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 is the DC component of discrete fourier transform not the same as the signal's arithmetic mean?
In this question we have a mathematical proof that the DC component of normalized discrete Fourier transform should be the same as the signal's arithmetic mean. However, in the following example I ...
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3D convolution product with Fourier transforms, FFTW and MPI in C
My question is not really from the field of signal processing but I think you are the most suited for answering my question.
I am willing to compute the gravitational potential which can be written as ...
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Fourier Transform of a triangle function [duplicate]
Good afternoon,
I am having a question regarding the following function :
Let $g(x)$ be :
$$\begin{cases}\pi + \frac{\pi}{2}x \text{ if } -2 <x < 0\\
\pi - \frac{\pi}{2}x \text{ if } 0 <x &...
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Computing the DFT: how is the number of operations reduced by splitting the signal into even and odd parts?
On page 137 of "Understanding Digital Signal Processing" by R.G. Lyons I found that if I separate the standard DFT form:
$$X(m)=\sum_{n=0}^{N-1}x(n)e^{-j2\pi nm/N}\tag{4-11}$$
by odd and ...
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Why is the time domain low-pass filter the "sinc" shape?
Consider:
I'm looking at low-pass filters, and I see that the time domain representation of an "ideal" filter resembles the shape above whereas the frequency domain is a box. I also get the ...
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Understanding symmetry in DFT magnitude plot
I am trying to intuitively understand the mystery of why the DFT of a complex vector with real values produces apparent symmetry in the magnitude plot (see the second plot here for example). In DFT $X[...
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What is the frequency representation of nonuniform sampling?
Uniform sampling can be thought of as multiplication of a function $x(t)$ with a Dirac comb function: $$\text{III}_T(t) = \sum_{k=-\infty}^{\infty}\delta(t-kT)$$
Multiplication of $x(t)$ with $\text{...
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Scaling plots after FFT
I have multiplexed 32 signals into 1 signal with python.
Now I want to plot that signal in time domain and to plot it's amplitude spectrum.
Professor gave us his plots so we can look up to it.
These ...
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Reconstructing the original signal from its DFT
Hi I am a newbie to signal processing and I am trying to better understand how inverse DFT works under the hood. Consider this signal and its DFT:
(Source)
For the sake of this post, let's assume ...
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Bandwidth of Information Signal
I have trouble finding the bandwidth of a signal. Say I have an info bearing signal m(t)=sinc(2t/pi). I found the fourier transform of the sinc function and found that the angular frequency was 1/pi. ...
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Why is the CT inverse Fourier Transform also an integral?
Intuitively it seems that to get a function back that was integrated, you would take the derivative. Instead, with the Fourier Transform we take an area under a curve of a modified function, and to ...
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Implementation of dispersion compensation of lamb waves
I am trying to implement the method from Paul Wilcox paper "A rapid signal processing technique to remove the effect of dispersion from guided wave signals" on data from a lamb wave ...
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Spectrum and Fourier transform of a sine wave
Take for example: $x(t)=A\sin(2\pi f_0t)$. The Fourier transform of this signal is $\hat{s(f)}=\frac{A}{2i}(\delta(f-f_0)-\delta(f+f_0))$. If we want to represent the spectrum of the signal we would ...
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Conjugate symmetric: 3D fourier transform dirmension
I have a real value input 3D tensor with the shape of `(H,W,D)=[8,8,20]', where H, W, and D represent height, width and depth in (z dimension), respectively. When converting to the DFT, what will be ...
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Proof for the energy correction factor of DFT
I am looking for a mathematical proof for the energy correction factor in conteext of windowed discrete fourier transform.
In Spectrum and spectral density estimation by the Discrete Fourier transform ...
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How to obtain filtered impulse response from frequency response?
I am trying to find the reverberation time of a room using the Schroeder method (i.e., Reverse-time integration method). Therefore, impulse responses should be measured first.
There are many ways to ...
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How do I determine stationarity from a set of 50 complex values collected every 10 minutes?
I am trying to determine stationarity from a somewhat stochastic process. Every 10 minutes, I collect a set of 50 FFTs, i.e., 1 trial over $50$ seconds, so an FFT occurs every time second. I ...
