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

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Problem designing a specific filter

I have the next problem. $H_{c1}(j\omega )$ is the ideal antialising filter and $H_{c2}(j\omega )$ is a real one. I'm asked to design $H(e^{j\Omega })$ so that $y[n]$ in the second diagram (the one ...
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Is it possible exctract sinusoids from non periodic signal?

Digital signal UT1-UTC is not periodic but is including many sinusoids (periodic elements in IERS nomenclature) that are not multiples of some fundamental. For example tidal sinusoids are not ...
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Why this DTFT plot from CTFT? [closed]

I have a sample test with an answer but don't understand how they got to the answer: $x(t)$ has info only between $2 < |\omega| < 4$ $X^F(\omega) = 0$ for other frequencies. All ...
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Recovering phase from spectrogram

In this answer, an iterative algorithm is given for recovering a signal from its spectrogram, assuming overlapping windows: $$ x_{n+1} = \operatorname{istft}(S \cdot \exp(i \cdot ...
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How to implement Cross Spectral Density [duplicate]

I am writing a program to compute the cross spectral density of an image, and a template image, which is the image I am trying to find in other image. Reading wiki1,wiki2,wiki3 from wikipedia, and ...
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Is this a kind of aliasing error and how to make correction?

I use a Discrete Fourier Transform to have a representation in the frequency domain: Notice that, both the real and the imaginary line are softly reaching zero on the left, from 0.5 Hz to 0 Hz. ...
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62 views

Z-transformation

Hi everybody i'm a student. Yesterday i had a test about my Engineering subject about signal processing and there was this problem: You have the sequence $x(n) = N+1 - |n|$. With $|n|\leq N.$ ...
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Matlab fast fourier transform (FFT) function and time in hours frequency : hertz or millihertz?

I have a function of type $$C(t)=0.31 \left(e^{-0.267t}-e^{-1.72t}\right)$$ Where $t$ is in hours The function represents the evolution in a window of 24 hours of a drug blood concentration, after ...
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Fourier transform of ${\Pi}_{a}(t)\cos(2{\pi}f_{0}t)$

I want to find the following: $\mathfrak{F}[{\Pi}_{a}(t)\cos(2{\pi}f_{0}t)]$ I first did: $\mathfrak{F}[{\Pi}_{a}(t)] = 2a\text{ sinc}(2af)$ then: $\mathfrak{F}[\cos(2{\pi}f_{0}t)] = ...
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Complex signal transform

I must strengthen my knowledge in the mathematical properties of signals, and would like to know the following thing: if I have a complex signal, written in magnitude and phase, it's like if I have ...
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Eigenvalues of correlation matrix which have the form of an harmonic function [duplicate]

As a continuation to this question, I took the matrix $C_{2 \times 2}$ which is: $$C=\left[ \begin{array}{} a& ace^{-\frac{|\phi_1-\phi_2|}{2}}\\ ace^{-\frac{|\phi_1-\phi_2|}{2}} ...
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Keeping track of units for spectral energy of discrete signals

The energy of a continuous signal $x(t)$ recorded between $t=0$ and $t=T$ (namely $x(t \notin [0, T]) = 0$) is defined as $$E = \int_0^T |x(t)|^2\, dt.$$ In most signal processing texts I met, the ...
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1answer
35 views

How do I calculate the eigenvalues of a covariance matrix which contains harmonic functions

I have read that $ce^{-\frac{\phi_i-\phi_j}{\rho}}$ is a harmonic function of the form $e^{-in\phi_i}$, and therefore it's eigenvalues are one of the Fourier components of $(|\phi_i-\phi_j|)$. My ...
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what is the relationship between DFS, DTFT and DFT? [duplicate]

The concepts of DFS,DTFT and DFT are pretty confusing. I went through many blogs and books but still the concepts are not clear. However my understanding is that: For an aperiodic discrete time ...
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1answer
33 views

Relation between Laplace and Fourier transforms

I know that $\ X_L(s=j\omega)=X_F(\omega)$ if $\ x(t)$ is one sided and absolutely integrable and hence, imaginary axis of the Laplace transform is the Fourier transform. But Fourier transform also ...
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49 views

Output of an LTI System With Given Frequency Response?

