Does any discrete signal has z-transform (even noise)?











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1
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I faced to the following discrete



$$
y[n+1]=Ay[n]+Bx[n]+eta[n]
$$



where $y[n] in mathbb{R}^n$, $A$ and $B$ are matrices with appropriate dimension, and $eta[n]$ is noise.



I have no problem with the following



$$
y[n+1]=Ay[n]+Bx[n]
$$



Because I can write the z-transform as follows



$$
zY(z)-zY(0)=AY(z)+BX(z)
$$



Can I simply write the following even for noise?



$$
zY(z)-zy(0)=AY(z)+BX(z)+ H(z)
$$



Please answer it conceptually and explain it.










share|cite|improve this question
























  • If $sum_{n=0}^infty |w[n]| r^n < infty$ and $sum_{n=0}^infty |x[n]| r^n < infty$ and $y[n+1] = A y[n]+B x[n]+w[n]$ then for $0 < |z| < min(r,1/|A|) $ : $$z^{-1}(-y[0] +sum_{n=0}^infty y[n] z^n) = sum_{n=0}^infty y[n+1] z^n = Asum_{n=0}^infty y[n] z^n+Bsum_{n=0}^infty x[n] z^n+sum_{n=0}^infty w[n] z^n$$
    – reuns
    Nov 18 at 2:42












  • What is the meaning of the constraint that you have for the noise $w[n]$? I understand that it comes from definition of z-transformation but it is not tangible for me? which structure has be enforced to the noise to have that condition?
    – Saeed
    Nov 19 at 4:31










  • For example $E[ |w[n]|^2] < C < infty$ implies $E[sum_{n=1}^infty frac{|w[n]|^2}{n^2}] < infty$ so $sum_{n=1}^infty frac{|w[n]|}{n^4} < infty$ and $sum_{n=1}^infty |w[n]| r^n < infty$ for $r < 1$
    – reuns
    Nov 19 at 15:08

















up vote
1
down vote

favorite












I faced to the following discrete



$$
y[n+1]=Ay[n]+Bx[n]+eta[n]
$$



where $y[n] in mathbb{R}^n$, $A$ and $B$ are matrices with appropriate dimension, and $eta[n]$ is noise.



I have no problem with the following



$$
y[n+1]=Ay[n]+Bx[n]
$$



Because I can write the z-transform as follows



$$
zY(z)-zY(0)=AY(z)+BX(z)
$$



Can I simply write the following even for noise?



$$
zY(z)-zy(0)=AY(z)+BX(z)+ H(z)
$$



Please answer it conceptually and explain it.










share|cite|improve this question
























  • If $sum_{n=0}^infty |w[n]| r^n < infty$ and $sum_{n=0}^infty |x[n]| r^n < infty$ and $y[n+1] = A y[n]+B x[n]+w[n]$ then for $0 < |z| < min(r,1/|A|) $ : $$z^{-1}(-y[0] +sum_{n=0}^infty y[n] z^n) = sum_{n=0}^infty y[n+1] z^n = Asum_{n=0}^infty y[n] z^n+Bsum_{n=0}^infty x[n] z^n+sum_{n=0}^infty w[n] z^n$$
    – reuns
    Nov 18 at 2:42












  • What is the meaning of the constraint that you have for the noise $w[n]$? I understand that it comes from definition of z-transformation but it is not tangible for me? which structure has be enforced to the noise to have that condition?
    – Saeed
    Nov 19 at 4:31










  • For example $E[ |w[n]|^2] < C < infty$ implies $E[sum_{n=1}^infty frac{|w[n]|^2}{n^2}] < infty$ so $sum_{n=1}^infty frac{|w[n]|}{n^4} < infty$ and $sum_{n=1}^infty |w[n]| r^n < infty$ for $r < 1$
    – reuns
    Nov 19 at 15:08















up vote
1
down vote

favorite









up vote
1
down vote

favorite











I faced to the following discrete



$$
y[n+1]=Ay[n]+Bx[n]+eta[n]
$$



where $y[n] in mathbb{R}^n$, $A$ and $B$ are matrices with appropriate dimension, and $eta[n]$ is noise.



I have no problem with the following



$$
y[n+1]=Ay[n]+Bx[n]
$$



Because I can write the z-transform as follows



$$
zY(z)-zY(0)=AY(z)+BX(z)
$$



Can I simply write the following even for noise?



$$
zY(z)-zy(0)=AY(z)+BX(z)+ H(z)
$$



Please answer it conceptually and explain it.










share|cite|improve this question















I faced to the following discrete



$$
y[n+1]=Ay[n]+Bx[n]+eta[n]
$$



where $y[n] in mathbb{R}^n$, $A$ and $B$ are matrices with appropriate dimension, and $eta[n]$ is noise.



