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1 vote

This is using the time shift property of the fourier transform. A shift in the time domain by $a$ corresponds to a phase shift in the frequency domain. $$f(x-a)\xrightarrow{\mathscr{F}}e^{-ja\Omega}F(... 1 vote Accepted ### What Does "Reduced Modulo N" mean in this context? It just means that the number in the double parentheses is changed to be between 0 and M-1 by adding to or subtracting from it M an integer number of times. See equation (5) of the paper:$$ \...
1 vote

This depends a bit on how you actually want to model this. Time discrete or time continuous ? Differential equations or difference equations? Impulse excitation gives you directly the impulse response ...
1 vote

### When is the Fourier transform of a periodic discrete signal $\mathcal{F}x[k]$ the same as $x[k]$ up to a diagonal matrix

tl;dnr version: No nonzero vector can satisfy the requirement stated in the body of this question. The rest of this answer is a long-winded proof of the assertion above. The Discrete Fourier Transform ...
1 vote

### When is the Fourier transform of a periodic discrete signal $\mathcal{F}x[k]$ the same as $x[k]$ up to a diagonal matrix

Let $\newcommand{\F}{\mathbf{F}_{{}_N}} \F$ be the (unitary) DFT-Matrix of size $N$. Let $\newcommand{\x}{\mathbf x}\x$ be the vector $\x=(x,\ldots x[N-1])$. Your equation says: \begin{align}\...
1 vote
Accepted

### Recover Fourier Transform of flipped signal from the FFT of orignal signal

We can start with a the simple DFT relationship of the time reversal, i.e. If $\mathcal{F} (x[n]) = X[k]$, then $\mathcal{F} (x[-n]) = X'[k]$, where $'$ denotes complex conjugation. Now ...

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