This module describes discrete time aperiodic signals.
Introduction
This module describes the type of signals acted on by the Discrete Time Fourier Transform.
Relevant spaces
The Discrete Time Fourier Transform maps arbitrary discrete time
signals in
${l}^{2}(\mathbb{Z})$ to finite-length, discrete-frequency signals in
${L}^{2}(\left[0 , 2\pi \right))$ .
Periodic and aperiodic 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.
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 f[n], n = ..., -2, -1, 0, 1, 2, ...
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, however,
$f(n)$ does not repeat for
any$N\in \mathbb{R}$ ;
i.e. there exists no
$N$ such that
this equation holds. This broader class of signals can only be acted upon by the DTFT.
Suppose we have such an aperiodic function
$f(n)$ . We can construct a periodic extension of
$f(n)$ called
${f}_{\mathrm{No}}(n)$ , where
$f(n)$ is repeated every
${N}_{0}$ seconds. If we take the limit as
${N}_{0}\to $∞ , we obtain a precise model of an aperiodic signal for which all rules that govern periodic signals can be applied, including Fourier Analysis (with an important modification). For more detail on this distinction, see the module on the
Discete Time Fourier Transform .
Aperiodic signal demonstration
Conclusion
A discrete periodic signal is completely defined by its values in one period, such as the interval [0,N].Any aperiodic signal can be defined as an infinite sum of periodic functions, a useful definition that makes it possible to use Fourier Analysis on it by assuming all frequencies are present in the signal.
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Tarell
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Damian
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Tarell
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Harper
Do you know which machine is used to that process?