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11 votes
Accepted

Why is $A\cos(2\pi f_ct)$ a non-stationary process?

A random process is a collection of random variables, one random variable for each time instant. It is best to write the random process as $$\{X(t)\colon -\infty < t < \infty\} \tag{1}$$ where ...
Dilip Sarwate's user avatar
10 votes
Accepted

Understanding of Random Process, Random Variable and Probability Density Function

when we observe the Random Process at a specific time $t_k$, that is the value at $X(s_1,t_k), X(s_2,t_k),\ldots,X(s_n,t_k)$, if we denote them by $(a_1,a_2,\ldots, a_n)$. Now the mapping between ...
AlexTP's user avatar
  • 6,595
9 votes

Understanding the definition of mean/autocorrelation

The definition of the autocorrelation function $R_x(\tau)$ depends on the nature of your $x$. If $x$ is a deterministic signal with finite energy then: $$R_x(\tau)=\int_{-\infty}^{+\infty}x(t)x^*(t-\...
Learn_and_Share's user avatar
6 votes

Processes: Orthogonal, Uncorrelated, Statistically Independent

You got some definitions wrong. It's correct that orthogonality means that $E[XY]=0$. Uncorrelated means that $X-\mu_X$ and $Y-\mu_Y$ are orthogonal, i.e., $E[(X-\mu_X)(Y-\mu_Y)]=0$. If you work that ...
Matt L.'s user avatar
  • 90k
5 votes

What's the meaning of ergodicity?

If you sample a random process for a specific t, you will get one realization of a random variable. For another t, you get another realization of that random variable. This random variable has its ...
QMC's user avatar
  • 784
5 votes
Accepted

generating white gaussian noise in matlab using two different functions

A white noise sequence is one for which each (random) element is uncorrelated from every other element: $$ E[y[n]y[m]] = \left \{ \begin{array}{ll} 0 & \mbox{for } n\not=m\\ \sigma_y^2 & \mbox{...
Peter K.'s user avatar
  • 25.7k
5 votes
Accepted

Relationship between the autocorrelations of X(t) and X(nt)

The answer to the OP's question is more straightforward than rb-j's comments make it out to be. $\{X(t)\colon -\infty < t < \infty\}$ is a continuous-time WSS random process with ...
Dilip Sarwate's user avatar
5 votes
Accepted

Higher-order moment of output of LTI system

For the case of input process $\{X(t)\}$ being white Gaussian noise with two-sided power spectral density $\frac{N_0}{2}$, the output process $\{Y(t)\}$ is a strictly stationary zero-mean Gaussian ...
Dilip Sarwate's user avatar
4 votes

Ergodicity of joint process

Joint behavior cannot always be deduced from individual behavior. For example, if $X$ and $Y$ are (nondegenerate) random variables with finite means, then $P\{X<E[X]\}$ and $P\{Y<E[Y]\}$ both ...
Dilip Sarwate's user avatar
4 votes
Accepted

LTI filtering for wide-sense stationary process

It's basically just about using the definitions and doing the math: $$\begin{align}R_{WW}[n]&=E\{W^*[k]W[k+n]\}\\ &=E\left\{\sum_mU^*[k-m]h^*[m]\sum_lU[k+n-l]h[l]\right\}\\ &=\sum_m\...
Matt L.'s user avatar
  • 90k
4 votes

Why is $A\cos(2\pi f_ct)$ a non-stationary process?

In easy words: A process is stationary if its stochastic properties are independent of the time you look at it. Think of it like this: A stochastic process is just a Random Variable (RV) that, ...
Marcus Müller's user avatar
4 votes
Accepted

Intuition about independent signals

Think it this way; assuming $x[-1] = 0$, then recusively compute the output $x[k]$ for $k \ge 0$ such as $$\begin{align} x[0] &= v[0] \\ x[1] &= a x[0] + v[1] \\ x[2] &= a x[1] + v[2] = a^...
Fat32's user avatar
  • 28.2k
4 votes
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How to find the output mean and autocorrelation of a non-linear system

The OP's updated working is incorrect. Following up what Hilmar suggested gives \begin{align} Y(t) &= a\left(X(t)\right)^2\\ &= a\left(S(t) + N(t)\right)^2\\ &= a\left(S(t)\right)^2 + 2aS(...
Dilip Sarwate's user avatar
4 votes
Accepted

Bandpass Stationary Stochastic Process

I've been a bit hesitant about adding another answer to the existing ones, but since no answer has been accepted yet, and since from Dan's own answer it seems to me that there might still be a few ...
Matt L.'s user avatar
  • 90k
3 votes

