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
$N$ -periodic), discrete time
signals in
${L}^{2}$ to finite-length, discrete-frequency signals in
${l}^{2}$ .
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:
$f(n)=f(n+mN)\forall m\colon m\in \mathbb{Z}$
where
$N> 0$ 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:
We can think of
periodic functions (with period
$N$ ) two different ways:
as functions on
all of
$\mathbb{R}$
or, we can cut out all of the redundancy, and think of them
as functions on an interval
$\left[0 , N\right]()$ (or, more generally,
$\left[a , a+N\right]()$ ). 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
$f(n)$ does not repeat for
any$N\in \mathbb{R}$ ;
i.e. there exists no
$N$ 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].
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