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Fourier transform magnitude of the sum of two signals
Let
$$\mathscr{F}\Big\{x_1(t)+x_2(t)\Big\}=X_1(f)+X_2(f)$$
I think that in general
$$\big|X_1(f)+X_2(f)\big|^2\leq\big|X_1(f)\big|^2+\big|X_2(f)\big|^2$$
but I was wondering if
$$X_1(f)X_2(f)=0,\qquad\...
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What is the sparse Fourier transform?
MIT has been making a bit of noise lately about a new algorithm that is touted as a faster Fourier transform that works on particular kinds of signals, for instance: "Faster Fourier transform ...
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Indexing in DFT (from an old paper)
There is a nice paper on explaining DFT from the 1960s in IEEE A guided tour of the fast Fourier transform. The author uses the following definitions of DFT
DFT
$$
X(j)=\sum_{k=0}^{N-1} x(k) \exp \...
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Frequency content of a noisy signal
To find the frequency content of a noisy signal (PSD), there are two methods below:
#1 Take the fourier transform of its power signal (square the noisy signal)
#2 Find the autocorrelation function of ...
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Why does multiplying a real signal by a random complex phase term result in "spreading" in the Fourier domain?
Suppose I have some real-valued signal $x\mapsto f(x)$. The amplitude of its Fourier transform $\mathcal{F}[f]$ then looks like a peak around the DC-term, decaying as we move towards higher ...
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On FFT, interpolating signal vs extending signal in time
When we interpolate, then FFT the output will have more bins.
When we extend the signal in time, Then FFT output will have more bins too but:
Interpolation increases max bin frequency but time ...
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Relationship between fourier transform and fourier series
Let
$$x(t) = A\sin(2 \pi f_0 t + \alpha)$$
its Fourier transform is given by $$ X(\omega) = \frac{A \pi}{i}(e^{ia}\delta(\omega-2\pi f_0) - e^{-ia}\delta(w+2\pi f_0)). $$
the Fourier series complex ...
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Finding Discrete Fourier Transform (DFT) for different DFT size
$N$ is an even integer, $x[n]$ is a finite length signal over the interval $n \in [0,N-1]$, and $X[k]$ is the $N$-point DFT of $x[n]$. Analytically find the DFT of sequence below in terms of $X[k]$. ...
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Is it useful to think of a Fourier Transform as writing out a signal in terms of a basis?
The (modified) trigonometric functions $\{0, \cos(kx), \sin(kx)\}$ serves as a basis for periodic function. I have also seen (but not rigorously) that the Fourier transform can also be seen as an ...
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Alternative way to find fourier transform
Let $$x(t) = A \text{rect}_T({t-\tau})$$I calculated its fourier transform through the direct way: $$X(\omega) = Ae^{-i\omega \tau} \int_{-T/2}^{T/2} e^{-i \omega t} dt = Ae^{-i\omega \tau} \frac{\sin(...
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Calculating k-space coverage from antenna positions
I work with radar and I want to understand which spatial frequencies I can measure, i.e. the k-space coverage, given a set of coordinates of transmitter-receiver combinations. The ideal coverage for a ...
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Is `fft` always the best choice?
I hope this is the right place to ask this question since it is partially a note which might help others. Until recently, I always used the fft-algorithm ...
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How to Use Convolution Theorem to Apply a 2D Convolution on an Image
How do I actually apply the convolution theorem? I have my fourier transformed image matrix, and a Fourier transformed kernel, but how do I actually multiply these together to achieve the intended ...
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How to identify square wave in audio signal
I'm interested in identifying a square wave signal from recorded audio. This subject may be a complex problem so I would like to present the problem where I'm currently stuck at.
I recorded the audio ...
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Steering Vectors and Bluetooth Low Energy for computing Angle of Arrival
I am new to signal processing, but I have background in mathematics. I am trying to use Bluetooth Low Energy (BLE) on three mobile devices, where one device is being tracked and the other two act as ...
