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Questions tagged [random-process]

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How to find a variance of sample sequence

I have a sequence such as $$r[n] = y[n]v[n]$$ y[n] and v[n] are zero-mean and statistically independent. I need to find a variance of $r[n]$ and show that it is white and equal to $\sigma ^2_y\...
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10 views

Rate distortion function for a Gaussian process with a squared exponential kernel

This is probably a question whose answer should be available in some paper or textbook, but my searching for it hasn't helped me find a result that I could use. The question is basically just what ...
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1answer
14 views

Particular Correlation formula

I'm reading a book where the autocorrelation of white noise is expressed as: What is the term $Q(k)$ and why is is expressed as an average value of a dot product ?
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1answer
60 views

Simulate time series given temporal auto-correlation functions

Given a random process $x[n] \in \mathbb{R}$ (say of length $N$) and all correlation functions such as: \begin{align} \langle x[i]\rangle\\ \langle x[i]x[j]\rangle\\ \langle x[i]x[j]x[k]\rangle\\ \...
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1answer
17 views

response of LTI system to a Random Input Signal

what is LTI filter? what is the output when x(t) is input? let x(t) be the input signal to the system and y(t) denote the output signal. The output of the system may be expressed in terms of ...
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3answers
62 views

Band-limited random signal with arbitrary distribution?

I'd like to generate a random discrete-time signal that is band-limited to some bandwidth B (by means of a digital filter, ie in MATLAB). The catch is that I'd like this signal to have an arbitrary ...
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1answer
41 views

Cramér-Rao lower bound

I have been trying to implement the Cramér-Rao lower bound from the paper - A reference-free time difference of arrival source localization using a passive sensor array (eq. 6 and eq. 7). $$ \...
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2answers
70 views

PSD of complex white gaussian noise

It may be a really simple question, but I'm not sure about this one: Given a complex white Gaussian noise process with iid real and imaginary parts and a double sided power spectral density of $N_0/2$...
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1answer
28 views

expected value of two LTI output signals multiplied (cross correlation)

I have an input signal x (assumed to be iid Gaussian with $\mu=0$, $\sigma^2$) which is fed into two linear systems: $y_1 = h_1 * x$ $y_2 = h_2 * x$ Now I would like to calculate $\mathbb{E}[y_1 y_2]...
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2answers
48 views

Random Process at a particular time instance

I was studying Random Process and I thought I understood what it was all about until I came across this example. Consider a random experiment of tossing a coin with sample space S = {H, T} The sample ...
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1answer
35 views

Converting a non-stationary random process into a WSS process by adding a random phase

Here is an example where this method has been implemented. We were trying to calculate the spectrum of a transmitted signal(Random signal/weighted pulse) The auto correlation function of the pulse ...
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2answers
40 views

Fourier-Analysis of Stationary Random Signals

Let's say we have discrete-time stationary random signals with Gaussian PDF of mean value 0 and variance 1, whose individual signal values are uncorrelated. For such a signal, how can we determine ...
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1answer
80 views

Physical interpretation of 4th-order correlations

BACKGROUND: Let's say we have samples of a random process $X(t)$ at two different times, $t_1$ and $t_2$, denoted $X(t_1), X(t_2)$. The values of $X(t)$ represent some voltage-like quantity (i.e. a ...
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1answer
95 views

Random Signals - statistical properties are time dependant?

I'm taking a course on DSP and we're being introduced to the random signals, in particular continuous time and discrete time random signals. We're told that if we repeat a single random experiment at ...
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1answer
83 views

Are two jointly stationary white noise processes independent?

I am currently dealing with a problem concerning beamforming, where two "jointly stationary zero-mean white noise processes" form the input of an adaptive system. One of those processes resembles the ...
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1answer
56 views

Covariance matrix associated with random DC level in Gaussian noise

Given a signal $x[n] = A + w[n]$ where $A$ is a Gaussian random variable and $w[n]$ is Gaussian white noise, then the covariance matrix of the signal is given by $[C(\sigma^2_A)]_{ij}=E[x[i-1]x[j-...
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1answer
32 views

Null autocorrelation function and stationary

I can show that a process $X(t)$ is Wide Sense stationary (WSS) by showing that $E[X(t)]$ is constant and that its autocorrelation function is in function of $\tau=t_1-t_2$, that is, $R_X(t+\tau,t)=...
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2answers
134 views

Understanding PSD: Why Does Power at High Frequencies Affect Low Frequencies?

