16 votes
Accepted

What are the statistics of the discrete Fourier transform of white Gaussian noise?

Math tools We can do the calculation using some basic elements of probability theory and Fourier analysis. There are three elements (we denote the probability density of a random variable $X$ at ...
  • 1,006
16 votes
Accepted

What is a good example of an ergodic process?

Just suppose I give you a series of numbers, and I tell you they were picked randomly. And you know I am not trying to deceive you. Numbers are: $3$, $1$, $4$, $1$, $5$, $3$, $2$, $3$, $4$, $3$. I ...
14 votes

What is a good example of an ergodic process?

From the wikipedia article: a stochastic process is said to be ergodic if its statistical properties can be deduced from a single, sufficiently long, random sample of the process. In other words: ...
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 ...
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 ...
  • 5,865
10 votes

How can a signal be both periodic and random?

Most realistic signals are both random and periodic. For example, you can modulate a harmonic oscillator with a slow enough random signal that moves its frequency around a $\mu_{f}, \sigma_f$. This ...
  • 10.2k
8 votes

If the mean of a random process is constant, does it imply the process is first order stationary?

In the usual sense of the term, first-order stationarity means that the first-order distribution of all the random variables is the same: each $X_t$ has the same CDF, and so the same pdf (or pmf) too ...
8 votes

Gaussian White Noise - Relation Between Distribution and Correlation

When we say the stochastic process $X_{t_1}X_{t_2}\cdots X_{t_i}$ is a Gaussian Process with mean $\mu$ and variance $\sigma^2$ it only means $$X_{t_i}\sim\mathcal{N}(\mu,\sigma^2)$$ So still we know ...
  • 4,135
8 votes
Accepted

Autocorrelation of Addition of Two Independent Signals

You're correct as the Cross Correlation function vanishes. This has the implicit assumption the process has zero mean (Actually, at least one of them). Namely, in order to have $ {R}_{XY} \left( \tau \...
  • 42.6k
7 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-\...
7 votes
Accepted

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

Calculate the autocorrelation of the process. $$\begin{align} R_{xx}[n] &=\mathbb{E}[(W[k] + c W[k-1])(W[k-n] + c W[k-1-n])] \\ &=\mathbb{E}[W[k]W[k-n]]+ \mathbb{E}[cW[k]W[k-1-n]]+\mathbb{E}[...
  • 4,890
7 votes
Accepted

Correlation of independent random processes

No. Quoting Wikipedia's article Independence (probability theory): If $X$ and $Y$ are independent random variables, then the expectation operator $\operatorname{E}$ has the property $$\...
6 votes
Accepted

Why is $\sin(t)$ a stationary process?

$\sin(t)$ is no random process, because there's nothing random about it. You could add a random amplitude to get a random process: $$x(t)=A\sin(t)\tag{1}$$ This is a random process because $A$ is a ...
  • 81k
6 votes

What is the practical meaning of the variance, covariance, mean value?

Adding to Zeeshan's useful answer, here are some additional comments toward some their practical use (not limited to this but may helps add intuition into their use depending on your background): ...
  • 38.4k
6 votes
Accepted

Capacity of cascade binary symmetric channels

A binary symmetric channel (BSC) can be characterized by its complemented probability $p$. Its well-known capacity is $$C = 1 - H(p) = 1 - (-p\log(p) - (1-p)\log(1-p))$$ where $H(p)$ is binary ...
  • 5,865
6 votes
Accepted

How can a signal be both periodic and random?

If you are talking about a given signal as "a deterministic realization of a phenomenon", it can be periodic, but not really random. However, some physical systems are prone to produce randomness ...
5 votes
Accepted

Power spectral density of $\left(x(t)\right)^2$?

Since the question has been raised as to whether the hint that I had given to the OP in a comment on the original question was appropriate for a newcomer to signal processing, here goes. Stripped of ...
5 votes
Accepted

explanation of correlation of stationary stochastic processes

You are mixing up two different notions. Your random process is a collection of random variables $\{X(t)\colon -\infty < t < \infty\}$, one random variable for each time instant. The ...
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 ...
  • 764
5 votes

What is the practical meaning of the variance, covariance, mean value?

Mean of a signal can be practically visualized as the dc average value present in the signal (for a complete sinusoidal period), for e.g Variance of a signal is the difference between the normalized ...
  • 346
5 votes
Accepted

Random signals as power signals

Note that the condition $$\int_{-\infty}^{\infty}|f(t)|^2dt<\infty\tag{1}$$ (i.e., that the signal $f(t)$ has finite energy) is very restrictive when we try to model signals, even though ...
  • 81k
5 votes

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

The spectrum allows you giving another complementary look on the data. It is the other side of the same coin. In Frequency Domain we are looking on the energy of each spectral component of our data. ...
  • 42.6k
5 votes
Accepted

Random Signals - statistical properties are time dependant?

The answer to the question (a counterexample) Properties of random processes will in general be time-dependent. They are not only when talking about stationary processes. Another related concept (not ...
  • 4,890
5 votes

PSD of complex white gaussian noise

With reference to $N_o$ this usually is the symbol for the power spectral density of thermal noise, where $N_o = kT$, where k is Boltzmann's Constant and T is the temperature in Kelvin. With regards ...
  • 38.4k
5 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 ...
  • 81k
5 votes
Accepted

Conceptual Questions on Colored Noise Process

Once you talk about the Spectrum of noise / process you implicitly says it is stationary in the wide sense. What does it mean to have a signal with uncorrelated samples? Do you understand it means you ...
  • 42.6k
4 votes

Gaussian White Noise - Relation Between Distribution and Correlation

We say that random signals can be Gaussian white noise, but for me if the signal has a Gaussian distribution it is necesseraly a white noise because as soon as we know the distribution then $\mathbb{E}...
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, ...
4 votes

Interpretation of Histogram in Statistical Image Processing

Does it assume that each pixel in images obey the same probability distribution for the histograms of images? Images of different scenes will definitely not obey the same probability distribution of ...
  • 23.1k

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