Signal Processing

Questions tagged [fourier]

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Approximation of Periodic Parabolic Function by Fourier Series!

I've just tried to approximate the periodic-parabolic signal by Fourier Series. I know, this sounds a bit strange. I am just trying to figure out relationship between Fourier Series and Taylor ...
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44 views

Prove Discrete Time Fourier Series Multiplication property

Note: This is not a homework problem. I'm just stalled at a point because I think I might be interpreting the duality property incorrectly. If $x_1[n]$ and $x_2[n]$ are periodic with period N, then if ...
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23 views

Gabor uncertainty and time-frequency resolution

I have a question about Gabor's uncertainty theorem, and how it relates to time and frequency resolutions. As I understand it, Gabor's uncertainty theorem states that the standard deviations of a ...
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1answer
70 views

How is this given impulse response of infinite duration? Isn't it just from -π to +π?

How is hd(n) infinite duration when it is from -π to +π. Book says as it is infinite duration, we in the next step take-: h[n]=hd[n] from n=-(N-1)/2 to (N-1)/2 and 0 otherwise. I can't see how this ...
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3answers
68 views

Convolve with a box filter in time domain

To low pass filter a signal, we have to convolve it with a sinc function because it's the same as multiplying the Fourier transform of the signal with a rect function, resulting in the high frequency ...
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56 views

Can FFT tells us existance of same frequencies with different phases?

So I know that applying FFT on a time-domain signal, shows which frequency components exists and what amplitudes each frequency signal has. My question is, suppose the signal contains same frequency ...
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24 views

Inverse discrete Fourier transform or inverse Fourier transform of composite function?

I collected spectrometric data which produced a graph with the intensity of each frequency of light. What more do I need to perform an inverse fourier transform of this data? Should I attempt an ...
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82 views

Why do we use the DFT instead of the DTFS? Or, why was the FFT algorithm built for the DFT instead of the DTFS?

As we know, both the DTFS (discrete-time Fourier series) and the DFT (discrete Fourier transform) are used to represent discrete-time periodic signals for all time (or the periodic extension of ...
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119 views

Some signals whose Fourier transform are a particular rotation of its self

Let $N\in \mathbb{N}$. I am looking for a non-zero scalar $\lambda$ and a nonzero vector $$f=(f(0),f(1),\cdots,f(N-1)) \in \mathbb{C}^N$$ satisfying the following equations for $l=0,\cdots,N-1$: $$\...
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1answer
99 views

Fourier Transform: $\omega$ vs $f$ as frequency variable

I try to understand how the Fourier transform changes when I try to compute $X(\omega)$ or $X(f)$. Can someone work me through the maths please for two examples, $x(t) = \exp(j\omega_0 t)$ and $x(t) = ...
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190 views

Where to start with DSP?

I have a DSP exam coming up this summer and I got an abysmal mark on an assignment earlier this year. Obviously, the lecture slides are not enough for me to understand DSP, so I am wondering where can ...
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64 views

How to compute correct phase in FFT even after applying phase unwrap and zeroing round off error?

I am converting a time-domain synthetic 1D signal to frequency domain using MATLAB fft. In the frequency domain, the amplitude vs frequency plot is coming reasonable which means it is showing the ...
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72 views

Calculating cross-correlation using Walsh-Hadamard transform

I am trying to implement MLS method of measuring impulse responses. There is an article describing the method: http://www.commsp.ee.ic.ac.uk/~mrt102/projects/mls.... As I understand, to get an impulse ...
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1answer
22 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 ...
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212 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 ...
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18 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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560 views

Fourier series of cycloid

What is the Fourier series representation of a cycloid? The parametric representation of the curve is as follows. $$ t=\dfrac{\theta-\sin\theta}{\pi}\\ x=\dfrac{1-\cos\theta}{\pi} $$ The period is $2$...
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54 views

MIT 6.003 HW#8 Problem 4 - Fourier Coefficients of Triangle Wave

In the mentioned homework, part of the solution involves finding the Fourier coefficients of the triangle wave. The solution mentions that we can express this function as follows: What does that ...
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49 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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How to compose a Discrete Prolate Spheroid (DPSS) dictionary?