I'm trying to wrap my head around power spectral density on a conceptual level, but I am having some difficulty. Suppose I have a communication system where I am receiving and sampling white Gaussian ...
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1answer
127 views

Autocorrelation of Addition of Two Independent Signals

Given a random signal $ Z \left( t \right) $ which is addition of two independent signals $ X \left( t \right) $ and $ Y \left( t \right) $ with constant parameters $ a $ and $ b $: $$ Z (t) = aX(t) +...
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24 views

Estimating a Function from its Covariance Function

I am trying to solve this question: Suppose that $n(t)$, $−∞ < t < ∞$ is a stationary Gaussian random process with covariance function $E\{n(t)n(t-\tau)\} = \delta(\tau) + {5 \over 4}e^{-\left|\...
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2answers
56 views

Applying the CUSUM algorithm to a correlated random process

As far as I know, the CUSUM algorithm is meant for detecting change points on discrete-time uncorrelated random processes. For instance, to apply the CUSUM algorithm to a discrete Gaussian process, ...
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39 views

Computing Power spectral Density

Acronyms: Power spectral density, PSD Autocorrelation, AC Hey so I'm in my first ever DSP class and thoroughly enjoy the material, but absolutely suffer in the HW. I have this question ...
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1answer
54 views

Does a collection of Gaussian random variables necessarily constitute a Gaussian Process?

If $\{X(t)\}$ is a Gaussian Process then the random variables $X(t_k)$ where $k = 1,2,3...n$, are jointly Gaussian. If each random variable $X(t)$ is a Gaussian variable, then will the random ...
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1answer
204 views

Mean Square Continuity of Random Process

Show that a stochastic process $X(t)$ is mean square continuous if and only if its autocorrelation function $R_X(t_1,t_2)$ is continous $\Rightarrow$ Proof: We have $E[(X(t)-X(t_0))^2]=R_X(t,t)-R_X(...
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Energy Detection in Presence of Colored Gaussian Noise

Before asking my question, let me introduce the context: For spectrum sensing based on energy detection, which has been widely studied in presence of AWGN, the optimal detection threshold is computed ...
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20 views

Random Vibration Parameters & Values

I have been searching similar questions and am not having much luck. I currently have a program setup for random vibration already and a request for random vibration to different settings.(chart and ...
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2answers
158 views

Applications of Power Spectral Density [closed]

I have a class covering Power Spectral Density but I have no idea why it matters. Could someone provide some examples of its use? Thanks
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1answer
197 views

Is the sum of white noise and shifted white noise white noise again?

Let $W[k]$ be a stationary white noise with variance = 1 Question: Is $X[k] = W[k] + c \cdot W[k-1]$ white noise? $c$ is a real number.
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2answers
75 views

Characteristic and moment generating function of a random variable interpretation

I have been studying about moments and cumulants of a random variable. Even though the definitions of characteristic and moments generating function are very similar (only the sign in the exponential ...
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0answers
38 views

A wide sense stationary random process that is not second order stationary [duplicate]

I have been reading Peebles Probability, Random Variables, and Random Signal Principles and it claims that second-order stationarity is sufficient to guarantee: $E[X(t)]$ is a constant $R_{XX}(t1,t2) ...
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1answer
121 views

Approximating a Gaussian Process

Suppose that $\theta_t$ is an exogenous variable with known Gaussian process. Next, suppose that for any index $i\in [0,1]$, $$ a_{i,t} = (1-\beta)\mathbb E[\theta_t|\mathcal I_{i,t}]+\beta \mathbb E[...
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1answer
70 views

Dimensional analysis of integrated white noise process

This question is somewhat related to this post. Let us consider we have a white noise current source $i_n(t)$, with a variance $\sigma_i^2$, and mean, $\mu_n=0$. Assume that this current is passed ...
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1answer
2k views

variance in the time domain versus variance in frequency domain

Hi All: I'm trying to better understand the connection between variance of a time series and the integral of the spectral density over all frequencies. Rather than going through all of the relations, ...
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2answers
744 views

Definition of average power?

There are two kind of average power I encountered in random signal class and textbook: definition 1: average power =$$E[|x(t)|^2]=R_{xx}(0)=\int^\infty_{-\infty} S_{xx}(f)\,df$$ definition 2: average ...
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1answer
172 views

Autocorrelation and PSD

Let $X(t)$ and $Y(t)$ be two orthogonal processes with power spectral densities $$S_{xx}(f) = S_{yy}(f)=\begin{cases} 1-\lvert f\rvert, & \lvert f\rvert<1 \\[1ex] 0,& \text{otherwise} \end{...
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1answer
63 views

Is there a way to obtain the original signal (stationary process) from its combination through filtering (matlab) and crosscorrelation?

