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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38 views

Laplace Transform: zeros and corresponding impulse response $h(t)$

Poles and the impulse response If our impulse response is in the form : $$h(t) = e^{-\sigma_0 t}\cos(\omega_0 t) \, u(t)$$ (where $u(t)$ is the unit step function) And its Laplace transform is : $$H(s)...
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How to compute autocorrelation with exponential IIR filter efficiently

I have a discrete dataset (called V). I want to compute the autocorrelation of V at multiple time horizons. Normally I know that I can use the IFFT of the PSD. But in my case I need it with ...
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3answers
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Interpreting N in DFT as the number of points vs. number of intervals

The "N" is DFT is understood to be the number of data points in a given sequence or in other words the length of the sequence. We recently have had discussions here Indexing in DFT (from an ...
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Zero-ISI Filter Bandwidth, Raw Data Bandwidth and Symbol Rate Relation

Please can you assist I am confused about the relation between Sinc and Rectangle transform pair and how that relates the Bandwidth of Pulses, Bandwidth of Zero-ISI Filter and the Symbol Rate. My ...
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1answer
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DFT: Why is the total time equal to $N\cdot T_s \quad \text{and not}\quad (N-1)\cdot T_s$? [duplicate]

In the definitions of the DFT DFT $$ X(j)=\sum_{k=0}^{N-1} x(k) \exp \left(-i 2 \pi\left(\frac{j}{N}\right) k\right) $$ Let us say, if we have $10$ points, $N=10$, each sampled at $0.2$ seconds, why ...
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Laplace transform : integral vs poles & zeros

If Laplace transform is expressed as : $$\int_{-\infty}^{+\infty} h(t)e^{-st}dt $$ with : $$s = \sigma + j\omega$$ and $h(t)$ an impulse response expressed as : $$h(t) = Ae^{-\sigma_0t}\cos(\omega_0t+\...
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1answer
30 views

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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Evaluate a complex exponential at negative infinity [duplicate]

I am learning about the properties of the Fourier Series (FS), which is defined by: $$x(t) = \sum_{k=-\infty}^{\infty}c_{x}[k]e^{j2\pi kt/T}\tag{1}$$ where $$c_{x}[k] = \frac{1}{T}\int_{T}x(t)e^{-j2\...
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36 views

N, N-Discrete fourier transform

What is $N$,N-point Discrete fourier transform? Is it different from 2D Fourier transform? and how to compute $N$, N-point Discrete Fourier Transform of a given laplacian filter kernel? say for a ...
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1answer
36 views

How can I get rid of this unexpected minus sign on my inverse Fourier transform of two impulse functions?

I'm trying to find the inverse Fourier transform of two impulse functions, which correspond to the Fourier transform of the function $h(t)=A\sin(2πf_0t)$. The Fourier transform of the above sine ...
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56 views

On accurately computing continuous Fourier transfoms using FFT

Suppose I am given a bounded function $f$ with support on an interval $[0,A]$ and I want to compute the continuous Fourier transform $$\hat{f}(\omega_k) = \int_0^A f(t) e^{-i \omega_k t}\,dt, \quad k =...
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NUFFT of non-uniformly sampled signal

I am trying to understand how to use nufft from the Matlab doc. My goal is to compute a FFT of an image (2D) which has missing points (not sampled). I have a list x and y of coordinates of the points ...
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1answer
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Why do we scale bins in FFT in this code?

Hi I am learning FFT I am confused about this bit of code: what is the reason for scaling the sampling frequency and what is bin scale and why and when do we use it? thank you ...
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Identifying sounds patterns from wav file

I have an audio recording stored as a .wav file. My goal is to identify exact sounds patters from it using Python. The problem I'm facing is there's a clear "ting" sound in the audio file, ...
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1answer
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Image Restoration by Solving Constrained Least squares in Frequency Domain (Frequency Domain Filtering)

I am trying to implement the constrained least squares filtering as described in Rafael C. Gonzalez, Richard E. Woods - Digital Image Processing 3rd Edition Section 5.9. The equation (...
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1answer
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One integral inverse CWT

MATLAB's icwt docs state inversion to be done by a single integral: $$ f(t) = 2 \Re e\left\{ \frac{1}{C_{\psi, \delta}} \int_0^\infty \left< f(t), \psi(t) \right> \frac{da}{a} \tag{1} \...
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Fourier transform diagonalizes time-invariant convolution operators

I got the following paragraph from the book "A wavelet tour of signal processing" chapter one, page 2. The Fourier transform is everywhere in physics and mathematics because it diagonalizes ...
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PSD of linearly modulated signal using autocorrelation?

