Appendix: Table of Laplace and z Transforms
Introduction
The table below is like Table 3.1 in Chapter 3, except there are more transforms here than in Table 3.1 and the column with h(t) is placed between the two transform columns, as if one were beginning with a system function, H_{a}(s), then finding the impulse response, h(t), and finally obtaining the z-transform, H(z). The lettered and numbered lines have the same use as in Table 3.1.
Line |
Laplace Transform |
h(t) |
z-Transform |
---|---|---|---|
A |
${H}_{a}\left(s\right)={\displaystyle \underset{0}{\overset{\infty}{\int}}h\left(t\right){e}^{-st}}dt$ |
h(t) |
$H\left(z\right)={\displaystyle \sum _{m=0}^{\infty}{h}_{m}}{z}^{-m}$ |
B |
AH_{a}(s) |
Ah(t) |
AH(z) |
C |
H_{a}(s) + G_{a}(s) |
h(t) + g(t) |
H(z) + G(z) |
D |
sH_{a}(s) − h(0^{+}) |
$\frac{d}{dt}h\left(t\right)$ |
− |
E |
$\frac{{H}_{a}\left(s\right)}{s}$ |
$\underset{0}{\overset{t}{\int}}h\left(\text{\tau}\right)\text{\hspace{0.17em}}d}\text{\tau$ |
− |
F |
$-\frac{d}{ds}{H}_{a}\left(s\right)$ |
th(t) |
$-Tz\frac{d}{dz}H\left(z\right)$ |
G |
H_{a}(s + a) |
e^{−at}h(t); a > 0 |
H(ze^{aT}) |
H |
e^{−nsT}H_{a}(s) |
h(t − nT |
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