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 ...
- 5,780
7
votes
Accepted
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....
- 41.2k
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
$$\...
5
votes
Accepted
Random signals as power signals
Note that the condition
$$\int_{-\infty}^{\infty}|f(t)|^2dt<\infty\tag{1}$$
(i.e., that the signal $f(t)$ has finite energy) is very restrictive when we try to model signals, even though ...
- 80.4k
5
votes
Accepted
Autocorrelation of the product of deterministic and random signal
As correctly pointed out in the comments, in general the process $Y(t)=x(t)M(t)$ is not wide-sense stationary (WSS), i.e. its autocorrelation function depends not only on the time difference parameter ...
- 80.4k
5
votes
Accepted
WSS Ergodic Process with Power Spectrum
You can compute the power of the process from its power spectrum as well as from its PDF. Equating the two gives you a relation between the constants $A$ and $B$. More specifically you get
$$\int_{-B}...
- 80.4k
5
votes
Accepted
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] = \...
- 41.2k
5
votes
Accepted
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{...
- 10.5k
4
votes
Accepted
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 ...
- 80.4k
4
votes
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 ...
- 80.4k
4
votes
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-...
- 80.4k
4
votes
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 ...
- 22.9k
4
votes
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 ...
- 37.6k
3
votes
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) =...
- 5,780
3
votes
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 ...
- 1,306
3
votes
Accepted
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 ...
- 18.9k
3
votes
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 ...
- 19.1k
3
votes
Accepted
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, ...
- 13.8k
3
votes
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 $...
- 18.9k
2
votes
Accepted
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 ...
- 10.5k
2
votes
Accepted
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 ...
- 4,051
2
votes
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,...
- 13.8k
2
votes
Accepted
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 ...
- 26.6k
2
votes
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). ...
- 4,721
2
votes
Accepted
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 ...
- 80.4k
2
votes
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),...
- 681
2
votes
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. ...
- 3,559
2
votes
Accepted
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 ...
- 3,272
2
votes
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 ...
- 22.9k
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 ...
- 1,306
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