Consider a signal $v(t)$ given by $$v(t)=\sum_{n=-\infty}^{\infty} b[n]p(t-nT).$$ Assume that $b[n]$ is uncorrelated with zero mean, i.e. $\mathbb{E}[b[n]b^*[m]]=\mathbb{E}[|b[n]|^2]\delta[n-m]$ and $\...
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Scipy fourier transform zero frequency spike (from DC offset) - de-meaning and hanning window have no effect

I am trying to plot the FFT of essentially a random signal that has a non-zero mean shown below. The FFT of the signal is peaked over the zero frequency which usually indicates a DC offset. Although ...
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matlab code question with fractional fourier tranform

I want to estimate the parameters of a chirp signal using the fractional fourier transform. Following the paper estimating chirp parameters, the chirp rate $\mu$ can be obtained with (minus) the ...
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Taking FFT of 10 concatenated time traces with a time difference between each trace?

My goal - I'm interested in the white noise spectral region at higher frequencies, especially the phase information. In this data acquisition instrument, digitised time data transfer is much faster ...
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1answer
46 views

Fourier transform of a signal and its autocovariance function

While I do know the difference between the two, in theory, I am not very sure about why we look at the Fourier transform of the auto-covariance function. What extra information does it give us over ...
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1answer
84 views

Optimally approximating the sign function by functions with compactly supported Fourier transform

I'm re-posting a question of mine from math.stackexchange in hopes that folks here might have the right kind of expertise. I'm looking for a systematic way to approximate the sign function $$\...
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1answer
25 views

Finding the equivalent filter H(u,v) in the frequency domain of a 3x3 spatial mask

I'm trying to find the equivalent frequency domain filter, $H(u,v)$, of a 3x3 spatial mask that averages all neighbours of a point $(x,y)$ in said 3x3 neighbourhood excluding the point itself. So far, ...
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1answer
41 views

Inverse Continuous Wavelet Transform derivation?

Wiki writes iCWT as $$ f(t) = C_{\psi}^{-1} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} W_f(a,b) \frac{1}{|a|^{1/2}} \tilde\psi \left(\frac{t - b}{a}\right) db \frac{da}{a^2}, \tag{1} $$ where $\...
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2answers
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What is unit of time average of product of two signals?

Consider the signals $e(t)$ and $h(t)$ with units $\left[\frac{V}{m}\right]$ and $\left[\frac{A}{m}\right]$ and their Fourier transforms $E(\omega)$ and $H(\omega)$ with units $\left[\frac{Vs}{m}\...
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Why DFT is used for approximating CTFT when you can approximate CTFT-integral itself?

I was using MATLAB for approximating FTs. Why DFT is used if we can approximate the transform-integration using summation.
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23 views

Impulse invariance vs. DT representation of a CT system: Where is the inconsistency?

Suppose you have a continuous-time (CT) system $h_c(t)$, bandlimited to $B$. Your goal is to represent the system as a discrete-time (DT) system $h[n]$, sampled at $f_s \leq 2 B$. Clearly $h[n]$ won't ...
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26 views

Log derivative interpretation

In the origin paper on Synchrosqueezing Wavelet Transform, the phase transform, used to extract the instantaneous frequency of a signal $f(t)$, is defined as $$ \omega (a, b) = -j[W_\psi f(a, b)]^{-1} ...
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19 views

Calculation of eigendecomposition of a signal in its Fourier domain?

I want to find the eigendecomposition of a 3-dimensional discretely sampled signal $X$, where each sample $X_{i,j,k}$ is treated as a vector $\langle i, j, k\rangle$ (with the origin at the middle of ...
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19 views

Right discrete cepstrum implementation in python

I know that the cepstrum is mainly computed as follow: $ C_{r}={\mathcal {F}}^{-1}\left\{\log({\mathcal {|{\mathcal {F}}\{f(t)\}|}})\right\} $ What I am wondering is if I should take the whole fourier ...
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2answers
78 views

Unclear time-to-frequency integration step

From here; $\hat f=\mathcal{F}(f)$, bar = complex conjugate: Time-shift property: $x(t-b) \Leftrightarrow e^{-j\omega b}{\bf X} (\omega)$, so why is it $+$ (red)? What at all is happening? Looks like ...
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2answers
95 views

Output of a stable LTI system

Let $\mathcal{L}$ be a stable LTI system. Is it true that if input has finite energy then output also has finite energy? I'm not sure about that. We know that $$\int_{-\infty}^{+\infty}|h(t)|dt\lt\...
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2answers
114 views

Autocorrelation for periodic signals

Autocorrelation for power signals is defined by $$R_x(\tau)=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^Tx(t)x^*(t-\tau)dt\tag{1}$$ Is it true that for periodic signals $(1)$ can be computed by $$R_x(\tau)=...
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FFT followed by SVD leads to topography?