I have a model of signal as $$ Y=AX + N $$ where $Y$ is received data in a linear array, $A$ is steering matrix, $X$ is data of sources and $N$ is noise. If $A$ has the form of $A=\exp(\alpha \sin(\...
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1answer
38 views

Is it common to impose the sparsity on the Fourier coefficient itself?

In compressive sensing, I see many works to impose the sparsity on the wavelet coefficients (e.g., by minimizing the L1 norm of such coefficients.) Another example in MRI is to impose sparsity on the ...
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235 views

Normalization factor in the convolution theorem

Maybe it's a trivial question, but I couldn't find any explanation for this. According to the convolution theorem, in the continues case we add normalization factor, i.e. $$ \mathcal F\left\{h\star g\...
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1answer
69 views

what does frequencies in non periodic signals mean?

What do the frequencies in the a Fourier transform of a non-periodic signal mean physically? Are there another definition of frequency that doesn't include the FT?
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92 views

DFT Signal DFT Length N , FFT

If we sample a signal let say sine(2 * pi * f) with f=1Hz and a sampling Frequency of Fs = 8Hz, is it right that the length of the data should be N = Fs/f or multiple of Fs/f like N= d*(Fs/f) with d=...
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1answer
77 views

If a time-series has odd number of samples does it have no energy at Nyquist frequency?

Suppose I have real time series A with n samples and time-spacing dt and I want to analyze ...
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36 views

Something like Hilbert Transform to obtain arbitrary Phase Shift? [duplicate]

I was wondering if there is something like a Hilbert Transform but that can implement an arbitrary phase shift to every frequency component. I mean, I know that the magnitude response of a "...
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36 views

even symmetry of magnitude and odd symmetry of phase [closed]

I'll appreciate it if any of you guys could help me with this question: Suppose that x(t) is a real signal, prove that the magnitude of its Fourier transform has even symmetry and its phase has odd ...
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50 views

Are there standards/references for determining spatial scale in photographic images?

EDIT, 12/12/20: Images below are of the radial sinusoid pattern. Left side is the "unrotated" version. Right side is what the a photograph of the pattern would look like if the disc was ...
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86 views

IDTFT of convolution in the frequency domain

I have tried everything. If you actually know how to solve this could you provide a hint? $$ e^{-2j\Omega}\frac{ \sin\left( \frac{7\Omega}{2}\right)}{ \sin\left( \frac{\Omega}{2} \right)}\star \frac{\...
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61 views

Fourier transform and energy of a convolution

Hi guys i have to find the fourier transform of the convolution: $$ sinc(t/2T)*\sum\limits_{n-\infty}^{+\infty} (-1)^{n}\delta(t - nT) $$ i was thinking of express the summatory as : $$\sum\limits_{n-\...
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1answer
33 views

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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38 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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112 views

Can different Discrete-Time-Fourier-Series(DTFS) coefficients have the same discrete sequence in the time domain?

Please, check the following discrete periodic sequence when the period $N=2$. $x[k]=\exp(j\frac{2\pi}{N}k), N=\text{period}$ $..., x[0]= 1, x[1]= -1, x[2]= 1, x[3]= -1, ... , N=2$ According to my ...
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1answer
79 views

What is the variance of DFT of Fourier coefficient of difference of a vector of white noise?

Consider $\big\{x[0], x[1], \ldots, x[N-1]\big\}$. Suppose, \begin{cases} x[n] \sim \mathcal N\left(0, \sigma^2\right)\\ \big\langle x[n], x[n-1]\big\rangle = \frac12 & \forall \ n\\ \big\langle x[...
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1answer
98 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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202 views

Why does DFT have only $N$ components?

