Mathematics - Signals And Systems - Continuous-Time Periodic Signals & Fourier Series

in StemSocial2 months ago

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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 Continuous-Time Periodic Signals & Fourier Series.

I highly suggest reading the previous article about Discrete-Time, as the representation is quite similar, in some sense.

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


Input Decomposition

Let's start simple, by basically refreshing our knowledge around linear systems, and why they are so useful...

First of all, due to the properties of linear systems, and mainly due to the principle of superposition, its possible to decompose any signal into a set of basic signals. The expected outcome of this procedure is a linear combination of basic signals for which the response can be easily computed.

Consider a signal x(t) in continuous-time. For linear systems, this signal can be decomposed to as a linear combination of φk 's in the following way:

In the end, what matters is that the response of the system will be some combination of easy to compute ψk 's:



Those ψk 's are the result of convolution for each of the basic φk 's. In continuous-time, the convolution integral can be represented as:


Basic Signals and the Concept of Fourier Analysis

As we already mentioned in some previous article, the main concept of Fourier analysis is using complex exponentials as the set of basic signals. Thus, the basic signals will be of the form:



Substituting sk with k, its easier to see the "complex" part:



When sk is complex the transform is also known as a Laplace transform.

The system response for such a signal is:



For periodic signals the result will be represented by a Fourier Series, whilst for aperiodic signals (which will be covered next time) the outcome is a Fourier Transform.


Fourier Series - Sum of Complex Exponentials

A Fourier Series is basically a weighted sum of complex exponentials, which are harmonically related. Its known that a signal is periodic only when the following relation is true:



where To is the period.

The period To can be substituted using the following equation:



which allows us to replace the ωk's by the product k ωo :

In this notion, a periodic signal x(t) can be decompsed into a sum of complex exponentials as such:


Fourier Synthesis

The coefficients, ak, can be calculated using:



for which I will skip the proof.

Trigonometric form

In the article about complex exponentials, it was shown how Euler's Relation can be used to get a trigonometric form. In this case, the trigonometric form looks as follows:



which results in the following complete decomposition of x(t) :



where ak = Bk + jCk and the harmonic relation between the complex exponentials is also taken into consideration.

The coefficients, ak, can be still calculated using the same equation as before, with ao having the special property of being the average value of x(t) during a period.

This is basically an alternative way of representing the complex exponentials that form the Fourier Series, where each exponential is a linear combination of sines and cosines, or sinusoids in general, of different phase angles.


Signal Approximation

Incorporating only 2N + 1 terms of the Fourier Series, its possible to end up with an approximation of x(t) in the form of signal xN(t). This signal is represented by the sum:

Doing that we introduced an error eN(t), which can be calculated by:

As N increases, the error eN(t) is generally decreasing, as we end up with a better approximation of the signal x(t).


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


Final words | Next up

And this is actually it for today's post!

Next time we will cover Aperiodic Signal in Continuous-Time, which are represented using something known as the Fourier Transform.

See Ya!

Keep on drifting!

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This is a good reference for those who are currently studying signals and systems or digital signal processing. By the way, your reference (Oppenheim) is a good notes for the subject matter. I used that one when I had my signals class.

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