The fourier tag has no usage guidance.
-1
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0answers
15 views
impulse response from frequency and phase spectrum
Is there a generally accepted way of calculating an impulse response of an LTI system from its frequency and phase spectra? Of course, the obvious thing to do is an inverse Fourier transform, but ...
0
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0answers
18 views
From Fourier (k space) to wavelet domain in MRI sensing
In compressed sensing MRI (cSENSE MRI) technology the idea seems to entail sampling from the Fourier domain (k space) in a way that, when transformed to the wavelet domain ("sparsification"), the ...
0
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0answers
30 views
Cosine Fourier Transform and Phase
If we do FT of cosine wave, then sine wave will be orthogonal. So, the imag parts of FT will be 0. This is my think. But, the result isn't show that. Like this.
Figure 1. Real and Imag parts (Y label ...
1
vote
3answers
68 views
Deconvolution Using Complex Division in The Frequency Domain
Consider these two signals:
a = [1 1 0 0 0 0 0 0]
b = [1 0 1 0 0 0 0 0]
their convolution is
c = a * b = [1 1 1 1 0 0 0 0]
...
0
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0answers
20 views
Finding essential bandwidth of the given function
Let the function to be $x(t) = e^{-2t}u(t)$.
I want to find the essential bandwidth of the function to be the frequency at which $|X(jw)|$ drops to $1\%$ of its peak value. With a search, I found that ...
1
vote
2answers
201 views
Hilbert transform of unit step function
How to calculate Hilbert transform, if it exists, of the signals like $u(t)$, $sgn(t)$.
What properties should a function satisfy for existence of Hilbert transform.
Absolute integrability of a ...
1
vote
1answer
49 views
Averaging magnitude squared coherence across multiple time series
In a previous post, A. Donda had suggested that, in order to calculate the average magnitude squared coherence of more than one pair of time series (e.g. y1 and x1, and y2 and x2), one ought to follow ...
1
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2answers
88 views
Resampling with and without replacement for estimating significance of spectral components
In order to test the significance of spectral components, it seems reasonable to randomly sort the data in order to destroy all the serial correlations / spectral order e.g. 100,000 times, and then ...
0
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3answers
123 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 $x(n)$:
1
vote
1answer
504 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 ...
1
vote
2answers
72 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 ...
1
vote
1answer
74 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 ...
1
vote
2answers
83 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\...
0
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1answer
53 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 ...
2
votes
3answers
144 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 ...
0
votes
1answer
44 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 ...
0
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2answers
53 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 ...
0
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1answer
38 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.
0
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0answers
71 views
Understanding Subband Filtering?
I fully understand how a Fourier Transform and FFT work, however, something I've seen before, which confuses me is subband filtering.
I've heard of application of subband filtering with audio where ...
5
votes
2answers
84 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 $$\mathcal{F} \left( \int_{-\infty}^{t} x(\tau) \mathrm{d} \tau \right) = \frac{1}{j\...
0
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1answer
40 views
Pulse wave question
Wikipedia, fount of all knowledge (Ha! LOL), gives a formula for a pulse wave here:
The formula is:
$$f(t)=\frac{\tau}{T}+\sum_{n=1}^{\infty}\frac{2}{n\pi}\sin\left(\frac{\pi n \tau}{T}\right)\cos\...
0
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0answers
56 views
Equation for impulse train
I am looking for a formula (Fourier series) to generate an impulse train waveform - a spike-wave with amplitude and period both $1$ – so that $f(x)$ has value $1$ at $x = 1,2,3,4...$ and $f(x)$ has ...
2
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1answer
172 views
struggling to understand why Fourier basis is orthogonal
Studying DSP on my own time on Coursera.
Was given a proof to why the Fourier basis is orthogonal, but I can't figure it out. Here is how it is proof goes.
Consider the Fourier basis
$$
\left\...
0
votes
1answer
42 views
Integral of the Fourier spectrum?
The integral
$$\int_{-\infty}^{\infty}|X(f)|^2df$$
of the absolute Fourier spectrum squared is the energy in the signal, but what about the integral of the 'simple' absolute Fourier spectrum?
$$\...
0
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0answers
29 views
Suppressing a frequency using filtering Matlab [duplicate]
Below I have the following frequency representation of a signal. Using Matlab, I need to suppress the two peaks because they represent a beep. I added the notch filter to suppress one peak. But my ...
0
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0answers
42 views
Fast fourier of exponential factor
I read in this post:
Online DFT Algorithm
That:
The following algorithm will compute the N-point DFT, $X_2[k]$ of the
new data set $x_2[n]$ from that of the already computed and stored
N-point ...
0
votes
2answers
97 views
Convolution effects width of the signal?
Let's say there are two signal with different frequency:
\begin{align}
X_1(\omega) &= 0\quad\text{for}\quad \lvert \omega\rvert > 1000\pi\\
X_2(\omega) &= 0\quad\text{for}\quad\lvert \omega\...
