Jointly WSS Processes: ϕxy(τ) & ϕxy(ω)

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In summary, the problem is to show that x(t) and y(t) are jointly WSS, and to determine an expression for ϕxy(τ) and ϕxy(ω) in terms of the input statistics and the system parameters. To do this, we expand R_{XY}(t,t+\tau) using the definition of x(t) and y(t) and then simplify the expression. This will show that ϕxy(τ) only depends on τ, proving that x(t) and y(t) are jointly WSS. We can then use this to find an expression for ϕxy(ω) and determine if it is always non-negative for all ω. Finally, we can specialize this formula to
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SeriousNoob
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Homework Statement



ϕ(t) = R(t) = autocorrelation
ϕ(f) = S(f) = power spectral density
Inputs v[t] and w[t] are zero mean, jointly WSS processes with auto-correlation se-
quences ϕvv(τ ) and ϕww(τ ), cross-correlation sequence ϕvw(τ ), power spectrums ϕvv(ω)
and ϕww(ω), and cross power spectrum ϕvw(ω). Let x(t) be the output of a LTI system
h1(t) with input v(t), i.e. x(t) = h1(t)v(t), and let y(t) be the output of a LTI system
h2(t) with input w(t), i.e. y(t) = h2(t)  w(t). Assume all quantities to be complex.

(a) Show that x(t) and y(t) are jointly WSS. Determine an expression for ϕxy(τ ) and
ϕxy(ω) in terms of the input statistics and the system parameters.

(b) Is the cross power spectrum always non-negative, i.e. is ϕxy(ω) ≥ 0, for all ω? Justify
your answer.

(c) Specialize this formula to obtain an expression for ϕxx(ω).


Homework Equations


ϕxy(τ) = E[x(t)y(t+τ)]
ϕx(f) = ϕv(f)|H|²
ϕx(f) = Fourier Transform( ϕx(t) )

The Attempt at a Solution


I'm at a complete loss. For jointly WSS, I need to prove x(t) and y(t) are WSS which they are because of v(t) and w(t) going through an LTI system. But I also need to show ϕxy(τ) only depends on τ.
I feel that I'm missing a crucial point and don't know how to start.
 
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  • #2
I suppose that it is trivial to show X(t) and Y(t) are WSS :wink: So the problem is only to prove X(t) and Y(t) are jointly WSS.
To do this, expand [itex]R_{XY}(t,t+\tau)[/itex]:

[itex]X(t) = \int_{-\infty}^{+\infty}V(t-\alpha)h_1(\alpha)d\alpha[/itex]

[itex]Y(t+\tau) = \int_{-\infty}^{+\infty}W(t+\tau-\beta)h_2(\beta)d\beta[/itex]

Hence:

[itex]R_{XY}(t,t+\tau) = \int_{\alpha=-\infty}^{+\infty} \int_{\beta=-\infty}^{+\infty} E[V(t-\alpha)W(t+\tau-\beta)]h_1(\alpha)h_2(\beta)d\beta d\alpha[/itex]

It is easy now :)
 
Last edited:

Related to Jointly WSS Processes: ϕxy(τ) & ϕxy(ω)

1. What is the purpose of studying Jointly WSS Processes?

Jointly WSS Processes refer to jointly wide-sense stationary processes, which are mathematical models used to analyze and understand random signals. These processes are important in fields such as signal processing, communication systems, and control systems. By studying these processes, scientists can gain insights into the behavior of complex systems and develop more efficient algorithms for signal processing and communication.

2. How do ϕxy(τ) and ϕxy(ω) relate to Jointly WSS Processes?

ϕxy(τ) and ϕxy(ω) are two common representations of the cross-correlation function of jointly WSS processes. The former represents the correlation between two processes at different time lags, while the latter represents the correlation at different frequencies. These functions are essential in characterizing the statistical properties of jointly WSS processes.

3. Can Jointly WSS Processes be applied in real-world scenarios?

Yes, jointly WSS processes have numerous applications in real-world scenarios. For instance, they can be used to analyze the performance of communication systems, predict stock market trends, and model weather patterns. These processes are also used in engineering and scientific research to study the behavior of complex systems.

4. What are the assumptions in studying Jointly WSS Processes?

The main assumption in studying jointly WSS processes is that the processes are wide-sense stationary, which means that their mean and autocorrelation functions are time-invariant. Additionally, the processes are assumed to be jointly WSS, meaning that their cross-correlation functions are also time-invariant. These assumptions allow for the simplification of mathematical calculations and make the analysis of these processes more feasible.

5. How can Jointly WSS Processes be modeled and analyzed?

Jointly WSS Processes can be modeled and analyzed using various mathematical techniques, such as time-domain methods, frequency-domain methods, and state-space methods. These methods involve the use of mathematical tools such as Fourier transforms, autocorrelation functions, and power spectral densities. Additionally, computer simulations and data analysis techniques can also be used to model and analyze these processes.

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