frequency folding due to aliasing in FFT

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promach

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Simply consider that the spectrum of a time discrete signal is cyclic with n*fs.
 

FvM:

1) I understand the periodicity of 2pi which results in X[k+N] = X[k] , but how does the periodicity concept places negative frequencies to the right of negative frequencies ?

2) Besides, I do not get why is it that the Nyuist frequency component at X[N/2] separates the positive and negative frequencies.
 

X[k] = \[\sum_{n = 0}^\N-1\]x[n]\[{e }^{-j*2*pi*k*n/N }\], k=0 to N-1. From this definition, if you evaluate the exponential for n>N/2 the frequencies will come out to be positive. And vice-versa for n<N/2. The upper half of the unit circle is +ve freq. and the bottom half represents -ve freq.
 

@promach:

consider a sine/cosine signal that increments by -170 degrees per sample. Now consider a signal that increment by +190 degrees per sample. They are the same.

This leads to different ways to represent the signal. In one representation, the signal is limited to the 0 to 2*pi range. In this case, the negative frequencies are interpreted as frequencies above pi radians per sample.
 

Hi,

In the wikipedia article....with this text: "Two different sinusoids that fit the same set of samples".
Imagine:
* the red line as input signal
* the black dots as samples
--> then the blue line results as "alias" signal.

Klaus
 

Above seem to be the algorrithm programming , quite long coding writing .
 

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