The Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies, known as a frequency spectrum.

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How do optical anti-aliasing filters work from a frequency domain perspective

To prevent aliasing, caused by the finite number of pixels on a sensor, a blurring filter is commonly used. How does that work from a frequency domain perspective? What is the transfer function of ...
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23 views

Fourier transform of moving image sensor

Consider an image sensor that is moving with some velocity $\mathbf{v}(t) \in \mathbb{R^3}$, and an orientation $\mathbf{R}(t) \in SO(3)$. Treat those vectors as fixed. At a specific time $t$ the ...
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28 views

Why can FFT only operate on images with specific properties?

Can FFT only operate on Grayscale images? If Yes, why? Can FFT only operate on images with dimensions of power of two? If Yes, why?
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24 views

Finding the deterministic autocorrelation function (ACF) from its power spectrum

The power spectrum of a stationary discrete-time random signal is $$\Phi_{xx}(e^{j\omega})=\begin{cases} 1 & |\omega|<\pi/2 \\ 0 & \pi/2 <|\omega| \le\pi \end{cases} $$ (a) ...
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39 views

How can the given two equations of linear canonical transform equate?

I am new to linear canonical transform and its uses in signal processing, my doubt arises from two different equations that i got from two different sources for linear canonical transform one is from ...
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44 views

Upsampling and windowing

Let's assume that $k$ represent a tone (i.e. frequency) index and $n$ represents time index. After upsampling the signal $V$ times,i.e., adding $V-1$ zeros between each two samples I get a following ...
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What's the meaning of the continuity in spectrum analysis? [closed]

We all know that for any, suitable, kind of signal $f(t)$, there corresponds a Fourier transform function $F(j\omega)$ such that $$ f(t) = \frac{1}{2\pi} \int_{-\infty}^{+\infty} F(j\omega) e^{i\...
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48 views

What is difference between terms $X(j \omega) ,X(\ e^{j \omega })$ and $ X(\omega)$?

While studying frequency transforms ,I get confused with the terms like $X(j \omega) ,X(\ e^{j \omega })$ and $ X(\omega)$ ,where $ \omega = 2 \pi f $. So what is the difference between them ?
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59 views

Very simple question regarding $X(f)$ vs. $X(j\omega)$

Let's take a time domain function $x(t) = \cos( 2 \pi f_0 t) $. Its Fourier transform can be represented as $$X(f) = \frac{1}{2} \left[ \delta(f - f_0) + \delta(f + f_0) \right]\tag{1}$$ as well ...
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Filtering and Differentiating phase-modulated signals

For a project work, I need to demodulate data from a Laser Doppler Vibrometer. The distance information is phase-modulated, therefore the velocity information is frequency-modulated: $$x \propto \...
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246 views

Is my tranform the essence of DFT?

I'm someone just learning DSP, and want understand its essence. My transform is the simplest possible. Input signal is just one frequency: $256\textrm{ Hz}$. Sampling frequency is $2560\textrm{ ...
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60 views

How can I implement convolution in frequency domain?

Suppose, we have a bitmap image represented as a 2D integer array, int [,] image2D; whose FFT is Complex[,] fftImage2D; ...
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42 views

midpoint smoothing spline

Suppose we have a time series which have peaks and troughs.(The red curve below) I would like to get an algorithm which is able to identify the peaks and troughs locations, then find the midpoint ...
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3answers
176 views

Can use of Fourier transform be minimized completely with the help of Laplace and Z transform?

Fourier transform has different types like continuous Fourier transform (CFT), Discrete time Fourier transform (DTFT) and Discrete Fourier transform ( DFT). CFT can be used in case of continuous ...
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45 views

Repeated Fourier transform - what happens? [duplicate]

I have a Fourier transformable complex function that is a function of independent real variable a. Now I take the Fourier transform of it, giving me a complex function of real variable b. Now I ...
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26 views

Autocorrelation function and energy spectral density

Find the energy autocorrelation function $\phi^e_{ff}(\tau)$ and the energy spectral density $\Phi^e_{ff}(\nu)$ of the signal $f(t) = e^{-\gamma |t|},$ where $\gamma>0$ is a real constant. ...
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27 views

FFT in different coordinate systems

Is there a transformation that will enable one to calculate the FFT in an arbitrary coordinate system? What I am interested is the following two cases: The space is Euclidean and infinitely ...
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1answer
39 views

How do I interpret the result of a Fourier Transform?

