5 votes
Accepted

Can we recover $|X(k)|$, given $|x(n)|$?

No: $$ \begin{align} \text{FFT}(A) &= \text{FFT}([1, -1, 1, -1]) = [0, 0, 4, 0]\\ \text{FFT}(B) &= \text{FFT}([1, 1, 1, 1]) = [4, 0, 0, 0] \end{align} $$ so $|A| = |B|$ while $|\text{FFT}(A)| ...
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3 votes
Accepted

Axes of Discrete Fourier Transform

Actually there are more than two answers for the DFT’s x-axis. I’ve seen spectral plots where an N-point DFT’s positive-frequency axis range is labeled: • Zero -to- N/2 (Frequency axis value is ...
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2 votes
Accepted

Constructing an input signal whose response is determined by the impulse response

You're right that you generally can't find a sequence $x[n]$ such that $$y[n]=\sum_{k=-\infty}^{\infty}\big|h[k]\big|\tag{1}$$ is satisfied for all values of $n$, but that's also not necessary. It is ...
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  • 80.2k
2 votes

A problem on FIR filter design given a difference equation

The filter order is the minimum number of (unit) delay elements necessary to implement the filter, or, equivalently, the order of the corresponding transfer function. In the case of causal FIR filters,...
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  • 80.2k
2 votes

What are the criteria for a change-of-basis transform to be doable in $O(n \log(n))$?

It sounds like you want to design a process that will more or less automatically arrive at the FFT for its functional and computation cost trade-off when that is optimal. And any other related (or not)...
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  • 2,186
2 votes
Accepted

How can I express the flipped output of multiplication in function of original inputs?

You just need to conjugate the matrix $D$ as well as the vector $x$, and flip the rows of $D$ upside down, i.e., the first row becomes the last row, etc. I.e., you need to introduce a new matrix $\...
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  • 80.2k
1 vote

FFT Artifacts and their cause beyond frequency resolution

The issue is probably that you're just zeroing out the FFT coefficients and then taking the inverse FFT. This is equivalent to multiplying the frequency domain by a response that is 1 everywhere ...
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1 vote

Basis Pursuit Denoising (BPDN) algorithm for Direction of Arrival Estimation

Check out SPGL1. It's probably the one most often used for basis pursuit/compressive sensing.
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  • 598
1 vote

What are the criteria for a change-of-basis transform to be doable in $O(n \log(n))$?

The usual reason you see $O(n \log(n))$ computations is when the $n^2$ direct-approach can be decomposed into two $n/2$ problems, and those can be decomposed into four $n/4$ problems etc. So the thing ...
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1 vote

What is the best way to decide whether a periodic signal is present in a noisy environment with a limited number of samples?

The problem you're running into seems to be that you have conflicting wants, specifically the fact that you want to operate at low SNR and use a small amount of samples. Integration gain needs to come ...
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  • 2,762
1 vote

Constructing an input signal whose response is determined by the impulse response

A more detailed proof based on Matt's answer. Let $n_0$ be the argument to $\max\{|y[n]|\}$, i.e. $\max\{|y[n]|\} = |y[n_0]|$. We can always find $x[n]$ as a series of +1's and -1's, s.t. $$|y[n_0]| = ...
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  • 43
1 vote
Accepted

Problem with the existence of inverse DTFT

You're absolutely right, the given system doesn't have a frequency response, at least not as a conventional function. There's a pole on the unit circle, so the system is not stable (actually, it is ...
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  • 80.2k
1 vote
Accepted

If $y(n)-y(n-1)=bx(n)$ is an LTI system, find $b$ such that $\left|H(e^{i\omega})\right|=1$ for $\omega=0$

After changing the given difference equation in your updated question, I don't see a problem anymore. Your answer is correct. If someone claims that $b=1$ is the correct solution, then have a look at ...
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