1
$\begingroup$

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?

$\endgroup$

1 Answer 1

3
$\begingroup$

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

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

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