0
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0answers
27 views

Parseval's theorem.

We consider two signal $h(t)$ and $g(t)$ such that $$\int_{-\infty}^\infty |g(t)|^2dt<+\infty$$ $$\int_{-\infty}^\infty |f(t)|^2dt<+\infty$$ Parseval's theorem states that: ...
0
votes
0answers
14 views

Top and bottom power spectral density of a height profile

Imagine I have a simple 1D height profile which is NOT symmetric. Now, what is truly important for me is to know what are the frequency content of the top profile (i.e. a cut profile above the ...
0
votes
0answers
36 views

Covariance between real and imaginary parts of Fourier transform of a stationary time series

Since Fourier transform of a random stationary time series(in the case of existence) is not necessarily real, my question is what is the relation between the covariance of real and imaginary parts of ...
0
votes
1answer
28 views

Discrete Time Fourier Transform of a real signal

I want to prove that if we have a real signal x[n] then for the DTFT it is applied that we have an even symmetry: | X(Ω+1/10) | = | X(-(Ω+1/10)) | (I mean the ...
3
votes
1answer
36 views

Image Reconstruction:Phase vs. Magnitude

Figure 1.(c) shows the Test image reconstructed from MAGNITUDE spectrum only. We can say that the intensity values of LOW frequency pixels are comparatively more than HIGH frequency pixels. $$ ...
0
votes
1answer
40 views

Fourier and $Z$ transform of a signal?

We have $$X(k)=4[u(k-2)-u(k)* d(k-3)]$$ I need to find the Fourier transform,$Z$ transform,as well as dhe magnitude and phase spectra. First of all I think that I need to convert the $u(k)$ and ...
1
vote
1answer
37 views

How do we determine the duration of a fundamental frequency using the DFT (or FFT)?

I'm still in the process of learning the details of the DFT (and FFT) and I've just made a test .wav file in Audacity by joining 3 one-second sine waves together. .wav file 1 = 440 Hz, sample rate ...
0
votes
0answers
33 views

Plotting the frequency spectrum of a signal

I've found this algorithm here on Mathematica.SE to plot the frequencies of a signal using Fourier. It works beautifully, but I'm having some trouble understanding ...
0
votes
1answer
12 views

Representing a real sampled signal with N samples as a complex sampled signal with N/2 samples

I am studying the discrete Fourier transform, and in its most basic definition it is an invertible linear transformation on the complex numbers. From Wikipedia: The sequence of $N$ complex numbers ...
2
votes
1answer
24 views

Problem about average of cos square (nt) where n is arbitrary

I often see people just say time average of cos^2(nwt) is 1/2, I want to know in what cases this is not valid? w is just the frequency, can be assumed as a constant. Assuming you are always ...
1
vote
1answer
16 views

How do I deal with a seemingly fractional delays in discrete time fourier transforms?

Is a transfer function of a discrete time system is $H(e^{j\Omega})=e^{-j\Omega/4}$ and I feed it an impulse, what will be it's response? I know that technically a transfer function of ...
4
votes
1answer
56 views

Finding the period of the solution to $y'(x) = y(x) \cdot cos(x + y(x))$ with Fourier transform; how to interpret complex result?

A question elsewhere on this site asks about detecting the frequency of oscillations in a system defined by differential equations. The equation is $y'(x) = y(x) \cdot cos(x + y(x))$. The solution ...
0
votes
0answers
13 views

Is there a better approach than DFT for identifying known pitches in a sound sample?

I've been using DFT to extract information about which notes are being played in a sample of music. My understanding of it is fairly basic but what I have so far works a at least in principle. ...
2
votes
3answers
165 views

Any good introductory book/tutorial on Fourier Transform (up to FFT) with plenty of exercises and solutions?

I wonder what could be a good book to start learning in depth all aspects of the Fourier transform up to the FFT algorithm, and beyond. I am going to dedicate quite some time on the subject, so I ...
1
vote
1answer
62 views

Wavelet or FFT for Transient signal analysis?

For now I use FFT to analyze the response of an electrical system to some transient signal. The transient signal is $x(t)$, which translates to $X(w)$ in the frenquency domain. On the other hand I ...
0
votes
2answers
57 views

Fourier Transform: why do all segments generate the same magnitude response?!

I'm working on a DTMF program, and what I've done is to break the one long input signal I initially receive into a bunch of smaller components. I perform an FFT on each of the small components and ...
1
vote
2answers
63 views

What's the point of Dirac delta function?

I have heard that The main useful property of Dirac delta function is it's fundamental property that $$ \int_{-\infty}^{\infty}f(x)\delta(x-a)dx=f(a) $$ I don't understanding why this equation is ...
0
votes
1answer
38 views

Why divide by N (length of input sequence) during IDFT?