Understanding the definition of mean/autocorrelation

In addition to stationarity, your process must be ergodic to relate these two definitions. Ergodicity tells us joint probability of your signal's value at two instant of time (which only depend on ...
Mohammad M's user avatar
  • 1,327
3 votes
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Understanding of Random Process/Random Variable

Let me explain it in another way. Consider you have 6 different function of time. You only throw your dice once and regarding the outcome you choose on of six functions and the chose one is one ...
Mohammad M's user avatar
  • 1,327
3 votes

Autocorrelation and Power Spectral Density (Discrete)

First of all lets state more correctly that a discrete-time 1D auto-correlation sequence (ACS), $\phi_{XX}[\kappa]$, of a single parameter $\kappa$, of a discrete-time random process $X[n,s]$ will ...
Fat32's user avatar
  • 28.2k
3 votes
Accepted

Response of Linear System to Stochastic Process

Consider the following LTI system with impulse response $h[n]$ $$ \{v[n]\} \longrightarrow \boxed{H(z)} \longrightarrow \{u[n]\} $$ From the analysis of LTI systems with WSS random inputs, the ...
Fat32's user avatar
  • 28.2k
3 votes
Accepted

What really means stochastic in field of signal processing

Well, getting a bit linguistic, according to the Oxford dictionary: stochastic (adj.): Having a random probability distribution or pattern that may be analysed statistically but may not be ...
Tendero's user avatar
  • 5,020
3 votes

Is there any computational method to prove whether a series is stationary or not?

1 Proof You're misusing the word proof. Remember that a proof, even one led with stochastic methods, always leads to an absolute "If A, then B, no doubt" statement. Since your $x$ is a realization ...
Marcus Müller's user avatar
3 votes
Accepted

Doubt about wide sense stationary random process

The mean of $X(n)$ is always $\mu_X =0$, because the noise has zero mean. Thus we should check whether the autocorrelation corresponds to a WSS process. $$\begin{align} R_X(n_1,n_2) &=\mathbb{E}[...
Tendero's user avatar
  • 5,020
3 votes
Accepted

Null autocorrelation function and stationary

If E[X(t)] is constant and RX(t+τ,t)=0 can I say the process is WSS? Can I say RX(t+τ,t)=0=RX(τ) and therefore is WSS? Two times the same question. It fulfills the definition (as you noticed yourself)...
Marcus Müller's user avatar
3 votes
Accepted

What is an "innovation filter"?

An innovations filter of a WSS process $X(t)$ is a causal and stable minimum phase filter that can be used to generate $X(t)$ from a white noise input $N(t)$: $$X(t)=\int_0^{\infty}h(\tau)N(t-\tau)d\...
Matt L.'s user avatar
  • 90k
3 votes

Band-limited random signal with arbitrary distribution?

Consider this approach: General a white Gaussian random sequence. Filter the white Gaussian sequence. The output, which we'll call $z$, will be Gaussian because a linear combination of Gaussians is ...
Ill-Conditioned Matrix's user avatar
3 votes
Accepted

When deriving the power spectral density of stochastic processes, why does taking an expectation allow the $T\rightarrow\infty$ limit to be taken?

It is not the expectation operator that makes sure that the limit exists. The expectation just results in an ensemble average, which we need to obtain a deterministic function $S(f)$ for the power ...
Matt L.'s user avatar
  • 90k
3 votes
Accepted

Moving from deterministic signals to stochastic signals in s-domain (Power Spectral Density)

You have to look at the autocorrelation function of $y(t)$: $$R_y(\tau)=E\{y(t)y(t+\tau)\}\tag{1}$$ with $$y(t)=(h_A\star a)(t) - (h_B\star b)(t)\tag{2}$$ where $\star$ denotes convolution. If you ...
Matt L.'s user avatar
  • 90k
3 votes
Accepted

Noise from irregular sampling pattern

The core is the sentence directly before the one you cited from the second link: The coherence of the samples interferes with the coherence of the image to produce errors called aliasing. The ...
Max's user avatar
  • 2,368
3 votes

Bandpass Stationary Stochastic Process

Work in progress Please hold all comments, complaints, questions, brickbats and downvotes unti I am done and have removed this request. Synthesis: Suppose $\{X(t)\colon -\infty < t < \infty\}$ ...
Dilip Sarwate's user avatar
2 votes
Accepted

What does the frequency axis of a Power Spectral Density mean?

The fact that the frequency variable of a power spectral density (PSD) equals the one of a Fourier transform of a "normal" time-domain signal can be seen more easily by considering the ...
Matt L.'s user avatar
  • 90k
2 votes

generating white gaussian noise in matlab using two different functions

wgn() is specifically meant to create a white noise with a predefined power levels while randn() is meant to generate normally distributed random numbers WITHOUT specifying the power. You will have to ...
Amal's user avatar
  • 353

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