# Questions tagged [random-process]

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

Consider a white Gaussian noise signal $x \left( t \right)$. If we sample this signal and compute the discrete Fourier transform, what are the statistics of the resulting Fourier amplitudes?
1answer
37 views

### PSD from autocorrelation in MATLAB

I am trying to simulate a simple stochastic process defined by the equation: \begin{equation} \frac{1}{v}\frac{db}{dt} +\Gamma_0 b= \sqrt{\sigma}R(t), \end{equation} where $R(t)$ is a zero-mean white ...
2answers
37 views

### Why is there only one integration in the solution if there is two integral in the formula?

In this problem the random variable is theta and according to the formula there should be two integrations but in the solution there is only one . Nor am i able to understand the meaning of x1 and x2 ...
0answers
50 views

### Proof of weak stationary random process autocovariance always goes to zero?

Professor told me that if a random process is weak stationary, and it does not feature any periodic component, then its autocovariance always goes to zero. I can intuitively understand it, however, ...
1answer
25 views

### Question regarding AC power of ergodic process

We know Ergodic process is the subset of Weakly stationary process which permits us to substitute time average for ensemble Average My teacher said If $X(t)$ is Ergodic random process then following ...
1answer
79 views

### Correlation of independent random processes

Suppose $X(t)$ and $Y(t)$ be two independent random processes. Is $E(X(t_1)Y(t_2))$ necessarily zero?
1answer
27 views

### Power contained in a random process $X(t)$

How do we calculate the AC and DC power of random process $X(t)$ , provided we have $R_x (\tau)$, and $S_x(f)$ ?
3answers
283 views

### Derivation of PSD of sampled bandlimited random process

When a bandlimited random process whose PSD \begin{equation} S(\omega) = \begin{cases} \frac{N_0}{2} & -10B<\omega<10B\\[2ex] 0 & \text{otherwise.} \end{cases} \end{equation} is ...
0answers
20 views

### Is the expectation of a random process $X(t)$ with zero DC component necessarily zero?

Is the expectation of a random process $X(t)$ with zero DC component necessarily zero? Or can it be non-zero depending upon the process?
1answer
30 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 ?
0answers
34 views

0answers
23 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 ...
0answers
59 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 ...
2answers
132 views

1answer
68 views

0answers
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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 ...
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, ...
2answers
192 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 ...
1answer
42 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)=...
0answers
96 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 ...