Questions tagged [fourier-transform]
The Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies, known as a frequency spectrum.
2,158
questions
3
votes
1
answer
58
views
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 ...
- 51
1
vote
0
answers
40
views
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 ...
- 11
12
votes
3
answers
2k
views
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 ...
- 123
1
vote
1
answer
77
views
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
$$X_1(f)X_2(f)=0,\qquad\...
- 31
1
vote
1
answer
51
views
Frequency content of a noisy signal
To find the frequency content of a noisy signal (PSD), there are two methods below:
#1 Take the fourier transform of its power signal (square the noisy signal)
#2 Find the autocorrelation function of ...
- 21
2
votes
1
answer
80
views
Finding Discrete Fourier Transform (DFT) for different DFT size
$N$ is an even integer, $x[n]$ is a finite length signal over the interval $n \in [0,N-1]$, and $X[k]$ is the $N$-point DFT of $x[n]$. Analytically find the DFT of sequence below in terms of $X[k]$. ...
- 21
1
vote
1
answer
58
views
Relationship between fourier transform and fourier series
Let
$$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)). $$
the Fourier series complex ...
- 123
1
vote
1
answer
61
views
Is it useful to think of a Fourier Transform as writing out a signal in terms of a basis?
The (modified) trigonometric functions $\{0, \cos(kx), \sin(kx)\}$ serves as a basis for periodic function. I have also seen (but not rigorously) that the Fourier transform can also be seen as an ...
- 113
0
votes
2
answers
74
views
Alternative way to find fourier transform
Let $$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(...
- 123
1
vote
0
answers
48
views
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 ...
- 111
1
vote
0
answers
27
views
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 ...
- 33
0
votes
0
answers
39
views
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 ...
-1
votes
0
answers
51
views
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 \...
- 123
0
votes
0
answers
39
views
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, ...
1
vote
0
answers
10
views
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 ...
- 11
0
votes
1
answer
45
views
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 ...
- 111
3
votes
1
answer
81
views
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 ...
- 31
2
votes
1
answer
156
views
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 $...
- 726
0
votes
1
answer
28
views
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 ...
- 93
1
vote
1
answer
56
views
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 ...
- 25
2
votes
1
answer
59
views
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:
...
- 51
1
vote
1
answer
143
views
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 ...
- 13
1
vote
0
answers
25
views
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. ...
- 111
1
vote
1
answer
37
views
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 ...
- 161
1
vote
2
answers
87
views
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 ...
- 363
0
votes
0
answers
19
views
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. ...
- 363
1
vote
1
answer
47
views
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 ...
- 95
2
votes
1
answer
167
views
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 ...
- 95
2
votes
1
answer
46
views
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 ...
- 314
2
votes
0
answers
38
views
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....
- 21
1
vote
1
answer
29
views
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 ...
- 123
0
votes
1
answer
92
views
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 ...
3
votes
1
answer
180
views
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....
- 101
1
vote
2
answers
258
views
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 ...
- 113
4
votes
1
answer
779
views
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 ...
- 296
1
vote
0
answers
80
views
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-...
2
votes
2
answers
1k
views
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 ...
- 143
1
vote
1
answer
86
views
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 <...
- 113
0
votes
0
answers
15
views
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 "...
- 143
0
votes
0
answers
38
views
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)...
- 101
0
votes
1
answer
30
views
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, ...
- 296
1
vote
2
answers
90
views
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 (...
1
vote
3
answers
86
views
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 ...
- 181
1
vote
1
answer
107
views
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 ...
- 63
1
vote
1
answer
63
views
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 ...
- 113
1
vote
1
answer
143
views
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}$, ...
- 113
1
vote
1
answer
52
views
Periodicity problem with FFT for generating "starburst" from aperture
I am following a paper on computationally modeling optical phenomena within a lens. (https://resources.mpi-inf.mpg.de/lensflareRendering/)
At some point they use the Fourier power spectrum of an ...
- 13
0
votes
0
answers
39
views
Discrete time Fourier transform of an exponential decaying sigal [duplicate]
I have a fundamental question about the discrete-time Fourier transform. I used two methods but got two results.
Background knowledge
The discrete time signal is given by:
$$x[ n ] = x[ {n{T_s}} ] = x(...
- 1
2
votes
2
answers
126
views
Fourier transform of a delayed LFM chirp
I know that for a given signal $s(t)$ and a given delay $ \tau $, by the shift theorem:
$$ \mathcal{F}\{s(t-\tau)(f) \} = e^{-j 2 \pi f \tau} \mathcal{F}\{s(t)(f) \} \tag{1} $$
However, when I try to ...
- 21
0
votes
0
answers
52
views
Derivation of 9 point Laplacian filter
I'm reading a paper on how construct isotropic laplacian filter, and perhaps because it's an old paper, the notation in it really bothers me a lot. So can someone please explain it to me? For example, ...
- 1