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Discrete Time Periodic Signals

Module by: Stephen Kruzick, Dan Calderon. E-mail the authors

Summary: This module contains information on discrete time periodic signals.

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.

DTFSreg.png

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∈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:
Figure 1: Notice the function is the same after a time shift of N
discrete-time periodic signal
discrete-time periodic signal (DTPeriodic.PNG)

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

  1. as functions on all of R
    Figure 2: discrete time periodic function over all of R where f⁢ n 0 =f⁢ n 0 +N f n 0 f n 0 N
    Figure 2 (per_fxn1.png)
  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.
    Figure 3: Remove the redundancy of the period function so that f⁢n f n is undefined outside 0 N 0 N .
    Figure 3 (per_fxn2.png)

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:

Figure 4: Interact (when online) with a Mathematica CDF demonstrating a discrete periodic sinusoidal signal with various frequencies, amplitudes and phase delays.
sinDrillDiscreteDemo

Conclusion

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

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