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Solving a difference equation via z-transforms
I'm trying to figure out if there is a "nice" solution to this difference equation
$$alpha p_t = p_{t-1} + beta x_t(x_t - x_{t-1}).$$
Using a $z$ transform, I get
$$P_z(alpha - z^{-1}) = beta X_z^2z^{t}(1 - z^{-1}) $$
$$Rightarrow P_z = beta X_z^2z^{t}frac{(1 - z^{-1})}{(alpha - z^{-1})} $$
Is there any further simplification possible ? Specifically is there any explicit non-recursive formula expressing $p_t$ as a function of $x_t$ ?
sequences-and-series time-series z-transform
asked Nov 21 '18 at 0:17
firdaus
388112
add a comment |
I'm trying to figure out if there is a "nice" solution to this difference equation
$$alpha p_t = p_{t-1} + beta x_t(x_t - x_{t-1}).$$
Using a $z$ transform, I get
$$P_z(alpha - z^{-1}) = beta X_z^2z^{t}(1 - z^{-1}) $$
$$Rightarrow P_z = beta X_z^2z^{t}frac{(1 - z^{-1})}{(alpha - z^{-1})} $$
Is there any further simplification possible ? Specifically is there any explicit non-recursive formula expressing $p_t$ as a function of $x_t$ ?
sequences-and-series time-series z-transform
asked Nov 21 '18 at 0:17
firdaus
388112
$z$ transform is a linear transformation...
– Cesareo
Nov 21 '18 at 1:55
add a comment |
I'm trying to figure out if there is a "nice" solution to this difference equation
$$alpha p_t = p_{t-1} + beta x_t(x_t - x_{t-1}).$$
Using a $z$ transform, I get
$$P_z(alpha - z^{-1}) = beta X_z^2z^{t}(1 - z^{-1}) $$
$$Rightarrow P_z = beta X_z^2z^{t}frac{(1 - z^{-1})}{(alpha - z^{-1})} $$
Is there any further simplification possible ? Specifically is there any explicit non-recursive formula expressing $p_t$ as a function of $x_t$ ?
sequences-and-series time-series z-transform
asked Nov 21 '18 at 0:17
firdaus
388112
I'm trying to figure out if there is a "nice" solution to this difference equation
$$alpha p_t = p_{t-1} + beta x_t(x_t - x_{t-1}).$$
Using a $z$ transform, I get
$$P_z(alpha - z^{-1}) = beta X_z^2z^{t}(1 - z^{-1}) $$
$$Rightarrow P_z = beta X_z^2z^{t}frac{(1 - z^{-1})}{(alpha - z^{-1})} $$
Is there any further simplification possible ? Specifically is there any explicit non-recursive formula expressing $p_t$ as a function of $x_t$ ?
sequences-and-series time-series z-transform
sequences-and-series time-series z-transform
asked Nov 21 '18 at 0:17
firdaus
388112
asked Nov 21 '18 at 0:17
firdaus
388112
asked Nov 21 '18 at 0:17
firdaus
388112
asked Nov 21 '18 at 0:17
firdaus
388112
asked Nov 21 '18 at 0:17
firdaus
388112
388112
$z$ transform is a linear transformation...
– Cesareo
Nov 21 '18 at 1:55
add a comment |
$z$ transform is a linear transformation...
– Cesareo
Nov 21 '18 at 1:55
$z$ transform is a linear transformation...
– Cesareo
Nov 21 '18 at 1:55
$z$ transform is a linear transformation...
– Cesareo
Nov 21 '18 at 1:55
add a comment |
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$z$ transform is a linear transformation...
– Cesareo
Nov 21 '18 at 1:55