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2
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1answer
4k views

Plotting a one-sided amplitude spectrum

I have a continuous signal $x(t)$ such that $$x(t)=12\cos(6\pi t)+6\cos(24\pi t)+3\cos(30 \pi t)$$ and is asked to sketch a $1$-sided Amplitude Spectrum of the signal $x(t)$ if sampled above the ...
1
vote
1answer
915 views

Finding Fourier series with function not centered at the origin

I am trying to find both Fourier cosine and sine series which represent the function F(t) in the interval $(0, \pi)$ where $F(t)=\begin{cases} \frac{\pi}{2} & \ \ 0<t< \frac{\pi}{2}\\ 0 ...
1
vote
1answer
99 views

Is that a counterexample to sampling theorem?

Sampling points are $\mathbb{Z}$. Sampling theorem tells us that functions with bandwidth lower than $\frac{1}{2}$ will have no aliases. Take functions with pure frequency $\frac{1}{4}$ as an ...
0
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2answers
147 views

Z-Transform Identity

I've come across an identity and would like to know if it has some sort of formal name or derivation or explanation or something! Also, I'm curious as to whether others are aware of such an identity. ...
1
vote
1answer
1k views

Visualize a difference equation with Matlab [closed]

I have a difference equation for a Single Pole Infinite Impulse Response Filter, defined on a discrete time-series: $y[n]-(1-\alpha)*y[n-1]=\alpha*x_n$ While the []s brackets refer to a position n ...
3
votes
3answers
124 views

Detecting significant decreases in a signal

I'd like to find a way to detect a significant drop/decrease in a signal. Below is an actual example of what I'd like to accomplish, with the arrow denoting the change that I'd like to detect (only ...
2
votes
1answer
9k views

$y(n) = x(-n)$ , causal or not , memory or memoryless?

$y(n) = x(-n)$ , causal or not , memory or memory-less ? it's a question in digital signal processing course . My guess it's memory less , causal because $x(-n)$ is only the inverse of the ...
1
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0answers
125 views

Worst-case error related to Cramer-Rao bound

I would like to understand the relation (if any) between the Cramer-Rao Lower Bound of estimation theory and the following simple definition of "reconstruction accuracy" which doesn't use any ...
3
votes
1answer
634 views

95% of energy of Bessel Functions

How can we determine what which Bessel function amplitudes contain the majority of the the energy? Similar to the Carson bandwidth rule, I want to determine which sidebands help make up the 95% of ...
0
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0answers
622 views

proof that orthogonal signals not interrupting each other

I know that signals that are orthogonal do not disturb each other. What I am curious is what is the proof behind why orthogonal signals in a single signal (i.e. a single signal can be broken down ...
2
votes
1answer
126 views

MA process ACF proof - don't understand it

I've got the proof but I don't understand a small detail. As you know for an MA process: $X_n = \sum _{i=0} ^q \beta_i Z_{n-i}$ where $Z_n$ is WGN (pure Gaussian random process). Then the ACF is: ...
2
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0answers
197 views

How to solve the recursive relation in Kalman filter?

I was wondering how to solve the Kalman filter's recursive equation (also see the appendix at the end of this post) for the estimated state $\hat{\textbf{x}}_{n|n}$ at time $n$, over discrete times ...
1
vote
2answers
436 views

Nyquist-Shannon sampling theorem - original version and modern version difference?

I just found out that the original version of Nyquist-Shannon sampling theorem differs from the modern version.. In the original version, it states that the bandlimited signal $x(t)$ can be ...
4
votes
1answer
1k views

Derivative of a random variable w.r.t. a deterministic variable

I'm reading about time series and I thought of this procedure: can you differentiate a function containing a random variable. For example: $f(t) = a t + b + \epsilon$ where $\epsilon \sim N(0,1)$. ...
1
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0answers
40 views

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 ...
0
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2answers
130 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 ...
1
vote
1answer
3k 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 ...
1
vote
1answer
254 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 ...
5
votes
6answers
588 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 ...
0
votes
1answer
54 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: ...
1
vote
1answer
357 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 ...
2
votes
1answer
186 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: ...
1
vote
2answers
2k 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. ...
1
vote
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 ...
1
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2answers
104 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, ...
2
votes
2answers
252 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 $R_X(t)$, ...
0
votes
1answer
108 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 ...
1
vote
1answer
366 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) = ...
1
vote
2answers
218 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 ...
0
votes
2answers
2k 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|\\ ...
2
votes
0answers
84 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 ...
7
votes
3answers
843 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, ... ...
12
votes
2answers
10k 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 ...
0
votes
1answer
119 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 ...
1
vote
0answers
359 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 ...
20
votes
8answers
4k 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 ...
0
votes
1answer
510 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?
1
vote
0answers
377 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) = ...
12
votes
3answers
15k 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 ...
5
votes
3answers
565 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 ...
0
votes
1answer
679 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) = ...
1
vote
1answer
323 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 ...
1
vote
1answer
941 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!
0
votes
1answer
276 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 ...
2
votes
0answers
702 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 ...
1
vote
1answer
826 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 ...
2
votes
0answers
168 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 ...
1
vote
0answers
116 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 $L$ ...
2
votes
1answer
116 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 ...
1
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0answers
31 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 ...