Autocorrelation Function of a Signal

In summary, the solution set the other three terms to different autocorrelation functions and I'm not sure how these other three terms are autocorrelation functions as well based off of the definition.
  • #1
GreenPrint
1,196
0

Homework Statement



I'm posting this question here at this point. I am having difficulty understanding autocorrelation in terms of solving the problem below. I don't seem to understand the math behind this.

A white noise process W(t) with unity (N_0/2 = 1) power spectral density is input to a linear system. The output of the linear system is X(t), where

X(t) = W(t) - W(t - 1)

Determine the autocorrelation of X(t) and sketch it.

Homework Equations



We can define the autocorrelation function of a stochastic process X(t) as the expectation of the product of two random variables X(t_1) and X(t_2), obtained by sampling the process X(t) at times t_1 and t_2 respectively. So we can write

proxy.php?image=http%3A%2F%2Fs12.postimg.org%2Fohb4fg819%2FCapture.png


f_(X(t_1),X(t_2))(x_1,x_2) is the join probability density function of the process X_(t) sampled at times t_1 and t_2

M_(XX)(t_1,t_2) is used to emphasize the fact that this is a second order moment. For M_(XX)(t_1,t_2) to dependent on the time difference t_2 - t_1, we have R_(XX)(t_2 - t_1)

Two different symbols for the autocorrelation function M_(XX)(t_1,t_2) and R_(XX)(t_2 - t_1) to denote that R_(XX)(t_2 - t_1) is the autocorrelation function specifically for a weak stationary process.

Let τ denote a time shift; that is, t = t_2 and τ = t_1 - t_2

proxy.php?image=http%3A%2F%2Fs7.postimg.org%2Fihwilnjij%2FCapture.png


The Attempt at a Solution



proxy.php?image=http%3A%2F%2Fs28.postimg.org%2Fov8x2e7al%2FCapture.png


I understand that the first term on the last line is indeed equal to
proxy.php?image=http%3A%2F%2Fs13.postimg.org%2Fhh56qatnn%2FCapture.png
. I'm however unsure what to do with the other three terms. The solution sets the other three terms to different autocorrelation functions and I'm not sure how these other three terms are autocorrelation functions as well based off of the definition.

Here's what the solution is. I don't understand how it went from the second line to the third line.

proxy.php?image=http%3A%2F%2Fs2.postimg.org%2Fdy111sucp%2FCapture.png


Any help would be greatly appreciated. I also don't understand how how the solution goes from the third line to the fourth line. It seems to just simply replacing the autocorrelation functions with dirac delta functions. I'm not sure how these are equal in any way.
 
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  • #2
GreenPrint said:

Homework Statement



I'm posting this question here at this point. I am having difficulty understanding autocorrelation in terms of solving the problem below. I don't seem to understand the math behind this.

A white noise process W(t) with unity (N_0/2 = 1) power spectral density is input to a linear system. The output of the linear system is X(t), where

X(t) = W(t) - W(t - 1)

Determine the autocorrelation of X(t) and sketch it.

Homework Equations



We can define the autocorrelation function of a stochastic process X(t) as the expectation of the product of two random variables X(t_1) and X(t_2), obtained by sampling the process X(t) at times t_1 and t_2 respectively. So we can write

proxy.php?image=http%3A%2F%2Fs12.postimg.org%2Fohb4fg819%2FCapture.png


f_(X(t_1),X(t_2))(x_1,x_2) is the join probability density function of the process X_(t) sampled at times t_1 and t_2

M_(XX)(t_1,t_2) is used to emphasize the fact that this is a second order moment. For M_(XX)(t_1,t_2) to dependent on the time difference t_2 - t_1, we have R_(XX)(t_2 - t_1)

Two different symbols for the autocorrelation function M_(XX)(t_1,t_2) and R_(XX)(t_2 - t_1) to denote that R_(XX)(t_2 - t_1) is the autocorrelation function specifically for a weak stationary process.

Let τ denote a time shift; that is, t = t_2 and τ = t_1 - t_2

proxy.php?image=http%3A%2F%2Fs7.postimg.org%2Fihwilnjij%2FCapture.png


The Attempt at a Solution



proxy.php?image=http%3A%2F%2Fs28.postimg.org%2Fov8x2e7al%2FCapture.png


I understand that the first term on the last line is indeed equal to
proxy.php?image=http%3A%2F%2Fs13.postimg.org%2Fhh56qatnn%2FCapture.png
. I'm however unsure what to do with the other three terms. The solution sets the other three terms to different autocorrelation functions and I'm not sure how these other three terms are autocorrelation functions as well based off of the definition.

Here's what the solution is. I don't understand how it went from the second line to the third line.

proxy.php?image=http%3A%2F%2Fs2.postimg.org%2Fdy111sucp%2FCapture.png


Any help would be greatly appreciated. I also don't understand how how the solution goes from the third line to the fourth line. It seems to just simply replacing the autocorrelation functions with dirac delta functions. I'm not sure how these are equal in any way.
From the second line to the third is a simple matter of subtracting the arguments, i.e:

[tex]E(W(t)W(t+\tau-1) = R_{ww}(t+\tau-1-t) = R_{ww}(\tau-1)[/tex]
The rest are done in the same way.

As for the deltas, look at the definition of white noise and its autocorrelation function.
 