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Amplitude and Phase spectrum from fourier transform of sawtooth [duplicate]
Let the following $T$-periodic signal :
then $$\begin{align}F(x(t))(\omega) =& \int_{-\infty}^\infty x(t) \exp(-i\omega t) \mathrm{d}t = \sum_{k=-\infty}^\infty \int_{kT}^{(k+1)T} x(t) \exp(-i \...
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What is the reason of existence of Fourier transform? (Why we use Fourier transform?)
I'm currently trying to understand Fourier transform and I've got curious about why Fourier transform exists.
Let's suppose that we have a 10 seconds of non-periodic wave. For example:
As far as I ...
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“The Fourier transform cannot measure two phases at the same frequency.” Why not?
I have read that the Fourier transform cannot distinguish components with the same frequency but different phase. For example, in Mathoverflow, or xrayphysics, where I got the title of my question ...
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Determine reflections from received signal
I have a reference signal $r(t)$ and the correlation between that reference signal and the received signal : $C_{XR}(\tau)$. The signal I receive contains reflections on walls. I have to build a ...
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Fourier Transform and Music Analysis
I am a senior in high-school and am currently trying to conduct an exploration on Fourier Analysis, specifically using the Discrete Fourier Transform to analyze a chord played on my piano. Basically, ...
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CFO Estimation in LoRa Chirp Signal (Preamble part)
I am trying to estimate CFO in LoRa chirp signal (preamble part). I have seen the discussion about CFO on this forum but it is mainly related to CFO estimation in OFDM.
I want to estimate the CFO in a ...
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Difference between DC component and zero frequency component of signal
We know that Fourier Transform of a signal exists if it is absolutely integrable
and it exists for periodic signals if impulse functions are allowed.
If we consider the fourier transform of $\text{...
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Fourier transform of $|x_\mathrm{a}(t)|^2$
Let $x_\mathrm{a}(t)$ be the analytic signal for real signal $x(t)$. I want to find an expression for $\mathscr{F}\{|x_\mathrm{a}(t)|^2\}(f)$ in terms of $x(t)$. The analytic signal can be written as $...
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Frequency response of $\mathrm{sinc}[n]$
In this image the frequency response of a discrete time filter given as $h[n]$. Can someone explain how the magnitude of the frequency response is found ?
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Why does applying Fourier Transform on point Spread Function yield h(t) which is complex-valued
I wanted to understand why this text talks about applying the Fourier transform on H(f) to obtain h(t). I view Fourier transform as moving from the time or spatial domain to the frequency / spatial ...
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How to get the scale bar of an FFT for a 2d image
I've been analyzing some images from a transmission electron microscope, including their FFTs, and I'm not sure how to apply a scale bar to the FFT images. I have calibrations for the real space image,...
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How to determine the nature (real or complex valued) of a signal?
A signal X(t) is a real valued time domain signal and Y(t) is a signal that only contains the non-negative spectral components of X(t). How do I determine whether Y(t) is real-valued or complex?
I ...
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FMCW radar: understanding of doppler fft
I am using fmcw radar to find out distance and speed of moving objects using stm32l476 microcontroller.
I transmit the modulation signal as sawtooth waveform and I read the recieved signal in the ...
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How can I plot a sinc function correctly?
I am generating a rectangular pulse using a piecewise function on Matlab. I have listened to some advice to use a normalization coefficient and the amplitude appears correct now. However, my issue is ...
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PSD of the sum of two zero-mean white noise signals
I am trying to solve the following exercise, where $y(t)$ is the sum of two signals $x_1(t)$ and $x_2(t)$ with each of them being the product of the convolution of $e_i(t)$ with $h_i(t)$.
So far I ...
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What is the phase of $Y(\omega)$ in relation to the phase of $X(\omega)$ where $x(t)$ modulated with an exponential carrier that has no phase shift?
A homework problem of a free online course I am taking, asked to draw the magnitude and phase of $Y(\omega)$ where $y(t) = x(t) c(t)$ where $c(t)$ is $e^{j 3 \omega_{c} t}$ and where $X(\omega)$ is:
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How can sound waves be modeled in a manner that distinguishes individual voices but also recognizes words?
More specifically, I'm curious how we can represent sound waves in a way that would both distinguish between individual voices and also recognize, e.g. that recordings of different people saying the ...
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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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