Questions tagged [stationary]
The stationary tag has no usage guidance.
47
questions
2
votes
1answer
115 views
Autocorrelation of a uniform random process
i am currently learning the basics of signal processing.
As you may know the definition of the autocorrelation is different if you look at a random process or for example a deterministic signal
My ...
1
vote
1answer
46 views
Questions about the stability (and stationarity) of a system and state space representations
i'm pretty new to the topic and I'm trying to understand how to determine the stability of a process. I'm giving this discrete-time stochastic system:
$$
\cases{ s_t = 2s_{t-2} + 3w_{t-2} \\
y_t = ...
1
vote
1answer
97 views
Energy definition for Autocorrelation lag 0 and lag 1 for complex signals
I am studying the role of an auto-correlation matrix for random signals and the difference of energy between a lag 0 and lag 1 matrix.
Consider a complex input signal $x(k)=[x1,x2]^T$ and $x(k-1)=[x0,...
1
vote
1answer
72 views
Any experiences for plotting a stationary wavelet transform?
I am experimenting with wavelets for my thesis and am currently working with the stationary WT pywavelets provides.
There are very nice plots for CWTs, but does anyone know a technique for producing ...
0
votes
0answers
52 views
Proof of weak stationary random process autocovariance always goes to zero?
Professor told me that if a random process is weak stationary, and it does not feature any periodic component, then its autocovariance always goes to zero.
I can intuitively understand it, however, ...
0
votes
1answer
26 views
Is a pulse of white noise still properly described as stationary?
I understand that a signal that is white noise is stationary (or more properly that the process generating it is stationary).
What if the white noise is delivered as a single pulse or a series of ...
0
votes
1answer
34 views
Radio Signal Stationarity
A radio signal recording of a wireless communication system (e.g: Wi-Fi traffic) is beaconized, channelized and subject to noise.
When working with such an RF signal, numerically transformed to a ...
3
votes
2answers
135 views
How to find a variance of sample sequence
I have a sequence such as
$$r[n] = y[n]v[n]$$
$y[n]$ and $v[n]$ are zero-mean and statistically independent.
I need to find a variance of $r[n]$ and show that it is white and equal to $\sigma ^2_y\...
0
votes
1answer
67 views
Converting a non-stationary random process into a WSS process by adding a random phase
Here is an example where this method has been implemented.
We were trying to calculate the spectrum of a transmitted signal(Random signal/weighted pulse)
The auto correlation function of the pulse ...
1
vote
2answers
23 views
Decimator effect on wide sense stationary input
I've seen that the output of a decimator when a WSS process is passed through remains WSS. I am not able to immediately see why this is. What is a good explanation of why the signal maintains ...
1
vote
2answers
50 views
Fourier-Analysis of Stationary Random Signals
Let's say we have discrete-time stationary random signals with Gaussian PDF of mean value 0 and variance 1, whose individual signal values are uncorrelated.
For such a signal, how can we determine ...
1
vote
2answers
116 views
Conversion from stationarity to non-stationarity
Is there any way to convert a non-stationary signal to a stationary one, perform operations on it meant for a stationary signal and then convert it back to the non-stationary one?
0
votes
1answer
62 views
Why doesn't law of large numbers apply to this stationary time-series?
There's a paragraph in Wikipedia that states the following:
Let Y be any scalar random variable, and define a time-series $\{X_t\}$, by
$$X_{t}=Y\qquad {\text{ for all }}t$$
Then $\{X_t\}$ is a ...
0
votes
1answer
88 views
Why does it mean that the process/signal is not stationary when its variance varied with time? [closed]
Why does it mean that the process/signal is not stationary when its variance varied with time? that is,
$VAR[X(t)]= \alpha \times t$,$t$ is time,and $\alpha$ is a constant,then $X(t)$ is not the WSS ...
2
votes
1answer
92 views
Identifying whether or not a cyclostationary signal is noisy or not using the cyclic autocorrelation
I am trying to determine whether or not a given signal has been corrupted by Gaussian noise, either bandlimited (with a filter) or not. The signal in question is a BPSK or PAM signal that is upsampled ...
0
votes
1answer
393 views
Autocorrelation of Addition of Two Independent Signals
Given a random signal $ Z \left( t \right) $ which is addition of two independent signals $ X \left( t \right) $ and $ Y \left( t \right) $ with constant parameters $ a $ and $ b $:
$$ Z (t) = aX(t) +...
