# 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:

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

1. as functions on all of $\mathbb{R}$
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.

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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The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
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2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
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Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22
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Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
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x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
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Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
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