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The bandwidth of the signal $x(t)$.

The bandwidth (B) of the signal $x(t)$ is the range of frequencies (measured on the positive semi-axis) in which $X(\omega)$ takes values ​​different from $0$. Very often $X(\omega)$ is different from ...
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0answers
83 views

Using l1 magic toolbox for compressive sensing : Positive definite matricies.

I'm trying to use l1 magic to reconstruct an image from a single pixel camera I've developed. The test functions used are random binary patterns projected onto the object scene, so each pattern is ...
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1answer
32 views

Is there a way to modify the exponential smoothing function to account for varying sample rates?

I am using a simple exponential smoothing formula to smooth a signal. X(n) = a * S(n) + ( 1 - a ) * X(n-1) However on certain setups, the sample rate is much ...
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1answer
40 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 ...
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15 views

Is there a standard way for modeling a Kalman filter where the measurements are obtained from differences?

Consider for simplicity a Kalman filter applied to the one-dimensional state space model $x_{n}=f_{n}x_{n-1}+q_{n}$ $y_{n}=h_{n}x_{n}+r_{n}$ with white noise errors. Assume that $r_n=e_n-e_{n-1}$ ...
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23 views

find period of discrete cosine

let us consider following we should find period of this discrete signal,for periodicity we should have $x[n+kN]=x[n]$ or $10\cos(0.088\pi(n+kN) +\phi)=10\cos(0.088\pi n+\phi)$ or $0.088\pi ...
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21 views

Total change of a signal overtime

I have some signals whose analytic type I do not know. I can only sample them every 0.1 secs. I want to pick that signal that changes as little as possible. For example, between sin2t and sint I ...
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0answers
47 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 ...
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28 views

Do 2 timeseries represent the input better than one?

I only have a very basic familiarity with signal processing and information theory so I'm sorry if this is a very straight forward question. I have a very brief input signal and two timeseries as ...
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1answer
36 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 ...
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1answer
121 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. $$ ...
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69 views

Expectation and convolution question.

I am learning in an image processing course, and the professor did the following: As part of a derivation, has this: What I do not understand, is how he was able to remove $r(i,j)$ to the ...
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1answer
60 views

Blind deconvolution of a function convolved with itself

I have a function/vector $f$ that I know is the result of an unknown function $g$ convolved with itself: $f = g \ast g$ Is there any way to do a blind deconvolution on $f$ with this constraint?
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1answer
34 views

A relationship among multiple periodic arrays

There are N periodic arrays ai[n] with period Ti, respectively, where i=1, 2, … , N. Each array has a property that a[n]=1 when n=k*T where k is integer, otherwise a[n]=0. Then a new array is created ...
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49 views

On the truncation of spherical harmonics

Suppose there is a function $f(\theta,\phi)$ defined on the surface of a sphere, and $\theta$ and $\phi$ are the polar and azimuthal angles respectively. Similar with the fact that a function defined ...
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1answer
39 views

Signal processing and properties question

We have the relationship between input and output: $Y(k)=|x(k+1)| + x(k)+ kx(k-5)$. Find the output of the system when $x(k)=d(k)$. What does this output represent? Show if the system is linear, ...
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26 views

Solution to iterative equation with floor operation

This question is motivated by the following signal processing problem. Suppose there is a source, which produces vectors of data of length $N_s$, and a filter (or other subsystem) that accepts ...
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1answer
38 views

How can I plot this signal

What will be the value of this signal (specified as impulse train) , say for values of t from 0 to 6 $$g(t)=\sum_{k=- \infty }^\infty \delta(t-2k)$$
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132 views

Approximate Periodic Function by shifting Basis Functions

Given a periodic "Target Function" $F(t)$ a set of $N$ periodic "Basis Functions" $B_i(t)$ of arbitrary shape All functions are defined on the same interval $T$. I am allowed to shift ...
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41 views

Function with bounded derivative as ODE

Given a function $x(t)$, I am looking for a function $y(t)$ which closely follows $x(t)$ except that its derivative must be bounded by a constant $c$, i.e. $\dot{y} \leq c$. Is there a way to describe ...
3
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1answer
67 views

Moving average as ODE

Is it possible to represent or approximate the moving average $m(t) = \frac{1}{w}\int_{t-w}^t x(\tau) d\tau$ of a function $x(t)$ as a set of ordinary differential equations $\dot{y} = \ldots$? I am ...
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1answer
50 views

Trigonometric regression

What methods are performed for regression with trigonometric functions? E.g. : Sequence: $$-1, 0, 1, -1, 0, 1, \text{.....}$$ Regression: ...
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1answer
46 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 ...
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18 views

How to evaluate the difference between two classes of data which are highly overlapped

I’m trying to implement a classifier based on a dataset comprising two classes of high dimensional time-series observations (the values of the two classes of observations are highly similar). I ...
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1answer
48 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 ...
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41 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 ...
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1answer
52 views

Impulse response and z transform question?

