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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Is there a name for the procedure of taking the FT over separate consecutive small time-blocks?

Suppose we have a continuous time-interval $I=[a,b]$, and a signal $x \colon I \to \mathbb{R}$. A procedure that is sometimes carried out (e.g. when doing bispectral analysis) is to partition $I$ ...
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Centered Fourier transform

What is the difference between the non-centered and centered Fourier transforms? In other words, when should you use one instead of the other? Non-Centered: $\quad \displaystyle X_1(f)=\sum\limits_{n=...
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Can you decimate / downsample a signal in frequency domain just like you can interpolate / upsample it?

To interpolate a signal I can just zero pad it in the frequency domain. If I want to decimate the signal, can I just discard some part of the frequency domain? So in MATLAB this works: ...
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what is nyquist rate of $h(t)\cdot h(t)$ and $h(t)\circledast h(t)$

Let's say we have $h_c(t)$ as a continuous-time signal with bandwidth $B$ and we would like to sample it. To be able to reconstruct it correctly, the sampling rate must be greater than $2B$. Now ...
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Derive DTFT of $x[2n]$

If the DTFT of discrete sequence $x[n]$ is $X(e^{j\omega})$, what is the DTFT of $g[n] = x[2n]$? I see the textbook answer is \begin{align*} G(e^{j\omega}) &= \frac{1}{2} \left( X(e^{j\omega/2}...
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Conceptual question on FFT and chirp signal

If I take the FFT of a sinusoid I will get a plot whit all the energy of the signal concentrated at the sinusoid frequency. But what happens if I have a signal in which the frequency keeps changing?(...
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DSP interview question: use of the identity in development of a significant transform

I'm preparing interview and found this question. But I don't really understand what is the question. Does it ask about Fourier transform or Z transform? How the simple identity $$xy=\frac{1}{2}x^2 + ...
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What does an image of Fourier Transformation of an image tell us?

First time studying image processing... I just don't understand what does fourier transformed image of an image describe? For example consider given following pictures, The first one is the image, and ...
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Right algorithm for fourier transform on physical heights

I have data from a LIDAR unit that I would like to get the spectral density of. Unfortunately, the only thing I remember from my Fourier analysis class are the methods that I know will not work. The ...
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Interpreting the inverse fourier transform from a graph

I'm given a graph of the fourier transform of some function $x(t)$. The graph is labelled $F(X(\frac{\omega}{\pi}))$ on the y-axis and $\frac{\omega}{\pi}$ on the x-axis. The graph is plotted only ...
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The Fourier transform of a damped cosine and the units of the result

If I take a simple transient voltage signal of the form $$f(t) = V_p e^{-t/\tau} \cos(\omega_0 t)$$ and take the Fourier transform in the normal way $$F(\omega) = \frac{V_p}{\sqrt{2 \pi}} \int^{+\...
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How to calculate the Fourier transform of absolute-valued sinusoid?

The signal $x(t) = |1+a\sin(\omega t)|,(a>0)$ is a continuous waveform. In order to extract the frequency parameter $\omega$, I conduct the FFT of it and obtain its spectrum showed as follows. The ...
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2D Fourier transform of Sobel kernel

Can someone explain me the highlighted text parts regarding this image ? Here is a pseudo-code of how it was created: ...
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Zero-padding the middle of a signal

I sample a signal at a certain frequency for a finite amount of time to get a sequence $$(x_n)_{n=1}^N = (x_1, x_2, ... , x_N)$$ with the intention of analyzing its power spectral density by ...
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why do we use $X(e^{j\omega})$ instead of $X(j\omega) $ in Discrete Time FT

I am studying DT-FT. But I cannot figure out why we use $X(e^{j\omega})$ instead of $ X(j\omega) $ in DT FT Thanks in advance..
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How moving part pixel intensity values of video frames becomes dominant compared to stationary part intensities in reconstructed frames?

Hello everyone i want to do dynamic texture video sementation using the Fourier transform in MATLAB. I am applying 3-D fft on dynamic texture video frames (using matlab function 'fftn') and ...
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FFT of color images incorporated into an Object Recognition method?

