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Introduction

This module describes the type of signals acted on by the Discrete Time Fourier Series.

Relevant Spaces

The Discrete Time Fourier Series maps finite-length (or $NN$-periodic), discrete time signals in $L2 L2$ to finite-length, discrete-frequency signals in $l2 l2$.

Periodic signals in discrete time repeats themselves in each cycle. However, only integers are allowed as time variable in discrete time. We denote signals in such case as x[n], n = ..., -2, -1, 0, 1, 2, ...

Periodic Signals

When a function repeats itself exactly after some given period, or cycle, we say it's periodic. A periodic function can be mathematically defined as:

$f⁢n=f⁢n+m⁢N ∀m :m∈Z f n f n m N m m$
1
where $N>0 N 0$ represents the fundamental period of the signal, which is the smallest positive value of N for the signal to repeat. Because of this, you may also see a signal referred to as an N-periodic signal. Any function that satisfies this equation is said to be periodic with period N. Here's an example of a discrete-time periodic signal with period N:
discrete-time periodic signal

We can think of periodic functions (with period $NN$) two different ways:

1. as functions on all of $R$
2. or, we can cut out all of the redundancy, and think of them as functions on an interval $0 N 0 N$ (or, more generally, $a a+N a a N$). If we know the signal is N-periodic then all the information of the signal is captured by the above interval.

An aperiodic DT function $f⁢n f n$ does not repeat for any $N∈R N$; i.e. there exists no $N N$ such that this equation holds.

SinDrillDiscrete Demonstration

Here's an example demonstrating a periodic sinusoidal signal with various frequencies, amplitudes and phase delays:

Conclusion

A discrete periodic signal is completely defined by its values in one period, such as the interval [0,N].