I have no problem with the following



$$
y[n+1]=Ay[n]+Bx[n]
$$



Because I can write the z-transform as follows



$$
zY(z)-zY(0)=AY(z)+BX(z)
$$



Can I simply write the following even for noise?



$$
zY(z)-zy(0)=AY(z)+BX(z)+ H(z)
$$



Please answer it conceptually and explain it.







z-transform






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share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 18 at 23:25

























asked Nov 17 at 17:47









Saeed

444110




444110












  • If $sum_{n=0}^infty |w[n]| r^n < infty$ and $sum_{n=0}^infty |x[n]| r^n < infty$ and $y[n+1] = A y[n]+B x[n]+w[n]$ then for $0 < |z| < min(r,1/|A|) $ : $$z^{-1}(-y[0] +sum_{n=0}^infty y[n] z^n) = sum_{n=0}^infty y[n+1] z^n = Asum_{n=0}^infty y[n] z^n+Bsum_{n=0}^infty x[n] z^n+sum_{n=0}^infty w[n] z^n$$
    – reuns
    Nov 18 at 2:42












  • What is the meaning of the constraint that you have for the noise $w[n]$? I understand that it comes from definition of z-transformation but it is not tangible for me? which structure has be enforced to the noise to have that condition?
    – Saeed
    Nov 19 at 4:31










  • For example $E[ |w[n]|^2] < C < infty$ implies $E[sum_{n=1}^infty frac{|w[n]|^2}{n^2}] < infty$ so $sum_{n=1}^infty frac{|w[n]|}{n^4} < infty$ and $sum_{n=1}^infty |w[n]| r^n < infty$ for $r < 1$
    – reuns
    Nov 19 at 15:08




















  • If $sum_{n=0}^infty |w[n]| r^n < infty$ and $sum_{n=0}^infty |x[n]| r^n < infty$ and $y[n+1] = A y[n]+B x[n]+w[n]$ then for $0 < |z| < min(r,1/|A|) $ : $$z^{-1}(-y[0] +sum_{n=0}^infty y[n] z^n) = sum_{n=0}^infty y[n+1] z^n = Asum_{n=0}^infty y[n] z^n+Bsum_{n=0}^infty x[n] z^n+sum_{n=0}^infty w[n] z^n$$
    – reuns
    Nov 18 at 2:42












  • What is the meaning of the constraint that you have for the noise $w[n]$? I understand that it comes from definition of z-transformation but it is not tangible for me? which structure has be enforced to the noise to have that condition?
    – Saeed
    Nov 19 at 4:31










  • For example $E[ |w[n]|^2] < C < infty$ implies $E[sum_{n=1}^infty frac{|w[n]|^2}{n^2}] < infty$ so $sum_{n=1}^infty frac{|w[n]|}{n^4} < infty$ and $sum_{n=1}^infty |w[n]| r^n < infty$ for $r < 1$
    – reuns
    Nov 19 at 15:08


















If $sum_{n=0}^infty |w[n]| r^n < infty$ and $sum_{n=0}^infty |x[n]| r^n < infty$ and $y[n+1] = A y[n]+B x[n]+w[n]$ then for $0 < |z| < min(r,1/|A|) $ : $$z^{-1}(-y[0] +sum_{n=0}^infty y[n] z^n) = sum_{n=0}^infty y[n+1] z^n = Asum_{n=0}^infty y[n] z^n+Bsum_{n=0}^infty x[n] z^n+sum_{n=0}^infty w[n] z^n$$
– reuns
Nov 18 at 2:42






If $sum_{n=0}^infty |w[n]| r^n < infty$ and $sum_{n=0}^infty |x[n]| r^n < infty$ and $y[n+1] = A y[n]+B x[n]+w[n]$ then for $0 < |z| < min(r,1/|A|) $ : $$z^{-1}(-y[0] +sum_{n=0}^infty y[n] z^n) = sum_{n=0}^infty y[n+1] z^n = Asum_{n=0}^infty y[n] z^n+Bsum_{n=0}^infty x[n] z^n+sum_{n=0}^infty w[n] z^n$$
– reuns
Nov 18 at 2:42














What is the meaning of the constraint that you have for the noise $w[n]$? I understand that it comes from definition of z-transformation but it is not tangible for me? which structure has be enforced to the noise to have that condition?
– Saeed
Nov 19 at 4:31




What is the meaning of the constraint that you have for the noise $w[n]$? I understand that it comes from definition of z-transformation but it is not tangible for me? which structure has be enforced to the noise to have that condition?
– Saeed
Nov 19 at 4:31












For example $E[ |w[n]|^2] < C < infty$ implies $E[sum_{n=1}^infty frac{|w[n]|^2}{n^2}] < infty$ so $sum_{n=1}^infty frac{|w[n]|}{n^4} < infty$ and $sum_{n=1}^infty |w[n]| r^n < infty$ for $r < 1$
– reuns
Nov 19 at 15:08






For example $E[ |w[n]|^2] < C < infty$ implies $E[sum_{n=1}^infty frac{|w[n]|^2}{n^2}] < infty$ so $sum_{n=1}^infty frac{|w[n]|}{n^4} < infty$ and $sum_{n=1}^infty |w[n]| r^n < infty$ for $r < 1$
– reuns
Nov 19 at 15:08

















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