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every continuous signal being modelled as a function

Can every coninuous signal be modelled as a function, which then can be converted into a series of sine and consine functions with unique frequencies? And let us say that we have some arbitrary ...
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2answers
102 views

wave separation

Say there is a wave of sines and cosines. (<- one can think of Fourier theory.) A) There is a wave that has the same frequency all the time. However, amplitude (- shape) of each period differs. Is ...
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1answer
2k views

Nyquist–Shannon sampling theorem shannon's proof

In Wikipedia, there is Shannon's proof on Nyquist-Shannon sampling theorem. ( http://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem#Shannon.27s_original_proof ) The original proof ...
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1answer
202 views

Complex Numbers and polar form

I am given the following information: $$x[n]= s^n,\qquad n=0,\pm 1,\pm 2,\ldots$$ where $s=\sigma + j\omega = re^{i\theta}$ is a complex number in general. I was wondering how the following is ...
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6answers
505 views

How to generalise the Fourier transform

The Fourier transform approximates a signal using a bunch of sine and cosine waves. The inverse Fourier transform then reconstructs the original signal from this information. I am told that it's ...
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1answer
41 views

Differences of distributions inside Kalman filter.

I am studying the Kalman filter algorithm but i can't understand one point. The k factor has to be chosen in order to minimize the variance of the signal. This lead to following equation: ...
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1answer
221 views

Is there an autocorrelation function with a constant integral whose absolute value integral diverges?

Suppose a function $g:\mathbb{R}\rightarrow\mathbb{R}$ such that: $|g(x)|\leq g(0)$; $g(x)=g(-x)$, i.e. $g(x)$ is even; $\int_{-\infty}^{\infty}g(x)dx=C$; There exists a Fourier transform of ...
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1answer
126 views

How to remove the boundary effects arising due to zero padding in discrete fft?

I have made a python code to smoothen a given signal using the Weierstrass transform, which is basically the convolution of a normalised gaussian with a signal. The code is as follows: ...
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1answer
1k views

time derivative of discrete data in simulink

I'm implementing a program in Java that was delivered in Simulink. My expertise is limited, and I'm stuck on converting a derivative block. The simulink code applies a du/dt block to the input data. ...
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1answer
1k views

constructing Feature Vector from given values

I am fairly new to signal analysis, so pardon any noobish questions, but I couldn't find a clear answer by googling. I am using PyLab to calculate certain values from a given data. My data is a 3D ...
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2answers
102 views

How to simplify the product of two $\exp$ functions

It's been a while since I did any of this. I have the following product: $\exp(-j2 \pi u|k|x) \cdot \exp(-j2 \pi v |k|x)$. This seems like it is something that can be simplified, but how? Note, ...
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2answers
231 views

An example of a “pathological” power-spectral density function?

Suppose that we are given a wide-sense stationary random process $X$ with autocorrelation function $R_X(t)$. Power spectral density $S_X(f)$ of $X$ is then given by the Fourier transform of $R_X(t)$, ...
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1answer
86 views

Signal fundamentals

I just finished reading the fundamentals chapter about signals (linearity,causality,memory and time invariance). I wanted to solve some exercises and I found this one. We have a signal with output ...
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1answer
239 views

Signal to odd and even

I have a signal that is described below $$x(t) = \begin{cases} -1, & t<0 \\ 2t-1, & 0\leq t<1 \\ 2-t, & 1\leq t<2 \\ 0, & t\geq 2 \end {cases}$$ $$x(-t) = ...
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2answers
180 views

Compressed sensing, approximately sparse, Power law

An x in $\mathbb{R}^n$ is said to be sparse if many of it's coefficients are zeroes. x is said to be compressible(approximately sparse) if many of its coefficients are close to zero.ie Let ...
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1answer
1k views

DTFT of a triangle function in closed form

I am sampling a continuous signal $x_c(t)$ that follows a triangle function in the time domain, meaning: $$x_c(t)=\left\{\begin{array}{rl}1-|t/a|,&|t|<|a|\\ ...
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0answers
80 views

