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41 views

Find the maximum of an integral function with respect to another function

I'm facing this statistical data analysis problem, where I have to maximize a certain statistic in order to find the optimal filtering function. I'm a little bit out of practice with the mathematics ...
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1answer
317 views

Calculate phase and amplitude of a sampled sine wave

I have an electronics project where I sample two sine waves. I would like to know what the amplitude (peak) and difference in phase is. Actually I just need to know the average product of the two ...
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1answer
79 views

How to derive this frequency response?

Given this difference equation $y(k)$ ... $$y(k) = \frac{1}{K^2} \sum_{m = k-K+1}^k \; \sum_{n = m-K+1}^m x(n) - \frac{1}{L^2} \sum_{m = k-L+1}^k \; \sum_{n = m-L+1}^m x(n)$$ ... how does one derive ...
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1answer
77 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 ...
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1answer
50 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 ...
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32 views

Integration containing a complex number

Folks, Can I treat the complex number in the following integral: $$\frac1{2\pi}\int\frac1{(1+jw)^2}dw$$ as a constant and move it outside of the integral, like this: ...
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56 views

problem integrating a dirac comb

Let: $$h(t)=\frac{\sin(\pi t(2N+1))}{\sin(\pi t)}$$ $$I=\int_\frac{-1}{2}^\frac{1}{2} h(t) dt$$ when $N\rightarrow\infty$ , obviously (with a change of variable $v=\pi t(2N+1)$ ): ...
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1answer
45 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 ...
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72 views

Reconstructing sine wave from samples

Suppose there is a sine wave signal, like the following: $$V(t) = M * sin(\phi_0 + \omega*\Delta t)$$ I can have it sampled and obtain $V_1$, $V_2$ and $V_3$ at $t_1$, $t_2$ and $t_3$ such that ...
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1answer
45 views

Multiple Characteristic Function and the Dirac Comb

Given the impulse train(Dirac comb): $$\Delta_T(t)=\sum_{k\in\mathbb{Z}}\delta(t-kT)$$ where $T$ is the signal period, $\delta(t)$ is the Dirac delta function and $\mathbb{Z}$ is the set of integers ...
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1answer
38 views

CT Fourier Transform

I need to find the Fourier Transform of the given signal below; $$ x(t) = \frac{\sin(\pi t)}{\pi t} \frac{\sin(2\pi t)}{\pi t}.$$ I know that if $ x(t) = \frac{\sin(Wt)}{\pi t} $ , then $ X(w) = ...
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3answers
374 views

Fourier Series coefficients/Trigonometric functions

I need some help about finding the Fourier Series coefficient of the given signal; $$ x(t) = \sin(10\pi t + \frac {\pi}{6} ) $$ I know that, $$ a_{k} = \frac{1}{T}\int_{0}^{T} x(t)e^{-jkw_{0}t}dt $$ ...
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2answers
53 views

Generating points from a standard Gaussian

I'm new to Gaussian distributions and I'm trying to generate say, $ N$ points from a $ M$ dimensional standard gaussian. What does this mean? How would I do this in matlab?
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2answers
23 views

Data preprocessing

How would you preprocess 2 dimensional data to have 0 mean? Say you have a matrix $M $ that is $p \times q $. Would you calculate the mean of each row, get a vector of length $q $ and subtract each ...
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0answers
97 views

Whitening matrix for Fast ICA

I have a matrix $X $ with dimension say $ m \times n $ with $ m> n $. I am trying to whiten this matrix in matlab by first taking the $C= \operatorname{covariance}(X)$ followed by eigenvalue ...
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0answers
39 views

Relationship between 2 sinusoidal signal data sets?

I'm trying to relate a near shore tidal signal (point A) to 3 points along a long model boundary (points B C D). I want to possibly have a relationship between B C D with which we can convert A ...
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2answers
113 views

Fourier Series Coefficient of a given signal

$$ {\rm x}\left(t\right) = \sum_{k = -\infty}^{\infty}\left[\delta\left(t-\dfrac{k}{3}\right) + \delta\left(t-\dfrac{2k}{3}\right)\right] $$ I need to find the Fourier series coefficient of x(t). I ...
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1answer
22 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. $\sum_{k=0}^\infty a^k*\delta(t-kT), |a|<1$ ...
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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}, .., ...
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88 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 $2N$ ...
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1answer
42 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) ...
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1answer
123 views

Why arctan equal to -90 degrees?

