Proof of First Difference Property for Fourier Series

Question

I am having trouble with deriving a proof for the first difference property for the Fourier Series.

Here is my attempt at the derivation:

$$ y[n] = x[n] - x[n-1] $$

Fourier Series Representation:

$$ a_k - a_ke^{-jk\omega_0} $$

Fourier Series:

$$ y[n] = \sum_{k=<N>}(a_k-a_ke^{-jk\omega_0})e^{jk\omega_0n} $$

I have set up the summation for the Fourier Series, however I have been having difficulty to compute the summation.

The summation should equal this:

$$ a_k(1-e^{-jk\omega_0}) $$

How would I evaluate the summation?

Answer 1

You have the result already written down in your question. If $a_k$ are the Fourier coefficients of $x[n]$, then your third formula is the Fourier series of $y[n]=x[n]-x[n-1]$. So the Fourier series coefficients of $y[n]$ are

$$b_k=a_k-a_ke^{-jk\omega_0}=a_k(1-e^{-jk\omega_0})$$

meaning that

$$y[n]=\sum_{k=0}^{N-1}b_ke^{j\omega_0 nk},\qquad\omega_0=2\pi/N$$