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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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Time Domain Behaviour of Thermal Noise

Let's consider the thermal noise (or Johnson–Nyquist noise): It is white noise, that means that its power spectral density does not depend on frequency. Now my question is: Which is a typical time ...
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Understanding the FFT phase spectrum with a simple example

I'm trying to compute the DFT using scipy's functions. I don't understand why the phase spectrum of a simple sine wave with 2 Hz frequency doesn't show $\pm\pi/2$ at the $\pm 2Hz$ frequencies. Instead,...
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Is it possible to reduce the complexity of radix-2 FFT if the input vector contains identical elements?

my question is about reducing the complexity of radix-2 FFT when the input vector has a specific structure. For an input vector of x with N elements, the complexity is given by O(N log2 N). My input ...
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Approximation with DTFT question

Let's suppose $x(t)$ has its Fourier Transform so that $ \hat{x}(f) = 0 $ when $|f| > 16$. This function is sampled with $f_m =64$ and the only samples saved are those $x(\frac{n}{f_m})$ with n = ...
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1answer
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Orthogonal Basis for a 2D Signals (Compressive Sensing)

I have a 2-D signal that is (1536x128) and that is sparse in the Fourier domain (after applying fft2). I want to apply compressive sensing to recover the signal using fewer random elements, but I am ...
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49 views

Complex output after inverse FFT of a real signal

I have a real one dimensional signal s (light absorbance in a flow cell), which has significant noise and some periodic noise after performing a deconvolution of $S$ from $S_o$. Basically fft($S$) was ...
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1answer
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Advantages of the Rotation Translation Operation Before Doing FT Smoothing

I was reading a relatively old paper from the 1970s on smoothing by FT methods (chemistry applications), where the authors show that if we do rotation translation operation on the signal (y- values) ...
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1answer
39 views

Finding the impulse response of a system

I have the following transfer function. $$ H(j\omega) = \frac{1+0.5 e^{-j\omega}}{1-1.8 \cos(\frac{\pi}{16}) e^{-j\omega}+0.81 e^{-j2\omega}}$$ I'm trying to find the impulse response of the system. ...
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How To Find The Lowest Note of a Chord Being Played

The goal of my project is to find the lowest note in a signal coming from a guitar being played. I would like to do this programmatically and have sourced the equipment (raspberry pi, USB interface, ...
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Is it possible to recover the time-domain signal after manipulating it in the frequency-domain? [duplicate]

I've worked a fair amount with EEG signals, though I've never had formal training in signal processing, so please excuse my ignorance. The problem is this: my signal has noise at many, many ...
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$x[n]$ after sampling of $cos(16\pi t+\phi)$ at 12kHz

I'm not sure what the question really means, so this is just guesswork. I think options 1 and 4 can be ruled out as $w_0<\pi$. The CTFT of $cos(16\pi t+\phi)$ has two spikes at $16\pi$ and $-16\...
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Discrete Time Fourier Transform (DTFT) cross correlation property

I came across this property of the Discrete Time Fourier Transform (DTFT) and I am having a tough time proving it. In general, consider two real signals $x[n] \: \& \: y[n]$. If $$ x[n] \...
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Find integral of DTFT after sampling (Graph of CTFT given)

So for the first question: If this is sampled at 10kHz, then the amplitude is scaled by 10000. In the DTFT, the frequency 3.5kHz gets mapped to 3.5/10* 2pi=0.7pi. So this point lies outside the range ...
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Which frequency bins give the best interpolation for the derivative of a function?

A function $u:[0,2\pi]\to\mathbb R$ sampled over $N$ equidistant points $\theta_j=(2\pi/N)j,\, j = 0, \dots, N-1,$ can be interpolated by a set of functions $\{u_{k_0}\}$ enumerated by integers $k_0\...
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2answers
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Why is a circular mask appropriate for Fourier filtering rectangular images?

Suppose I apply 2D DFT to an image with dimensions $H{\times}W$ where $H \neq W$, then shift the DC component to the center. Why does a circular mask capture the lowest frequency components, i.e. why ...
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Derivation of Nyquist Frequency and Sampling Theorem

I have been looking through different sites and questions over the internet about Sampling theory, but couldn’t find the clear definition of how nyquist frequency condition is derived? It would be ...
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Fourier Transform of finite time series

I have some signal 𝑠(𝑡) which is real data i.e. finite. The time runs from −𝑇 to +𝑇. The signal amplitude is large at 𝑡=0 and small (→0) at the ±𝑇 limits. I can do a finite (discrete) Fourier ...
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What is the form of the spectral derivative in the all-positive-frequency notation in DFT?