Can somebody show me why $$-\arctan\left(\frac{2\pi}{1-\cos(2\pi)}\right)$$ equals to $-90^\circ$ degrees? Thanks.
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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 ...
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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 ...
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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) ...
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37 views

how can evaluate this integral?

how can calculate this integral: $$A=\int _{-\infty}^{\infty}exp(j2\pi ft)df$$ its answer is $\delta (t)$(impulse function), however how can I get this answer? Thanks in advance.
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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} ...
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1answer
30 views

Elimination of complex variable in integral

I have the equation: $$\frac{1}{\tau}\intop_{0}^{\tau}A\sin\left(\Omega t\right)\cdot A\sin\left(\Omega\left(t-\lambda\right)\right)\mathrm{d}t$$ for which the attempted solution is to convert the ...
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1answer
90 views

3-bit sensors — A question about Hamming distance in signals

I came across this question on Willy Wu's riddle site You have two 3-bit sensors, A and B, that measure the same thing, whatever it is -- temperature of the room, radioactivity levels, whatever. ...
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85 views

Differential equations and Kalman filters

I have been told that every differential equation has an associated Kalman filter. How do we get the Kalman filter of a given differential equation. For example let's say we have $$my''+cy'+ky=f(x)$$ ...
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1answer
111 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 ...
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1answer
64 views

Using the FFT to align two instances of the same signal

I'm working on a program that has a software oscilloscope-like viewer for audio signals. The scope basically takes in blocks of signals at a regular rate and adds them to its existing signal data. ...
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3answers
579 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 ...
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2answers
642 views

How can a unit step function be differentiable??

Recently, I am taking a Signal & System course at my college. In all of the signal & system textbooks I have read, we see that it is written " When we differentiate a Unit Step Function, we ...
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2answers
151 views

Mathematical explanation for image edge detection and denoising

I am trying to understand why the convolution kernel, $$\left[\begin{array}{rrr} -1&-1&-1\\ 2&2&2\\ -1&-1&-1 \end{array}\right]$$ detects the edges in an image. If anyone has a ...
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1answer
161 views

Can the measurement matrix used for compressive sensing be a sparse matrix?

I am interested in analyzing Compressive Mechanism: Utilizing Sparse Representation in Differential Privacy. In my research, the measurement matrix $A\mathbb \in R^{m \times n}$ needs to be sparse. ...
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1answer
467 views

DSP, discrete time, how to calculate the frequency

this is from Digital Signal Processing, 4th ed, Sanjit K Mitra, problem 2.39b. The question is: Determine the fundamental period of $x[n] = cos(0.6n\pi + 0.3\pi)$ Since x[n], is in square brackets, ...
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1answer
436 views

Regarding $x^2-a^2$ inside the argument of dirac delta

My undergraduate system textbook has this property in the appendix $$\delta(x^2-a^2)=\frac{1}{2|a|}[\delta(x-a)+\delta(x+a)]$$ and I can't seem to derive the result I tried the following: ...
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2answers
2k views

Is impulse response always differentiation of unit step response of a system?

I was trying to solve a question in which the transfer function of a system was asked, its unit step response being given: c(t) = 1-10exp(-t) The method that the book followed was to first find out ...
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2answers
106 views

Why are these two delta function equal

In my system textbook it claims that $$\delta(x)=\delta(-x)$$ I understand the proof as follow $$\int_{-\infty}^\infty f(x)\delta(-x)\,dx$$ let $u=-x\,\:,\: du=-dx$ $$\int_{-\infty}^\infty ...
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3answers
983 views

Proofs of dirac delta property

How would I formally prove this property of dirac delta? $$\int \delta(a-x) \delta(x-b) \,dx = \delta(a-b) $$ I attempted to use the definition of a dirac delta $$\int ...
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1answer
35 views

understanding bases and frames for Gabor transform

For the 2D discrete Gabor transform, why is it that we cannot use a set of orthonormal basis for its representation, instead we have to use frames for representing it?
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353 views

Wavelets: Cone Of Influence

While reading this paper I came across the term Cone of Influence which is described as ...
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1answer
63 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 ...
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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 ...
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21 views

showing identities involving instantaneous frequency of signal

If the short-time Fourier transform of a signal $f(t)$ is given by $$ S_f(u,\xi)=\int_{-\infty}^{\infty}f(t)g(t-u)\exp{(-i\xi t)} dt $$ and $$ f(t)=\exp[i\phi(t)], $$ how do I show that $$ ...
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1answer
53 views

Periodic Fuctions - Signals -

If, in the periods, the two half's signal periodic have the same form and opposite phases, the periodic signal has symmetry of half wave. If the periodic signal $g(t)$, of period $T_0$, satisfy the ...
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1answer
3k views

The definition of NMSE (normalized mean square error)

Many papers use the NMSE function without ever explicitly defining it. I have always assumed that $$MSE(x,y)=\frac 1N \sum_i (x_i-y_i)^2$$ and $$ NMSE(x,y)=MSE(x,y)/MSE(x,0) = \frac{\| x-y\|_2^2}{\| ...
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1answer
49 views

Interpret convolution diagram

How do I interpret this "do convolutions" diagram? 1) How are the results computed? 2) When looking at this part: "x[n-k]" Do you interpret convolutions as delays or time reversals? $ y[n]= ...
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1answer
597 views

Undecimated Wavelet Transform (a trous algorithm) - how to determine 'anchor'/'center' of convolution filter

i am currently implementing the 'Undecimated Wavelet Transform' with the 'a trous' algorithm. See e.g. http://www.znu.ac.ir/data/members/fazli_saeid/DIP/Paper/ISSUE2/04060954_2.pdf, section II-A. As ...