I will have my final exam this week. While I was studing for it I coudn't solve this problem. I think that I should work more about Fourier Transforms. I m thinking that the solution can be obtain by ...
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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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Formulas of the Fourier transform family

It has annoyed me that there doesn't seem to be a source online where the complete complex Fourier transform family is presented with every variable defined. The lack of definitions can be a nuisance ...
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42 views

Question regarding transfer functions and prerequsities for finding the real impulse response

The transfer function of a system is given by: $$\large H(s) = \huge \frac{V_{out}(s)}{V_{in}(s)}$$ In digital domain the principle is of course the same, just replace laplace transform with ...
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Given a discrete time signal, what is the sequence of possible frequencies I can get from DTFT?

I know that when I have a discrete time signal, let's say: The definition of the DTFT is given by: Now, my question is regarding Omega(n). I know the frequencies will be discret because we can't ...
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Response of a system to a step function (heaviside)

I'd like to compute the response to a step function of a electrical/thermal system. Generally I can "easily" compute the transfer function $H$: $$H(\omega) = \frac{V_{out}(\omega)}{V_{in}(\omega)}$$ ...
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Different sampling rates causing unexpected results

I am trying to calculate the Modulation Transfer Function of a radio graph. A line is drawn from region of high attenuation to low attenuation and these attenuation values are sampled at the rate of ...
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1answer
34 views

Phase spectrum: $\mathrm{arg}\mathcal{F}(\omega)$ or $\mathrm{-arg}\mathcal{F}(\omega)$?

Do we need minus or not? I need to extract phase spectrum from that thing $$f(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{i\omega x}d\omega\int_{-\infty}^{\infty}f(t)e^{-i\omega t}dt$$ I can do it ...
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How can spectrum of DTFT of sampled band-limited analog signal be in $[-\pi,\pi]$

Say the analog signal $x(t)$ and its spectrum $X_a(\Omega)$. After sampling with frequency $F_s$ or sampling period $T$ we get $$x[n] \triangleq x(nT) = x\left(\frac{n}{F_s}\right)$$ and its ...
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Relation between two k-spaces phase-frequency and spatial frequencies in

When I see MRI explained, two types of 2D k-space images seem to be described as if they were the same. Axes are the two spatial frequencies. This images is directly fourier-transformed into the ...
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36 views

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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How does Adobe After Effects generate its “audio spectrum” effect?

I'm trying to replicate the "audio spectrum" effect from Adobe After Effects. An example can be seen in this video: Obviously, it has to be some variant of a fourier transform, but I've tried ...
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$\int^{\frac12}_{-\frac12}X(f)e^{j2\pi nf}df=\frac{1}{F_s}\int^{\frac {F_s}{2}}_{-{\frac {F_s}{2}}}X(F)e^{j2\pi nF/F_s}dF$ for $f=\frac{F}{F_s}$?

I am using Digital Signal Processing Principles, Algorithms, and Applications 4th edition and by Proakis. Here is what I don't understand say the signal $x_a(t)$ has its Fourier Transform $X_a(F)$ ...
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Trouble understanding discrete Fourier Transform

In the paper Calculation of a constant Q spectral transform - J.C.Brown it is mentioned The conventional linear frequency representation given by the discrete Fourier transform gives rise to a ...
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Inverse Sliding DFT

From paper: Bradford R., Dobson R., ffitch J. - Sliding is Smoother than jumping In chapter 6 - Signal Reconstruction, the inverse of the sliding DFT can be achieved by this formula: ...
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What's a “Fourier filter”?

