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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questions
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What is the significance of the linear phase in the phase response of an M-point moving average filter?
I have plotted the magnitude and the phase responses of an M-point moving average filter, the following are the plots when M = 10:
We can observe that corresponding to every lobe in the magnitude ...
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1answer
40 views
Circular wrapping of an asymmetric function (in DFT calculations)
Convoluting a signal (using discrete FT) for a given interval [a, b] with a Gaussian can be done by circular wrapping as shown in Numerical Recipes. I found a shortcut for circular wrapping so that ...
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1answer
50 views
The magnitude spectrum of a sharpening filter
I'm trying to derive an expression for the magnitude spectrum of the following sharpening filter.
$$
g(m,n) = \delta(m,n)+\lambda (\delta(m,n) - h(m,n))
$$
where $\lambda$ is some positive constant ...
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3answers
3k views
Deriving the integration property of the Fourier Transform
I want to derive the property of the Fourier Transform that states that if $X(j\omega) = \mathcal{F} (x(t))$ then $$\mathcal{F} \left( \int_{-\infty}^{t} x(\tau) \mathrm{d} \tau \right) = \frac{1}{j\...
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3answers
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What is the correct solution for Fourier transform of unit step signal?
The unit step signal defined as
$$
u[n]= \lbrace 1; n>=0; \\
\qquad0; n<0 \rbrace
$$
has three possible solutions for its Fourier domain representation depending on the type of ...
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1answer
45 views
Processes/Transforms involved to get brainwave data from raw EEG? (Autocorrelation confusion)
Not clear on what the autocorrelation function of raw EEG means physically
why can't you take the FT of a the EEG itself and get frequency data?
With BCI & basic electrode setups you can ...
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1answer
25 views
RMS in frequency domain using Plancherel theorem
I have an acceleration measurement in x, y and z direction, i.e. three vectors each of length 3000.
$x = [x_{1}, x_{2}, x_{3},...,x_{3000}]$
$y = [y_{1}, y_{2}, y_{3},...,y_{3000}]$
$z = [z_{1}, z_{2},...
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3answers
50 views
how to compute the signal passing from the low pass filter?
I am currently trying to solve this question.
Let $x[n]=\cos(\frac{\pi}{2}n)$ and $h[n]=\frac{1}{5}\text{sinc}(\frac{n}{5})$.
Compute the convolution $y[n]=x[n]āh[n],$ and write the value of $y[5].$
...
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0answers
11 views
Discrete Wavelet Transform Time Series
My problem is to cluster some time series together. But due to a huge length I was interested in using some methods to reduce the dimensionality. I was thinking of Discrete Wavelet Transform since the ...
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0answers
22 views
How do I calculate the inverse Fourier transform of this function [closed]
How do I compute this integral to find the inverse Fourier transform
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1answer
64 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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4answers
69 views
What is the correct gain of an RRC Filter?
Breaking my brain all morning with this reading previous questions and googling ...
I have made an RRC filter from the equation on wikipedia. It works fine and I compared it to commpy library in ...
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0answers
17 views
Deconvolution of sidelobes in a point spread function?
It seems that most deconvolution algorithms mainly handle the main lobe of a point spread function (PSF) and assume that sidelobes can be safely neglected.
For a direct algorithm trying to perform a ...
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3answers
36 views
Summation limits in DFT
Assume a discrete time signal $(x_n)$ is given. Some texts define the DFT as
$$ X[k] = \sum_{n=-N}^N x_n\exp\left(\frac{-2\pi j k n}{N}\right) $$
while others define it as
$$X[k] = \sum_{n=0}^{N-1} ...
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1answer
8k views
Understanding the meaning of amplitude in FFT
I am recording data with a magnetometer of the background magnetic field in a building. I have applied the FFT algorithm to the data in order to look for the frequencies that appear in it. I would ...
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3answers
588 views
Phase information of a signal when creating a spectrogram
Sorry for the uninformed question but I need some help understanding this. I'm trying to understand what phase information is and it would be very helpful if someone could correct my understanding of ...
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1answer
33 views
Does it matter if the scaling term is in the DFT versus the inverse DFT?