The Discrete Fourier Transform (DFT) of a function $u:[0,2\pi] \to \mathbb R$ sampled over $N$ equidistant points $\theta_j = 2\pi j/N,\, j = 0, \dots, N-1,$ is defined by $$ \tilde U_k = \frac1N \...
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Solving equation with convolution

I have the measured signal $y(t)$ that can be modeled in the frequency domain as $Y(f)$: $$Y(f) = X(f)\cdot A(f) - [X(f)\cdot B(f)] \ast C(f)$$ where $\ast$ is the convolution. I know $A(f)$, $B(f)$,...
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Inverse Fourier Transform Dirac impulse with scaled argument

Currently, I am dealing with the sampling problems and I don't understand how to calculate inverse Fourier transform of a scaling impulse function $\textrm{IFT}\{\delta(\Omega T)\} = ?$, $T$ is ...
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Sampling of frequency response

Let's consider any physical quantity depending on the frequency. For example, the impedance of a certain electrical component: $Z(f)$. Now, imagine to measure it in a continuous interval of ...
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How to perform spectral inversion in the frequency domain to convert a low-pass filter into a high-pass filter?

To convert a linear-phase FIR low-pass filter into a high-pass filter with the same cut-off frequency, we can invert the sign of the low-pass filter's impulse response $h(n)$ and then add one to the ...
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Fourier transform of a periodic/aperiodic signal

Generally speaking, I know that periodic signals (continuous-time domain signals) with period 2pi/wo have a spectrum with equidistance Delta-impulses of distance w0. My question is that, if we have ...
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1answer
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Palmprint Identification - Why do we align the images before we use the Fourier Transform?

I have been reading the paper PALMPRINT IDENTIFICATION BY FOURIER TRANSFORM by WENXIN LI, DAVID ZHANG and ZHUOQUN XU about identifying persons based on an image of their palm, one version of the paper ...
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Relationship between DFT and DTFT

Let $x[n]$ be an $N$-point sequence, i.e. $x[n] = 0$ for $n < 0$ and $n \ge N$. Let $X[k]$ be the $N$-point DFT of $x[n]$. Let $$y[n]= \left \{ \begin{array}{ll} \displaystyle\sum_{l=-\infty}^{+\...
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FFT freqency bin center in R

I'm trying to do a spectral analysis in R. I learned it in Python from Allen Downey's ThinkDSP book. What is the R equivalent of the Python numpy function, numpy.fft.fftfreq? If you provide a window ...
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Two Consecutive Inverse Fourier Transforms [duplicate]

What happens to a function F(w) if you take two consecutive inverse Fourier transform of it?
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How does one interpret an element of the “transfer matrix” used to calculate frequency domain granger causality (via VAR models)?

I am attempting to gain a better mathematical understanding for how autoregressive models can be used to infer frequency-domain granger causality. All freq. domain measures of causality that utilize ...
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Frequency Domain Signal to Noise Ratio

I am doing some research on low-cost air pollution sensors. I'm measuring the "ground truth" with a single low-noise sensor, and I'm trying to use it to calibrate a low-cost sensor that has high noise....
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If the cosine function is periodic, why does it have a Fourier Transform? [duplicate]

As far as I understand Fourier Transforms are for non-periodic signals and Fourier Series for periodic signals. So why is it we can take the Fourier Transform of a cosine when it is a periodic ...
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How do I make sense of the cosine wave having Fourier Transform coefficients which have infinite magnitude?

To illustrate my question better, consider the Fourier Transform of an aperiodic (as a periodic cosine wave has a Fourier Transform not Fourier Series) cosine wave $$f(x) = \begin{cases} \cos(2\pi ...
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1answer
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How does minimum-latency partitioned convolution reverb work when you receive input samples in chunks, rather than one at a time?

