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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 *[-N _{1}, N_{1}]* 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, *N _{1} = 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 *x _{2}[n]* is:

and so the output *Y(Ω)* is equal to:

Taking the Inverse Fourier Transform gives us the final result:

## RESOURCES:

### References

### Images

Mathematical equations used in this article were made using quicklatex.

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

## 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
- Exercise on Convolution → Discrete-Time Convolution Example with the help of visualizations
- Exercises on System Representation using Difference Equations → Simple Block Diagram to LCCDE Example, Direct Form I, II and LCCDE Example
- Exercises on System Representation using Differential Equations → Equation to Block Diagram Example, Direct Form I to Equation Example

### 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
- Exercises on Continuous-Time Fourier Series → Fourier Series Coefficients Calculation from Signal Equation, Signal Graph
- Exercises on Continuous-Time Fourier Transform → Fourier Transform from Signal Graph and Equation, Output of LTI System

## 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!