Questions tagged [fourier-transform]

The Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies, known as a frequency spectrum.

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Why aren't negative frequencies folded in reconstruction of the aliased signal?

I'm working on the problem 1.9 from the book Introduction to Signal Processing by Sophocles J. Orfanidis. The pdf version and solution is freely available here. This is the solution for part a of the ...
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Impact of STFT window function and FFT length on computation time

I have been doing a study which part of it includes a comparison of computation time vs window type and length (among some other things in the computation time, however I speak in terms of relative ...
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Phase extraction from Fourier transform

Is it possible in principle to correctly extract the phase from Fourier transform? I just tried to do so using Python, here some attempts: ...
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FFT of a gaussian signal in Python

I've been trying to get the FFT of a gaussian in Python. When I use the following parameters, the FFT goes hand in hand with the theoretical FT of the gaussian, but if I increase $\sigma$ they rapidly ...
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understand short time fourier transform

I am reading this paper for signal denoising. In the paper, the authors says The core concept in this paper is to compute a regression between a noisy signal frame and a clean signal frame in the ...
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Why is this Python implementation of trigonometric interpolation not working properly?

Consider a signal $u_j, j = 0, 1, \dots, N-1,$ sampled on an evenly-spaced grid of points, $x_j$. Define the discrete Fourier transform of $u_j$ by $$U_k := \frac{1}{\sqrt{N}} \sum_{j=0}^{N-1}u_je^{-(...
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3 answers
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Stanford EE 261 HW6 Q1 - Sampling below Nyquist Rate

The problem (taken from here) asks for possible sampling rates that will not cause aliasing in the following frequency spectrum: The range of possible values after some math is given as $B_2 < f_s ...
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Bandwidth visualization in frequency domain

Consider some signal in frequency domain: the maximum length of which corresponds to the half of the original signal ($N/2$), here $N=32$. It is known that the bandwidth of each sample is $2/N$, so ...
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3 votes
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Using spectogram to speed up a signal - Time Scaling/Phase Vocoder

Background About half a year ago, while learning about spectograms as part of an Image Processing course I took, I was told you can speed up audio using spectograms as follows: Calculate the ...
-1 votes
1 answer
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causal rectangular signal fourier transform example question

I have a discrete time fourier transform example for rectangular function. I understand most of the steps but I don't get how $e^{-j5\omega}$ came from when it tries to change the summation limit ...
1 vote
1 answer
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What Does "Reduced Modulo N" mean in this context?

I am trying to understand a piece of notation used in several papers, the simplest/shortest of which is this paper by Crochiere. The equation in question is Equation 7 on the second page: $x_m(sR) = ...
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1 answer
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In what cases can you get aliasing below the Nyquist frequency?

I took the one-sided FFT of a signal and plotted up until the Nyquist frequency. Then, I took the real part of this FFT multiplied by $i\omega$ following a calculation that I'm trying to do of a ...
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1 answer
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Sparse signal FFT

Say I had a time domain signal $x[k]$ wich is sparse: $\log(N)^2$ nonzero samples and the fourier transform has only a very (very!) small number of high frequency components. Are there any techniques ...
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Advantages/disadvantages of impulse excitation vs step excitation

i'm currently working on a circuit transient simulation, and I'm wondering how the type of input excitation can affect the final result. In particular the focus is on impulse input vs step input, to ...
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mp3 encoding in the frequency domain

Let's start with an arduino signal, which can be periodic over time. When this signal is converted from analog to digital it "turns" into a series of bits. At this point, is this signal in ...
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25 views

What can I do to improve the sound of a signal?

I want to improve the sound of my signal, I know I can do it by increasing or decreasing the amplitude of the signal itself. Are there any other ways to do this? How can I apply the Fourier transform ...
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2 answers
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When is the Fourier transform of a periodic discrete signal $\mathcal{F}x[k]$ the same as $x[k]$ up to a diagonal matrix

I am looking for all pairs $(x[n],q)$ where $x[n]$ is a periodic discrete signal with period $N$ and $q$ is a rational number satisfying the following identity: $$\mathcal{F}x[k]=e^{i(q-\frac{\pi k}{...
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Spectrum Y(f) of the signal $\DeclareMathOperator{\rect}{rect}\DeclareMathOperator{\sinc}{sinc}y(t) = \rect(t/T_0) \cos(2\pi f_0t)$

I have to compute the spectrum $Y(f)$ of the signal $y(t) = \rect(\frac{t}{T_0}) \cos(2\pi f_0t)$ with $f_0 = \frac{1}{T_0}$. What I have done so far is the following: I know from my notes that $\rect(...
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Confusion Understanding the mathematical expression of duality property of dft?

