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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Impulse response of a transfer function

Lets say we have a transfer function from an LTI-system that goes as follows, and we want the impulse response for it: $$\frac{10\cdot e^{\left(-\mathrm{i}\right)\cdot \pi\cdot f\cdot 8}}{5+\mathrm{i}\...
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Is a Fourier transform a sound way to analyse a transient signal?

I am currently working on a project that involves analysing transient signals from sensors. While not actually part of the analysis itself, I discussed it with the team, and they are using an fft to ...
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Convolution of a non-symmetrical window function by a cosine signal in the frequency domain

A have a time signal: The associated DFT spectrum of this signal: The time signal can be considered as a non-symmetrical rectangular window function multiplied by a cosine signal with a frequency $...
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turn circular convolution into linear convolution by zero padding: A special case

We know that, multiplying a kernel and signal spectrum in Fourier domain will lead to a circular convolution and not a linear convolution, so in order to it become linear convolution we must zero pad ...
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Fourier transform of discrete time unit step function

To obtain fourier transform of u[n], u[n] - u[n-1] = delta[n] , taking fourier transform of both sides of the equation results in : ...
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solution on the time domain becomes “periodic” after the inverse fourier transform

I was trying to solve european option pricing problem using Conv method (introduced by Lord in 2008 https://pdfs.semanticscholar.org/0632/460bd50b2151f74ac40028df4cc60e73a884.pdf). The final step of ...
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Understanding Fourier Transforms in abstract math terms

I am having a hard time implementing a method that computes Fourier transform coefficients for the complex form using the trapezoid rule. I have floated questions in the math and stackoverflow ...
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1answer
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Analytically determine a PSD from a transient function

This question is related to a series of questions I have asked about the units of PSD and ESDs. I include it as a separate question as it may have worth in isolation. As I understand it to compute ...
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Bin sizes for non-uniform discrete Fourier transforms

For a non-uniform discrete Fourier transforms, do the specified frequencies – i.e., $f_k$ in – refer to the midpoint of the bin or the lower bound? I read the answer here, but that stated that ...
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Transfer function and deconvolution

Forewords This question is about methodology references and numerical application. I am posting on Signal Processing because I think this question belong to this place. I am new to the stack, feel ...
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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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Why the unilateral Laplace transform?

Why is the Laplace transform commonly taught as the unilateral Laplace transform? I mean, for the Fourier transform, we commonly have the bilateral transform... if the signal is 0 for $t<0$, then ...
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DTFT based Frequency Sampling

H($e^{jw}$)= 1, |w| < $\pi/2$ and 0, $\pi/2$ <= |w| <= $\pi$ I took M equally spaced frequencies from 0 to $2\pi$. If we assume h[n] to be causal, $H(e^{jw})$ should have some phase and it'...
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Why does an anti-symmetric function has zero amplitude at the center of an even length window

I am performing FFT on a real odd function and the resultant transform has zero amplitude in the last bin. Essentially if Y= rfft(X), then Y[-1] is always zero. I stumbled on this answer which says ...
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Real Fast Fourier Transformation (FFT) in 2D changes with changing the axes order

I am using Python to calculate the real FFT for a 2D array. I found that the real FFT function does not return an array with the same size as the input array, rather, it returns an array with the same ...
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Multitaper F statistics

I'm having some problems interpreting the F-statistics output from multitaper analysis. To illustrate, the following code-snip in R performs multitaper analysis on the same sine-frequency but with ...
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Scaling factor for comparing spectrums obtained via Continuous Fourier Transform and Discrete Fourier Transformation?

Essentially I am trying to calculate the Bremsstrahlung spectrum numerically for magnetized plasma and want to compare the resultant spectra with the standard textbook spectrum formula for ...
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How to visualize high-pass filtered images?

When I try visualizing a high-pass filtered image, all I see is gray, similar to the middle subfigure in the attached figure from a related paper. The authors claim they normalize the image to have ...
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Confusion in deriving formula for fourier tansform of impulse train

I was trying to derive fourier transform for impulse train : I know how to solve for this using using properties of fourier transform. But now I wanted to use a brute force approach to it so I did ...
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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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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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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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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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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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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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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 [closed]

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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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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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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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 $\delta[n]$?