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Questions tagged [signal-processing]

Questions on the mathematical aspects of signal processing. Please consider first if your question might be more suitable for http://dsp.stackexchange.com/

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Are $\delta(x-5)$ and $\delta(5-x)$ the same function?

As the title says, I'm confused whether $\delta(x-5)$ and $\delta(5-x)$ are the same function. Similarly, are $\delta(x-5,y-6)$ and $\delta(5-x,6-y)$ the same function? To, me it seems they are! But ...
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0answers
16 views

Inverse of a ramp function [on hold]

If an upward ramp as shown in this picture function's formula is (t-1)1(t-1) where 1 is the delay, then what would be the function for a downward ramp? Say, if the signal formed a triangle instead of ...
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0answers
10 views

Why does the mixing matrix in independent component analysis have to be square?

In ICA, we have a set of observations X which are made of a linear combination of independent components S. They are related in the following way: X = AS Where A ...
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1answer
15 views

Compute the DTFT of this signal, geometric series [closed]

$$ x[n] = 2^n u[-n]$$ Can someone explain to me the Geometric series of this signal and the relevant workings? Thanks a bunch!
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0answers
13 views

Uniqueness of solutions when input is discontinuous at $t_0$.

I have recently been reading Signal Processing and Linear Systems by B. P. Lathi, but I also have some general knowledge of differential equations from reading parts of Elementary Differential ...
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0answers
7 views

How to compute the amplitude of a sampled wave in three dimensional space [closed]

I have a sine or cosine wave in a random direction in 3D space and in a random orientation. I want to calculate the amplitude and axis for such a wave. I am sampling the wave at a high rate, compared ...
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1answer
27 views

Comparability of Analytic Signal with (Discrete) Fourier Analysis

I am working with a finite signal response from an experiment. Basically, I feed in a uniform amplitude sine wave which ramps from 20Hz to 20kHz over the course of 50 seconds, and I read the output, ...
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0answers
12 views

Magnitude spectrum of a sampled time continuous signal

I have a signal $x(t)=\cos(2 \pi 200 t)+2\cos(2 \pi 400 t)+\cos(2 \pi 600 t)$. With sample frequency $F_s=1000 Hz$. How do I draw the magnitude spectrum of the sampled signal $x(n)$? I've tried ...
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0answers
17 views

Is the Shannon capacity for a baseband signal different than its modulated counterpart?

I understand the Shannon-Hartley theorem to define channel capacity $C$ as $$C=W\log\left(1+\frac{S}{N_0 W}\right)$$ where $W$ is the channel bandwidth, $S$ is the signal power, and $N_0$ is the ...
1
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1answer
32 views

Fourier coefficients double half-wave rectifier

I’d like to find the Fourier coefficients of $$ x(t)=|A \cos(2 \pi f_0 t )| $$ The period is $ T_0 / 2 $ so i applied the definition $$\frac{1}{T}\int_{0}^{L} |A \cos(2\pi f_0 t )| \cos \left(\frac{k ...
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0answers
15 views

Please help me understand this paper about Discrete Wavelet Transform!

I am trying to understand this paper says. I have found this paragraph to be vague: (a) Preprocessing Stage. The first step in this stage was to apply the Discrete Wavelet Transform (DWT). This ...
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1answer
26 views

What information could be extracted from this Discrete Wavelet Transform of a time series data?

I have decomposed this time series data in 5 levels using Discrete Wavelet Transform Daubechies 4. I am trying to understand what kind of information one can extract from applying DWT to a time ...
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0answers
9 views

What frequencies do the coefficients of a wavelet decomposition represent?

Every time I decompose a time series signal using Discrete Wavelet Transform I get a signal with half the frequency, right? Suppose I have a time series signal sampled at 60 Hz. The first discrete ...
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0answers
9 views

How exactly Wavelet Transform will be helpful to analyze a signal?

I have a time series noisy ECG signal data. My whole intention is to get that data and extract the heartbeat signal from it and discover the BPM rate. I have managed to remove the noise from it ...
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1answer
18 views

What is the mathmagic involved to convert a gain equation in terms of discrete formula

Sorry my poor knowledges about advanced math. Consider this post that talks about biquad filters. At the beginning of the post the author shows the filter's transfer equation is $$ H(z) = \frac{Y(...
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0answers
36 views

Multiplications between four convolutions (discrete time). With partial solution.

This part of the thesis, PAM4 signal filtering. p - convolution noise x - input signal (PAM4) $$\ E(p^4[n]x^2[n]) = $$ Where E - is Expected Value and p[n] = x[n]*e[n], * - denotes convolution. ...
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0answers
18 views

Dose there exist a map with high capacity(i.e. can be modeled by a lot of parameters) that maps a function to another?

