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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why does the spectral envelope of human speech not change w.r.t. pitch when taking a Fourier transform?

In the context of speech recognition (recognizing individual speech sounds), the pitch of a certain person can change at different times. Excerpt from Statistical Signal Processing by Steven Kay: ...
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39 views

Energy of a sinc signal

My book give me two signals to demonstrate that the temporal translation does not alter the energy and area. It gave me $$ x(t)=\operatorname{sinc}(t) $$ and $$ s(t)=x(t-T)$$ and I found that ...
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Fourier transform properties

I have to find the Fourier transform of $$ x(t)= \frac{1}{T}e^{-\frac{t-T}{T}}u(t-T) $$ First I applied traslation property , so $$ F[x(t-T)] = X(f) e^{-i 2 \pi f T} $$ after I applied time scaling ...
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Hand implementation of Fourier transform have small peaks unseen in Python package

I've implemented the basic version of discrete Fourier Transform and I'm testing it using a pure sinusoid. However, small bumps show up in addition to the large peak. I tried Numpy.fft for this and I ...
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Finite sequence input to DTFT

i'm studying the practical utility of Fourier transforms and i have some questions. I hope to receive answers in layman terms. 1) Does the DTFT take only infinite input sequences? 2) If i apply the ...
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Autocorrelation to diagnose faults

I'm attending a very practical course on signals and i have some doubts, i hope to receive answers in layman terms. 1) My prof said i can use the autocorrelation of the output of a process to ...
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1answer
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Aliasing and DTFT of a real signal

We are analyzing a real signal with the DTFT. Since we are using a limited number of samples it's like we are transforming a finite signal. As I remember, the FT of a finite signal has an infinite ...
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37 views

Fourier transform of a noisy signal

Let's suppose i have a sensor that measures the output of a process and this cause a not negligible noise that affects my signal. My goal is to analyze the process signal in order to find faults. How ...
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Spectral Leakage in layman's terms

I'm trying to understand the concept of spectral leakage for the DFT, without going deep to the mathematical intricacies (it's for practical purposes). I've read from the book "Introduction to Digital ...
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22 views

Linearity and time-shifting of $\mathcal{F}\{0.8^n\cos(0.1πn)u[n]\}$

To preface, this is not a homework related question but purely for self-study purposes. Hi there, I try to calculate $\mathcal{F}\{0.8^n\cos(0.1πn)u[n]\}$ by using the properties of Discrete time ...
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ROC of the function in the problem 9.14 of Oppenheim's Signals and Systems textbook

I have solved the problem 9.14 in Oppenheim's Signals and Systems textbook, but my solution and the one in Slader is different. Problem is given above. And Slader solution is here. I have also ...
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1answer
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Fourier coefficients of odd and even part of a signal

I have this signal and I have to find the Fourier coefficients of the odd and even part. First I found that $$ x_p(t) = \frac{1}{2} ( x(t) + x(-t) ) $$ and I made the graphic of this part and I ...
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Fourier coefficients of 1/(1+it)

I have to find the Fourier coefficients of $$ \frac{1}{1+ t^{2}} $$ I tried with $$ \frac{1}{T}\int_{0}^{T} \frac{1}{1+it}e^{-i 2 \pi f_0 T } $$ but I should do at least two integrals by parts , so I ...
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Response of an ideal integrator to a cosine wave

It sounds like a very elementary question on system theory but I got quite confused about it, so hopefully you guys can enlighten me. I'm considering an ideal analog integrator, i.e., a system with ...
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Fourier transform of $\sum_{n=-\infty}^\infty(-1)^n\delta(t-nT_0)$

Given $x(t)$ and $h(t)=\sum_{n=-\infty}^\infty(-1)^n\delta(t-nT_0)$, I have to compute $Y(f)$, where $y(t)=x(t)h(t)$. I have thought about using that, in this case, $Y(f)=X(f)*H(f)$. I know that $\...
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Frequency response of each component of a system given its global response

Given the following block diagram, find the frequency responses $H1(f)$ and $H2(f)$. The frequency response of the whole system has to be $H(f)=(\alpha_0+\alpha_1e^{-j2\pi T_1f}+\alpha_2e^{-j2\pi T_2f}...
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E of a signal using Rayleigh

