33
votes
Accepted
Stationary vs non-stationary signals?
There is no stationary signal. Stationary and non-stationary are characterisations of the process that generated the signal.
A signal is an observation. A recording of something that has happened. A ...
- 10.4k
19
votes
Stationary vs non-stationary signals?
@A_A's good answer misses one point: stationarity or nonstationarity are generally only applied to stochastic signals, not deterministic signals.
In general, when statistical tests are applied for ...
- 24.3k
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 ...
- 19.7k
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 \...
- 50.4k
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 ...
- 19.7k
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 ...
- 85.8k
6
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 ...
- 50.4k
5
votes
Accepted
Cross-correlation or cross-covariance of non-zero mean signals
What are reasons to choose for cross-correlation or cross-covariance when comparing signals with non-zero mean?
Well, part of the issue is that cross-correlation as defined in your equation:
$$(f \...
- 24.3k
5
votes
If $X(t)$ is a WSS process with mean 5, what is the mean of $X(2t)$?
First, the mean of a random / stochastic process is:
$$ \mu \left( t \right) = \mathbb{E} \left[ X \left( t \right) \right] $$
For WSS we know that $ \mu \left( t \right) = \mu $ but the important ...
- 50.4k
4
votes
Accepted
LTI filtering for wide-sense stationary process
It's basically just about using the definitions and doing the math:
$$\begin{align}R_{WW}[n]&=E\{W^*[k]W[k+n]\}\\
&=E\left\{\sum_mU^*[k-m]h^*[m]\sum_lU[k+n-l]h[l]\right\}\\
&=\sum_m\...
- 85.8k
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, ...
- 28.3k
4
votes
Wiener filtering/deconvolution for non-stationary noise
The Wiener Filter can written as a Least Squares problem (See How Is the Formula for the Wiener Deconvolution Derived?).
In you case, Since noise is basically with different STD per sample you may use ...
- 511
4
votes
Accepted
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(...
- 19.7k
3
votes
Accepted
Why doesn't law of large numbers apply to this stationary time-series?
I believe that you are thinking that each value of $X_t$ is determined by a different realisation of $Y$, which in this example is not true.
Suppose that $Y$ is the value that comes out from a dice ...
- 4,960
3
votes
Stationary signal: time-domain vs frequency domain
Your definitions are not correct.
For a Strict Sense Stationary process (signal) the joint distribution of your process' value for all instants of time must be independent of time, in other words if ...
- 1,328
3
votes
Accepted
Doubt about wide sense stationary random process
The mean of $X(n)$ is always $\mu_X =0$, because the noise has zero mean. Thus we should check whether the autocorrelation corresponds to a WSS process.
$$\begin{align}
R_X(n_1,n_2)
&=\mathbb{E}[...
- 4,960
3
votes
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 ...
- 28.3k
3
votes
Accepted
Is $A\cos(\omega t+\theta)$ a Gaussian random process?
I'm pretty sure that even algebraically, $Z$ will not be Gaussian. You're multiplying a Gaussian by a random variable that has a distribution that is effectively the histogram of a (co)sine function.
...
- 24.3k
3
votes
Accepted
Z-transform of difference equations and stability of a process
The paper deals in the lag operator $L$, rather than $z^{-1}$ as DSP people tend to use.
As a result, all the DSP results you know are inverted. So "inside the unit circle" becomes "outside the unit ...
- 24.3k
3
votes
Accepted
How to find a variance of sample sequence
$\sigma_r^2=\sigma_y^2\sigma_v^2$ iff
1) $\mathbb{E}\{y\} = \mathbb{E}\{v\}=0$.
AND
2) $y$ and $v$ are independent, OR uncorrelated and their squares ($y^2$ and $v^2$) are also uncorrelated.
Proof:...
- 178
3
votes
Why is a random process strictly stationary when its joint Probability density function is time invariant?
Why is a random process strictly stationary when its joint probability density function is time invariant?
This query, taken from the title of the question cannot be answered because there is no Why.
...
- 19.7k
2
votes
What is the difference between wide sense and strict sense stationary processes?
A process is stationary if:
its mean is a constant value: $\mu_x(t)=\mu x$
its MSV(mean square value) is a constant value.
its variance is a constant value. $\sigma^2_x(t)=\sigma^2_x$
its ...
- 29
2
votes
What is the difference between Kalman filter algorithm and stationary Kalman filter algorithm?
I’m not sure what you mean by the stationary Kalman filter, but it seems to be what I would call the steady-state Kalman filter.
If that is the same thing, then you just solve for the Kalman gain at $...
- 24.3k
2
votes
Decimator effect on wide sense stationary input
The decimator by integer $M$ can be shown to be the following block:
$$ x[n] \longrightarrow \boxed{ \downarrow M } \longrightarrow y[n] = x[Mn] $$
Assuming that the input $x[n]$ is WSS it has the ...
- 27.6k
2
votes
Accepted
Converting a non-stationary random process into a WSS process by adding a random phase
It is not true that you can convert any non-stationary random process into a WSS random process by just adding a random phase. What is true is that you can make a (wide-sense) cyclostationary process ...
- 85.8k
2
votes
Accepted
Is a pulse of white noise still properly described as stationary?
The pulsed process is not stationary. You can use the term "quasi-stationary" to describe a process that looks stationary in short time scales (e.g. during silence, or in the middle of a pulse).
- 14.5k
2
votes
Accepted
Understanding kurtogram parameters
[Given figures are from A new improved Kurtogram and its application to planetary gearbox degradation feature analysis, Xianglong Ni et al.]
This $k$ in the graph (and the $K$ in the Table) ...
- 31.3k
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