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 of a triangle function [duplicate]
Good afternoon,
I am having a question regarding the following function :
Let $g(x)$ be :
$$\begin{cases}\pi + \frac{\pi}{2}x \text{ if } -2 <x < 0\\
\pi - \frac{\pi}{2}x \text{ if } 0 <x &...
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Understanding symmetry in DFT magnitude plot
I am trying to intuitively understand the mystery of why the DFT of a complex vector with real values produces apparent symmetry in the magnitude plot (see the second plot here for example). In DFT $X[...
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Scaling plots after FFT
I have multiplexed 32 signals into 1 signal with python.
Now I want to plot that signal in time domain and to plot it's amplitude spectrum.
Professor gave us his plots so we can look up to it.
These ...
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Reconstructing the original signal from its DFT
Hi I am a newbie to signal processing and I am trying to better understand how inverse DFT works under the hood. Consider this signal and its DFT:
(Source)
For the sake of this post, let's assume ...
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Why is the CT inverse Fourier Transform also an integral?
Intuitively it seems that to get a function back that was integrated, you would take the derivative. Instead, with the Fourier Transform we take an area under a curve of a modified function, and to ...
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Spectrum and Fourier transform of a sine wave
Take for example: $x(t)=A\sin(2\pi f_0t)$. The Fourier transform of this signal is $\hat{s(f)}=\frac{A}{2i}(\delta(f-f_0)-\delta(f+f_0))$. If we want to represent the spectrum of the signal we would ...
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Why is the time domain low-pass filter the "sinc" shape?
Consider:
I'm looking at low-pass filters, and I see that the time domain representation of an "ideal" filter resembles the shape above whereas the frequency domain is a box. I also get the ...
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Fourier transform magnitude of the sum of two signals
Let
$$\mathscr{F}\Big\{x_1(t)+x_2(t)\Big\}=X_1(f)+X_2(f)$$
I think that in general
$$\big|X_1(f)+X_2(f)\big|^2\leq\big|X_1(f)\big|^2+\big|X_2(f)\big|^2$$
but I was wondering if
$$X_1(f)X_2(f)=0,\qquad\...
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Frequency content of a noisy signal
To find the frequency content of a noisy signal (PSD), there are two methods below:
#1 Take the fourier transform of its power signal (square the noisy signal)
#2 Find the autocorrelation function of ...
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Finding Discrete Fourier Transform (DFT) for different DFT size
$N$ is an even integer, $x[n]$ is a finite length signal over the interval $n \in [0,N-1]$, and $X[k]$ is the $N$-point DFT of $x[n]$. Analytically find the DFT of sequence below in terms of $X[k]$. ...
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Relationship between fourier transform and fourier series
Let
$$x(t) = A\sin(2 \pi f_0 t + \alpha)$$
its Fourier transform is given by $$ X(\omega) = \frac{A \pi}{i}(e^{ia}\delta(\omega-2\pi f_0) - e^{-ia}\delta(w+2\pi f_0)). $$
the Fourier series complex ...
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Is it useful to think of a Fourier Transform as writing out a signal in terms of a basis?
The (modified) trigonometric functions $\{0, \cos(kx), \sin(kx)\}$ serves as a basis for periodic function. I have also seen (but not rigorously) that the Fourier transform can also be seen as an ...
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Alternative way to find fourier transform
Let $$x(t) = A \text{rect}_T({t-\tau})$$I calculated its fourier transform through the direct way: $$X(\omega) = Ae^{-i\omega \tau} \int_{-T/2}^{T/2} e^{-i \omega t} dt = Ae^{-i\omega \tau} \frac{\sin(...
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Is `fft` always the best choice?
I hope this is the right place to ask this question since it is partially a note which might help others. Until recently, I always used the fft-algorithm ...
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Calculating k-space coverage from antenna positions
I work with radar and I want to understand which spatial frequencies I can measure, i.e. the k-space coverage, given a set of coordinates of transmitter-receiver combinations. The ideal coverage for a ...
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How to identify square wave in audio signal
I'm interested in identifying a square wave signal from recorded audio. This subject may be a complex problem so I would like to present the problem where I'm currently stuck at.
I recorded the audio ...