I have to find The energy of a signal using Rayleigh th. the signal is $$ x(t) = A e^{-At } u(t) $$ assuming A>0 Using the classic definition of E , I found that it should be $$ \frac{A}{2} $$ Using ...
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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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Frequency response of a system given its block diagram

Given the following block diagram, I am asked to find the frequency response $H(f)$ of the system. This is what I have done: The output of the first block is $j2\pi fX(f)$, and after going through $...
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Fourier Transform of an acceleration signal containing engine orders

I am trying to understand how to evaluate this equation in the context of acceleration data which contain engine orders $a^{f_{e}^{crit}}(f)=\sum_{o}^{K}A^{o,f_{e}^{crit}}\mathscr{F}(cos(2\pi \cdot ...
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Different representations of frequency space of 2D image FFT

I'm learning images processing using FFT. In my test example provided below the input pixel values are clamped 0-1 (0-255), but I do eventually want to process floating point heightfield pixel values....
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Why is DTFT of $e^{jn\omega_0}$ an impulse train?

update : After asking the question, I figured out that DTFT result is an impulse train. Now my question evolved to, how it is derived in this way? Using the DTFT formula seems not to be working, ...
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Why does the frequency window affect the inverse fourier transform oscillation frequency?

I have an oscillating pulse in the frequency domain that I would like to inverse Fourier transform. My signal looks like: Which was coded in MATLAB using the following code: ...
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How to design FFT for 2000 points?

How should I design FFT with fixed samples - always 2000, sampling frequency is also 2000, memory is external, there is no need to get sorted array. As far I know it may go like factoring 2000 into $2^...
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223 views

Sawtooth wave Fourier coefficients

I have to calculate the Fourier coefficients of this signal. I found that signal equation is $$ y = \frac {A(2t-T)}{T} $$ To find Fourier coefficients I wrote $$ x_k = \frac{2A}{T} \int_{0}^{T/2} \...
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What is the unit for power spectrum of acceleration signals in log scale

I have a signal recorded by an accelerometer in m/s^2. It is basically accelerations over a period of time. I have calculated the power spectrum of the signal over in a certain frequency band and then ...
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1answer
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DFT practice question

This is probably basic but as I am new to the field it confuses me a bit. While looking at some solutions provided to a problem in the final step following happens: $$ \frac{1}{10}\sum_{l=-\infty}^{\...
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Relationship between real and imaginary part of a real-valued and causal system

I have one question about the real part of a real-valued and causal system with the imaginary part of its Fourier transform given by $$\textrm{Im}\big\{X(e^{j\omega})\big\}=3\sin(2\omega)-2\sin(3\...
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Proof that module of FT of 2 independent signals is sum of modules

I found on these posts (PSD subtraction and PSD of a sum of two stationary real signals) what I expected : that, just like the variance of the sum of 2 independent signals is the sum of the variances ...
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Fourier coefficients of |A cos (x)|

I have to find Fourier coefficients of $x(t) = |A \cos (2 \pi f t )|$. The problem also give me $y(t) = |x(t)|$. To find Fourier coefficients I wrote $$ \frac{1}{T_0} \int_{0}^{T_0} |A \cos (2 \pi f ...
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The expectation in power spectral density

I'm a bit confused with the definition of the power spectral density (PSD). From Wiki https://en.wikipedia.org/wiki/Spectral_density , I found the definition is: $$ S_{xx}(\omega) = \lim_{T\...
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Interpreting the amplitude of signals in fourier transform

I totally understand the concept of fourier transform, but one thing thats bothering me is the amplitude that we plot in the frequency domain. What does that amplitude of each frequency signifies? Is ...
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inverse discrete FFT in python, multiple times?

