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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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Fourier Transform Signals - Time Transformations

I was going over some review problems and came across an interesting one. Using the techniques of (linearity, time shifting, and time scaling) what are some approaches I could use to turn the ...
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751 views

Correct magnitude spectra of a cosine DFT?

I've just started my course on DSP and haven't laid my hands on MATLAB yet. I was wondering if the plot of the magnitude spectra was correct for the below shown $x(n)$:
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Why edge sharpening produces high frequency?

I have a low-resolution image in which the high frequencies are missing. When I apply an edge sharpening filter some of the missed high frequency is recovered. I am wondering why this edge sharpening ...
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190 views

Wiener Filter Additive Noise Uncorrelated

I have been trying to understand for a while now why a non-causal Wiener filter that has a frequency response of the type H(jω)=S_yx(ω)/S_xx(ω) where ...
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When Is a Power of 2 FFT Slower than Smaller Sized Non Power of 2 FFT in MATLAB

Knowing that computing an FFT is faster if the amount of samples is a power of 2 I have always tried to pad the inputs to Matlab's FFT with zeros until the next power of 2 is achieved. Matlab's ...
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3k views

Impulse response of ideal filters

I am aware that an ideal low-pass filter in both continuous time and discrete time has a $\mathrm{sinc}$ impulse response. What would the impulse response of an ideal high-pass or band-pass filter ...
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227 views

Simple $\sin(2\pi 1000t)$ Fourier transform in PSpice not behaving as expected

So I have this very basic circuit show below which I am simulating with PSpice. Now, when doing the Fourier transform of $\cos(2\omega1000t)$, I expect to see two impulses(one at -1kHz and one at ...
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477 views

BIBO Stability and the convergence of the frequency response of a system

It is my understanding that an LTI system is BIBO stable if and only if its impulse response $h(t)$ is absolutely integrable. This also happens to be one of the Dirichlet conditions for the ...
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1answer
430 views

Time scaling of discrete-time sequences and the DTFT

In the second edition of Signals and Systems by Alan Oppenheim, he discusses the DTFT of a "time-expanded" sequence that is effectively a slowed down version of the original sequence and can be ...
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1answer
595 views

Computing the center of mass of a signal using the Fourier transform

I am having difficulty with a homework problem which asks: I have some ideas in mind but I have no clue as to whether they are correct or not. Below is my attempt:
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516 views

Fourier transform of even/odd parts of a complex signal

Why does Oppenheim state the following properties: \begin{align} \mathcal F\big\{x_e (t) \big\} &= \Re\big\{ X(j\omega) \big\}\\ \mathcal F\big\{x_o (t) \big\} &= j \Im\big\{ X(j\omega) \big\...
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116 views

Uniqueness of Fourier Series Representation and the Fourier Transform of Periodic Signals

If we are given a signal of the form $$x(t) = \sum_{k = -\infty}^{+\infty} a_k e^{j k \omega_0 t},$$ can we call it a Fourier Series representation of $x(t)$ right away? Suppose we are given the ...
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776 views

Calculating SPL from pressure signal - Amplitude vs Power method

I have a pressure signal from a Fluent FFowcs-Williams Hawkings acoustics analysis. I converted this pressure signal into the frequency domain in order to get SPL values, using Matlab. I used the ...
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3answers
545 views

Periodicity of the discrete-time Fourier Transform

The DTFT of a sequence $x[n]$ can be written as $$X(e^{j\omega}) = \sum_{n = -\infty}^{\infty} x[n] e^{-j\omega n}.$$ Is the smallest (fundamental) period in frequency of the DTFT always $2\pi$? Or ...
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Result of inverse FFT is sometimes shifted in real space

I am using the Numpy fft2, ifft2, and related functions and I am sometimes running into a strange situation where the output ...
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181 views

What is the exact meaning of the output of the Discrete Fourier Transform

I'm fairly new to the subject, but so far my understanding that this would be a transform you could use to go from a discrete set of data, say [1, 0, 1, 2] to a continuous sinusoidal function in the ...
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126 views

Duality of the continuous-time Fourier transform - derivation and notation

Suppose we have the Fourier transform pair $x(t)$ and $X(\omega)$ such that $$X(\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} \mathrm{d}t$$ The duality property states that $X(t)$ and $2\pi ...
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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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114 views

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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199 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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112 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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448 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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230 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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696 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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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
84 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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619 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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56 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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184 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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128 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
955 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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3answers
3k 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
268 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
145 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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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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120 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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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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81 views

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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831 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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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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250 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
755 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
319 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
68 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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976 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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6k 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?