Is there any instance where Fourier transforms of color images have been used in conjunction with other object recognition method? Any instance of usage of Fourier transforms in color images? I ...
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Fourier Transform of Alternating Periodic Rectangular Pulse

I'm having trouble determining Fourier transform of signal. I have 2 ideas on how to solve this problem. Given the signal is periodic I could use formula for Fourier transform of periodic signals: $$...
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Mathematical relationship between highpass and lowpass filtering

Let $g, h_{HP}, h_{LP}: \mathbb{R} \rightarrow \mathbb{R}$ and $G, H_{HP}, H_{LP}$ denote their continuous Fourier transforms under the Fourier operator $\mathcal{F}$. Let $*$ denote the continuous ...
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Discrepancy between Gaussian FFT and its Fourier Transform

I am trying to do the FFT of a Gaussian signal and comparing it to the theoretical Fourier transform. For infinitely small time step $dt$ and infinitely long signal length $T$, the 2 should become ...
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Fourier transform exercise

I have this signal: $$ X(f)= 2\delta(f)+ \delta\left(f-\frac 1{T_0}\right)+\delta\left(f+\frac 1{T_0}\right)+\textrm{rect}\left(\frac{f-\frac 4{T_0}}{\frac 2{T_0}}\right)+\textrm{rect}\left(\frac{f+\...
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Find state space model from transfer function

Let's suppose we have: G(s) = (s+1)/(s^2-2s+1) how can we find the state space representation of the transfer function: x_dot = x2 x2_dot = 2*x2-x1+u where u is an arbitrary input. I am very new ...
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Fourier transform of $\cos(n\omega t)$

My question is probably very stupid, but I've been strugling for a while on it now... In need to find the Fourier transform of $1+\cos^3(2\pi ft)$. I wrote that : $$\cos^3(2\pi ft)=\frac{\cos(6\pi ...
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Difference between Fourier Transform and DFT? - Example

I have read many excellent answers to similar questions, but never one this specific. Here is another way to ask it. Why is the modulation transfer function (MTF) of $\textrm{rect}(x/5) = \textrm{...
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FFT of SIN waves with different phase delays

I have come across a peculiarity of FFTs which has got me somewhat baffled. I've simply summed up 101 sine waves and taken the FFT using this matlab script : ...
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Fourier transform 4 times = original function (2D and higher)

The Signal Processing SE post linked below shows how the Fourier Transform applied 4 times to a 1D function returns the original function, i.e. F{ F{ F{ F{ g(x) } } } } = g(x) Link to 1D case: ...
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Why is the last value of an RFFT always real?

I am using numpy to do FFTs of real-valued data. And I don't understand why the Nyquist frequency is always real (or has zero phase). So, say ...
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“Flanging” in frequency domain?

Flanging is defined as a mix of two identical signals where one signal is delayed in time by a small and gradually changing period, around 10-20 milliseconds. Since delay in the time domain is ...
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Store a FFT with a minimal amount of data

I have an array x of length 1024 (stored as 16 bits integers, named for example np.int16 in numpy/python), i.e. the size of x is ...
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Interpreting FFT Coefficients from System Matrix That Is Originally Toeplitz (Not Circulant)

If I have a measured signal $y$, true signal $x$, and a convolution matrix $A$ that is a Toeplitz but not circulant matrix, I can write the convolution as \begin{equation} y = A x \ . \end{equation} ...
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What is the frequency of $\cos x -\sqrt 2\cos\sqrt 2 x$?