We have $g(k)=\{ [(1/5)^k]u(k)\text{ for $1 \le k\le3$ and $0$ for other }k\}$ The input is $x(k)=\delta(k) +3\delta(k-1)+ \delta(k-2) $ Using Z transform we have to find the output $y(k)$ and the ...
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1answer
19 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 ...
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1answer
27 views

Differential equations signal theory question?

We have the system in the figure.I have to find the differential equation that describes the system.Now,the equation is x(k-1)-y(k-1)=y(k),but I don't understand why.I mean I have studied systems and ...
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1answer
76 views

Essential mathematics for Image Processing

What are the most essential mathematical concepts one has to be familiar with for succeeding in the field of Image/Signal Processing and Machine Learning. I am somewhat familiar with Tensors, ...
2
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1answer
27 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 ...
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34 views

Expected Value With Signum

So I am trying to show convergence of a filter, and in order for it to converge, I need the following condition to hold: $ E\{ \mathbf{s} x sgn(\mathbf{h}^{T} \mathbf{s} x) \} \; \alpha \; E \{ ...
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1answer
60 views

If $tf(t)=tg(t)$, where $f(t), g(t)$ are distributions, then $f(t)=g(t)+\lambda \delta (t)$

Using Dirac distribution properties, prove that if $tf(t)=tg(t)$, where $f(t), g(t)$ are distributions, then $f(t)=g(t)+\lambda \delta (t)$ for some $\lambda\in \mathbb R$. If someone knows please ...
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1answer
40 views

Discrete Function Approximation Error - Which type? (Applied math, signals)

I have two functions, one derived via software, and we can call it the exact function, $f_{exact}$. The other is a result I got through hardware, and we can call it the approximation, $f_{approx}$. ...
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1answer
17 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 ...
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1answer
27 views

Causality of a discrete recursive system

I am new in this site and this my first question. How can I mathematically prove that the system with a transfer function like the below one is causal? P.S: I know that for a system to be causal ...
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1answer
233 views

Finding an impulse response to an inverse system of that of which an impulse reponse is given

I need to solve a bunch of these questions and would appreciate a solid direction. I have an impulse response $h[n]=\left(\frac{1}{5}\right)^nu[n]$. Firstly we are to find a whole number A which ...
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1answer
30 views

Hilbert transform pair

We know the Hilbert transform is defined as: $$ y(t)=\mathcal{H}(x(t))=\mathrm{P.V.}\ \{x(t)*\frac{1}{\pi t}\} $$ In the frequency domain this is equivalent to: $$ Y(f)=X(f)\times (-j\mathrm{sgn}(f)) ...
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1answer
65 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 ...
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1answer
109 views

Rational approximation of $\tanh\,(\sqrt[4]{s}$)

I'd like to find a rational representation of $$f(s) = \frac{\tanh\,\sqrt[4]{s}}{\sqrt[4]{s}}= \frac{a_0 + a_1 s + a_2 s^2 + ... + a_n s^n}{b_0 + b_1 s + b_2 s^2 + ... + b_m s^m} $$ For the case ...
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2answers
50 views

Compressive Sensing matrix

I am working with compressive sensing recovery with image and I test with various sensing matrices: Case 1: Sensing matrix A of size MxN is i.i.d Gaussian matrix. Case 2: Sensing matrix A is size of ...
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1answer
36 views

BIBO stable system

I would like to ask if this system is Bounded Input Bounded Output stable : $$y[n] = r^nx[n],\quad r\in \mathbb{R}$$ And why? I think this system is stable because $$| x[n] | ≤ B,\quad B < ...
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0answers
14 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. ...
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16 views

Phase reference of a periodic signal

Assume an arbitrary (discrete) signal that is periodic and known over a whole period. I need a way to select a characteristic point along the signal such that I can always retrieve it even when the ...
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1answer
85 views

Sketching the spectrum of a signal

The figure below shows Fourier spectrum of a signal $g(t)$ Sketch the spectrum of the signal $2g(t)\cos^2(100\pi t)$. Show value in sketch.
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1answer
45 views

Query regarding phase of analytic signal.

Using the following matlab commands, phase of analytic signal can be obtained. sig_a = hilbert ( sig ) ; ph = atan(angle(sig_a)) ; As for as my understanding is angle(sig_a) will give phase of ...
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0answers
12 views

number of possible component in sinusoidal model

suppose that we have following model $y[t]=A_1(sin(\omega_1*t+\phi_1)+A_2*sin(\omega_2*t+\phi_2)+....+A_p*sin(\omega_p*t+\phi_p)$+$z(t)$ my question is not related how to determine number of ...
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3answers
234 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 ...
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1answer
124 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 ...
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1answer
53 views

laplace transform of a sine function

I'm a little confused about how to find Laplace transforms of a sine function when it is a function of time. As in, suppose the function is $x(t)=\sin(at)$ , then I can proceed to get ...