5 votes

Stanford EE 261 HW6 Q1 - Sampling below Nyquist Rate

The expression $B_2 < f_s < 2B_1$. is only part of the correct answer. The full correct answer is: $${2B_2\over m+1} ≤ f_s ≤ {2B_1\over m}$$ where $m$ is any positive integer ensuring that $f_s ≥...
user avatar
4 votes
Accepted

Stanford EE 261 HW6 Q1 - Sampling below Nyquist Rate

No aliasing means that the original information of the continuous-time signal (or higher rate discrete-time signal) is still contained unambiguously in the sampled signal. The original signal can - at ...
user avatar
  • 79.2k
2 votes
Accepted

Sparse signal FFT

Partial FFT Sparse FFT Can also subsample the input, which will alias (fold) the high frequencies onto lower, then take FFT at the lower length, and then shift the result back onto higher frequencies ...
user avatar
2 votes

Phase extraction from Fourier transform

You need to evaluate the DFT at the frequency of the input signal. I recommend doing this experiment with the DTFT instead of the DFT. You can implement it in just a few lines of code, and unless the ...
user avatar
  • 13.5k
2 votes

FFT of a gaussian signal in Python

There are two issues: The time axis is not long enough to capture a sufficient length of the Gaussian. The FFT is not properly scaled. For the first item mentioned regarding the time axis, the ...
user avatar
  • 36.1k
2 votes
Accepted

understand short time fourier transform

To understand: Using these features allows us to abstract many of the phase uncertainties and to focus on ``turning off" parts of the input spectral frames that are purely noise. check out ...
user avatar
  • 21.7k
2 votes
Accepted

Phase Spectrum 0 Phase

I was working on the zero-phase Ricker wavelet. Chances you were working on the "linear-phase" wavelet. Assuming a real input, zero-phase requires time symmetry, i.e. $x[-n] = x[n]$. If you ...
user avatar
  • 30.6k
1 vote
Accepted

Impact of STFT window function and FFT length on computation time

I think this is just measurement noise. You would not expect the window to make any difference (if it's precomputed). However the window length should make a small but consistent difference and that's ...
user avatar
  • 30.6k
1 vote

Bandwidth visualization in frequency domain

As a complement to Dan's answer: don't think in terms of the bandwidth per sample; this concept makes no sense. I recommend studying the discrete-time Fourier transform (DTFT) before the DFT. If you ...
user avatar
  • 13.5k
1 vote
Accepted

Bandwidth visualization in frequency domain

The length of the DFT is equal to the length of the time samples used ($N$ samples in time results in $N$ samples in frequency), and this frequency corresponds the the frequency range from $f=0$ to $f=...
user avatar
  • 36.1k
1 vote

causal rectangular signal fourier transform example question

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(...
user avatar
  • 1,619
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: $$ \...
user avatar
  • 21.7k
1 vote

Advantages/disadvantages of impulse excitation vs step excitation

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 ...
user avatar
  • 30.6k
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 ...
user avatar
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 $\x=(x[0],\ldots x[N-1])$. Your equation says: \begin{align}\...
user avatar
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 ...
user avatar
  • 30.6k

Only top scored, non community-wiki answers of a minimum length are eligible