I was wondering what really happens when taking the inverse discrete FFT on some set of numbers, for 3 times? Because looking at it, it looks like we're getting an output that is identically with the ...
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Fourier Transform of the full Morlet wavelet

In 2014 someone asked here the Fourier transform of the Morlet wavelet; link below: Fourier Transform of Morlet wavelet Function? However, it was the approximated Morlet wavelet not written with the ...
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Intuition behind image derivative using Fourier Transform for edges detection

This equation can be shown mathematically: $\frac{\partial f}{\partial x}=\frac{2\pi i}{N} \mathcal F^{-1}\left(u\cdot \mathcal F(f(x,y)\right)$ I am struggling to understand the intuition behind it ...
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What is the right way to downsample using Fourier method?

I want to know what is the right way to downsample a sampled signal using Fourier transform as the implementation in scipy.signal.resample confuses me. Reading ...
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How to determine sampling frequency for an x(t) signal avoiding aliasing?

The antitransform of the function is given: $ \hat{x}(f) = \frac{123 + i246\pi f}{246 - 24600 \pi^2 f^2 + i4920\pi f} $ I'm asked to determine which frequency can I sample using the function x(t) ...
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Minimum statistics noise estimate - how to calculate the underestimation factor?

I have implemented a basic noise estimator using the minimum statistics method. Noise power estimate is obtained as a minimum of the short time power estimate within a window of subband power samples. ...
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1answer
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Fourier antitransform using scaling property?

I'm trying to calculate the antitransform of: $\frac{1}{2\cdot(1+5w)^2}$ Now I know the antitransform of $\frac{1}{(1+5w)^2} = t \cdot e^{-5t} u(t) $ But in this case I got that divided by 2. I ...
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Invertibility of Time-Dependent Fourier Transform

I am reading Oppenheim & Schafer's (O&S) Discrete Time Signal Processing (2nd or 3rd edition, does not matter) and I find hard to understand a technicality behind the Time-Dependent Fourier ...
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Getting the right frequency (using FFT)

I am implementing the method from this paper: https://dspace.mit.edu/bitstream/handle/1721.1/66243/Picard_Noncontact%20Automated.pdf?sequence=1&isAllowed=y The main idea is cardiac pulse ...
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Fourier transform is an isomorphism…but we don’t get when each frequency appears?

Statistician here who wants to get some DSP knowledge for time series analysis. I’ve known for years that if we hit a function with a Fourier transform, we have an inverse Fourier transform that will ...
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Do $|s(t)|$ and $|S(f)|$ uniquely determine $s(t)$?

Consider a signal $s(t)$. My question is if you know $|s(t)|$ and $|\mathcal{FT}[s(t)](f)| = |S(f)|$ or equivalently $|s(t)|^2$ and $|S(f)|^2$ is it possible to determine $s(t)$? That is, is $s(t)$ ...
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Interpret periods from fourier transfrom graph of periodic impulse signals

I have obtained the fourier transform of a signal with following graph My question (though might be too junior for the professionals in signal processing field) is: how can we extract the period of <...
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How to know basics about convergence

I apologize if the post is incorrect. I'm a sophomore student studying Electrical engineering. As a part of an introductory course on signal and linear systems, I'm required to learn Fourier and ...
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Getting different spectrum from velocity, and position data using Omega arithmetic

I am solving a very long problem and one part of it requires me solving an ODE and computing FFT of the resultant data. Essentially I have a differential equation$\frac{dz}{dt}$ for velocity, from ...
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Limitations in Backing Out a Transfer Function

Suppose you have an LTI system for which the (complex) frequency response $H(j\omega)$ has been measured in some frequency window $[\omega_1,\omega_2]$. Now imagine that you want to provide an input $...
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FFT giving a huge magnitude of first frequency and pretty much zero after that

So I have a (visually) very noisy time series signal and I have applied the fast Fourier transform using numpy's fft function. I am wondering why I am seeing the ...
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Formula for PSD across an axis of a 2D output

Consider a 2D stationary input $e(x,y)$ and a 2D real convolution function $h(x,y)$. Let $S=h*e$ be the result of the convolution of $e$ by $h$. If needed, we may assume $e$ is isotropic (spectrum ...
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Phase difference measurement of a signal sampled with two different sampling frequencies

I am working on phase interferometry for locating a transmitter. The direction of arrival of an incident wave can be estimated from the phase difference caused by the antenna separation as shown ...