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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1answer
56 views

Is there a name for the procedure of taking the FT over separate consecutive small time-blocks?

Suppose we have a continuous time-interval $I=[a,b]$, and a signal $x \colon I \to \mathbb{R}$. A procedure that is sometimes carried out (e.g. when doing bispectral analysis) is to partition $I$ ...
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fourier transform of smart-phone accelerometer in matlab

I'm new to matlab. I want to process my smart-phone accelerometer data in matlab. I know Matlab let's you connect your phone via USB cable to see accelerometer data in realtime. But according to some ...
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1answer
56 views

How to model a generic low frequency signal?

I'm trying to apply Fourier analysis to a specific problem I have. I have essentially an integral like the following $$ \int_{\Omega} f(t) g(t) dt $$ And I'm trying to assume that $g$ is a narrow ...
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130 views

Inverse Discrete-Time Fourier Transform of $X(Ω)=jΩ$

I am trying to solve it by using the properties but I can’t seem to find the same solution as on my book.
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108 views

Use samples of Fourier transform as DFT?

Consider the LTI system given by: $H(z) = 1 - \frac{1}{2}z^{-1}+\frac{3}{4}z^{-2}$ Let $x[n] = (\frac{1}{2})^nu[n]$ be the input to the system. We want to find the output for $n = 0,1,...,N_a$, using ...
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370 views

Why averaging in Spectral magnitude domain not in complex domain to estimate spectrum of a process

Consider we need the magnitude spectrum of the signal. Signal is recorded in $N$ trials to a certain stimuli. Signal varies from each trial because of noise. So, to estimate the actual magnitude ...
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2answers
116 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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566 views

Why doesn't this complex multiplication in the frequency domain produce my expected phase shift?

I know how to change the phase of a complex number by multiplying by $\cos \theta + i \sin \theta$. And I understand that the phase of a sine wave is reflected in its Fourier transform. So, I am ...
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2answers
2k views

How do i select my frequency range to plot when fft is made on time domain

I have a vibration signal that i need to convert from time domain to frequency domain using fft in python. ...
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1answer
76 views

How to differentiate a time domain signal in the complex transfer function?

I have an input-output data set where the input is current and the output velocity. I am interested in the transfer function from current to acceleration though. So suppose: $H(s) = \frac{I(s)}{V(s)}$ ...
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1answer
589 views

How to understand FFT results of scipy.fftpack?

I calculated FFT for a speech wav-file using scipy.fftpack. How do I read (understand) the return of FFT? I have read that it supposed to be like so: y[0] is 0Hz loudness, y[1] is 1Hz loundess, ... y[...
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54 views

IDTFT of $\sum_{k=-\infty}^{+\infty}(u(\Omega+\pi)+u(\Omega+\frac{\pi}{4})-u(\Omega-\frac{\pi}{4})-u(\Omega-\pi))\star \delta(\Omega-2k\pi)$

Compute the IDTFT of the following signal: $$X(\Omega)=\sum_{k=-\infty}^{+\infty}\left(u(\Omega+\pi)+u\left(\Omega+\frac{\pi}{4}\right)-u\left(\Omega-\frac{\pi}{4}\right)-u(\Omega-\pi)\right)\...
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1answer
156 views

Inverse DTFT of $H_1(\Omega)=\begin{cases} 10,& \frac{\pi}{3} \leq |\Omega| < \pi\\ 0,& 0 \leq |\Omega| < \frac{\pi}{3}\\ \end{cases}$

What is the inverse DTFT of the $2\pi$-periodic extension of following function: $$H_1(\Omega)=\begin{cases} 10,& \text{for } \frac{\pi}{3} \leq |\Omega| < \pi\\ 0,& \text{for } 0 \leq ...
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1answer
109 views

Find Fourier Transform of signal that satisfies this equation

I have attempted it this way: $G(j\omega)$ is a rectangular pulse in frequency domain. Using results, inverse Fourier transform of $G(\omega)$ will be $$g(t)=\frac{\sin(2t)}{\pi t}$$ Now given that $...
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1answer
837 views

DCT vs DST for image compression

I am doing a research project for DST (Discrete Sine Transform) versus DCT (Discrete Cosine Transform) image compression and for my conclusion, my supervisor told me to discuss why the differences ...
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2answers
2k views

Calculating 1/3 Octave Spectrum from FFT / DFT

I am not often on this forum and I am not an expert on the subject. I struggle with the theory of FFT / DFT and the 1/3 octave spectrum. Assume I have a DFT analysis of a given signal. It (the DFT ...
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1answer
231 views

Fourier Transform frequency translation property in practical applications

From the material of my Signal Analysis and Processing course: The frequency translation property is extremely important for practical applications. It allows taking a signal from a portion of ...
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1answer
126 views

Relation between time domain, DTFT domain and frequency domain

Problem The sampling frequency of a continuous-time signal is $S$ kHz, what does $\frac{\pi}{4}$ radians/sample in DTFT domain represent in Hz in frequency domain? Prove the relationship. Doubts I ...
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9answers
448 views

Where is the flaw in this derivation of the DTFT of the unit step sequence $u[n]$?