Note: The following two paragraph have nothing to do with the question I ask, by they explain my motivation. Let $S = \{{\sum_{i}\delta (t-t_i)\ |\ t_i \in[0,T]\}}$, where $\delta$ is the Dirac ...
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0answers
25 views

Is there something similar to Fourier Series where the harmonics are division by n?

thanks for any help! Do there exist any techniques relating to writing a periodic function as a sum of sines or other functions where the inputs to those functions are x/n (for some n) instead of n*x (...
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0answers
29 views

Multiplications between four convolutions (discrete time).

Stuck on this, for long period. Need some help. Working on the thesis, PAM4 signal filtering. p - convolution noise x - input signal (PAM4) $$\ E(p^4[n]x^2[n]) = $$ Where E - is Expected Value and ...
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0answers
9 views

Constructing signal vectors for machine learning

I have daily stock returns related to sectors. At the end of each month I want to construct a vector of signals using the past data with different methods over different moving windows like EWMA over ...
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1answer
54 views

Determine Discrete-Time Fourier Transform of exponential or sine with time-shift?

Consider the following periodic and time-discrete signal: $$y[n] = \sin (\Omega_0 n+\phi)$$ How do I determine the discrete-time Fourier transform for $-\pi<\Omega<\pi$ for that signal? I ...
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1answer
29 views

Distance between the noise and the corrupted signal

How can one formalize the fact that the law of $X+Z$ where $X \in \mathbb{R}^d$ is any vector-valued random variable and $Z\sim \mathcal{N}(0, \sigma^2 \mathbf{I}_d)$ closely resembles the law of $Z$ ...
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0answers
11 views

Solving system of inequalities where individual variables already have constraints

I'm currently trying to come up with an alternative to the Griffin Lim algorithm in signal processing, and I have come to a stage where I need to solve a system of inequalities where the variables are ...
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0answers
22 views

Understanding the process for plotting $x(t)=(1/2)*δ(t+2)$

I have stumbled upon an exercise where I have to plot/calculate the output of different variations of input functions in a LTI system similar to the one given as a example below (which is supposed to ...
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0answers
8 views

Describing a discrete time LTI system by difference equation and impulse response

Consider a system where the output is a discounted summation of all previous inputs, which can be described by one of the following two input-output relationships; $y[n]=a y[n-1]+x[n]$ $y[n]=\sum_{k=...
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1answer
21 views

Cross-fading between 2 values on number lines.

Background: I'm working on writing a software application where I need to cross fade between 2 digital audio signals. In the software I have 4 variables that play a role: ...
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0answers
49 views

Discrete-time Fourier transform of $x[n] = 1$

How to calculate the DTFT of $1$? The sequence $x[n] = 1$ is not absolutely summable, so one can not compute the DTFT by using the definition $$X(\Omega) \space = \sum \limits_{n=-\infty}^{\infty}x[...
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1answer
35 views

Exponential, cosine, sine

To begin with, I am sorry if my question is pathetic but I cannot understand why sometimes exponential involves cos and sometimes both cos and sin. I know that ...
2
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1answer
118 views

Optimal estimator of Poisson distribution

As part of my homework, I was asked to solve the following: a signal and noise are stochastically independent and it is known that $n\sim Poiss(\mu_n)$ and $x\sim Poiss(\mu_x)$ we define $y=x+n$ ...
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0answers
24 views

Understanding scaling and zero padding in 1D convolution

I am trying to understand discrete convolution using scaling $S$ and zero-padding of a vector with the formula given according to: $$ R_{t} = \left\{ \begin{array}{ll} \frac{1}{S}\sum_{i=0}^{k-...
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1answer
21 views

Is this the correct way to perform filtering using Wavelet decomposition?

My math is a little bit rusty. I am learning a lot of stuff recently and some stuff is not clear yet. I ask anybody answering this, to not answer using complex mathematical notation because like I ...
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0answers
30 views

Fourier coefficients of a sum

I have to find the Fourier coefficients of $ x(t)= A_1 cos (2 \pi f_0 t + \gamma_1 ) + A_3 cos (6 \pi f_0 t + \gamma_3 ) $ First of all I applied the definition $ C_k = \frac{1}{T_0} \int_{0}^{T} {...
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0answers
17 views

Why is there noise for higher bandwidth intensity distributions?

