# Conversion from laplace transform to z-transform [closed]

I would like to know if

$$\text {Z-Transform ( } G(s)H(s) \text{ )} = \text {Z-Transform (}G(s) \text{)} \text { Z-Transform (} H(s) \text{) } = G(z)H(z)$$

where G(s), H(s) are the Laplace transform representations of g and h, and G(z) and H(z) are the Z-transform representation of g and h.

Is this relation true?

• The Laplace transform encodes a continuous signal, the Z transform encodes a discrete signal. How do you propose the two are related? – Lutz Lehmann Feb 18 '14 at 11:19
• The relation between s and z is $s = \frac{1}{T}ln(z)$ is it not? – KillaKem Feb 18 '14 at 15:01
• This assumes a change in the quality of the underlying signal, i.e., sampling with a sampling step of $T$. Then the variable replacement will not change the functional relationship. But use different symbols for the functions, $G(s)$ is a different function than what you named $G(z)$, which then really is $G(\tfrac 1T\ln(z))$. – Lutz Lehmann Feb 18 '14 at 15:11

Since $H(s)$ is a continuous function, you can't just calculate a Z-transform of $H(s)$ without first sampling it, to make it discrete. Also, it doesn't make much sense to do a time->spectrum transform (such as a Z-transform) on a spectral representation ($H(s)$)
I'm assuming that when you write "$Z-Transform(H(s))$", what you really want to do is to convert $H(s)\to H(z)$, meaning to calculate the Z-transform of $h[nT]$, where $h[nT]$ is $h(t)$, sampled at intervals of $T$, and $h(t)$ is the inverse Laplace transform of $H(s)$.
If I'm correct in my assumption, the transformation you are seeking is known as "star transform", which would provide a transform function $H^{*}(s)$, in terms of $e^{sT}$, which may be easily converted to $H(z)$ by way of the substitution $z=e^{sT}$.
Edit - some elaboration on the conversion process: what you need to do is calculate $H^{*}(s)$ from $H(s)$ using one of the two relations described in "Relation to Laplace transform" in the Wikipedia article, then do the substitution $z=e^{sT}$, to get $H(z)$.