I'm writing a reverb system where I receive an input block of samples 480 elements long, do some operation on them, and pass the block on to the next effect. I've been reading up on partitioned ...
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1answer
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DTFT of even and odd samples

Here to find DTFT of $h(2n)$ they have scaled omega, while in RHS to find DTFT $x(2n+1)$ they didn't, why is that?
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Inverse Discrete Time Fourier Transform of $1$

$\textrm{DTFT}(\delta[n]) =1$, but $\textrm{IDTFT(1)} = \frac{\sin(\pi n)}{\pi n}$. Why it is not equal to the unit impulse $\delta[n]$?
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What determines peaks in FFT?

I ran FFT on three audio files and found that the results for some have more peaks than the other. Could anyone give me any conceptual explanation as to what determines these peaks? Below are plots of ...
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Why zero padding the 2-d DFT interpolates images in spatial domain?

I was applying different image interpolation techniques and I came know to about interpolation in frequency domain. In this technique we first take 2d DFT of an image, padd it with zeros and take the ...
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1answer
50 views

Why the spectral coherence is unity for all frequencies between single-frequency time series and itself

In the example below, I am plotting the coherence between time series and itself. The time series do has one frequency.The coherence magnitude was one for all frequencies. I wonder why it is not zero ...
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How to find a Matched Filter Transfer Function from large signal sample

Lets say I have a system where I have a small sample of a signal with no noise $\hat{x}(t)$ and a lot of a similar signal with noise $y(t) = \hat{x}(t) + n(t)$, and from $\hat{x}(t)$ I want to create ...
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1answer
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Signal processing using numpy python

To process a .wav audio file with numpy (using fast Fourier transform algorithm). I want to process an audio signal at a particular interval with a sampling frequency 44100hz and sampling rate of 20ms ...
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Fourier Spectra : Significance of the Negative Amplitude [duplicate]

For example, for an aperiodic gate pulse, the Fourier Transforms for the continuous time case is a sinc function, while the discrete time case gives a sine over sine periodic kind of a function. In ...
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Is there a version of Welch's method that doesn't look for power?

Welch's method splits a time signal, $x(n)$ into $M$ periodograms $P_m$, $P_{x_m,M }(k) = \frac{1}{M}|F_k(x_m)|^2$ and averages them to give the Power Spectral Density (PSD), $S_{x}(k) = \frac{1}{K}...
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How do I decide which frequencies are signal and which are noise?

I have an arbitrary recorded digial signal, on which I have run a Fourier transform. I'm not sure what conventions are on a case like this, but I have 1024 frequency bins. Second bin is the highest ...
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Fourier Transform: interpretation of continuous spectrum at specific frequencies

B. P. Lathi in his book "Principles of Linear Systems and Signals" mentions in the Fourier Transform: When $x(t)$ is periodic, the spectrum is discrete, and $x(t)$ can be expressed as a sum of ...
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Aliasing when interpolating with DFT?

I'm coming from an understanding of the continuous-time Fourier Transform, and the effects of doing a DFT and the inverse DFT are mysterious to me. I have created a noiseless signal as: ...
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How to search for irregular signals: Fourier, DWT or k-means?

See my notebook here I want to search for irregular time signals in a data set of ~3 500 000 time signals. By eyeballing I have found hundreds of flat and oscillating signals, but just a few that are ...
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1answer
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What is Imaginary in Fourier transform?

How to plot graph of $e^{-t}$ in frequency domain. What would be the axis? If its Fourier transform is $1 /(1+j\omega)$, then how can we plot imaginary on frequency domain (amplitude vs frequency ...
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Explain Auto-Tune in a simple way

I have to do a presentation about Auto-Tune and its relation to the Fourier transform. What is a good explanation on how does Auto-Tune work?
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Impulse response of a 3x3 PSF - how to find analytical expression for fourier transform of a 3x3 matrix?

I have a filter $\mu[n_1, n_2]$ with taps: $$ (1/8) (1/4) (1/8)$$ $$ (1/4) (1/2) (1/4)$$ $$ (1/8) (1/4) (1/8)$$ How do I find an analytical expression for $\hat\mu(w_1, w_2) $? Since it looks so ...
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2answers
76 views

STFT on time varying signal with good time and frequency resolution

I am trying to determine the main frequency of a noisy signal that varies in frequency over time. Ideally I want to detect changes in the frequency as rapidly as possible - say 50Hz update rate, but I ...
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Is there any practical application for performing a double Fourier transform? …or an inverse Fourier transform on a time-domain input?

In mathematics you can take the double derivative, or double integral of a function. There are many cases where performing a double derivative models a practical real-world situation, like finding the ...