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Questions tagged [random-process]

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2
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2answers
105 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 ...
0
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2answers
43 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]...
1
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0answers
24 views

Relation between power spectral density and mean absolute value

The root mean square $$\sigma_{x} = \sqrt{\frac{1}{T}\int_0^T x^2(t) \, \mathrm{d}t}$$ of a finite zero-mean random signal $x(t)$ in the range $0 < t < T$ is related to the signal's power ...
0
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0answers
15 views

Analytical spectral density of a On/Off modulation defined by a Bernoulli process

Consider a narrow band signal (laser) that I can modulate digitally with a on/off switch controlled by a digital pseudo random number generator. The resulting signal features a linewidth broadened by ...
0
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1answer
123 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[...
0
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0answers
21 views

Images as Markov chains

I have seen literature on representing black and white images as probability distributions and then computing "distances" between them, for example, in optimal transport. I was wondering if there is ...
-1
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0answers
14 views

The variance of the correlation output of a White Gaussian Noise and a BPSK signal?

Suppose the White Gaussian noise is $n(t)$ with zero mean and power density $N_0/2$. The BPSK signal is $b(t)\sin(2\pi f_c t)$. And $b(t)$ is the rectangular waveform as $$ b(t) = \sum_n B_n P_{T_c}(...
0
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1answer
26 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 ?
0
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0answers
55 views

On the spectral representation of deterministic and random signals

I went back to many references in order to fix some of the confusions that I have on many concepts in signal spectral representation. I concluded that: 1) Deterministic signals may be represented ...
3
votes
2answers
130 views

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\...
0
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0answers
11 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 ...
3
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1answer
69 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\\ \...
14
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3answers
12k views

What is a good example of an ergodic process?

I'm trying to find simple examples of an ergodic process. What process comes to your mind as a good illustration of its properties? A quick research (Wikipedia, another answer) mainly gives examples ...
0
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1answer
27 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 ...
0
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1answer
46 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). $$ \...
5
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4answers
3k views

Gaussian White Noise - Relation Between Distribution and Correlation

Im a beginner in signal processing so my question may be obvious. A white noise has the property to have its autocorrelation function that is equal to $$\mathbb{E}[f(t+\tau)f(t)]=\sigma^2 \delta(\...
0
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3answers
494 views

How Are Images Considered Non Stationary Signal When They Are Invariant to Time?

I have read Wavelets are better than Fourier in dealing with non-stationary signals such as images, but I don't understand how images are considered stationary??
2
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1answer
1k views

Random process $X(t)$ with autocorrelation function given find the mean and the variance

Autocorrelation function is $$R_{xx}(\tau)=\frac{20}{1+2\tau^2}$$ So at $\tau=0$$$R_{xx}(0)=20=E[X(t)X(t)]=E[X^2(t)]$$ The variance is $$\mathrm{Var}[X(t)]=E[X^2(t)]-E^2[X(t)]=20-E^2[X(t)]$$ As $X(t)$...
0
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1answer
67 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)...
2
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2answers
195 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 $N_0/2$...
0
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1answer
226 views

Power Spectrum: Definition

I am new to the study of time series. Recently I have asked a question about the covariance of real and imaginary part of a real(in time domain) stochastic time series and I have received an answer ...
1
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2answers
53 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 ...
0
votes
1answer
49 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 ...
1
vote
2answers
45 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 ...
5
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1answer
83 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 ...
1
vote
1answer
138 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 ...
1
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1answer
104 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 ...
0
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1answer
100 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-...
3
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0answers
73 views

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 ...
1
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3answers
1k views

Sum of Sine and Cosine with Random Phase as LTI System

I have the following system: Where $ {H}_{1} \left( f \right) = {H}_{2} \left( f \right) $ and $ \theta \sim U[0, 2\pi]$ independent of any other factor in the system. Given the input is identical, ...
0
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2answers
160 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 ...
1
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1answer
38 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)=...
0
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0answers
95 views

What is SNR of Signal with Additive White Gaussian Noise [duplicate]

Calculating the power of AWGN should be equal to infinity as PSD is constant and its integration is infinity over all frequencies. Hence for any signal with finite power mixed with AWGN, SNR should ...
16
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3answers
56k views

Variance of White Gaussian Noise

It could seem an easy question and without any doubts it is but I'm trying to calculate the variance of white Gaussian noise without any result. The power spectral density (PSD) of additive white ...
0
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1answer
251 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) +...
1
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2answers
74 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, ...
2
votes
1answer
57 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 ...
3
votes
1answer
284 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(...
1
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2answers
91 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 ...
0
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2answers
219 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
1
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1answer
210 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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0answers
39 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) ...
3
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1answer
78 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 ...
3
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1answer
3k 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, ...
2
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2answers
1k 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 ...
0
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1answer
206 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{...
1
vote
1answer
319 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 $x(t-\tau)$, $\tau\ge0$ ...
2
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2answers
2k 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 ...
0
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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-...
1
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2answers
133 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 ...