1
vote
0answers
43 views
A wide sense stationary random process that is not second order stationary [duplicate]
I have been reading Peebles Probability, Random Variables, and Random Signal Principles and it claims that second-order stationarity is sufficient to guarantee:
$E[X(t)]$ is a constant
$R_{XX}(t1,t2) ...
1
vote
1answer
134 views
Doubt about wide sense stationary random process
I have white Gaussian noise $F[n]$ with zero mean and autocorrelation $R_F[n_1,n_2]=\delta[n_1-n_2]$.
If now I consider the random process defined as
$$X[n]=u[n]e^{-kn}F[n]$$ Is $X[n]$ a wide-ense ...
1
vote
2answers
468 views
What is the difference between Kalman filter algorithm and stationary Kalman filter algorithm?
I want to compute the stationary Kalman filter algorithm but I haven't found any information about that algorithm ( not even the pseudo code ) so, I wonder what is the difference between the Kalman ...
0
votes
1answer
1k views
Why is doing fft on a non-stationary signal a problem?
Why is it a problem to do frequency analysis on a non-stationary signal?
what makes the frequency interpretation incorrect?
0
votes
1answer
303 views
Auto-covariance of the product of deterministic and wide-sense stationary signal
Anybody give me an advice how to find the auto-covariance of the product of deterministic and wide-sense stationary signal. I couldn't find how to solve this, I have looked and searched the internet, ...
-1
votes
1answer
287 views
Stationary signal: time-domain vs frequency domain
If I understand correctly, a signal is stationary if:
time domain: it's generated from the same distribution at each instant time.
frequency domain: its frequency content does not change in time.
...
1
vote
1answer
673 views
If the mean of a random process is constant, does it imply the process is first order stationary?
If a random process is first order stationary, its mean is constant. However, if a random process has a constant mean say $3$ and an autocorrelation equal to $9 + 15e^{|-\tau|}$. The process is ...
0
votes
1answer
187 views
LTI filtering for wide-sense stationary process
Why is it that if $U[n]$ is wide-sense stationary and it is convolved with $h[n]$ to produce $W[n]$, the autocorrelation becomes $R_{WW}[n] = R_{UU}[n]*h[n]*h[-n]$?
I know that in general $R_{WW}[n_{...
4
votes
2answers
385 views
Why is $A\cos(2\pi f_ct)$ a non-stationary process?
I am studying analog communication and having Communication system - Simon Hykin as one of the reference.
There is a question
Consider the sinusoidal process$$X(t) = A\cos(2\pi f_ct)$$where the ...
2
votes
0answers
38 views
Is there any method/algorithm to estimate the magnitude of non-stationarity in a signal?
e.g. the global Lyapunov exponent can give sense of the level of chaos in the signal. Is there any reliable numerical technique to estimate "how" non-stationary (or how predictable) a signal is?
Also, ...
4
votes
1answer
2k views
Cross-correlation or cross-covariance of non-zero mean signals
Cross-correlation for uniformly sampled signals is defined as [1]
$$(f \star g)[n]\ \stackrel{\mathrm{def}}{=} \sum_{m=-\infty}^{\infty} f^*[m]\ g[m+n].$$
Cross-covariance for wide-sense stationary (...
0
votes
1answer
37 views
Testing for changes in auto-covariance
I am working with uniformly-spaced time series data where I am interested in knowing whether there are changes in temporal auto-covariance. The mean can be assumed constant. Visually, there are no ...
2
votes
2answers
544 views
Does the inverse Fourier transform only produce stationary time signals?
I have a question regarding the inverse Fourier transform and its relevance to non-stationary signals. And by non-stationary signal, I'm talking about a signals whose frequency content varies with ...
5
votes
1answer
76 views
Is there any computational method to prove whether a series is stationary or not?
I have a discrete series $x[n]$. It is extracted from real life and I do not have probability distribution of each value $x[n]$. Is there any computational method to prove whether the series is ...
1
vote
2answers
757 views
Why is $\sin(t)$ a stationary process?
I am trying to understand the meaning of the term Stationary Process.
For example, I was told that $\sin(t)$ is a stationary process.
Could someone try to explain, in simple words, why is $\sin(t)$ (...
0
votes
1answer
113 views
Z-transform of difference equations and stability of a process
According to this paper:
$y(t)$ is stationary if all of the roots (of characteristic equation) lie outside the unit circle
Here, $y(t)$ is causal.
To me it seems the case is exactly the opposite, ...