E.g. the constant Q-transform is built by adding so called "Fourier filters". What's a "Fourier filter"?
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Multiplication property DTFT

I was truing to solve an example of DTFT which is following multiplication property. The problem is $$ a^n \sin(\omega_0 n) u[n]$$ we know that the definition of DTFT is $$ X(j \omega) = \sum _ ...
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1answer
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Multiplier in digital signal processing?

whenever we start working on Discrete Cosine Transform(DCT), Loeffler algorithm a key role. Furthur if we approach hardware implementation of this algorithm, we find methods based on Distributed ...
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Standard deviation of the spectrum of white noise

I have a temporal signal which looks like $f(t) = t\eta(t)$, where $\eta(t)$ is a white noise with the mean $\eta_m$ and STD $\sigma$. I want to calculate the corresponding spectrum, $F(\omega)$, take ...
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Fourier transform of certain noisy function

So, I have a noisy signal in time domain, $f(t) = t \eta(t)$ where $\eta(t)$ is white noise with variance $\sigma$ and mean zero, and that it has the property $\langle \eta(t)\eta(t') \rangle = ...
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Principal Component Analysis as a step between STFT and MFCC

For computing Mel Frequency Cepstral Coefficients you can use already calculated STFTs as a basis and perform the Mel frequency mapping on it. My question: Does it make a significant difference if I ...
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What is the phase of the Fourier Transform of $h(n)={\frac13,\frac13,\frac13}$ (zero position is at 2nd element)

I am using Digital Signal Processing Principles, Algorithm and Applications, 4th edition written by Proakis and Manolakis. Here is s a part that I don't understand. $h(n)={\frac13,\frac13,\frac13}$. ...
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65 views

Calculating original signal from Discerete Fourier Transform

I am trying to calculate the original equation using a DFT. I start with a equation, generate values from this equation and then get the dft of these values. The aim is to generate the original ...
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Frequency shift property of Fourier transform

Which one of the following is actually a definition of frequency shift property $$e^{j\omega_0 t}x(t)\leftrightarrow X(j\omega - \omega_0) \tag{1}$$ $$e^{j\omega_0 t}x(t)\leftrightarrow X(j(\omega - ...
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Can you use Fourier transformations (or other) to read multiple superimposed barcodes?

If you printed bar codes on tracing paper/acetate etc. and then positioned several in front of one another, could you extract the individual codes from the aggregate overlaid image? I feel intuitively ...
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1answer
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How would I generate these figures?

I was given the original image and I need to produce Image 1. Does anybody know how I would go about this? I am on MATLAB, but I am more interested in the theory ...
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2answers
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Unable to obtain correct correct fourier transform for homework

Okay so I have this question for my homework which I am asked to find the fourier transform of a signal. However I am unable to obtain the correct answer, the question and my working is shown below. ...
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Is the following equation established?

Is the following equation established? $$\int_{-\infty}^{\infty}s(t)r^*(t)dt=\int_W S(f)R^*(f)df $$ $$s(t)\xrightarrow{\text{Fourier Transform}}S(f) $$ $$r(t)\xrightarrow{\text{Fourier ...
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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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1answer
59 views

Fourier Transform of exponential

While solving Example 4.1 of Signals and Systems by Alan Oppenheim. Example 4.1 is: $$ x(t)=e^{-at}u(t), a>0$$ and the transform I get is: $$ X(j\omega)\frac{1}{a+j\omega}, a>0$$ The problem is ...
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Multi-Time Window FFT

One can achieve better resolution results by taking FFT of different sizes of the input signal. FFT size decreases as frequency increases, i.e. longer FFT length for lower frequencies and shorter FFT ...
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Are Fractional Fourier transform applicable to non linear signals

I am aware that generally fourier analysis is applicable to linear signals from the literature papers. I wanted to know if fractional fourier transforms are also applicable to only linear signals ...
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why $-$ sign in DTFT pair for constant

In discrete time Fourier transform, The DTFT of constant 1 is $$\sum_{l=-\infty}^{+\infty} \delta(\omega-2\pi l) $$. I have confusion that why there is $-$ sign, why it can't be ...
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26 views

Numerical Fourier transform for exact frequency

Mathematically, suppose I have a function $f(t)=\sum_k c_k e^{-i \omega_kt}$, where $\omega_k$ may not fall in $[0,2\pi]$. With an analytical Fourier transform, I can get a sum of delta functions ...
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How to calculate the energy in the dirac delta function signal?

I'm having DSP for the first time, and after some classes I got confused about the following: Suppose I have a signal which its fourier transform in a frequency band $[ \omega_1,\omega_2] $ is just a ...