For example, I entered the following "equation" into Wolfram|Alpha: FourierTransform[Piecewise[{{sin[t],t > 0 and t < 2*pi}}, 0], t, \[Omega]] So as to ...
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55 views

FFT over a fixed and equal numbers

I've got a vector of $100 000$ numbers. All numbers are equal ($7000$ for example). If I perform FFT over this vector, what will I get? From my understanding, I should receive a fixed DC line. Is ...
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59 views

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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75 views

Does “keying on” a sine wave at a zero-crossing reduce its bandwidth?

I understand that a pure sine wave of infinite duration occupies no bandwidth, i.e. it is only the modulation of a carrier that gives it sidebands. Does the exact timing of a sudden modulation make ...
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185 views

Why are Fourier analysis and transform only applicable for LTI systems?

Why are Fourier analysis and transform only applicable for LTI systems? What if the system is not LTI, won't Fourier analysis or transform be possible?
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117 views

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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30 views

Fourier transform of $ne^{-an}u[n]$

I need to find the Fourier transform of the following signal: $$ne^{-an}u[n]$$ The answers start by using the rule of the basic signal: $$a^nu[n] \rightarrow \frac{1}{1-ae^{-j\omega}} $$ and then ...
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amplitude at exact frequency in wide band signal

Could anyone suggest the most computationaly efficient method for finding amplitude of exact frequency having a noisy wide band signal. To be more specific about a task. I have some physical ...
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27 views

Transfer functions from wavelet transfrom

So I have this problem where I need to measure the phase of a signal and correct for a delay associated with the travel time of the signal while simultaneously determining the transfer function of my ...
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669 views

Why real part of FFT converts image into rotation + original?

I have read this image: taken its FFT (2D) and then Inverse FFT to get exactly the image back. Code is provided for reference: ...
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28 views

How would Fourier and Cosine Transforms responds to summation of cosines with same frequency but different phases?

For example, if I have two signals, $\cos(2\pi ft+\frac\pi4)+\cos(2\pi ft+\frac\pi3)$, what would be different in both transforms (Fourier and cosine) how would the spectrum changes? And What would ...
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29 views

How to convert a spatial frequency in a 2D-DFT into the units radians per pixel?

Let's say I have a 2D image, and I take the discrete Fourier Transform (via FFT) of that image. In the frequency domain, I get the following image: In this image, let's just assume all the spatial ...
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52 views

Computation of Only Even or Odd Frequency Bins of DFT

I have an algorithm where I am computing the FFT of a large signal. However, I desire only the even or odd terms of the DFT of the signal, but not both. Currently, I discard these undesired terms. Is ...
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40 views

Hilbert transform linearity

Explain why the Hilbert transform of $f(t)=\operatorname{sinc}(at) \cos(2 \pi \nu_c t)$ is $$\hat{f} (t) = \operatorname{sinc} (at) \sin(2 \pi \nu_c t),$$ where $0<a<\nu_c.$ Attempt: I have ...
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36 views

Hilbert transform from analytic signal

Show that the Hilbert transform of $h(t) = m(t) \cos(2 \pi \nu_c t)$ is $$\hat{h} (t) = m(t) \sin(2 \pi \nu_c t),$$ where $m(t)$ is a real valued, band-limited function (i.e. we have Fourier ...
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25 views

How is Linear Canonical Transform a generalization of Fractional Fourier Transform?

