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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 ...
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Understanding Ergodicity and Ensemble Averaging

I will try to explain this practically using MATLAB notation. Yet before that I must say the ergodic property sometime is limited to a level of moment, namely ergodic in the 1st , 2nd, 3rd moment, etc....
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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 $$\...
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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 ...
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Autocorrelation of the product of deterministic and random signal

As correctly pointed out in the comments, in general the process $Y(t)=x(t)M(t)$ is not wide-sense stationary (WSS), i.e. its autocorrelation function depends not only on the time difference parameter ...
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WSS Ergodic Process with Power Spectrum

You can compute the power of the process from its power spectrum as well as from its PDF. Equating the two gives you a relation between the constants $A$ and $B$. More specifically you get $$\int_{-B}...
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The Standard Deviation of The Derivative of a Signal

Since the signal is discrete and the operation is linear it be formed using a Filter. Assuming the signal is given by $ x \left[ n \right] $. Then its derivative is given by: $$ y \left[ n \right] = \...
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Complex normal Gaussian noise

You should use the the standard formula: s = randn(m, n) + 1i*randn(m, n); And as pointed out by MBaz, the output should be scaled accordingly by $\frac{1}{\sqrt{...
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Filtering random Signal

The output signal will still be normally distributed, but its power spectrum, i.e. its frequency content, will obviously be different from the input signal. If $S_X(\omega)$ is the power spectrum of ...
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The Standard Deviation of The Derivative of a Signal

As pointed out in pichenettes' answer, you need to know more about the original signal $x(t)$. If the signal's power spectrum $S_x(\omega)$ is known then the power spectrum of its $n^{th}$ derivative ...
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Addition of signal variance and noise variance

If the signal and noise are uncorrelated then the variance of the sum of the two equals the sum of their variances. So, yes, you can add variances (of uncorrelated signals). However, you cannot add dB-...
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Random sampling vs uniform sampling

The key idea is that the random sampling approach enforces more constraints on the resulting signal than the uniform sampling approach does. The POCS (projections onto convex sets) algorithm used for ...
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Rayleigh Fading Simulation

Your approach will not model the actual channel but can be used to create the received waveform as having passed through a Rayleigh fading channel. I describe considerations to actually create the ...
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Random signals as power signals

I think simple. We want to model a random physical phenomenon for analysis purpose. One way is to model it by a stochastic process $X(t)$, i.e. a time series of random variables $\left\lbrace X(t_k) =...
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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 ...
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Random process $X(t)$ with autocorrelation function given find the mean and the variance

The limit $\lim_{\tau\to\infty} R_x(\tau)$, if it exists, equals $E^2[X(t)]$ and so $E[X(t)]=0$ in this case. More generally, the mean of a WSS process is nonzero only if the power spectral density ...
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The Standard Deviation of The Derivative of a Signal

It depends on the frequency content of the signal. Consider sinusoidal signals with an amplitude of 1 and frequencies of 1Hz and 1kHz. Both have the same standard deviation, but the standard ...
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Doubts and some confusion on variance for complex rv

If the imaginary and real components each has a variance of 0.5, then what would be the total variance? The variance of the complex RV in that case would be $0.5 + 0.5 = 1$. If on the other hand, ...
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Does a collection of Gaussian random variables necessarily constitute a Gaussian Process?

As you correctly say, if $\{X(t)\colon t \in \mathbb T\}$ is a Gaussian process, then for every possible choice of $n$ random variables $X(t_1), X(t_2), \ldots, X(t_n),$ where $t_k \in \mathbb T$ for $...
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Digital Gaussian White noise signal generation in C++

If you are generating sample of white noise (Normal distribution value) with frequency of 48 kHz, then you already have this in frequency range of 0 - 24 kHz (at least you should if generator is truly ...
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Uniform quantizer for gaussian input signal

The problem with quantizing Gaussian distributed signals (like the real/imaginary part of an OFDM signal) is that they can take any value in theory. It is thus necessary to clip such signals at ...
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What is the entropy for these cases?

Adding a bit of detail to Marcus' answer: Your question is about the "entropy of symbols" and the "entropy of real numbers". In information theory, only sources have entropy. A source has an alphabet,...
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What is the entropy for these cases?

The entropy $H(X)$ of a continuous random variable $X$ is infinite. Proof is trivial (note that we can, without loss of generality, use the natural logarithm, since any other logarithm is the same but ...
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Filtering random Signal

You will be creating a random band-pass signal. What you are supposed to see for such a signal if you plot the time sequence, is a varying sinusoid of frequency $0.3 \pi$ (midpoint of the pass band). ...
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Gaussian, Rayleigh, and Exponential RVs

So you have a complex impulse response, and the real and imaginary parts of each tap are modeled as i.i.d. zero mean Gaussian variables with variance $\sigma^2/2$ (correct me if I'm wrong). The DFT of ...
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Pink ($1/f$) pseudo-random noise generation

I have been using Corsini and Saletti's algorithm since 1990: G. Corsini, R. Saletti, "A 1/f^gamma Power Spectrum Noise Sequence Generator", IEEE Transactions on Instrumentation and Measurement, 37(4),...
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What is the value of the standard deviation $\sigma$?

randn gives you Gaussian noise with std. dev. of 1. (x/255)*(randn(size(image))) gives you a noisy image of std. dev. ...
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Detection of sine signals with random amplitudes

Since both the null hypothesis $\mathcal H_0$ and the alternative hypothesis $\mathcal H_1$ are signal + noise, this is not only the detection of a random signal in WGN, but also a discrimination ...
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Estimate changing time lag between signals

If the signals are as you've drawn them (flat, with abrupt changes), then perhaps this approach might work: Take the absolute value of the differences between successive time samples. The non-zero ...
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2 votes

Random signals as power signals

In addition to Marcus Müller comment, If a signal has finite energy then the signal value must reach zero after long enough time, but for random signals your signals generally don't have such ...
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