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This modules derives the Discrete-Time Fourier Series (DTFS), which is a fourier series type expansion for discrete-time, periodic functions. The module also takes some time to review complex sinusoids which will be used as our basis.

Introduction

In this module, we will derive an expansion for discrete-time, periodic functions, and in doing so, derive the Discrete Time Fourier Series (DTFS), or the Discrete Fourier Transform (DFT).

Dtfs

Eigenfunction analysis

Since complex exponentials are eigenfunctions of linear time-invariant (LTI) systems , calculating the output of an LTI system given ω n as an input amounts to simple multiplication, where ω 0 2 k N , and where H k is the eigenvalue corresponding to k. As shown in the figure, a simple exponential input would yield the output

y n H k ω n

Simple LTI system.

Using this and the fact that is linear, calculating y n for combinations of complex exponentials is also straightforward.

c 1 ω 1 n c 2 ω 2 n c 1 H k 1 ω 1 n c 2 H k 2 ω 1 n l c l ω l n l c l H k l ω l n

The action of H on an input such as those in the two equations above is easy to explain. independently scales each exponential component ω l n by a different complex number H k l . As such, if we can write a function y n as a combination of complex exponentials it allows us to easily calculate the output of a system.

Dtfs synthesis

It can be demonstrated that an arbitrary Discrete Time-periodic function f n can be written as a linear combination of harmonic complex sinusoids

f n k N 1 0 c k ω 0 k n
where ω 0 2 N is the fundamental frequency. For almost all f n of practical interest, there exists c n to make [link] true. If f n is finite energy ( f n L 0 N 2 ), then the equality in [link] holds in the sense of energy convergence; with discrete-time signals, there are no concerns for divergence as there are with continuous-time signals.

The c n - called the Fourier coefficients - tell us "how much" of the sinusoid j ω 0 k n is in f n . The formula shows f n as a sum of complex exponentials, each of which is easily processed by an LTI system (since it is an eigenfunction of every LTI system). Mathematically, it tells us that the set ofcomplex exponentials k k j ω 0 k n form a basis for the space of N-periodic discrete time functions.

Dft synthesis demonstration

HarmonicSinusoidsDiscreteDemo
Download or Interact (when online) with a Mathematica CDF demonstrating Discrete Harmonic Sinusoids. To download, right click and save as .cdf.

Dtfs analysis

Say we have the following set of numbers that describe a periodic,discrete-time signal, where N 4 : 3 2 -2 1 3 Such a periodic, discrete-time signal (with period N ) can be thought of as a finite set of numbers. For example, we can represent this signal as either a periodic signal or asjust a single interval as follows:

Periodic Function
Function on the interval 0 T
Here we can look at just one period of the signal that has a vector length of four and is contained in 4 .

The cardinalsity of the set of discrete time signals with period N equals N .
Here, we are going to form a basisusing harmonic sinusoids . Before we look into this, it will be worth our time to look at the discrete-time,complex sinusoids in a little more detail.

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Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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