I have studied that Fourier transform changes the domain of a signal from time to frequency, and in that way it is a 90 degree shift. When it comes to Fractional Fourier Transform a generalization of ...
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72 views

Hilbert transform pair proof

I am looking for the proof that the Hilbert transform of $\displaystyle\frac{\sin(at)}{at}$ is given by $$\frac{\sin^2(at/2)}{at/2}.$$ How do we prove this? This is a $\operatorname{sinc}(at)$ ...
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34 views

What does it mean for the DFT phase to be relative to a cosine wave?

The following paragraph from Understanding Digital Signal Processing got me puzzled: The answer is: The DFT phase at the frequency $mf_s/N$ is relative to a cosine wave at that same frequency of ...
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29 views

Deriving the DFT magnitude of $A\cos(2\pi nk/N)$

Given that $$x(n) = A\cos(2\pi nk/N),$$ the $N$-point DFT of $x(n)$ can be expressed as follows—the derivation can be found in here: $$X(m) = \color{red}{\frac{A}{2}\sum_{n=1}^{N-1}e^{-j2\pi n(m-k)/N}}...
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31 views

Where does $\frac{N}{2}$ came from in approximating an N-point DFT?

I've came across the author saying that ... for a real cosine input having k cycles in the N-point input time sequence, the amplitude response of an N-point DFT bin in terms of the bin index m is ...
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48 views

Why is N-point DFT approximated by the sinc function?

While looking into DFT leakage, I've came across the author saying that "..., the amplitude response of an N-point DFT bin in terms of the bin index m is approximated by the sinc function." ...
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21 views

Should S-Matrix components for scattering response in a Polarimetric Radar be Real or Complex?

I am trying to implement some polarimetric decomposition algorithms on my MIMO polarimetric radar. At first, I will try the Pauli Decomposition algo. But instead of doing signal processing on the ...
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46 views

Why does Fourier zero-fill interpolation pad the middle frequencies?

In this: http://dspguru.com/dsp/howtos/how-to-interpolate-in-time-domain-by-zero-padding-in-frequency-domain the author says that we can interpolate a signal by zero-padding the middle frequencies in ...
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69 views

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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1answer
20 views

The signal in MRI

I am trying to understand the signal formation in MRI and have a confusion. I understand that in the presence of the external magnetic field $B0$, the protons are precessing at the frequency given by ...
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21 views

How to analyze baby speech time series

I've been collecting speech data for my baby brother (who is now 6 months old) with the intention of doing computational analysis of the development of his speech patterns. I haven't much deep ...
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66 views

How to interpret output of matched filter with complex input?

I have implemented a matched filter based on the Fourier Transform approach. In the real numbers domain that means that I use as the coefficients of my filter (B) the inverted time-samples of the ...
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36 views

Bandwidth range for Fast Fourier vs principal component analysis?

I've read somewhere that the Fast Fourier is only applicable to those processes exhibiting bandwidth. Where as principal component analysis can be applied to a process exhibiting any finite bandwidth....
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35 views

Calculate 2D Windowed Sinc Kernel

I am implementing a Windowed Sinc Filter for Scaling images by the factor of 2. Therefore I need to calculate a Filter Kernel. I already know how to calculate the kernel including a Blackman Window ...
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Fourier transform of a Fourier transform

I generate a Gaussian noise and then I filter it with a passband FIR Kaiser window filter. When I perform the Fourier transform of the output of the filter and plot its magnitude spectrum, it is ...
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29 views

3D wiggle plot for an analytic signal: Heyser corkscrew/spiral

Just reading The Analytic Impulse, A. Duncan, 1988, I met the name "Heyser corkscrew" for the first time in my DSP life, for a cisoid. This representation is quite common for analytic signals, but ...
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Why Fourier transform and Stockwell-transform retain the absolute phase information of one signal?

Hello friends am studying the topic of signal processing and the Fourier transform and the s-transform and in most books as for example "Time-Frequency Signal Analysis and Processing. 2nd" of Boashash ...
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29 views

Autocorrelation of a noisy linear map

I am interested in calculating the autocorrelation function of a linear map with some noise (model given below) but am slightly confused in doing so. At first, I did not realize there were two ...