Solve Autocorrelation of WSS RP and Modulated Carrier Cos(w0t+Θ)

[ahamdiheme]

I am given the autocorrelation of a wss rp which modulates a carrier cos( w₀ t + $$\Theta$$ ). I am to find the autocorrelation of the output Y(t).
I have found the autocorrelation of the carrier and I'm not sure what to do next.
If it were possible to get the rp from its autocorrelation, it would have been straight forward but now, i don't know what to do.
Any valuable suggestions?
Thanks

 

[berkeman]

I am given the autocorrelation of a wss rp which modulates a carrier cos( w₀ t + $$\Theta$$ ). I am to find the autocorrelation of the output Y(t).
I have found the autocorrelation of the carrier and I'm not sure what to do next.
If it were possible to get the rp from its autocorrelation, it would have been straight forward but now, i don't know what to do.
Any valuable suggestions?
Thanks

This is a homework/coursework question, so it should be moved to the Homework Help forums. But it's a complicated enough question that I'd like to leave it here for now. But the HH rules should still apply, so please only reply with hints and suggestions as the OP is asking for. Thanks.

 

[oswaler]

Based on the provided information, it seems like you are trying to find the autocorrelation of the output Y(t) given the autocorrelation of a wide-sense stationary random process (WSS RP) that modulates a carrier cos(w0t+Θ). In order to find the autocorrelation of Y(t), you will need to use the properties of autocorrelation and the relationship between the WSS RP and the carrier signal.

First, it is important to understand that the autocorrelation of Y(t) will be a function of both the autocorrelation of the WSS RP and the carrier signal. This is because the WSS RP is modulating the carrier signal, so the resulting output will have components from both signals.

One approach to finding the autocorrelation of Y(t) is to use the definition of autocorrelation, which states that it is the expected value of the product of two signals at different time instants. In this case, the two signals are the WSS RP and the carrier signal, which can be expressed as X(t) and C(t) respectively. Therefore, the autocorrelation of Y(t) can be written as E[Y(t)Y(t+τ)] = E[X(t)C(t)X(t+τ)C(t+τ)]. Using the properties of expected value and the fact that the WSS RP is uncorrelated with the carrier signal, we can simplify this expression to E[X(t)X(t+τ)]E[C(t)C(t+τ)]. This means that the autocorrelation of Y(t) is the product of the autocorrelation of the WSS RP and the autocorrelation of the carrier signal.

Alternatively, you can also use the relationship between the WSS RP and the carrier signal to find the autocorrelation of Y(t). Since the WSS RP is modulating the carrier signal, the output can be written as Y(t) = X(t)C(t). Using this relationship and the properties of autocorrelation, you can express the autocorrelation of Y(t) as R_Y(τ) = E[Y(t)Y(t+τ)] = E[X(t)X(t+τ)C(t)C(t+τ)] = E[X(t)X(t+τ)]E[C(t)C(t+τ)].

In conclusion, there are several ways to find the autocorrelation of Y(t) given the autocor