Why does the DFT have only $N$ components in it? I can see that after N components the frequency component is periodic and repeats with the same values but that does not seem to explain why we can ...
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1answer
234 views

frequency domain to time domain with magnitude and phase

I have a signal magnitude and phase in frequency domain. I need to have it in time domain but I really have no idea how to do it. I heard something about mirroring the signal but I'm kinda new at this....
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146 views

How to plot thermal noise in the “time domain”?

$P = 4kT$ (where $k$ = Boltzmann’s constant, and $T$ = temperature of the instrument ($K$)) And the mean voltage is thus, of course, $V^2/R = 4kT$ And the voltage is distributed as a Gaussian around ...
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47 views

2D Fourier Transform [closed]

Is it possible to get a 2D Fourier Transform by first taking 1D Fourier transform in first dimension and then performing another 1D Fourier Transform in the other dimension? If yes, can you explain
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100 views

Prove a property using shift theorem and duality

I'm reading Lectures on the Fourier Transform and Its Applications and I'm going to prove shift theorem for the inverse Fourier transform using duality. According to the mentioned source, the duality ...
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21 views

Fundamental frequency in DFT [duplicate]

I was looking into the fundamental frequency in DFT and I noticed many sources mention that the fundamental frequency is $1/N$, where $N$ is the number of samples. (When doing the DFT, we have $k/N$, ...
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2answers
376 views

Amplitude after Fourier transform

How to obtain the correct amplitude after the numerical Fourier transform of a signal? Example: consider an exponential decaying wave $y(x)=e^{-x}\sin(100\pi x)$ with Fourier transform $y_f(x_f)$ ...
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1answer
137 views

Time scale and Fourier transform

Consider the Fourier transform $F(\omega)$ of the function $f(t)$. The magnitude of $F(\omega)$ depends on $\omega$ and thus also depends on the scale of the $t$-axis. For example, when $f_1(t)$ is a ...
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1answer
67 views

What's the relation between frequency band of $X(j\omega)$ and $\Phi_{xx}(j\omega)$?

in which: $x_{c}(t)$ is a continuous-time signal $X(j\Omega)$ is the Fourier Transform of $x_{c}(t)$ $\Phi_{xx}(j\Omega)$ is the Power Spectrum Density of $x_{c}(t)$ which defined as Fourier ...
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1answer
129 views

Frequency response and sampling theorem for triangular function

The triangular function is defined as follows: $h_l(x) = \begin{cases}1-|x|,&|x|<1;\\0&\text{otherwise}.\end{cases}$ According to ccrma.stanford.edu: "If the output of the interpolator ...
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38 views

How to downsample a fourier transformed signal?

I have a signal of length 100000 timestamps sampled at a frequency of 25kHz. First I apply a high pass filtering at (300Hz) and then do the Fast Fourier Transformation. Then the absolute values are ...
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2answers
108 views

Why the Nyquist frequency is 0.5 of Fs, why not 0.55 or 0.65?, brief explanation [duplicate]

This my elaboration of the aliasing issue: a continuous signal can be represented by factors of : $e^{(i2{\pi}ft)}$ if we sample this signal then I will get: $e^{(i2{\pi}fk/N)}$ where $k=0,1,2.., N-1$ ...
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1answer
75 views

Receiver function, frequency domain deconvolution not giving logic results

I'm working on some code for receiver function method in seismology. For anyone one not into the topic, it's just a deconvolution of two time series (seismograms). This can be done in the time domain ...
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2answers
101 views

Question on N point DTFT - Fourier transform

I have been trying to use the logic that both X and Y should have same Z transform, but according to the definition, Y is not anti causal.
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1answer
428 views

Conjugate symmetry of the DFT of real-valued sequences

I have read about Fourier transformation that real signals are "mirrored" in the real and negative halves of the Fourier transform because of the nature of the Fourier transform. For ...

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