  • #3
Hey MathematicalPhysicist,

Thanks for the reply. I didn't realize this was so simple. My new attempt at a solution

proxy.php?image=http%3A%2F%2Fs28.postimg.org%2Fov8x2e7al%2FCapture.png

Capture.png


I know that
Capture.png
. But I don't understand how this equation is true
Capture.png
. It reminds of the simple fact that if

LHS = RHS
LHS + 1 = RHS +1

But that doesn't seem to be the case here. Adding one to both functions inside the expectation operator doesn't change the value of the expectation?

E[W(t - 1)W(t + tau)] = E[W(t - 1 + 1)W(t + tau + 1)] = E[W(t)W(t + tau + 1)] = R_(WW)(tau + 1)?

I don't understand this. Is it some property of the expectation operator:

E[F(t)(F(t)] = E[F(t + 1)F(t + 1)]?
 
  • #4
For [tex]E(W(t-1)W(t+\tau))=R_{ww}(t+\tau-(t-1))=R_{ww}(\tau+1)[/tex]

Remember you always take the aboslute value of the difference of times.
 
  • #5
MathematicalPhysicist said:
For [tex]E(W(t-1)W(t+\tau))=R_{ww}(t+\tau-(t-1))=R_{ww}(\tau+1)[/tex]

Remember you always take the aboslute value of the difference of times.

I actually didn't know this

[tex]E[F(f(t))F(g(t))] = E[F(t)F(g(t) -f(t))] = R_{FF}(g(t) - f(t))[/tex]?

If I were to try and find the power spectral density of [tex]X(t)[/tex] I would make use of [tex]S_{XX}(f) = F[R_{XX}(\tau)][/tex]. Were [tex]F[][/tex] is the Fourier Transform. [tex]S_{XX}(f) = F[2*delta(\tau) - delta(\tau - 1) - delta(\tau + 1)][/tex] I know that [tex]F[delta(t)] = 1[/tex]. So the Fourier Transform of the first term is simply just two. I however am having difficulty understanding how [tex]F[- delta(\tau - 1) - delta(\tau + 1)] = cos(2\pi*f_{0})[/tex]

[tex]F[delta(\tau - 1) - delta(\tau - 1)][/tex]
[tex]F[-e^{i} - e^{-i}][/tex]

I'm not sure what variable is included in the exponential. But I do know that

[tex]-e^{ix} - e^{-ix} = -(e^{ix} + e^{-ix}) = -\frac{2}{2}(e^{ix} + e^{-ix}) = -2cos(x)[/tex]

Thanks for all of your help.
 
  • #6
GreenPrint said:
I actually didn't know this

[tex]E[F(f(t))F(g(t))] = E[F(t)F(g(t) -f(t))] = R_{FF}(g(t) - f(t))[/tex]?

If I were to try and find the power spectral density of [tex]X(t)[/tex] I would make use of [tex]S_{XX}(f) = F[R_{XX}(\tau)][/tex]. Were [tex]F[][/tex] is the Fourier Transform. [tex]S_{XX}(f) = F[2*delta(\tau) - delta(\tau - 1) - delta(\tau + 1)][/tex] I know that [tex]F[delta(t)] = 1[/tex]. So the Fourier Transform of the first term is simply just two. I however am having difficulty understanding how [tex]F[- delta(\tau - 1) - delta(\tau + 1)] = cos(2\pi*f_{0})[/tex]

[tex]F[delta(\tau - 1) - delta(\tau - 1)][/tex]
[tex]F[-e^{i} - e^{-i}][/tex]

I'm not sure what variable is included in the exponential. But I do know that

[tex]-e^{ix} - e^{-ix} = -(e^{ix} + e^{-ix}) = -\frac{2}{2}(e^{ix} + e^{-ix}) = -2cos(x)[/tex]

Thanks for all of your help.
You can do what I wrote provided W is WSS (wide sense stationary).

Read about it here:
http://en.wikipedia.org/wiki/Stationary_process#Weak_or_wide-sense_stationarity
 

What is the Autocorrelation Function of a Signal?

The Autocorrelation Function of a Signal is a mathematical tool used to measure the similarity between a signal and a time-shifted version of itself. It is commonly used in signal processing and statistics to analyze and identify patterns in data.

How is the Autocorrelation Function calculated?

The Autocorrelation Function is calculated by multiplying a signal by a delayed version of itself and then integrating the product over a given time interval. This process is repeated for different time delays, resulting in a function that describes the correlation between the signal and its delayed version at different time lags.

What does the shape of the Autocorrelation Function tell us about a signal?

The shape of the Autocorrelation Function can provide information about the periodicity and frequency components of a signal. A sharp peak in the function indicates a strong correlation between the signal and its delayed version at a specific time lag, while a flatter function suggests a weaker correlation or no correlation at all.

What are the applications of Autocorrelation Function?

The Autocorrelation Function is used in a variety of fields such as signal processing, time series analysis, and machine learning. It can be used to detect periodicity in a signal, identify similarities between different signals, and remove noise from a signal.

Can the Autocorrelation Function be negative?

Yes, the Autocorrelation Function can take on both positive and negative values. A positive value indicates a positive correlation between the signal and its delayed version, while a negative value indicates a negative correlation. A value of zero indicates no correlation between the two signals at a specific time lag.

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