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Today we continue with my mathematics series about Signals and Systems in order to cover Discrete-Time Fourier Series & Transform.
So, without further ado, let's dive straight into it!
Getting into Discrete-Time
In the previous three articles we covered how periodic and aperiodic continuous-time signals can be represented through the use of the Fourier Series and Fourier Transform correspondingly. In order to represent signals in Discrete-Time a similar principle is applied. In other words, such signals are again represented using linear combinations of complex exponentials. The complex exponentials are the eigenfunctions of LTI systems, which in turn simply allow for a complex amplitude change. Therefore, an LTI system is characterized by a spectrum of scale factors which are applied to each frequency.
Discrete-Time Fourier Series
In Discrete-Time, a Fourier series is again constructed using harmonically related complex exponentials. The fundamental frequencies of those complex exponentials are multiples of the fundamental frequency of the periodic sequence that's to be represented.
In the article about complex exponentials we mentioned that complex exponentials are only unique when the frequency variables spans a range of 2π.
Beyond this range, the complex exponential is repeated over and over.
This is a very important difference that continuous- and discrete-time complex exponential signals have.
As such, a periodic sequence with period N can be easily represented as a linear combination of complex exponentials of the form:
as there are only N unique complex exponentials available.
Complex exponentials of this form are periodic in k with period N. Additionally, they are also periodic in n with period N.
Its easy to conclude that this simplifies the analysis part for Discrete-Time quite a bit, as only N Fourier series coefficients need to be determined. The whole problem is simplified into solving N equations of N unknowns.
Using Fourier Series, a Discrete-Time period signal x[n] is represented as follows:
where k : <N> is used for denoting a quantity of N coefficients.
As the number of coefficients used is increased, a signal can be approximated better and better, as more frequency scale factors are "discovered".
The GIF that follows demonstrates this for a square wave:
The analysis part is about calculating the correct ak coefficients.
In total there are N equations of the form:
Discrete-Time Fourier Transform
In order to represent an aperiodic signal, a similar approach to Continuous-Time is used. So, we construct a periodic signal of x[n], which has a Fourier Series. As the period of this periodic signal increases its Fourier Series turns into the Fourier Transform of the original signal x[n].
Similar to Continuous-Time, the coefficients will be taken as samples of an envelope. The envelope is determined by the behavior of a sequence over one period, but not by the specific value of the period. Increasing the period of the sequence increases, the nonzero values inside of one period remain the same. So, its easy to conclude that the Fourier series coefficients are samples of the same envelope function, but with increasingly finer spacing along the frequency axis. This envelope function allows for the representation of an aperiodic signal in one period, giving us the Fourier Transform of the aperiodic signal.
Fourier series of periodic x[n]
The periodic x[n] can be represented using a Fourier Series, as follows:
As N → ∞ the first sum turns into an integral, and converges to x[n], giving us the Fourier Transform synthesis equation:
As N → ∞ in the coefficient calculation, the equation gives us the Inverse Fourier Transform or analysis equation:
The Fourier Transform turns any aperiodic discrete-time signal x[n] into its envelope samples X(Ω), and vise versa:
Fourier Transform of periodic signal
The Fourier Transform can also be defined for periodic signals.
Similar to Continuous-Time, the Fourier Series coefficients are 1 / N times the samples of the Fourier Transform (in CT it was 1 / To) and defined using an impulse train:
- Alan Oppenheim. RES.6-007 Signals and Systems. Spring 2011. Massachusetts Institute of Technology: MIT OpenCourseWare, License: Creative Commons BY-NC-SA.
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)
- 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
Final words | Next up
And this is actually it for today's post!
Next time we will cover the Properties of the Discrete-Time Fourier Transform!
Keep on drifting!