This question is related to this other question of mine where I ask for derivations of the discrete-time Fourier transform (DTFT) of the unit step sequence $u[n]$. During my search for derivations I ...
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3answers
113 views

Complexity of FFT derivation

I am confused regarding the complexity of the Fast Fourier Transform (FFT). The Discrete Fourier Transform is: $$\qquad X\left [ k \right ]=\sum_{n=0}^{N-1}x[n]W_{N}^{kn}\quad \text{where}\quad W_{N}...
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1answer
43 views

DTFT fourier transform (modified property)

I know there are 3 properties of DTFT that help with my problem $$ a^{n}u[n]=\frac{1}{1-ae^{-jΩ}} $$ $$ (n+1)a^{n}u[n]=\left(\frac{1}{1-ae^{-jΩ}}\right)^{2} $$ $$ \frac{(n+r-1)!}{n!(r-1)!}a^{n}u[n]=\...
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Discrete-time Fourier Transform of the unit step sequence $u[n]$

From text books we know that the DTFT of $u[n]$ is given by $$U(\omega)=\pi\delta(\omega)+\frac{1}{1-e^{-j\omega}},\qquad -\pi\le\omega <\pi\tag{1}$$ However, I haven't seen a DSP textbook that ...
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0answers
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How to derivate in the frequency domain

I have two Time Domain functions, $f_1(t)$ and $f_2(t)$. I have both Fourier Transforms, $F_1(\omega)$ and $F_2(\omega)$. Functions $f_1$ and $f_2$ are not independent and, in fact, $f_1$ is also a ...
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1answer
698 views

Fourier transform of function division in time domain

$F_1(\omega)$ is the Fourier Transform of $f_1(t)$. $F_2(\omega)$ is the Fourier TRansform of $f_2(t)$. Can I obtain the Fourier Transform ($F_3(\omega)$) of $$ f_3(t) = \frac{f_1(t)}{f_2(t)} $$ ...
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2answers
927 views

Amplitude of first harmonic of a square wave, $2/\pi$ or $4/\pi$?

I am getting really confused about the value of the first harmonic of a $50\%$ duty cycle $-1$ to $1$ square wave. By doing the math I found $\frac{2}{\pi}$, in my lesson and Wikipedia it's $\frac{4}{...
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1answer
198 views

Does the DTFT of $\frac{u[n-1]}{n}$ exist?

I have started learning DSP on my own and I have this doubt. I have done some googling but haven't found an answer. I hope that someone here would give the answer. It will be of great help.
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Reconstructing the song using its dominant frequencies

I have found out 20 dominant frequencies of a song by splitting it into 20 sound-clips and applying Fourier transform on each one of the sound-clips. Now I am trying to reconstruct the song back from ...
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1answer
664 views

2D Fourier transform of Sobel kernel

Can someone explain me the highlighted text parts regarding this image ? Here is a pseudo-code of how it was created: ...
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1answer
2k views

Time-Bandwidth Product

The following text is cited from a textbook, "Spotlight Mode Synthetic Aperture Radar: A Signal Processing Approach", I would like to ask if anyone knows the proof to the following statements, as the ...
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3answers
275 views

Transfer function of a frequency shifting system

There is a system which shifts frequencies of input by -Fc such that: Y(S) = X(S).H(S) But X(S) has value zero from 0 to Fc. I am confused on how the product of X(S) and H(S) becomes a positive ...
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1answer
59 views

Integral of the Fourier spectrum?

The integral $$\int_{-\infty}^{\infty}|X(f)|^2df$$ of the absolute Fourier spectrum squared is the energy in the signal, but what about the integral of the 'simple' absolute Fourier spectrum? $$\...
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1answer
856 views

How does MATLAB recover picture from magnitude spectrum alone?

