# Discrete time periodic signals

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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: Notice the function is the same after a time shift of N

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

1. as functions on all of $\mathbb{R}$ discrete time periodic function over all of where f n 0 f n 0 N
2. 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. Remove the redundancy of the period function so that f n is undefined outside 0 N .

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: Interact (when online) with a Mathematica CDF demonstrating a discrete 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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There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
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da
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Preparation and Applications of Nanomaterial for Drug Delivery
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Application of nanotechnology in medicine
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I think
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scanning tunneling microscope
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The nanotechnology is as new science, to scale nanometric
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nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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biomolecules are e building blocks of every organics and inorganic materials.
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