Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am learning fourier transform and I came across this question in which author right away says the given equation is "even". How does this equation become "even"?

$$x[n]=\begin{cases}A & -M\le n\le M\\0 & \text{elsewhere}\end{cases}$$

Regards

share|improve this question
1  
Graphically, even functions are those which are symmetric about the y-axis. This function is obviously symmetric about the y-axis. –  rschwieb Jan 2 '13 at 18:31
add comment

2 Answers

up vote 6 down vote accepted

A function $x(n)$ is even if $x(n) = x(-n)$.

For your function there are two cases:

Case 1: $|n| \leq M$. Then $x(n) = x(-n) = A.$

Case 2: $|n| > M$, and $x(n)=x(-n) = 0.$

In both cases, $x(n) = x(-n)$, so $x$ is even.

EDIT: Incidentally, the words "even" and "odd" come from the fact that if $x(n)$ is analytic, all of the terms in its Taylor series have even powers of $n$ if $x$ is even, or odd powers of $n$ if $x$ is odd. But functions can be even or odd even if they're not differentiable, such as in your case.

share|improve this answer
    
so if the limit were 0<= n <= M, the equation would have been odd? –  Umer Farooq Jan 2 '13 at 18:30
    
No. Odd does not mean "not even" (I know it's confusing). Rather, odd means $x(-n) = -x(n)$, which would not be the case for your modified function. $x(n) = n$ or $x(n) = \sin n$ are examples of odd functions. –  user7530 Jan 2 '13 at 18:32
    
so in which case this equation would be odd? –  Umer Farooq Jan 2 '13 at 18:36
    
$x(n) = -A$ on $-M \leq n < 0$, $x(n) = A$ on $0 < n \leq M$, and $x(n) = 0$ elsewhere. –  user7530 Jan 2 '13 at 18:38
    
@UmerFarooq: Another point to observe is that if $A=0$, then the function is both even and odd. Your proposed modification is neither even nor odd. –  Cameron Buie Jan 2 '13 at 19:04
add comment

For $-M\leq n\leq M$, $x(-n)=A=x(n)$. Elsewhere, $x(-n)=0=x(n)$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.