Scale invariance and $1/f^2$ power spectrum

In the paper Occlusion Models for Natural Images : A Statistical Study of a Scale-Invariant Dead Leaves Model; Lee, A. B. Mumford, D. B. Huang, J.; International Journal of Computer Vision I read ...
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3answers
569 views

Looking for a Calculus Textbook

I want to start signal processing and I need a book that satisfies my mathematical requirements: I am in the third grade of high school and I don't know any useful thing about limit, differential, ... ...
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2answers
5k views

How do I - exactly - project a vector, onto a subspace?…

I am trying to understand how - exactly - I go about projecting a vector onto a subspace. Now, I know enough about linear algebra to know about projections, dot products, spans, etc etc, so I am not ...
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1answer
99 views

Signal Analysis/Processing Textbook

Can anybody recommend me a decent Signal Analysis/Processing textbook. If possible one that deals a little with MATLAB. I have an little knowledge of Real Analysis and fourier transforms. Wavelets i ...
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0answers
292 views

Fast Walsh–Hadamard transform generalization for non-power-of-two orders?

I have to process vectors through a Hadamard matrix of order N. If N is a power of 2, I can use the Fast Walsh–Hadamard transform; but if N is not a power of two (for instance, N=12), it is not ...
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8answers
2k views

Rapid approximation of $\tanh(x)$

This is kind of a signal processing/programming/mathematics crossover question. At the moment it seems more math-related to me, but if the moderators feel it belongs elsewhere please feel free to ...
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1answer
323 views

How to extract module and phase from this transfer function?

I have this transfer function: $$H(x)= \frac{1}{x+i(1+x)}$$ How can I extract module and phase and represent them?
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0answers
295 views

How to Find Phase Lead/Lag

I have the transfer function $$ H(s) = \frac{s+1}{0.1s+1} $$ I apply the Bilinear Rule with a sampling time T =.25 sec to the transfer function and get a z-domain representation of $$H(z) = ...
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3answers
11k views

What does the Fourier Transform mean in the context of images?

This is clearly a very important equation with tonnes of properties that I see come up a lot in image processing literature, but I don't understand why this equation is important, and what it is ...
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3answers
465 views

Period of a product of $\sin$ and $\cos$

I want to find the period of $\sin(t) \cos(\pi t)$. I started off by transforming that into $\frac{1}{2}\left [ \sin((\pi +1)t) - \sin((\pi - 1)t\right ]$, but then I get stuck. How do I find the ...
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1answer
559 views

Explanation of sinusoidal function: $f(x,y) = A\cos(2\pi(ux + vy) + \phi)$

I never got to take a signals and systems course and this has come up in the math of my image processing review. Can this be explained? The equation for a sinusoidal signal is $f(x,y) = ...
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1answer
309 views

Fourier transform of Kronecker deltas

I have a binary 2D image that consists of 95% black pixels with a few white pixels scattered about, and I want to convolve it with a 2D gaussian kernel. I'm hoping to exploit its sparsity to improve ...
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1answer
715 views

Inverse Z-transform of $\frac{1}{(1-z^{-1})^2}$?

What is the inverse Z-transform of $\frac{1}{(1-z^{-1})^2}$? Title says it all. I have a one line solution but can't work out how to get there from tables or first principals. Thanks!
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1answer
190 views

How to apply the solution of $y(n) = (0.85)y(n-1) + x(n)$ to data

I learned how to solve difference equation $y(n) = (0.85)y(n-1) + x(n)$ using z Transform, and inverse z Transform, I get $h(n) = 0.85^n u(n)$ where $u(n)$ is unit step sequence. Now my ...
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0answers
401 views

How can I use the time-frequency uncertainty principle?

I have a signal composed of the summation of a set of sine waves of different frequencies. The amplitude of these sub-signals can change so many times a second. I have been told that, if I want to ...
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1answer
493 views

Finding the poles of a system from a difference equation in MATLAB

I have a system tha is described by the following difference equation: $y(n) + 0.3y(n-1) - 0.3y(n-2) = 0.5x(n) - x(n-1)$ How can i compute, using MATLAB (e.g. with ...
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0answers
149 views

Singular Value Decomposition

I want to decompose an image $A$ using the Discrete Wavelet Transform and then find the singular values, $S$, such that $A=USV$. I will then do the same to another image such that $B=USV$. I will ...
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0answers
105 views

Signal filtering and moving averages

Background Given a signal $x_n$ for $n=1,2,\dots$ we can consider its filtered values: $$y_n = \frac{b(L)}{a(L)}x_n$$ where $a(L)=a_0 + a_1L + a_2L^2 + \cdots + a_nL^n$ (similarly for $b$) and $L$ ...
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1answer
113 views

How can one classify (match within a certain confidence interval) gestures based on accelerometer readings?