From Wikipedia, the equation for the 1D DFT is
From a separate source, the equation for the 2D DFT is
Notice how the 2D DFT definition features a scaling term while the first definition does not. Is ...
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1answer
70 views
Phase rotation without Fourier Transform
Is there any known math equation or other method to perform phase rotation on signal without decomposing it to frequency domain?
In frequency domain it is obvious. You just need to perform phase shift ...
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1answer
63 views
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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1answer
61 views
Bluestein's algorithm to evaluate the DFT from $f_o$ to $f_o + k\Delta_F$
Briefly, the convolution between $x(nT) e^{-j2\pi f_o nT} e^{-j \pi \Delta_F Tn^2}$ and $c(nT) = e^{j \pi \Delta_F T n^2}$ multiplied $e^{j \pi \Delta_F T k^2}$ allows me to find the DFT $X(f_k = f_o +...
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1answer
36 views
Derivation of Inverse Fourier transform from forward Fourier transform
Consider the Fourier pairs:
$$\psi(x,t) \stackrel{\mathrm{FT}}{\longleftrightarrow} \Psi(k,t)$$
$$\text{If } \quad \quad\Psi(k,t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \psi(x,t) e^{-ikx} \, dx \...
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1answer
20 views
DTFT Pairs confusion
When I am in the DT Fourier Domain, and I want to come back to the time domain, which pair do I use? Asking because both pairs have the exact same "form" in the Fourier domain, and that is ...
3
votes
1answer
175 views
Non Equispaced / Non Uniform DFT Bandwidth
I need to construct Fourier transform of non-equispaced data.
That is, I have signal $s(t)$, $t\in[0,T]$ sampled at non-equispaced points $t_k$, $k=0...N-1$ with sample values $s_k = s(t_k)$. For ...
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1answer
36 views
How to accurately compute the Winger-Ville Distribution of an exponential
I am using MATLAB for this question so hopefully you can help me out.
I am trying to compute the Wigner-Ville distribution (WVD) of a sinusoidal signal defined as
\begin{equation}
x(t) = e^{-i\omega_0 ...
4
votes
2answers
119 views
Why Is the Total Time Equal to $ N \cdot {T}_{s} $ and Not $ \left( N - 1 \right) \cdot {T}_{s} $ In the Context of DFT?
In the definitions of the DFT
DFT
$$
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 ...
1
vote
1answer
65 views
Fourier transform of time division
I know that Fourier transform of $t^n f(t)= i^n \frac{d}{d\omega^n} F(\omega)$.
But does this work when $n<0$?
Is there any direct relation to compute the Fourier transform of $\frac{f(t)}{t}$?
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1answer
34 views
multiplication of a function with a Fourier-transformed equals to Fourier-transformed with a function
I already showed b item using the fact that it is $h\left(0\right)=\int \:f\left(t\right)g\left(0-t\right)dt$
I struggle a lot of hours trying to find the trick in item C.
Can anyone help please ?
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24 views
Find corresponding wavenumber from FFT - Python
I have a set of data taken from a high speed camera. I've done some image processing which results in getting a pixel location at each frame. This oscillates with time and so I have performed an FFT ...
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2answers
325 views
Why does Hilbert filter distorts the shape of the signal?
If all the harmonics composing a the signal are shifted by the same amount, this would be the same as sampling later or earlier in time. I think.
Take a simple pulse train as an example.
If Fourier ...
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2answers
606 views
What does it mean for a function to have frequencies?
In a lecture, my professor mentioned that $\cos$ has two frequencies. I see that using the inverse Euler's formula we can express $\cos$ as a some linear combination of complex exponentials, each with ...
2
votes
1answer
507 views
Covariance between real and imaginary parts of Fourier transform of a stationary time series
Since Fourier transform of a random stationary process in time (in the case of existence) is not necessarily real, my question is what is the relation between the covariance of real and imaginary ...
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1answer
33 views
How do I find the Energy Density Function of $g(t)$ if i am not given an input or impulse response?
$$g(t)=\frac{12a}{t^2+a^2}$$
I need to find the Energy Density Function of the signal, but everywhere I look has an input and an impulse response. Does anyone know how to solve this. Would I just take ...
