Continous Time Gaussian Distribution

[EngWiPy]

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

 

[chiro]

Hey S_David.

If w(t) is the only random component, then the mean will be a multivariate normal with mean P(t)d and a covariance matrix S.

The distribution of a multi-variable Normal is given by:

http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Density_function

Again the assumption is that P(t)d is not a random variable and also that is independent of w(t).

 

[EngWiPy]

Hey S_David.

If w(t) is the only random component, then the mean will be a multivariate normal with mean P(t)d and a covariance matrix S.

The distribution of a multi-variable Normal is given by:

http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Density_function

Again the assumption is that P(t)d is not a random variable and also that is independent of w(t).

Thanks for replying,

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.

 

[chiro]

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

 

[EngWiPy]

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

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.

 

[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.

 

[EngWiPy]

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.

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.

 

[chiro]

Then you want something like this:

http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Density_function