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Amplitude and Phase spectrum from fourier transform of sawtooth [duplicate]
Let the following $T$-periodic signal :
then $$\begin{align}F(x(t))(\omega) =& \int_{-\infty}^\infty x(t) \exp(-i\omega t) \mathrm{d}t = \sum_{k=-\infty}^\infty \int_{kT}^{(k+1)T} x(t) \exp(-i \...
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Fourier Transform and Music Analysis
I am a senior in high-school and am currently trying to conduct an exploration on Fourier Analysis, specifically using the Discrete Fourier Transform to analyze a chord played on my piano. Basically, ...
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CFO Estimation in LoRa Chirp Signal (Preamble part)
I am trying to estimate CFO in LoRa chirp signal (preamble part). I have seen the discussion about CFO on this forum but it is mainly related to CFO estimation in OFDM.
I want to estimate the CFO in a ...
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Why is the DC component of discrete fourier transform not the same as the signal's arithmetic mean?
In this question we have a mathematical proof that the DC component of normalized discrete Fourier transform should be the same as the signal's arithmetic mean. However, in the following example I ...
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Why does multiplying a real signal by a random complex phase term result in "spreading" in the Fourier domain?
Suppose I have some real-valued signal $x\mapsto f(x)$. The amplitude of its Fourier transform $\mathcal{F}[f]$ then looks like a peak around the DC-term, decaying as we move towards higher ...
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Fourier transform of $|x_\mathrm{a}(t)|^2$
Let $x_\mathrm{a}(t)$ be the analytic signal for real signal $x(t)$. I want to find an expression for $\mathscr{F}\{|x_\mathrm{a}(t)|^2\}(f)$ in terms of $x(t)$. The analytic signal can be written as $...
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Why does applying Fourier Transform on point Spread Function yield h(t) which is complex-valued
I wanted to understand why this text talks about applying the Fourier transform on H(f) to obtain h(t). I view Fourier transform as moving from the time or spatial domain to the frequency / spatial ...
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PSD of the sum of two zero-mean white noise signals
I am trying to solve the following exercise, where $y(t)$ is the sum of two signals $x_1(t)$ and $x_2(t)$ with each of them being the product of the convolution of $e_i(t)$ with $h_i(t)$.
So far I ...
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What is the phase of $Y(\omega)$ in relation to the phase of $X(\omega)$ where $x(t)$ modulated with an exponential carrier that has no phase shift?
A homework problem of a free online course I am taking, asked to draw the magnitude and phase of $Y(\omega)$ where $y(t) = x(t) c(t)$ where $c(t)$ is $e^{j 3 \omega_{c} t}$ and where $X(\omega)$ is:
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How can I plot a sinc function correctly?
I am generating a rectangular pulse using a piecewise function on Matlab. I have listened to some advice to use a normalization coefficient and the amplitude appears correct now. However, my issue is ...
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Noise color in multidimensional frequency space - is there an angular effect or dependence?
I understand that if there is a non-constant level of noise in one-dimensional frequency space, e.g. noise variance increasing linearly with frequency gives "blue noise" in the iFFT space. ...
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Conjugate symmetric: 3D fourier transform dirmension
I have a real value input 3D tensor with the shape of `(H,W,D)=[8,8,20]', where H, W, and D represent height, width and depth in (z dimension), respectively. When converting to the DFT, what will be ...
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On FFT, interpolating signal vs extending signal in time
When we interpolate, then FFT the output will have more bins.
When we extend the signal in time, Then FFT output will have more bins too but:
Interpolation increases max bin frequency but time ...
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Impulse response aquired by ifft seems to require circshift
On the system below:
I have found W by inputing 20-1000 frequency and found amplitude and phase for each frequency. I've tested for minimum phase and none min S. But the answer is not logical for me. ...
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What is the effect of multiplication on the value of discontinuities?
I see that the values of a distribution at discontinuities are essentialy equal to the mean of the right-hand and left-hand limit values when dealing with distributions in relation to the Fourier ...
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What is the Fourier Transform of $\operatorname{sgn}(t) \cdot \operatorname{sgn}(t)$?
I am wondering what the Fourier Transform of $\operatorname{sgn}(t) \cdot \operatorname{sgn}(t)$ will be, where $\operatorname{sgn}(t)$ indicates the signum function. It would seem obvious that this ...
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Why a windowed signal is considered to evaluate the Power Spectral Density?