I have a stationary process $w_1(t)$, white in band $B=[-2, 2] KHz$, and another process: $x(t)=w_1(t)-w_1(t+t_0)$, where $t_0=250\mu s$. I want to re-obtain $w_1(t)$ by filtering $x(t)$ through $h(t)...
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1answer
260 views

Solving Wiener Hopf integral equation for causal filter of predictor

Given a stochastic signal $x(t)$ with autocorrelation function $R_{xx}(\tau)=\mathrm{exp}(- \alpha|\tau|)$, $\alpha>0$. I want to predict $x(t+\lambda)$,$\lambda>0$ by $x(t-\tau)$, $\tau\ge0$ ...
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33 views

Bandwith of the Gaussian Random Process

$X(t)$ is a zero-mean WSS Gaussian random process. Its power spectral density is $$S_x(f)= (4 \times 10^{-5}) \text{tri}(\frac{f}{10^{5}})$$ Where $\text{tri}$ is the triangle function. What is the ...
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2answers
1k views

Capacity of cascade binary symmetric channels

Let's imagine that we have interconnected in cascade $L$ binary symmetric channels each with the same transition probability $p(y|x) \in \{p, q=1-p\}$, where the output of each BSC is connected to the ...
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1answer
30 views

log-likelihhood function for N sample of data

if $x(t)=b A e^{ j\omega t} + e(t)$ for $t= 1,2,...,N$ where $b$ is a parameter, $A$ is a vector $M \times 1$, $e(t)$ is a white Gaussian noise with covariance matrix of $Q$ theh what is log-...
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119 views

Autocorrelation function $R_{yy}(t_1,t_2)$?

If $x(t)$ is a zero mean stationary Gaussian process and if $y(t)=x^2(t)$,then $\{y(t)\}$ is called a square law detector process. Now i want to find autocorrelation function $R_{yy}(t_1,t_2)$,that is ...
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27 views

Power Spectrum of Linear Mixtures of Processes

Suppose we have $n$ WSS stochastic processes $x_i(t)$ , and we construct another process from them using a linear mixture, $y(t) = (\sum_{i=1}^n w_ix_i(t)) + b$. If these were deterministic, then the ...
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2answers
70 views

What is the distribution of it?

If $\theta$ is uniformly distributed in $(0, 2\pi),$ then what is the distribution of $e^{i\theta},$ where $i = \sqrt{-1}?$ And what are the statistical properties of $\left[e^{i0\theta}\, e^{i1\theta}...
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1answer
123 views

PDF of a Shifted Rectangular Pulse

I wanted to determine the PDF of a Stochastic Process. I am familiar with the concept of PDF for a Random Variable which maps the outcomes to its probabilities but I am not able to find the PDF of a ...
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1answer
142 views

Spectral flatness or Wiener Entropy for AR(1) and AR(2)

I'm sudiying compressibility of random processes by using Spectral flatness aka Wiener Entropy I would like to know if there is any reference which derives this quantity, for autoregressive ...
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1answer
478 views

Average Power Spectral Density of PAM signals

I am reading through the PAM transmission scheme and about the power spectral density of the signals. Given that the Average Power Spectral Density of PAM Signals is: $$ \Phi_{ss}(f)=\Phi_{aa}\left(e^...
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1answer
95 views

Testing whether a process is a Wiener process

Ideally I would like links to code implementations (eg. Matlab ) or book references, but I would appreciate suggestions on various methods. We start with sampled process $X_{t}$. A straightforward ...
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1k views

Random signals as power signals

Why are random signals considered as power signals (i.e. signals with infinite energy and finite average power)? Does this make any sense? What does it mean for random signals to have infinite energy ...
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2answers
321 views

Understanding the definition of mean/autocorrelation

I was studying about the definitions of mean, expected value and autocorrelation. I wanted to verify my understanding the evaluation of mean, expected value and autocorrelation. At the same time to ...
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1answer
108 views

Understanding of Random Process/Random Variable

At a simpler level to my previous question, I wanted to confirm my understanding on Random Process based on Random Variables using an example. So, I took this example: If we consider a dice, which ...