:

Using Fourier transforms, let's evaluate y[n] for each of the following inputs:

### Solution

First of all, let's take the Fourier transform of both sides of the LCCDE:

As such H(Ω) is equal to:

#### a.

For x[n] = δ[n], X(Ω) = 1, which makes the output equal to the impulse response:

and, so the Inverse Fourier Transform gives us the output y[n] and h[n] in the same time:

#### b.

The Fourier Transform of x2[n] is:

and so the output Y(Ω) is equal to:

Taking the Inverse Fourier Transform gives us the final result:

## RESOURCES:

### Images

Block diagrams and other visualizations were made using draw.io and GeoGebra

## Final words | Next up

And this is actually it for today's post!

From next time we will start getting into concepts like Filtering, Modulation, Sampling, Interpolation etc.

See Ya!

Keep on drifting!

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