I am using an accelerometer-enabled device (mobile phone, to be specific) that enables sampling acceleration at a rate of about 20 samples per second. The samples contain three values, each ...
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0answers
30 views

Name this concept: Comparing equal sized vectors vs. comparing features

If you obtain a vector by taking $n$ discrete samples over some underlying function, then it's easy to compare that vector with another of the same size. With a bunch of $n$-dimensional vectors, you ...
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1answer
82 views

Haar Basis in Signal processing

I want to help one of my friends who studies engineering. He has a homework at the signal processing course. I think I realize what I have to do, but since I don't have their course, I do not fully ...
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0answers
95 views

Solving multiple phase angles for multiple equations

I have several equations and each have their own individual frequencies and amplitudes. I would like to sum the equations together and adjust the individual phases, ...
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0answers
284 views

Moving Average System?

Can someone elaborate on what a moving average system is? I know that the system is defined as: y[n] = (1/3) [x[n] + x[n-1] + x[n-2]] How would we draw y[n] given that we have a graph with ...
2
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1answer
344 views

What is a cardinal basis spline?

Wikipedia says: the normalized cardinal B-splines tend to the Gaussian function and writes them as "Bk". Meanwhile, cnx.org Signal Reconstruction says: The basis splines Bn are shown ... ...
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1answer
53 views

Calculating Basis Functions for DFTs (64 Samples)

I am attempting to graph some 64 sample'd basis functions in MatLab, and getting inconsistent results -- which is to say, I'm getting results that are still sinusoidal, but don't have the frequency ...
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3answers
520 views

Describing a Wave

I have this wave in front of me, and I am to describe this into a math description such as its function that is equivalent to representing this wave. I have no idea how to start and could use some ...
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1answer
41 views

Function to take discrete sample of a continuous sinusoid function?

Say I have continuous sinusoid function such as: $$x(t) = \cos(2\pi f_0 t)$$ where $f_0$ is the frequency and $t$ is some time in the function. I want to take samples of this function at some sample ...
2
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1answer
646 views

Hilbert transform of white noise

What is the Hilbert transform of a white noise $\xi(t)$? By the Hilbert transform I mean: http://mathworld.wolfram.com/HilbertTransform.html Thank you.
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1answer
218 views

Signal extraction from multivariate normal

Define: $y= \theta + \varepsilon + a,$ where $a$ is a choice variable in a behavioral economic model, with equilibrium solution $a^e$, and $\theta$ and $\varepsilon$ are independently distributed ...
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1answer
874 views

Combining Sine Waves on Chart

I would like to combine multiple sine waves with differing amplitudes, frequencies and phases into a single curve that I can display as a graph. What formula will I need to create the points for the ...
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0answers
68 views

Means to classify data streams or compare similarity

Last year I converted some Matlab code into c to run on embedded Linux. I'm an engineer and normally shy away from maths, but this got me thinking about different ways to classify data or compare the ...
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2answers
228 views

Can it be proven that such functions don't exist?

We are given $x_1,x_2 \in \mathbb{R}$ and we want to find two functions $v_1(t),v_2(t)$ such that: $$x_1x_2 = \int_{-\infty}^{\infty} v_1(t)-v_2(t) dt$$ A very interesting restriction that we have ...
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0answers
1k views

When comparing different signals, how can I normalize their amplitude?

I am comparing two sound signals in the frequency domain. As they can have been recorded in different volumes I need to normalize them. My initial approach was to divide each sample by the mean ...
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2answers
81 views

Should I combine the negative part of the spectrum with the positive one?

When filtering sound I currently analyse only the positive part of the spectrum. From the mathematical point of view, will discarding the negative half of the spectrum impact significantly on my ...