Determining stochastic process to give a specified autocorrelation function
$begingroup$
I have a discrete-time real-valued process Y[n] whose autocorrelation function is: $$R_Y[0] = 3+u, R_Y[1]=R_Y[-1]=-2+u, text{ and } R_Y[k] = u, mid k mid > 1$$
Part A
Specify a choice of $u$, with $|u|>0$, for which there is a valid random process with the given autocorrelation function.
For a wide sense stationary (WSS) process, we have that the auto-correlation function (ACF) must be even symmetric and have its max at $R_Y[0]$. However, nothing in this problem specifies that $Y$ is WSS. Still, I believe the maximum must still be at $R_Y[0]$ so I choose
$$u=1$$
This gives me
$$R_Y[0]=4, R_Y[-1]=R_Y[1]=-1, text{ and } R_Y[t]=1 text{ for } |t|geq 2$$
Part B
Specify a random process with the given autocorrelation function (i.e. specify the stochastic generation mechanism for the process).
My thought on this is to utilize the Wiener–Khinchin theorem
$$S_Y(omega) = DFT(R_Y[t])$$
and then use the fact of
$$S_Y[omega]=|H(omega)|^2S_x(omega)$$
making $S_x=1$ via white gaussian process and solving for a suitable $|H(omega)|^2$. However, I believe the above equation only holds if $S_X$ is WSS. This is fine though as there are no restrictions on Y[t].
So, to summarize, we have
$$R_Y(t) = 3delta(t) - 2(delta(t+1) + delta(t-1)) + 1$$
giving the Fourier transform to be
$$S_y(omega)=3 + 4 cos(2pi f) + delta(f)$$
This is where i hit a snag. Making $S_X=1$, I cannot have the above equation as a power spectral density (PSD) because it contains negative values due to the $cos(2pi f) $ term.
I am not sure what to do. Any ideas?
Some thoughts:
- I could factor $cos$ into something non-negative but this only moves the problem around (and am not sure possible).
- Abandon the assumption $S_X=1$ by designing a process which gives $S_Y$ and simply make $H(omega)=1$. I am unsure if this is possible though.
- Stay in time domain and exploit the fact of $R_Y[tau]=E[Y[t]Y[t-tau]]$.
- Use Wold's decomposition theorem. This is a long shot.
- Model $R_Y$ as ARMA.
stochastic-processes fourier-analysis random-variables signal-processing
asked Jan 12 at 20:17
AvedisAvedis
476
$endgroup$
add a comment |
$begingroup$
I have a discrete-time real-valued process Y[n] whose autocorrelation function is: $$R_Y[0] = 3+u, R_Y[1]=R_Y[-1]=-2+u, text{ and } R_Y[k] = u, mid k mid > 1$$
Part A
Specify a choice of $u$, with $|u|>0$, for which there is a valid random process with the given autocorrelation function.
For a wide sense stationary (WSS) process, we have that the auto-correlation function (ACF) must be even symmetric and have its max at $R_Y[0]$. However, nothing in this problem specifies that $Y$ is WSS. Still, I believe the maximum must still be at $R_Y[0]$ so I choose
$$u=1$$
This gives me
$$R_Y[0]=4, R_Y[-1]=R_Y[1]=-1, text{ and } R_Y[t]=1 text{ for } |t|geq 2$$
Part B
Specify a random process with the given autocorrelation function (i.e. specify the stochastic generation mechanism for the process).
My thought on this is to utilize the Wiener–Khinchin theorem
$$S_Y(omega) = DFT(R_Y[t])$$
and then use the fact of
$$S_Y[omega]=|H(omega)|^2S_x(omega)$$
making $S_x=1$ via white gaussian process and solving for a suitable $|H(omega)|^2$. However, I believe the above equation only holds if $S_X$ is WSS. This is fine though as there are no restrictions on Y[t].
So, to summarize, we have
$$R_Y(t) = 3delta(t) - 2(delta(t+1) + delta(t-1)) + 1$$
giving the Fourier transform to be
$$S_y(omega)=3 + 4 cos(2pi f) + delta(f)$$
This is where i hit a snag. Making $S_X=1$, I cannot have the above equation as a power spectral density (PSD) because it contains negative values due to the $cos(2pi f) $ term.
I am not sure what to do. Any ideas?
Some thoughts:
- I could factor $cos$ into something non-negative but this only moves the problem around (and am not sure possible).
- Abandon the assumption $S_X=1$ by designing a process which gives $S_Y$ and simply make $H(omega)=1$. I am unsure if this is possible though.
- Stay in time domain and exploit the fact of $R_Y[tau]=E[Y[t]Y[t-tau]]$.
- Use Wold's decomposition theorem. This is a long shot.
- Model $R_Y$ as ARMA.
stochastic-processes fourier-analysis random-variables signal-processing
asked Jan 12 at 20:17
AvedisAvedis
476
$endgroup$
add a comment |
$begingroup$
I have a discrete-time real-valued process Y[n] whose autocorrelation function is: $$R_Y[0] = 3+u, R_Y[1]=R_Y[-1]=-2+u, text{ and } R_Y[k] = u, mid k mid > 1$$
Part A
Specify a choice of $u$, with $|u|>0$, for which there is a valid random process with the given autocorrelation function.
