# Continous Time Gaussian Distribution

1. Mar 15, 2013

### S_David

Hello all,

I have the following equation

$$\mathbf{v}(t)=\mathbf{P}(t)\mathbf{d}+\mathbf{w}(t)$$

where v(t) is a 2-by-1 vector, P(t) is 2-by-2N matrix, d is a 2N-by-1 vector, and w(t) is an 2-by-1 Gaussian process vector where each element is of zero mean and variance N0. What is the probability distribution function (p.d.f.) of v(t) given that P(t) and d? I know it is Gaussian with mean P(t)d and covariance matrix N0 I_2, but I am not sure how to write it. Is the following right:

$$p(\mathbf{v}(t)\Big|\mathbf{P}(t),\mathbf{d})=A \exp\left(-\frac{1}{N_0}\int_{-\infty}^{\infty}\Big\|\mathbf{v}(t)-\mathbf{P}(t)\mathbf{d}\Big\|^2\,dt\right)$$

where A is some constant, since I am concerned only for the exponential argument. I appreciate your help

Thanks

2. Mar 15, 2013

### chiro

3. Mar 15, 2013

### S_David

Actually P(t)d is also random, but I need the distribution given P(t)d, which basically means it is a constant. I know the distribution in the discrete-time, I need the equivalent in the continuous-time, since I have continuous functions.

4. Mar 15, 2013

### chiro

What is the distribution of P(t) and d?

5. Mar 15, 2013

### S_David

I rather not to go into details. Just I want to say, the above equation is the received signal over a wireless channel in a communication system, and w(t) is additive white Gaussian noise. In the detection process P(t) (the channel) is estimated, and d is chosen such that the conditional p.d.f is maximized.

6. Mar 15, 2013

### chiro

Since P(t)d and v(t) are independent then the joint distribution is P(P(t)d = X, w(t) = Y) = P(P(t)d = X(t))*P(w(t) = Y(t))

Now you can get the distribution for the sum using the convolution theorem since both variables are independent.

Once you have the distribution for the sum of the two independent variables, you can use this to calculate a conditional distribution using P(A|B) = P(A and B)/P(B).

Since you don't give details for the specifics, that is as far as my advice can go.

The convolution theorem integral is the same form as found in standard convolutions of signals and I have a feeling you know what to do.

If you want to find an optimal set of parameters for a parametric distribution family for P(t)d then look up the Expectation Maximization algorithm (or EM algorithm) which is used in a lot of applications similar to yours.

7. Mar 16, 2013

### S_David

I am sorry, but I just needed how to write the p.d.f of a continuous-time Gaussian process of mean P(t)d and covariance matrix N_0 I_2.

8. Mar 16, 2013