Question

how to find the variance of y(t) for a linear time invariant system for gaussian noise generalized process

Answer 1

To calculate the variance of y(t) for an LTI system with Gaussian noise as input, find the system's impulse response, compute the auto-correlation function of the input, convolve it with the impulse response, and evaluate the output auto-correlation at zero delay.

Step-by-step explanation:

To find the variance of y(t) for a linear time-invariant (LTI) system when the input is a Gaussian noise generalized process, you need to apply knowledge from signal processing and stochastic processes. Assuming you have the system's impulse response h(t), the output y(t) is the convolution of h(t) with the input noise x(t). If x(t) is Gaussian, then y(t) will also be Gaussian, as a convolution of a Gaussian process with an LTI system maintains the Gaussian statistical characteristics.

Here's a step-by-step approach:

1. Obtain the impulse response of the system, h(t).
2. Calculate the auto-correlation function of the input noise, Rx(τ).
3. Compute the output auto-correlation function, Ry(τ) = h(τ) * Rx(τ) * h(-τ), where '*' denotes convolution.
4. The variance of y(t) is obtained by evaluating the output auto-correlation function at zero, i.e., σ2y = Ry(0).

Remember that the variance is a measure of the spread of the values of y(t) around the mean, which for a zero-mean Gaussian process is directly given by the auto-correlation function at zero lag.