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].
Questions & Answers
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?