During DFT of a input sequence of length N, we find X(k). We find inner product with a basis vector to get the coefficient: X(k) = <x[n], e[k, n]>    |  k = 0, 1, 2, ... ...
0
votes
1answer
34 views

Question about the frequency domain and the fourier transform

if you have a signal say x(t) in continuous time and you transform it using the Fourier transform for continuous time you get X(w) which is the frequency domain representation of this signal x(t). ...
0
votes
1answer
38 views

Amplitude Spectrum, Nyquist Frequency, mixed/min/max wavelets

The problem is here. Now I know the definition of mixed/max/min phase wavelets, whether the roots lie within the unit circle or not. Starting from n = 1, let $$ x_t = ( 5, 6) $$ $$ X(z) = 5 + 6z $$ ...
0
votes
2answers
54 views

Fourier transform of 1 cycle of sine wave

Consider the signal: $\begin{align*} f(t) &= \sin(\omega t) \tag{$0 \leq t \leq 2\pi/\omega$}\\ &= 0 \tag{elsewhere} \end{align*}$ How to compute the Fourier transform of $f(t)$? I ...
0
votes
0answers
28 views

Fourier transform, what is the signal we're analyzing?

I studied Fourier Transform at university (very basic) and I know that it is a mathematical tool to get the frequencies out of a time signal (of some kind). There's something I have always wondered: ...
0
votes
1answer
34 views

What is the relationship between periodicity in a time domain signal and periodicity in the frequency domain representation of the same signal?

Is it true that the frequency domain representations of signals are always periodic? If so, is there intuition as to why? I'm having some trouble understanding what periodicity in the frequency ...
0
votes
2answers
29 views

Finding transfer function with Fast Fourier Fransform.

I have two signals with input = a(t) and output = b(t) that have been sampled every 0.01s and as such the fast Fourier transform has been used on both and utilised to produce a transfer function. The ...
0
votes
0answers
28 views

Determining valid frequency domain DTFT's

This question may or not be off-topic but it concerns Fourier Transforms so I'm assuming it's of some relevance here. One of my problem set questions is... Are the following frequency domain ...
1
vote
1answer
39 views

What is the meaning of a continuous curve in the frequency domain?

I am sorry for how rudimentary this question will sound. I approach the frequency domain thinking in discrete terms. The plane is frequency on the x axis and amplitude on they y (ignoring phases). ...
1
vote
1answer
44 views

How does this phase shift in x-space affect the position of a spectrum in k-space?

I'm working on a new form of signal detection with which I hope to recover both the amplitude and phase of a very small signal. However, doing this requires the use of some Fourier maths that I don't ...
0
votes
2answers
36 views

Very short theory question signals?

My teacher asked us this question yesterday in the lecture but it didn't make any sense to me. He asked: What do the coefficients of the exponential Fourier series represent? Also, what's the ...
0
votes
1answer
43 views

2-D Fourier Transform of complex exponential with 2-D quadratic phase

I've been looking around to see if there is either an exact transform pair or an approximation to either of the following but have not been able to find anything: $$ \mathcal{F}_{xy}\left( ...
2
votes
1answer
104 views

Need to learn wavelet, suggest steps and resources

I am looking for a good introduction to wavelets and wavelet transforms. that covers the following: Basics Vector Spaces – Properties– Dot Product – Basis – Dimension, Orthogonality and ...
1
vote
0answers
62 views

Fourier Transform of a Gaussian Signal?

As far as I know this is the formula for FT : On this question on part b) I fint on the answer the part with e^-jwt is changed with cos(wt) I have no idea how cos(wt) came in ... would you please ...
3
votes
0answers
79 views

Artifacts and low frequencies FFT.

I am working on analyzing a time signal and want to preform a FFT. However I run in to some artifacts at low frequencies. I have managed to reproduce the behavior in a test signal. Given by $S(t) = ...
1
vote
1answer
79 views

Given a Poisson-noisy signal, what is the noise distribution of its Fourier transform?

Disclaimer: I'm not a mathematician, but here's my attempt at a mathy version of my question Start with a noiseless, discretely sampled expected signal $I(x_n)$. Construct a Poisson-noisy measurement ...
1
vote
1answer
53 views

Fourier analysis of an exponential function review

I am working through and reviewing some of the examples presented on Fourier analysis from a Modern Digital and Analog Communication Systems book. In one of the examples, the author goes through the ...
0
votes
2answers
123 views

Why is the DTFT (Discrete Time Fourier Transform) unique to each input?