This is the transformation I did. The code fft2() the Lena picture than ifft2() it back to the original. Add some ...
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5k views

What are the advantages of Laplace Transform vs Fourier Transform in signal theory? [duplicate]

What are the advantages of Laplace Transform vs Fourier Transform in signal theory?
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1answer
789 views

How to convert dB back to manitude and then to rectangular format [duplicate]

I am experimenting with audio (wav files) using Short-Time Fourier Transform (STFT) in Python using scipy.signal.stft. As I ...
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0answers
248 views

Hilbert Spectrum vs. STFT

If I wanted to plot a spectrogram of a signal, I would by default use the Short Time Fourier Transform. However, the Hilbert transform does something similar: it lets us calculate instantaneous ...
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2answers
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Is there any optimised algorithm to calculate 2D fourier transform

I m trying to implement my own code for finding the 2D fourier transform of an image in MATLAB using the formula for it, but it take toooo much time to come up with the answer, is there a defined fast ...
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1answer
500 views

DFT of time reversed signal

I was looking into proof and find something strange: The last part we obtain from DFT definition. $$X[k] = \sum^{N-1}_{n=0}x[n]W^{kn}_N, \quad\text{Where}\quad W^{kn}_N = e^{-j\frac{2\pi}{N}nk}$$ ...
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Problem identifying the analytic expression of such determined signal

I came across this problem I am supposed to find the Fourier transform of $g(t)$, but I am not able to find the analytical expression of such signal. The teacher suggests that I should consider ...
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2answers
712 views

Convolution in spatial domain is multiplication in frequency domain

I have to prove convolution in spatial domain = multiplication in frequency domain using two matrices. $$ x(m, n) = \begin{bmatrix} 1 && 2 \\ 3 && 4 \end{bmatrix} $$ $$ h(m, n) = \...
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2answers
898 views

Discrete time inverse fourier transform of cosine squared

$$ X(\omega) = \cos^2(\omega)$$ I tried this problem, and I ended up getting $0$, which doesn't make any sense. I integrated: $$ x(n) = \frac{1}{2\pi}\int_{0}^{2\pi} \cos^2(\omega)e^{j{\pi}n} d\...
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1answer
187 views

How does a Hermitian FFT work in Numpy?

Say, I create a Hermitian complex signal using, import numpy as np t = np.arange(-4, 4) z = np.exp(1j * t) Here z should be a ...
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2answers
225 views

Mathematical relationship between highpass and lowpass filtering

Let $g, h_{HP}, h_{LP}: \mathbb{R} \rightarrow \mathbb{R}$ and $G, H_{HP}, H_{LP}$ denote their continuous Fourier transforms under the Fourier operator $\mathcal{F}$. Let $*$ denote the continuous ...
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Windowed Fourier Transform

I need some help with applying the properties of Fourier Transform. We define the windowed Fourier Transform of $f \in L^2(R)$ as $$Sf(\mu,\xi)=\int_\mathbb{R}f(t)g(t-\mu)e^{-i\xi t}dt$$ Prove ...
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1answer
113 views

FFT of resultant of signals

If one has two signals (say, two acceleromters mounted perpendicularly) and a piece-wise resultant acceleration signal is determined, it appears that frequency content information cannot be determined ...
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1answer
204 views

Finite moving average filter

I am trying to solve this problem but I need a lot of help. Below are my answers for the separate parts, please check and tell me where I am wrong because I am weak on the fundamental concepts of this....
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2answers
306 views

Relation between samplingrate and frequency

I am working on Fourier Transformation, and applying this for recognizing an audioclip. I have a 9 second long audio clip of a guitar strumming an A-Minor. The audioclip has a sampling rate of 44100 ...
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1answer
205 views

How to calculate multiplication of two discrete series

Short version: How to multiply two discrete sequences? Long version: Convolution of two discrete sequences is weighted sum. For instance, convolution of two sequences: ...
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1answer
85 views

Plotting a sampled signals DTFT using its CTFT

So I know the connection between the DTFT and the CTFT is the following: Where the left-hand side is the discrete time fourier transform. I need to choose a sampling rate which won't cause any ...
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1answer
304 views

Confused on the difference between the frequency spectrum of an entire song, and the frequency spectrum of a point in time

I am very much a beginner in this field - but find it really interesting. However I am a little confused on a certain area of knowledge. If I have understood correctly: At any point in an audiosignal,...
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1answer
232 views

Why do wavelets integrate to 0 and how do they match a signal?

I have been reading about the Wavelet transform recently and its relationship to the Fourier transform. From what I understand the wavelet transform represents signal data with many short-lived ...