17
votes
2answers
18k views
Stationary vs non-stationary signals?
There are nice technical definitions in textbooks and wikipedia, but I'm having a hard time understanding what differentiates stationary and non-stationary signals in practice?
Which of the following ...
3
votes
2answers
92 views
How to show that the signal $x_n = A\cos(\omega n)$ can be fully predicted by a system with two weights $w_1,w_2$
I am trying to solve the following exercise:
Show that the signal $x_n = A\cos(\omega n)$ can be fully predicted by a system with two weights $w_1,w_2$ (i.e. $x_n = w_1 x_{n-1} + w_2 x_{n-2}$). ...
3
votes
1answer
143 views
Is $A\cos(\omega t+\theta)$ a Gaussian random process?
$Z(t) = A\cos(\omega t+\theta)$ where $A$~$N(0,\sigma ^2) $ and $\theta $~$(0,2\pi)$ are independent.
I'm trying to figure out if $Z(t)$ is a Gaussian random process and whether it is strict sense ...
1
vote
1answer
432 views
How to calculate the noise power for a non-stationary noise?
With stationary noise we have constant mean and variance (let's assume it is Gaussian noise). My first question is, how is the noise power calculated and how it is related to the variance?
Now, I ...
0
votes
1answer
505 views
separating stationary and non stationary parts of univariate signal
Does anyone know if there is a procedure as to separate the stationary and non stationary parts of a univariate signal. I have seen signal source separation and blind separation algorithms (all of ...
2
votes
1answer
222 views
Is jointly wss (wide sense stationary) a transitive relation?
I've been try to either prove or find a counter-example to the idea of jointly-wss being transitive.
In other words: does ($x$ and $y$ are jointly wss) $\wedge$ ($y$ and $z$ are jointly wss) imply ...
1
vote
0answers
5k views
What is the difference between wide sense and strict sense stationary processes?
What is the difference between wide sense and strict sense stationary processes (SP) ?
According to the definition (by Heinrich Meyr, Marc Moeneclaey, Stefan A. Fechtel in "Synchronization, Channel ...
0
votes
1answer
51 views
Signal Plus Weakly Stationary Noise
I was reading the book "Spectral Analysis of Time Series" By Herman Koopmans.
On Page 55, he explains that a specific type of non-stationary signal which is the result of adding weakly stationary ...
2
votes
1answer
415 views
Covariance between real and imaginary parts of Fourier transform of a stationary time series
Since Fourier transform of a random stationary process in time (in the case of existence) is not necessarily real, my question is what is the relation between the covariance of real and imaginary ...
1
vote
0answers
136 views
The autocorrelation of a WSS process as a linear operator
If I'm given a autocorrelation matrix of a WSS process what interpretation should I put on the resulting vector.
More concretely the matrix takes the form
$\begin{bmatrix} x_1 & x_2 & \...
3
votes
0answers
260 views
Why does the Stationary Wavelet Transform shift this image?
I am running a stationary wavelet transformation on a brain image. I can't understand why it shifts with each level so that the image is no longer centered. You can see that the X component shifts to ...
2
votes
1answer
1k views
Colored noises: Stationary or non-stationary?
I know for sure that white noise is considered as a stationary sound, but is that true for the rest of them? http://en.wikipedia.org/wiki/Colors_of_noise
Also, can we meet some of these sounds in ...
2
votes
1answer
706 views
A special case of 2 jointly Weak-Sense Stationary (WSS) stochastic processes
I know that 3 conditions must be met in order a pair of stochastic processes $X(t)$ and $Y(t)$ to be characterized as jointly WSS:
1. $X(t)\;\; WSS$
2. $Y(t)\;\; WSS$
3. $R_{xy}(t_1,t_2) = R_{xy}(...
0
votes
0answers
50 views
How to show that an ergodic process must be a strict-sense stationary one? [duplicate]
I have trouble to distinguish these two concepts namely ergodic process and strict-sense stationary process. I look at one of the books about the signal processing that says an ergodic process must be ...
1
vote
0answers
66 views
Is a Stationary VAR Process with Zero Mean Gaussian Innovations a Gaussian Stationary Process?
Consider the stationary VAR process
$${\bf X}_t = \sum_{\tau = 1}^{L} A_\tau {\bf X}_{t-\tau} +{\bf \epsilon}_t$$
If the innovations $\epsilon_t \sim MVN({\bf 0},\Sigma)$ then is ${\bf X}_t$ a ...
