# The average of a random process

1. May 4, 2013

### EngWiPy

Hello all,

I have the following continuous-time random process:

$$v(t)=\sum_{k=0}^{K-1}\alpha_k(t)d_k+w(t)$$

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

2. May 4, 2013

### Office_Shredder

Staff Emeritus
That probably depends on what v is physically. Velocity?

3. May 4, 2013

### EngWiPy

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

4. May 5, 2013

### Office_Shredder

Staff Emeritus
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
$$\lim_{T \to \infty} \frac{1}{T} \int_{-T/2}^{T/2} |f(t)|^2 dt$$

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