Let's consider a real function of time $x(t)$ being a Fourier-transformable signal. The synthesis equation states that:
$$x(t)=∫_{-∞}^{+∞}X(f)⋅e^{2πjft} df $$
Meaning: $x(t)$ can be seen as the sum of ...
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In IDFT, can x[n] be reproduced if DFT is ranging from -1pi to 1pi?
In short, I can reproduce the original x[n] from DFT via IDFT. However this happens only when I take samples DFT from 0 to 2pi. When DFT is ranging from -1pi to 1pi, the reproduced x[n] is not correct....
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Quantifying stitching effects
I have many images like this with high stitching, medium stitching, low stitching, and no stitching effects (the adjectives high, medium, low and no are purely subjective). How to quantify the amount ...
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How to determine the nature (real or complex valued) of a signal?
A signal X(t) is a real valued time domain signal and Y(t) is a signal that only contains the non-negative spectral components of X(t). How do I determine whether Y(t) is real-valued or complex?
I ...
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How Best to Characterise a Window Function
How do I characterise my window function?
Do please forgive me here as I am more a practical than theoretical person. I have invented a window function which I use prior to discrete Fourier transforms....
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Inverse fft does not give back the same image
I tried to Fourier decompose my image using FFT and reconstruct it back using IFFT. While I did this, I noticed something peculiar:
The first image is the regular one, and the second is the image I ...
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Why is the phase of the FFT not exactly 0 degrees for a cosine and 90 degrees for sine wave?
Let's say I have two signals. The first is a cosine wave and the second is a sine wave. Each oscillates at 0.01 Hz. The sample rate is 1 Hz and the length of time series is 1000 seconds. Each has an ...
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Does an analytic function composed with another stay analytic?
Assume $s(t)$ is analytic, such that it has no negative frequency components.
Will $s(f(z))$ also be analytic, assuming $2≥f'(z)≥1$?
Concretely, I work with audio data that is mapped through a time-...
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do digital equalizers use Fourier transforms?
I’ve been reading up on approaches to pitch shifting, and have found some common approaches use fourier transforms to achieve a change of pitch.
I’m curious, is fourier transform technology used in ...
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Extracting the phase using FFT without detrending
I'm trying to use FFT to extract the phase of the following signal:
This is just a function in the form of: m*t + 5*sin(2*π*f*t - π/4)
Is it possible to get the <...
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Clarification on video about frequency deviation in phase modulation bandwidth
I am watching this video and at 7:12 he calculates the frequency deviation $\Delta f = f_{c}f_{m}A_{m} $ where the $c$ underscript is for "carrier" and the $m$ underscript is for "...
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Combining two time series with different cadences (different Nyquist frequencies)...?
If it helps, I want to do perform the below computation in python.
Two time-series - 30-min cadence (Nyquist of 24 cycles / day) lasting 27 days, followed by 10-min cadence (Nyquist of 72 cycles / day)...
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How to compute the direction/bearing of a vector time series in the frequency domain (i.e. as a function of frequency)?
I have a time-domain magnetic field, $\mathbf{b}(t)$, with components $b_x(t)$ and $b_y(t)$ where $x$ and $y$ denote orthogonal coordinate directions (i.e north and east, respectively). At each time, ...
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Understanding fourier transform normalization for continuous vs discrete
I intuitively understand what the DFT is doing and what the equation means. We are essentially adding up all the points of the input after projection onto the unit circle and then taking an average (...
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Why a differentiator is unstable from pole zeros view point?
A differentiator with frequency response $j2 \pi f$ is unstable because as frequency increases its response becomes out of bound.
But from a pole zero point of view a differentiator just have zeros ...
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Three dimensional Wiener-Khinchin formula for incompressible isotropic random field
When I am reading Uriel Frisch's book "Turbulence", on page 55 he claimed that the Wiener-Khinchin formula for an incompressible isotropic random field, such as the velocity field of ...
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Understanding FIR windows
We learnt about the various windowing techniques recently and I can't seem to wrap my head around why one would use anything other than a rectangular window.
I created a signal with 10 evenly spaced ...
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How does downsampling decrease the noise power?
I read one equation from a technical document, i.e., assuming the time-domain white Gaussian noise samples with variance $\sigma_{TD}^{2}$ in dB are sampled using ADC with sampling rate $f_{s}^{ADC}$, ...
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