Duality Property for DFT Above dsp.se question provides good understanding about dft duality property but i am having difficulty understanding its mathematical expression because on Google when i try ...
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1 answer
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Recover Fourier Transform of flipped signal from the FFT of orignal signal

I trying to recover the Fourier Transform of a flipped signal directly from the Fourier transform of the original signal. More precisly, let s be a random signal: <...
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Phase Spectrum 0 Phase

I am a beginner in signal processing. I am learning about the Fourier transform. I was working on the zero-phase Ricker wavelet. As I understand If I extract the phase spectrum I should get something ...
6 votes
1 answer
428 views

Proving Fourier transform pair with derivatives using duality

I want to use duality to prove the Fourier transform pair $t^nx(t) \overset{\mathscr{F}}{\longleftrightarrow} j^n\frac{d^nX(\omega)}{d\omega^n}$ but I am struggling. I learned that if $x(t) \overset{\...
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Can the Fourier Transform of the unit step be used as a filter?

Using the FT of the step function we have $H(\delta)=\pi\delta(\omega)+\frac{1}{j\omega}$, and it's magnitude is $\infty$ at $\omega=0$ and approaches $0$ as $\omega$ goes to both positive and ...
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What is the magnitude response and phase of this function?

Given the system/filter $H(\omega)=\frac{1}{5-j\omega}$, find $h(t)$, it's magnitude response and phase and identify what type of filter it is. Now clearly given it's form, $h(t)=e^{5t}u(-t)$, but I'm ...
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Inferring shape of information signal from its DFT?

I came across this question recently, and I am very confused by (b)(ii). b(i) gives $x_n$ = [0.5, 0, -0.5, 0]. My approach to (ii) was to recognise that $X_m$ represents the frequency content of the ...
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2 votes
1 answer
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FFT and spatial frequency basic knowledge

Using the following website: on how the Fourier transform works (Interested in the Basic Principle part). I found out that if you have 3 pixels closer in the spatial domain you'll get more spaced ...
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Why does fourier transforming an analytic signal gives me negative frequency components?

I consider an array: import numpy as np from scipy.fft import fft from scipy.signal import hilbert a=np.random.rand(5) First I manually compute the fourier ...
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1 answer
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Spectrum of triangular pulse - MATLAB

I am new in matlab.I was given the code for spectrum of rectangular pulse of amplitude 1V and duration 1ms, and now I have to find the spectrum of a triangular pulse of amplitude 1V and duration 1ms ...
-1 votes
1 answer
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How to inject a 2D plane sine wave to the array of seismic sensors?

I've several sensors positioned at various points in the $X$,$Y$-Cartesian coordinate system, and I've experienced a problem to inject a planar Sine wave to the spatially positioned sensors, the ...
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What is the Fourier convolution theorem range of application (example of Dirac comb times rectangular window)?

$\DeclareMathOperator{\sinc}{sinc}$ I have questions regarding the Fourier transform of the product of functions or distributions and the range of application of the convolution theorem. Context When ...
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1 answer
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Time shift and Phase Examples

Given are two cosines according to the following formula $x_i(t) = cos(2\pi f_i t)$ with $f_1 = 1Hz$ , $f_2 = 2Hz$ and $f_3 = 3Hz$ . The two cosines are delayed by $\tau=0.1s$ to yield $y_i(t) = cos(2\...
1 vote
2 answers
79 views

Multiplication term $ \frac{ 1}{T_s} $ in sampling theorem

\begin{equation} X(\Omega) = \frac{ 1}{T_s} \sum ^{\infty}_{k=-\infty} X_a\left \lbrace \frac{\Omega /( 2 \pi) - k}{T_s}\right \rbrace \end{equation} What is the purpose of multiplying sampled ...
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How to calculate impulse response from frequency response whose denominator has an imaginary roots

It is quite easy to solve when the denominator of the frequency response has no imaginary values..... But I do I solve something like this? $$H(\omega)=\frac{1+2e^{-2j\omega}}{1-\frac{1}{15}e^{-j\...
1 vote
1 answer
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Computing the frequency response using fourier transform of an unstable LCCDE system