Question Considering that $1\over{2\pi}$ is the frequency of $\cos x$ and also of $\cos x - 2\cos 2x$, what is the frequency of $\cos x - \sqrt 2\cos\sqrt 2 x$? Thoughts Perhaps "frequency" isn't ...
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Energy Spectrum of a signal after FFT in Matlab

I have a vector of signal x(t) with its time vector. I want to obtain a frequency representation of the signal, in particular the energy spectrum of x(t). Can someone please show some light into ...
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Signal Processing using Fourier Transform

So I'm trying to understand how MRI machines work. I understand all the concepts of it, the parts, what they do, how the machine works, etc. The part I'm having trouble with is the fourier transform ...
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Techniques to deriving DTFTs

Are there general techniques to derive DTFTs? Given a bandlimited function $x(t)$, how do I find $$X(\omega)=\sum_{n=-\infty}^\infty x[n]e^{-i\omega n}$$ Generally, it is easier to derive the ...
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How to 'interpret' the Fourier Transform (specifically, of a convolution kernel)

As part of a homework assignment, I had to take the Fourier transform of the kernel I was using to convolve a signal. The kernel was a constant rectangular function, that was 1 within the square $(-1,...
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Difference between Fourier-Transform and FFT of rectangular pulse

I'm trying to find a link between the Fourier-Transformation of aperiodic Signals and the FFT of them. So to start with a basic example, let's take a rectangular pulse with width 0.1s and amplitude of ...
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Basic difference between Fourier transform and laplace transform? [duplicate]

I have read few links about difference between Fourier transform and Laplace transform but still not satisfied Please correct me if i am wrong Simply put, the main difference between Fourier ...
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How to get around the circular shift property of Discrete Fourier Transform?

I understand that when we introduce a linear time shift using DFT on a finite sequence, the algorithm assumes that the signal repeats itself outside of the given range. Here is an example explaining ...
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Parseval's Theorem for discrete series

I need to use Parseval's Theorem to find \begin{equation} S\:=\:\sum _{n=-\infty }^{\infty }\:\left[\left(\frac{\sin\left(\frac{\pi }{4}n\right)}{2\pi n}\right)\left(\frac{\sin\left(\frac{\pi \:}{6}n\...
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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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Determination of periodicity in data and finding mean

I have to find whether there is any pattern (I mean periodicity or close to periodicity) and if there is, for one cycle i have to perform numerical integration to determine mean. In the first picture ...
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Practical cross-spectrum estimation using Blackman-Tukey approach

I would like to estimate the cross-spectrum of two signals using the (lag-windowed) Blackman-Tukey approach but I'm having difficulties with proper practical implementation. As defined in equation 2.8....
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Faster way of getting 2D frequency amplitudes than DFT?

I'm making a small program which gets the DFT of an image to get a general idea of the image's overall orientation. It does this by rotating a line in a radar sweep type pattern, keeping track of ...
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Why is discrete cosine preferred to FFT in neuroimaging GLM

High-pass filtering is often used in neuroimaging data analysis. Commonly, whenever a general linear model is fitted to the data (as for instance in statistic parametric mapping) a number of columns ...
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Instantaneous frequency vs fourier frequency [closed]

Lets consider a pure sine signal at $\nu$ that is chopped using square pulses (like a burst mode on signal generators). My understanding is that instantaneous frequency is $\nu$ when oscillations are ...
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What is the optimal adaptive grid for calculating a DFT using a fixed number of sampling points?

I'm currently facing the following problem: I want to approximate the Fourier transform $F(\omega)$ of a (let's say, $L^2(\mathbb R)$) function $f(x)$ by calculating the discrete Fourier transform, ...
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Am I handling offline FFT correctly?

I need some help clarifying FFTs and what they represent. I have a buffer containing compressed audio. Due to limitations, I can't handle the full uncompressed audio but can decompress small segments ...
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Fourier transform and anti-trasform--identity missing

I have a very silly doubt: If we define the power spectral density: S(f)=$\frac{1}{2\pi}\int exp(-i\tau2\pi f)r(\tau)d\tau$ (1) where $r(\tau)$ is the correlation coefficient. If we do the Fourier ...
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Given a plot of both the magnitude $|H(\omega)|$ and its angle, How can you find the $H(\omega)$?

I'm specifically trying to use an inverse Fourier Transform to find $h(t)$, but I'm finding it difficult to get $H(\omega)$ in the first place. I'm under the impression from my textbook that $H(\...
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FFT of resultant of signals

If one has two signals (say, two acceleromters mounted perpendicularly) and a piece-wise resultant acceleration signal is determined, it appears that frequency content information cannot be determined ...