Determining stochastic process to give a specified autocorrelation function












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$begingroup$


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:




  1. I could factor $cos$ into something non-negative but this only moves the problem around (and am not sure possible).

  2. 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.

  3. Stay in time domain and exploit the fact of $R_Y[tau]=E[Y[t]Y[t-tau]]$.

  4. Use Wold's decomposition theorem. This is a long shot.

  5. Model $R_Y$ as ARMA.










share|cite|improve this question











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    0












    $begingroup$


    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:




    1. I could factor $cos$ into something non-negative but this only moves the problem around (and am not sure possible).

    2. 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.

    3. Stay in time domain and exploit the fact of $R_Y[tau]=E[Y[t]Y[t-tau]]$.

    4. Use Wold's decomposition theorem. This is a long shot.

    5. Model $R_Y$ as ARMA.










    share|cite|improve this question











    $endgroup$















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      0








      0





      $begingroup$


      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:




      1. I could factor $cos$ into something non-negative but this only moves the problem around (and am not sure possible).

      2. 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.

      3. Stay in time domain and exploit the fact of $R_Y[tau]=E[Y[t]Y[t-tau]]$.

      4. Use Wold's decomposition theorem. This is a long shot.

      5. Model $R_Y$ as ARMA.










      share|cite|improve this question











      $endgroup$




      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:




      1. I could factor $cos$ into something non-negative but this only moves the problem around (and am not sure possible).

      2. 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.

      3. Stay in time domain and exploit the fact of $R_Y[tau]=E[Y[t]Y[t-tau]]$.

      4. Use Wold's decomposition theorem. This is a long shot.

      5. Model $R_Y$ as ARMA.







      stochastic-processes fourier-analysis random-variables signal-processing






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      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 12 at 22:26







      Avedis

















      asked Jan 12 at 20:17









      AvedisAvedis

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