Real Sequence As Rational Function in e^-jw

[dduardo]

How do I convert a piece wise function like this:

y[n]=

{1 -N<=n<=N
{0 otherwise

to something like this:

p0+p1e^(-jw)+...+pMe^(-jwM)
------------------------------
d0+d1e^(-jw)+...+dNe^(-jwN)

Basically, what formula do I need to use to calculate the coeffients?

[Edit] I'm trying to use the MATLAB function FREQZ, but requires the formula to be in the above format.

 

[learningphysics]

How do I convert a piece wise function like this:

y[n]=

{1 -N<=n<=N
{0 otherwise

to something like this:

p0+p1e^(-jw)+...+pMe^(-jwM)
------------------------------
d0+d1e^(-jw)+...+dNe^(-jwN)

Basically, what formula do I need to use to calculate the coeffients?

[Edit] I'm trying to use the MATLAB function FREQZ, but requires the formula to be in the above format.

Well, that looks the frequency response of a discrete time system.

I might be wrong, but I'd first get the z-transform of your sequence:

z^N + z^(N-1) + z^(N-2) + ...+ 1 + z^(-1) + z^(-2) + ...z^(-(N-1))+z^(-N)

then plug in z=e^(jwdt)

dt is the time between samples.

e^(jwdtN) + e^(jwdt(N-1)) + ...+ 1 + e^(-jwdt) + e(-jwdt2) + ... e^(-jwdtN)

So that would be your numerator, and the denominator is just 1. Is that form acceptable?

 

[tacman]

To convert a piecewise function like y[n] to a rational function in e^-jw, you can use the following formula:

y[n] = (p0 + p1e^(-jw) + ... + pMe^(-jwM)) / (d0 + d1e^(-jw) + ... + dNe^(-jwN))

To calculate the coefficients, you can use the discrete Fourier transform (DFT) or the fast Fourier transform (FFT) algorithm. These algorithms can be implemented in MATLAB using the functions fft() or fft2(). Once you have the Fourier transform of the piecewise function, you can use the coefficients of the transform to calculate the coefficients of the rational function in e^-jw.

For example, if the Fourier transform of y[n] is Y(e^-jw), then the coefficients of the rational function can be calculated as follows:

p0 = Y(0)
p1 = Y(1)
...
pM = Y(M)
d0 = Y(0)
d1 = Y(1)
...
dN = Y(N)

Note that the values of p0 to pM and d0 to dN may need to be scaled by a factor of 1/N, depending on the specific implementation of the DFT or FFT algorithm.

Once you have the coefficients, you can use the MATLAB function FREQZ to calculate the frequency response of the rational function in the specified format.