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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971 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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206 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
685 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
289 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
61 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
882 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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3answers
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?
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850 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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259 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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257 views

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
539 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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138 views

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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739 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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930 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
202 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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237 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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234 views

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
117 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
216 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
310 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
223 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
90 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
317 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
247 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 ...
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2answers
436 views

Combining spectrograms with different windows to getting arbitrary time and frequency resolution

Spectrograms convert an amplitude-time signal into a frequency-time signal using Short Time Fourier Transform (STFT). The window size L used in STFT is a design choice. As L increases, you get higher ...
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1answer
2k views

How to do simple extrapolation with Fourier transformation?

I have 1024 sample points, and I would like to do really simple extrapolation using Fourier transformation. First I apply Fast fourier transformation on the data. My first intuition was that I just ...
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80 views

Difference in Interpretation: $ω$ (rads/s) vs. $ω$ (rads) and $X(ω)$ vs. $X(e^{jω})$

The fourier transform of a continuous time signal $x(t)$ is $X(ω)$ where the unit of $ω$ is radians/second. And for a discrete signal $x(n)$, the DTFT is $X(e^{jω})$ where the unit of $ω$ is radians. ...
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1answer
247 views

Parseval's Theorem for discrete series

I need to use Parseval's Theorem to find \begin{equation} S\:=\:\sum _{n=-\infty }^{\infty }\:\left[\left(\frac{\sin\left(\frac{\pi }{4}n\right)}{2\pi n}\right)\left(\frac{\sin\left(\frac{\pi \:}{6}n\...
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3answers
83 views

How can a changing signal be Fourier transformed?

Fourier transformation is used to split a periodic signal into frequencies, and calculate the phase and shift for each frequency. But what if we would like to record an audio, which is changing, for ...
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1answer
476 views

Significance of an impulse in the frequency domain

I know that $X(f)$ gives the amplitude associated with the frequency component $f$ of a signal $x(t)$. Now, a sinusoidal signal in time $x(t) = A \cos (2 \pi f_0 t)$, has a Fourier transform $X(f) = \...
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203 views

Phantom harmonics when using cosine windows why do they appear and how to avoid them?

Given an L order cosine window, it is possible to show that the width of the main lobe is given by: $$\omega_w = \frac{2 \pi L}{(2N+1)}$$ Where $L$ is the order of the window, $N$ is the maximum ...
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1answer
55 views

Computing real signal with minimum absolute values from even magnitude spectrum

I want to derive a real audio signal from an arbitrary even magnitude spectrum. The phase spectrum affects the values of the signal in the time domain; for example, a phase of 0 for all frequencies ...
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3answers
1k views

Periodicity of a constant signal!

This can be a very silly question, but I'm quite confused: If we take the Fourier transform of any constant signal, we get an impulse at zero, which says that its frequency is zero and, hence, it is ...
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2answers
647 views

Signal reconstructed using IFFT: amplitude scale larger than input

I am trying to test reconstruction of signal from IFFT. Below is the python program. I break the signal into overlapping windows and do FFT after multiplication with window function. Then I apply IFFT ...
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2answers
478 views

Separation of overlapping frequencies

I have a signal with multiple frequencies, and two of them, one of which is my main frequency, overlap. Are there any techniques that could separate two frequencies that almost overlap? I can ...
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1answer
547 views

Discrepancy between Gaussian FFT and its Fourier Transform

I am trying to do the FFT of a Gaussian signal and comparing it to the theoretical Fourier transform. For infinitely small time step $dt$ and infinitely long signal length $T$, the 2 should become ...
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1answer
48 views

How to find out amplitude of harmonic after diff type of FT?

I am new in DSP, so I apologies if it's dump questions but I didn't find needed information on the Internet. So the questions is: Assume that we have simple harmonic signal described by next function ...
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1answer
184 views

Finding audio signal time delay: cross-correlation vs. impulse response?

I need to find a time delay between the two audio signals, one comming from one set of speakers, and the other being the source signal that is initialy sent to speakers. I send some noise through ...
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2answers
108 views

Does MatLab's `ftt` function gives a Fourier Transformed signal in terms of frequency $f$ or angular frequency $\omega$?

When we use MatLab's fft function to Fourier transform a time signal, do we obtain a Fourier space signal in terms of frequency $f$, or angular frequency $\omega$? ...
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1answer
82 views

Spectrum analysis of a function

I asked this on Math SE but received no replies. I hope this is a relevant forum to ask in. I would like to analyze the spectrum of the following function: $$f(t)=\cos(t\cdot a(1+b\cos(ct)))$$ with $...
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112 views

Fourier Transformed a signal $f(t) = \cos(2\pi*4t)*5\cos(2\pi*0.4t)$ but did not get expected results. What went wrong?

I created a signal with two sinusoidal components by specifying $$ f(t) = \cos(2\pi*4t)*5\cos(2\pi*0.4t) $$ I expected with I used MatLab's fft function on this that signal that I would see a peak ...
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0answers
211 views

Inverse Fourier Transform of the real part of fourier transform, and inverse transform of the imaginary part of fourier transform [closed]

How can I calculate the inverse fourier transform of the real part of the fourier transformation?
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1answer
897 views

Show That the Power Spectrum Density Matrix Is Positive Semi Definite (PSD) Matrix

Given a Wide Sense Stationary Multi Variate (Vector) Random Process $ \boldsymbol{x} \left[ n \right] $ it Auto Covariance Matrix Function is given by: $$ {R}_{x, x} \left[ m \right] = \mathbb{E} \...
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What will be the filtered output?

I tried to solve this question from basic Here is my work Image 1 Image 2 But the correct answer is Option $(B)$.What is the mistake i am doing?
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254 views

What is the physical meaning of Fourier Transform having only $\delta(\omega)$ functions?

While calculating the Fourier Transform of the signal $$x(t) = 1 + \cos(6 \pi t + \pi/8)$$ I found that its Fourier Transform is purely in terms of Dirac Delta $\delta(\omega)$, that is its transform ...
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2answers
442 views

A system that perfoms Fourier Transform operation - is it an LTI system?

If a system takes input as the time domain signal and outputs the frequency domain signal, is such a system an LTI system? For if the input time domain signal can be represented as a linear ...
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1answer
976 views

Prove the dirac delta contains all frequencies

I'm looking for a mathematical proof that the dirac delta contains all frequencies. I just read in a text book that the frequency spectrum of a dirac is just a horizontal line of amplitude 1, whereas ...
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1answer
555 views

Find the nyquist sampling rate?

I have a signal $x(t)$ for which I want to find the Nyquist frequency : $$ x(t) = \frac{\sin{\pi t/2}}{\pi t/2} \ast \sum^\infty_{n=-\infty}\delta(t-10n)$$ I am trying to solve this in the time ...