For a wide sense stationary (WSS) process, we have that the auto-correlation function (ACF) must be even symmetric and have its max at $R_Y[0]$. However, nothing in this problem specifies that $Y$ is WSS. Still, I believe the maximum must still be at $R_Y[0]$ so I choose
$$u=1$$
This gives me
$$R_Y[0]=4, R_Y[-1]=R_Y[1]=-1, text{ and } R_Y[t]=1 text{ for } |t|geq 2$$
Part B
Specify a random process with the given autocorrelation function (i.e. specify the stochastic generation mechanism for the process).
My thought on this is to utilize the Wiener–Khinchin theorem
$$S_Y(omega) = DFT(R_Y[t])$$
and then use the fact of
$$S_Y[omega]=|H(omega)|^2S_x(omega)$$
making $S_x=1$ via white gaussian process and solving for a suitable $|H(omega)|^2$. However, I believe the above equation only holds if $S_X$ is WSS. This is fine though as there are no restrictions on Y[t].
So, to summarize, we have
$$R_Y(t) = 3delta(t) - 2(delta(t+1) + delta(t-1)) + 1$$
giving the Fourier transform to be
$$S_y(omega)=3 + 4 cos(2pi f) + delta(f)$$
This is where i hit a snag. Making $S_X=1$, I cannot have the above equation as a power spectral density (PSD) because it contains negative values due to the $cos(2pi f) $ term.
I am not sure what to do. Any ideas?
Some thoughts:
- I could factor $cos$ into something non-negative but this only moves the problem around (and am not sure possible).
- Abandon the assumption $S_X=1$ by designing a process which gives $S_Y$ and simply make $H(omega)=1$. I am unsure if this is possible though.
- Stay in time domain and exploit the fact of $R_Y[tau]=E[Y[t]Y[t-tau]]$.
- Use Wold's decomposition theorem. This is a long shot.
- Model $R_Y$ as ARMA.
stochastic-processes fourier-analysis random-variables signal-processing
asked Jan 12 at 20:17
AvedisAvedis
476
$endgroup$
I have a discrete-time real-valued process Y[n] whose autocorrelation function is: $$R_Y[0] = 3+u, R_Y[1]=R_Y[-1]=-2+u, text{ and } R_Y[k] = u, mid k mid > 1$$
Part A
Specify a choice of $u$, with $|u|>0$, for which there is a valid random process with the given autocorrelation function.
For a wide sense stationary (WSS) process, we have that the auto-correlation function (ACF) must be even symmetric and have its max at $R_Y[0]$. However, nothing in this problem specifies that $Y$ is WSS. Still, I believe the maximum must still be at $R_Y[0]$ so I choose
$$u=1$$
This gives me
$$R_Y[0]=4, R_Y[-1]=R_Y[1]=-1, text{ and } R_Y[t]=1 text{ for } |t|geq 2$$
Part B
Specify a random process with the given autocorrelation function (i.e. specify the stochastic generation mechanism for the process).
My thought on this is to utilize the Wiener–Khinchin theorem
$$S_Y(omega) = DFT(R_Y[t])$$
and then use the fact of
$$S_Y[omega]=|H(omega)|^2S_x(omega)$$
making $S_x=1$ via white gaussian process and solving for a suitable $|H(omega)|^2$. However, I believe the above equation only holds if $S_X$ is WSS. This is fine though as there are no restrictions on Y[t].
So, to summarize, we have
$$R_Y(t) = 3delta(t) - 2(delta(t+1) + delta(t-1)) + 1$$
giving the Fourier transform to be
$$S_y(omega)=3 + 4 cos(2pi f) + delta(f)$$
This is where i hit a snag. Making $S_X=1$, I cannot have the above equation as a power spectral density (PSD) because it contains negative values due to the $cos(2pi f) $ term.
I am not sure what to do. Any ideas?
Some thoughts:
- I could factor $cos$ into something non-negative but this only moves the problem around (and am not sure possible).
- Abandon the assumption $S_X=1$ by designing a process which gives $S_Y$ and simply make $H(omega)=1$. I am unsure if this is possible though.
- Stay in time domain and exploit the fact of $R_Y[tau]=E[Y[t]Y[t-tau]]$.
- Use Wold's decomposition theorem. This is a long shot.
- Model $R_Y$ as ARMA.
stochastic-processes fourier-analysis random-variables signal-processing
stochastic-processes fourier-analysis random-variables signal-processing
asked Jan 12 at 20:17
AvedisAvedis
476
asked Jan 12 at 20:17
AvedisAvedis
476
edited Jan 12 at 22:26
Avedis
asked Jan 12 at 20:17
AvedisAvedis
476
asked Jan 12 at 20:17
AvedisAvedis
476
asked Jan 12 at 20:17
AvedisAvedis
476
476
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