Fourier series/ periodic function

[Brown Arrow]

Homework Statement

Suppose f is a periodic function of period 2pi and that g is a horizontal shift of f, say
g(x) = f(x + a). Show that f and g have the same energy.

Homework Equations

n/a

The Attempt at a Solution

i can see that if f(x) is shifted by 'a' that it does not make the function any difference.
but in not sure how should i go about showing it.

could someone help me out pls.

BA

 

[Dick]

If f(x) is shifted by a it does make a difference to the Fourier transform. Can you show that difference doesn't change the energy?

 

[Brown Arrow]

:/ that's the part I am not sure about.. can you give me a hit.

 

[Dick]

:/ that's the part I am not sure about.. can you give me a hit.

Probably. What's your definition of Fourier transform and energy? In the complex formulation the Fourier transform of f(x+a) is a complex number times the Fourier transform of f(x). What's the complex number?

 

[Brown Arrow]

well i did not learn it by complex #

my definition of energy is E=(a_k)² +(b_k)²

or E=(1/pi) ∫[f(x)]² ; where the integral is from -pi to pi

 

[Brown Arrow]

help

 

[Dick]

well i did not learn it by complex #

my definition of energy is E=(a_k)² +(b_k)²

or E=(1/pi) ∫[f(x)]² ; where the integral is from -pi to pi

So you want to prove the integral of f(x)^2 and f(x+a)^2 are the same from -pi to pi. Substitute u=x+a. So the u limits are -pi+a to pi+a. The integral of a periodic function over a single period is the same, no matter where the period starts.

 

[Brown Arrow]

i see thanks man