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vote
2answers
203 views
Difficulty with a Fourier Transform
What would be the best way to take the Fourier transform of
$$
f(t)\cdot \cos\big(\pi(t-1)\big)
$$
I'm aware that when you take the Fourier Transform of $\cos(kt)$ you get two impulse at the location ...
3
votes
1answer
95 views
The Fourier transform of sinusoids' products with possible other components
I know that in general it transforms the signal from the time to the frequency field but these specific cases seem pretty demanding. Do I calculate each part separately and then just leave them with ...
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1answer
2k views
Zero Padding of FFT
There are many question related to the zero padding a time domain signal to get more frequency bins after performing Fourier transform. As I understand this process is equivalent to trigonometric ...
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3answers
749 views
FFT of input length 1536
Does anyone knows can i find a FFT of 1536 length input. Its a specification given in 3gpp Lte and we need a transform of 1536 input size which is neither a power of any number i would say. I just ...
3
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1answer
2k views
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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2answers
5k views
Difference between CTFT and DTFT?
I have tried to read different articles but still confused in difference between continuous time Fourier transform and discrete time Fourier transform?
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1answer
35 views
Fourier transform as the integral of a parameter multiplied by an homogeneous wave
Can a Fourier transform in space be interpreted as the integral of a parameter multiplied by an homogeneous wave $\sigma$?
where $\sigma$ is:
$\sigma$=$e^{-ikx}$
Are there papers or book that ...
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0answers
23 views
Unit impulse fourier transform confusion
I just saw this question.
What is the correct solution for Fourier transform of unit step signal?
But i cant understand how the 3d form is equal to the 1st one since de 1st one is purely imaginary and ...
0
votes
1answer
49 views
How to find inverse Fourier transform of summ of delta functions?
I am practicing for my exam that I have this semester and I stumbled upon this one.
How can i find inverse Fourier transform given:
$$
X(j\omega) = \sum_{k=-\infty}^{\infty}\delta(\omega-2k+1)
$$
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How do I perform 2D Fourier domain multiplication if the filter mask doesn't match the image size?
Let's say I have an image that is 512 x 512 pixels. I've been tasked with creating two ideal half-band low-pass filters that will filter the image. The first filter is 8 x 8, and the second one is 16 ...
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1answer
117 views
Anyone explain to me this video?
I was watching a
video
in time 24:48
I would like to know where you got the value
.9 (1.14z + .941) and 1.0232 + .757
Does anyone explain how he got those numbers?
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2answers
73 views
How do you reduce $H\left(e^{j\frac{\pi}{2}}\right)$ further according to a textbook solution
I want to know how I could get from the first line to the second. I've been trying to figure it out for a while with no luck. Thank you in advance!
\begin{align}
H\left(e^{j0.5\pi}\right) &= \frac{...
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1answer
47 views
Finding set of orthogonal basis functions for composite signals
I've been given a list of 5 composite signals, where each is composed of 10 sinusoids of different frequencies. For instance, the first composite signal $S_1$ is given by
$$
S_1 = \sum_{i=1}^{10} A_i \...
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0answers
17 views
What is the requirement to reconstruct a spatial domain signal if we sample in the frequency domain?
I've come across an interesting question with regarding to signal reconstruction.
The sampling theorem states that a signal in the time or spatial domain must be sampled at twice a rate twice the ...
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0answers
27 views
How to find Integration of sinc function and then it's infinite summation?
I was doing this problem,i got answers of (a) and (b) by going following way,
first i computed F.T. of x(t) as follows which in turn will be used to estimate it's bandwidth which will come as 2Hz and ...
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0answers
52 views
Continuous vs. Discrete Fourier Transform for representing a physical phenomenon
This question is related to the differences between discrete and continuous Fourier Transform equations. Although I've found some good explanations about the difference in this forum, I cannot see how ...
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0answers
24 views
Why do we not encode frequency in 3 dimensions in MRI?
If I understand correctly, to select a particular slice (of an object, during an MRI scan) that is orthogonal to the direction of the main magnetic field applied, we apply a magnetic field that varies ...
3
votes
2answers
537 views
What does the frequency axis of a Power Spectral Density mean?
I have never really understood what the frequency axis meant when we plot the Power Spectral Density(PSD).
Does it correspond to frequency as we get after we take the Fourier Transform of a time ...