I am implementing the localizer method from this paper. One of the steps is not hard to understand and implement, but I don't understand why it is applied or what is the rationale behind it. Each data ...
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63 views

Fast Fourier Inversion: Functions of a Complex Argument $f:\mathbb{C} \rightarrow \mathbb{R}$

I originally posted this question on math stack exchange, but I think it may be better suited for this community. I'm interested in functions $f: \mathbb{C} \rightarrow \mathbb{R}$ with associated ...
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Discrete representation of a signal that has unevenly spaced samples in frequency

I'd like to describe a problem that I've been struggling with for a while. I want to apologize in advance due to the long text. I just want to be as clear as possible in my first post. Consider the ...
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Windowing function for Inverse Fourier Transform

It is a common practice to apply windowing function, such as Hann or Hamming, to a time domain signal before FFT, in order to reduce spectral leakage. Often, we do 1) Windowing, 2) FFT, 3) frequency ...
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115 views

Alternative convolution theorem?

Instead of padding $x_1[n]$ and $x_2[n]$ then taking $$ \text{iDFT}(\text{DFT}(x_1[n])\cdot\text{DFT}(x_2[n])), \tag{1} $$ assuming we know $x_1(t)$ and $x_2(t)$, and their FT's, what if we do $$ \...
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2answers
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DFT of pure sinusoidal wave

I'm writing a program in which you can synthesize waves by adding to a sound's Fourier transform, and then inverse the transform to get the modified sound. In order to do this, I need to know what to ...
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How do I calculate the correct amplitudes of a discrete Fourier transform (DFT)?

I have a set of samples values in time domain. I know they are uncorrelated and I have to extract the correct amplitudes. However, the values are only ~88% of what they should be. As a test see the ...
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1answer
94 views

How is signal to noise ratio actually measured by receiver equipment?

This sounds like quite a basic question but it surprised me, how is SNR actually measured? You have the incoming signal: It seems like the SNR is just the visual comparison of the peak signal '...
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2answers
66 views

Why is Fourier space not adequate for (theoretical or digital) filters?

As far as I have seen, almost all theoretical filter design occurs in Laplace or Z-space. Also, there is a pervasive connection to real life analog filters in the design. If one is just thinking in a ...
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32 views

DFT algorithm designed for a “sample-by-sample” senario

Suppose I have a system that wants to take an input signal (audio in this case) and wants to output a Discrete Fourier Transform of it in real time (ie every sample). My initial thought is that if you ...
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36 views

Find CT Fourier transform of $ \left[ \frac{ \sin(\pi ~ t) }{\pi ~t} \right] \left[ \frac{ \sin(2\pi ~ (t-1)) }{\pi ~(t-1)} \right] $using properties

Use properties of Fourier Transform to solve the question. The question is in the imgur link below. I got $f_t$ of $\frac {sin(pi \cdot t)} {pi \cdot t}$ as rectangular pulse with value $1$ from -pi ...
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1answer
56 views

What is the DFT of $[x_1 -x_2 x_3 -x_4…x_n]$

If DFT of $[x_1 x_2 x_3... x_N]$ is $Y(k)$, what is the DFT of $[x_1 -x_2 x_3 -x_4,....x_N]$ in terms of $Y(k)$? I have tried to formulate it but I cannot get a simplified expression for DFT of ...
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33 views

finding power spectral density from a vector

I have been given a vector: \begin{equation} v= \:\begin{pmatrix}\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{pmatrix} \end{equation} my job is to find the power spectral density from this vector \...
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1answer
39 views

Bandwidth of a bandpass signal

If the Fourier transform of an aperiodic continuous time signal has signal components between the minimum frequency w1 and the maximum frequency w2, but not all the frequencies between w1 and w2, is ...
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1answer
44 views

Why is the continuous time Fourier series of DC signal an impulse?

In case of continuous time Fourier transform(CTFT), I can easily calculate the Fourier transform of DC signal by using Fourier duality or inverse CTFT. But I don't know how to calculate the continuous ...
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3answers
123 views

Does zero-padding distort the spectrum?

It's said to "sample the DTFT", revealing what "DFT fails to see". And I fail to see how this sampling isn't distortion. The "spectrum" aims to provide the sinusoidal ...

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