As the title implies. I know the DFT of a signal is unique due to the matrix, but can anyone give a solid explanation as to why the DTFT is unique for each signal input? Thanks for your time!
2
votes
1answer
76 views

Paley Wiener Theorem on sinc function

Use the Paley-Wiener theorem to argue that, although ${\rm sinc}\left(t\right)$ is bandlimited, ${\rm sinc}\left(t^{3}\right)$ is not. Explain how the above result allows reconstruction of some ...
2
votes
1answer
48 views

Complex Integration of DTFT

Question A discrete-time signal $u \in \mathcal{l}^2(\mathcal{Z})$ has DTFT \begin{equation} \hat{u}(\omega) = \frac{5+3\cos(\omega)}{17+8\cos(\omega)} \end{equation} Use complex integration to find ...
0
votes
1answer
44 views

Understanding a step in the proof of the Inverse Fourier transform theorem

I'm trying to understand the proof of the Inverse Fourier Transform theorem in Stéphane Mallat's "A wavelet tour of signal processing". Near the end of the proof, we have: $ \lim_{\epsilon ...
1
vote
1answer
21 views

How do you compute the Fourier Transform of this Unit-Impulse Function?

I have been given this problem from a textbook (not homework, trying to study for an exam. The goal is to find the Fourier transform of this function. $\sum_{k=0}^\infty a^k*\delta(t-kT), |a|<1$ ...
0
votes
0answers
35 views

DFT one-dimensional vector

That's a way to define one-dimensional Discret Fourier Transformation (DFT)? If I have a signal $x \in \mathbb{R}^N$ and I take a rectangular window of lenght M: $y = (x_m, x_{m+1}, x_{m+2}, .., ...
1
vote
0answers
84 views

Fourier Transform over function depend on time and frequency

In my task I need to perform Inverse Fourier Transform from spectrum that depend on time and frequency arguments simultaneously. E.g., I have a discrete spectrum of some function $S(t, f)$ with $2N$ ...
0
votes
1answer
41 views

Omitting part of Frequency domain, Fourier Transform, Image Processing

In my Image and Signal Processing lecture, the Professor said that if every other column of the frequency domain of an image is zeroed out, then the reconstructed image is aliased. (along the x-axis) ...
1
vote
1answer
69 views

symmetric window for discrete windowed Fourier transform

The discrete windowed Fourier transform of a signal $f$ of period $N$ is given by $$ Sf[m,l]=\sum_{n=0}^{N-1}f[n]g[n-m]\exp\left(\frac{-i2\pi l n}{N}\right). $$ Why is it that the window $g$ must be ...
0
votes
0answers
66 views

How to prove Fourier inverse transform worked?

$$g(t)=\int\limits_{-\infty}^{\infty}g(f)e^{i\omega t}df$$ $g(t)$ is a function of time, $g(f)$ is a function of frequency, $e^{i\omega t}$ represent wave, and $\omega = 2\pi f$, the angular ...
3
votes
0answers
55 views

Recovery of Bandlimited Signals

Let $\Omega > 0$ and denote by $\mathcal{B}_\Omega$ the subspace of $L^2(\Bbb R)$ consisting of signals that are bandlimited to $(-\Omega, \Omega)$. Denote $\mathcal{L}_{\Omega} : L^2(\Bbb R) ...
0
votes
0answers
151 views

Q: Calculating Fourier Coefficients and Inverse Fourier Transform

Let $\Omega >0$ and $x \in \mathcal{B}_{\Omega/2}$ is continuous. Define $\hat{y}(\omega) = \sum_{n \in \Bbb Z} \hat{x}(\omega - n\Omega)$. If $\hat{y}$ is expressed as \begin{equation} ...
0
votes
1answer
98 views

How to interpret the results of 2D Fourier Transform on an image?

I have a class where we're studying signals processing (mostly filtering of sounds and images) and while I kind of understand the results of a Fourier Transform for sounds I don't really get the ...
17
votes
3answers
572 views

Fourier transform of $\left|\frac{\sin x}{x}\right|$

Is there a closed form (possibly, using known special functions) for the Fourier transform of the function $f(x)=\left|\frac{\sin x}{x}\right|$? $\hspace{.7in}$ I tried to find one using ...
0
votes
1answer
59 views

Amplitude versus time producing unexpected patterns.

I am writing a program to generate audio frequencies in multi-channel PCM format. This question may be more suited on an audio forum but I would like to know what is going on mathematically. My ...
0
votes
0answers
21 views

Does this property of a signal's DFT have a name? Or is it expected?

I have a sequence $x_n$ of $N\approx 2\cdot 10⁵$ elements (non-periodic, although there seems to be a noisy sinusoidal of period $10$ samples added somehow) and I take subsequences $x_m$ from this big ...