The average of a random process

[EngWiPy]

Hello all,

I have the following continuous-time random process:

[tex]v(t)=\sum_{k=0}^{K-1}\alpha_k(t)d_k+w(t)[/tex]

where d_k are i.i.d. random variables with zero mean and variance 1, alpha_k(t) is given, and w(t) is additive white Gaussian process of zero-mean and variance N_0.

Can we say that the average power of v(t) is E{|v(t)|^2}?

Thanks

 

[Office_Shredder]

That probably depends on what v is physically. Velocity?

 

[EngWiPy]

That probably depends on what v is physically. Velocity?

Thanks for replying.

v(t) is the received signal in a communication system.

 

[Office_Shredder]

The word average can be used to refer to two things here. Expectation of a random variable is often referred to as average - what you have written down is the expected value of the instantaneous power of the signal. The average power of a (deterministic) signal is often defined as
[tex] \lim_{T \to \infty} \frac{1}{T} \int_{-T/2}^{T/2} |f(t)|^2 dt [/tex]

This doesn't depend on t at all - if you want the expected value of this, then you need to do more work

 

[blue_raver22]

for your question. The average power of a random process is defined as the expectation value of the squared magnitude of the process. In this case, the process v(t) is a sum of random variables d_k and a Gaussian process w(t). Therefore, the average power of v(t) can be expressed as the sum of the average powers of d_k and w(t). Since d_k has a variance of 1, its average power is also 1. Similarly, the average power of w(t) is N_0. Therefore, we can say that the average power of v(t) is E{|v(t)|^2} = K + N_0.