0
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0answers
109 views
How to interprete the Power Spectral Density estimate plot
I extracted values for motion from a cartoon and used the pwelch function for MATLAB to calculate the Power Spectral Density estimate by Welch.
But I don't really know who to analyze this plot. My ...
0
votes
1answer
98 views
MATLAB's $\tt cpsd$ and $\tt pwelch$ - different results for cross spectral power density
I am wondering why the following code does not yield the same results for Sxy1 and Sxy2, where ...
3
votes
1answer
105 views
Time domain basis
I have some troubles with understanding time domain, not on the intuitive level, but in math terms.
For example I have a vector signal $$ x = [x_0,x_1,x_2,...,x_{N-1}]$$
I understand that generally ...
0
votes
1answer
58 views
How to estimate covariance matrix using Fourier representation?
So, I have multidimensional time-series $X \in R^{(d \times T)}$, and I want to determine the covariance matrix of that signal in a specific frequency band.
I might filter the signal to that specific ...
4
votes
2answers
223 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 ...
-1
votes
1answer
43 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 ...
2
votes
1answer
186 views
Bandwidth and Bit rate
I'm kind of confused about digital transmission.
Is the bandwidth occupied by some digital signal the Fourier spectrum of the bit (pulse) format,or the one determined by the bit rate?
1
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2answers
265 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 ...
1
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0answers
53 views
Fastest PC configuration for FFT [closed]
I'm currently implementing rotation-invariant phase correlation algorithm, which is a variant of phase correlation algorithm (https://en.wikipedia.org/wiki/Phase_correlation) to estimate relative ...
1
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1answer
37 views
Modulation and filtering
When I modulate a signal $x(t)$ with $\cos(2 \pi f t)$ and the modulated signal passes through a HPF, what output do i get in the frequency domain?
1
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0answers
127 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?
0
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0answers
86 views
Implementing a ram-lak filter in javascript implementation of fast fourier transform?
Having a non-mathematical background I'm trying to understand how to implement filtering of data in the frequency domain as part of FFT.
A javascript implementation of a discrete FFT is found here.
...
1
vote
1answer
355 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 ...
3
votes
1answer
382 views
How to “scale” the FFT when using it to calculate discrete convolution?
As you probably know, the discrete convolution $ H = F \ast G $ of some $ F \left[ x \right] $ and some $ G \left[ x \right] $ can be calculated using the Fast Fourier Transform (FFT).
To do this, ...
0
votes
2answers
115 views
Explanation of fundamental filtering's consequences on signal
Can anyone explain why exactly an "Overshooting" phenomena is observed when the fundamental harmonic is removed as seen on the figures? Is it technically right to call this "overshooting" at all ? If ...
0
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2answers
108 views
Non Periodic, Deterministic Power Signals
Any one know of work on non-periodic but deterministic power signals? Now one member in this class would be the quasi periodic signals. I wonder if there is a generalized Fourier analysis of non-...
-1
votes
1answer
88 views
Meaning of transform's area - Fourier
What is the graphic meaning of the transform's area?
$$\int_{-\infty}^{+\infty}{X(f)df}$$
Where $X(f)$ is the continuous Fourier transform of the signal $x(t)$.
Thank you very much.
0
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1answer
223 views
Difference between frequency components and harmonic components - Fourier
What is the difference between frequency components and harmonic components? The first concern the continuous domain of frequency, while the second concern the discrete domain of frequency ($f_{k}=kf_{...
6
votes
4answers
239 views
Fast & accurate convolution algorithm (like FFT) for high dynamic range?
It seems that FFT-based convolution suffers from limited floating-point resolution due to evaluating everything around the roots of unity, as you can see in the $10^{14}$-factor error in this Python ...
3
votes
1answer
126 views
Filter design to realize Cauchy product
I come from Computer Science so please pardon for my possibly wrong terminology.
I need to design a filter which has coefficients
$$h_0, h_1, \ldots, h_n, \ldots \quad\text{such that}\quad h_0 > ...
0
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1answer
271 views
Fourier Transform of a signal using direct integration and properties
Am trying to compute the Fourier Transform of a function using the properties of the Fourier Transform once and checking my answer using direct integration. My problem is that am not getting the same ...
3
votes
1answer
142 views
Fourier Slice Theorem - Reconstruction Fourier Space
I've stuck in one problem. I need to perform Fourier Slice Theorem on sinogram of medical image.
I read a lot about this theorem. I write a matlab code but results are always non-sense after inverse ...
4
votes
1answer
105 views
estimating spectral optimization
I'm relatively new to DSP so excuse my simplified words, and my detailed explanation.
if the signal have non-coherent sinusiod, it will induce energy spreading into the frequency domain.
One of the ...
