Mathematics - Signals And Systems - Continuous-Time Periodic Signal...

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 φ_k '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 ψ_k 's:

Those ψ_k 's are the result of convolution for each of the basic φ_k '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 s_k with jω_k, its easier to see the "complex" part:

When s_k 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 T_o is the period.

The period T_o can be substituted using the following equation:

which allows us to replace the ω_k's by the product k ω_o :

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

Fourier Synthesis

The coefficients, a_k, 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 a_k = B_k + jC_k and the harmonic relation between the complex exponentials is also taken into consideration.

The coefficients, a_k, can be still calculated using the same equation as before, with a_o 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 x_N(t). This signal is represented by the sum:

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

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

RESOURCES:

References

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

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

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 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)

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!

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