# Tag Info

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 ...
• 20.4k
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### 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 ...
• 6,595

• 25.7k
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 ...
• 20.4k
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 ...
• 20.4k

### 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 ...
• 20.4k
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• 28.2k
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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(...
• 20.4k
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### 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 ...
• 90k

### 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 ...
• 1,327
Accepted

### 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 ...
• 1,327

### 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 ...
• 28.2k
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### 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 ...
• 28.2k
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 ...
• 5,020

### 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 ...
• 30.7k
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• 90k

### 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 ...
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 ...
• 90k
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### 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 ...
• 90k
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 ...
• 2,368

### 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 $\{X(t)\colon -\infty < t < \infty\}$ ...
• 20.4k
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 ...
• 90k