Given LCCDE system. Is it possible to calculate the frequency response using Fourier transform?
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1 answer
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Calculating instantaneous power of a frequency band in a 1-d signal

I am trying to calculate the instantaneous power of a 1-d signal within a particular frequency band in an on-line, real-time, streaming application. I have tried using sliding window FFTs but these ...
1 vote
1 answer
186 views

is $y(t) = (x(t))^2$ non-linear and time-invariant system?

i was able to show that it is not linear but for time-invariant I am not sure. $y(t) = (x(t))^2$ Let $y(t)$ be the output corresponding to the input $x(t).$ Let $x_T(t) = (x(t-T))^2.$ Then the output $...
0 votes
1 answer
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Continuous Inverse Fourier Transform [closed]

How to find inverse Fourier transform of: $ X(j\omega) = \frac{cos(3\omega) \cdot cos(\omega)}{\omega^2} $ The answer to this question is: $ x(t) = \frac{1}{2} y(t + 1) + \frac{1}{2} y(t - 1) $ where $...
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1 answer
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Fourier transform of the energy

If I have the time history of the energy, Does it make sense to do the fourier transform of this energy? or if I want to see the energy in frequency the PSD is the only tool? and Why
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1 answer
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DFT magnitude of signal with changing amplitude vs constant amplitude

I'm fairly new to the DFT, and I'm trying to get a deeper understanding of what I'm looking at when I look at the polar magnitude of its output. I have two sinusoids. I generated sinusoid b by just ...
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How to think conceptually about the Fourier transform?

I am trying to understand the FFT (and Fourier transform more generally) and want to make sense of a sample calculation. Here I have the following function plotted on a log-log scale: And then I take ...
1 vote
1 answer
106 views

Classifying STFT from multiple signal samples

I have a collection of signals (IQ wav) split up into ~2s samples of sampling rate 2MHz, and can collect the STFT information from these samples through the following code: ...
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Fourier Transform: $x(t)=2\sin(2\omega_0t)\cos(3\omega_0t)$

I'm currently studying Fourier Transforms and do not understand the Fourier Transform of $x(t)=2\sin(2\omega_0t)\cos(3\omega_0t)$ My solution states that it is $X(\omega)=\frac{2}{2\pi}(-j\pi[\delta_0(...
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How do you prove that the bandwidth of a signal is inversely proportional to the length of the signal?

I am trying to prove the below identity where $f_c(x)=f(cx)$ such that c is a positive number. $F_c(\alpha)=\frac 1 c F(\frac\alpha c)$ F above represents the Fourier transformed $f(x)$. I attempted ...
5 votes
3 answers
806 views

A case that zero padding increase real resolution and extract more info than naive DFT?

It is widely accepted that zero padding cannot reconstruct more information that is originally present in the sample data, which I think is intuitive, because zero padding adds no more information. ...
2 votes
0 answers
29 views

Additional artefacts in limited angle Radon transform reconstruction using the Fourier Slice Theorem

I want to simulate the limited angle Radon transform reconstruction problem by employing the Fourier-Slice Theorem which states that $$ \mathbf{F}\left(\mathbf{R} f\right) (\theta, \sigma) = \mathbf{F}...
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102 views

Is it possible to classify signal samples using STFT without (image-based) spectrograms?

I am aiming to conduct a classification-based study using signals collected from various devices. I've researched other approaches which make use of STFT for producing a spectrogram for speech and EEG/...
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Converting signal to frequency domain then back to time domain - noise issue

I am trying to convert a signal into frequency domain using fft then construct a time domain signal (with any length) from the frequency response. With my matlab code, it works fine when there is no ...
1 vote
0 answers
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How to restore time domain signal after multiply cosine signal when cosine signal has unknown initial phase?

I use AD9954 as cosine signal generator,then use ad835 to multiply this cosine signal with antenna signal. As ,the antenna signal frequency shifted after ad835,then ...
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5 votes
1 answer
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The Matrix Form of a 2D Circular Convolution

I have 3 closely related questions regarding 2d convolutions and how they are represented in matrix form. 1. Miming what happens in 1d, I assume the product of a doubly block circulant matrix $A$ by a ...
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What means `crop` in FFT calculation?

In soapy power manual: Crop: -o PERCENT, --overlap PERCENT percent of overlap when frequency hopping (incompatible with -k) -k PERCENT, --crop PERCENT percent of crop when frequency hopping (...
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