I have been trying to computationally evaluate the following E-field computationally: It is basically the sum of waves of a given wavelength, weighted by the square root of a gaussian distribution ...
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2answers
57 views

Adding two cosine waves, result in the standard A*cos(ωt + B) form

Let's say I have the composite wave: $y(t) = \cos(40t) - 0.3 \cos(40t - 16)$ This combines two waves of the same frequency but different phases, and the objective here is to add them and present the ...
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0answers
14 views

The distribution of phase angle between identical zero mean Gaussian random signals?

I am trying to solve the 2013 paper set by ISRO for electrical engineers. I was hoping to get some clues on how to approach this problem - No 22. random signals which have gaussian distributions will ...
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1answer
19 views

Fundamental frequency of the sum of two sinusoids

I'm trying to find the fundamental frequency of $\cos(23t + \pi/2) + \sin(5t + \pi/5)$ Now the period for both are $2\pi/23$ and $2\pi/5$ respectively. So i need to find the LCM of these two ...
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0answers
24 views

Cross-correlation of a deterministic signal and white Gaussian noise

I'm trying to describe the cross-correlation of a finite length input signal x[n] with the same signal corrupted by white Gaussian noise. If the signal would be infinitely long, the noise would be ...
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0answers
14 views

Autoregressive process with random walk perturbation.

Suppose we have an autoregressive process, $$y_t=\phi y_{t-1} +u_t$$ where $|\phi|<1$. If $u_t$ is an i.i.d random variable this process is stationarity. What if $$u_t=u_{t-1}+g+\epsilon_t$$ where $...
1
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0answers
20 views

Fourier Series for Rectangular Pulse Train (Natural Sampling Pulse Amplitude Modulation)

i'm trying to calculate the frequency spectrum of a signal that has been sampled using natural sampling PAM (pulse amplitude modulation). The message signal is given as $s(t)$, and the sampling signal ...
0
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1answer
18 views

Mapping [nonlinear] sensor input to sensor output, and vice-versa

I am hoping someone can refer me to a text reference, forum, or other online resource describing the following question. I thought this forum might be a good place to start, but apologies if I was ...
3
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1answer
56 views

What theorem in topology was Claude Shannon referring to?

In 1949, in a classic paper, Claude Shannon wrote the following: As we change the message a small amount, the corresponding signal will change a small amount, until some critical value is reached. ...
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0answers
24 views

Iterative gaussian convolution and decimation when the decimation/convolution ratios are not divisible by two?

I have a bit of a mathematics problem. I have an image $I_1$ that I have previously blurred (from an image $I_0$) with a Gaussian of standard deviation $\sigma_1$. I would like to blur image $I_1$ ...
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1answer
47 views

Smoothness of a time series: relationship between ARMA model and signal derivatives

I have a discrete stochastic function $f(t_k)$ (in my case, it's a time signal related to atmospheric parameters). I know this function to have a certain smoothness. One way I can specify its ...
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0answers
28 views

Region of convergence in discrete system?

When dealing with a discrete system, it is a widely-accepted idea to use the $z$-transform to analyze properties of the system in frequency domain. However I'm confused of two questions, regarding the ...
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0answers
31 views

The significant role of dual frames in the progress of Frame theory.

For a given frame $\{\zeta_i\}_{i=1}^\infty$, any Bessel sequence $\{\eta_i\}_{i=1}^\infty$ satisfying in the following identity for every $\xi\in H$ $$\xi=\sum_{i=1}^\infty \langle \xi, \eta_i\...
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0answers
14 views

difference/increment in discrete time signal/sequence?

What should the difference between consecutive numbers/intervals on the $x$-axis in a discrete-time signal be? Should it always be $1$ or can we have spacing/gap between consecutive intervals as $0.1$ ...
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0answers
20 views

Discrete Representations of Continuous Surfaces

I apologize if this question is too general, but I'm an electrical engineering student currently taking a numerical methods course and I'm unsure about the terminology. My question is as follows, ...
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3answers
67 views

how to Find the period of complex exponential function?

How we can find period of this sequence? $x[n]= e^{jn2π/3}$ Is it equal to $T=2π/(2π/3) $ or not? I mean relation $T=2π/\omega$ will be valid in this case or not?
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0answers
17 views

Biased estimates of Hurst exponent in R/S analysis

I've used the standard R/S algorithm for estimating the Hurst exponent in Mathematica*, and tested it on fBm and fGn for $H\in\{0.05,0.1,\ldots,0.95\}$, generating 1000 time series for each $H$. The ...
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1answer
128 views

Recovering original image from its edges

Suppose we read an image $X$ with $1\times P$ dimensions (a single row and $P$ columns) and apply to it the simplest edge detector, that calculates the horizontal derivative say, $F = [1, 0, −1]$ to ...