Does any discrete signal has z-transform (even noise)?
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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.
z-transform
asked Nov 17 at 17:47
Saeed
436110
add a comment |
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.
z-transform
asked Nov 17 at 17:47
Saeed
436110
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
add a comment |
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.
z-transform
asked Nov 17 at 17:47
Saeed
436110
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
z-transform
asked Nov 17 at 17:47
Saeed
436110
asked Nov 17 at 17:47
Saeed
436110
edited Nov 18 at 23:25
asked Nov 17 at 17:47
Saeed
436110
asked Nov 17 at 17:47
Saeed
436110
asked Nov 17 at 17:47
Saeed
436110
436110
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
add a comment |
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
add a comment |
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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