Question A.1: Illustrate the use of the preceding equations to take a time...
Illustrate the use of the preceding equations to take a time domain waveform and transform it to the z domain.
For example, the unit step function u(t) can be transformed. In this case, f(kT) = 1 for all k > 0. Therefore,
F_{s}(z) = \sum\limits_{k=0}^{\infty }{} z^{-k} = (1+z^{-1}+z^{-2}+z^{-3}… ) (A.7)
We make use of the following mathematical identity:
\frac{x}{x-1}=1+x^{-1}+x^{-2}+x^{-3}… (A.8)
Therefore,
F_{s}(z)= \frac{z}{z-1} (A.9)
Other functions can be transformed in a similar way. For example, the natural exponential f(kT) = e ^{-akT} :
F_{s}(z) = \sum\limits_{k=0 }^{\infty}{}e ^{-akT}z^{-k} = 1 + e ^{-aT}z^{-1}+e ^{-a2T}z^{-2}+e ^{-a3T}z^{-3}… (A.10)
This can be rewritten as
F_{s}(z) = 1+ (e^{aT} z)^{−1} + (e^{aT} z)^{−2} + (e^{aT} z)^{−3} … (A.11)Again using the math identity (A.8),
F_{s}(z) =\frac{e^{aT} z}{e^{aT} z-1}= \frac{z}{z-e^{-aT} } (A.12)In this way, the third column in Table A.1 can be constructed.
So far, we have talked about ideal sampling (multiplying by impulse functions). In reality, we also ‘‘hold’’ the value for an entire clock period. Thus, each element in the infinite series in (A.6), is multiplied by a square wave of length of one period:
F_{s}(z) = \sum\limits_{k=0}^{\infty }{} f(kT)z^{-k} (A.6)
g_{hold} (t) = u(t) − u(t − T) (A.13)G_{hold} (s) = \frac{1 − e^{−sT} }{s}
Some care should be taken in general when manipulating and converting systems from the s to the z domains. Note especially that, in general:
Z[G(s)H(s)] ≠ G(z)H(z) (A.14)
| Table A.1 common laplace Transforms and z Transforms | |||
| Time Domain Function y(t)/Discrete Time Domain y(kT) | F(s) (Continuous Laplace Equivalent) | F(z) (Discrete Laplace Equivalent) | Comments and Additional Explanations |
| \delta (t) | 1 | 1 | Unit impulse function |
| \delta (kT) | |||
| u(t) | \frac{1}{s} | \frac{z}{z-1} | Unit step function; also an integrator |
| u (kT) | |||
| t | \frac{1}{s^{2}} | \frac{T_{z}}{(z-1)^{2} } | Ramp |
| Kt | |||
| t^{2} | \frac{2}{s^{3} } | \frac{T^{2}z(z+1)}{(z-1)^{3} } | Parabolic |
| (kT)^{2} | |||
| t^{n} | \frac{n!}{s^{n+1} } | \underset{a\rightarrow 0}{\lim } (-1)^{n} \frac{d^{n} }{da^{n} }(\frac{z}{z-e^{-aT} } ) | Exponential |
| (kT)^{n} | |||
| e^{-aT} | \frac{1}{s+a }
|
(\frac{z}{z-e^{-aT} } )
|
Natural exponential decay
|
| e^{-akT} | |||
| 1-e^{-aT} | \frac{1}{s(s+a) } | \frac{z(1-e^{-aT})}{(z-e^{-aT})(z-1) } | Natural exponential growth |
| 1- e^{-akT} | |||
| \sin (at) | \frac{a}{s^{2} +a^{2}} | \frac{z\sin (\omega T)}{z^{2}-2z\cos (\omega T)+1 } | Sine wave, no decay |
| \sin (akT) | |||
| \cos (at) | \frac{s}{s^{2} +a^{2} } | \frac{z\left[z-\cos(\omega T)\right] }{z^{2}-2z\cos (\omega T)+1 } | Cosine wave, no decay |
| \cos (akT) | |||
| e^{aT} \sin (bt) | \frac{b}{(s-a)^{2} +b^{2} } | \frac{ze^{-aT} \sin (\omega T)}{z^{2}-2ze^{-aT}\cos (\omega T)+e^{-aT} } | Sine wave with exponential decay/growth |
| e^{akT} \sin (bkT) | |||
| e^{aT} \cos (bt) | \frac{s-a}{(s-a)^{2} +b^{2} } | \frac{z^2 – ze ^{-aT} \cos (wT)}{z^2 – 2ze^{-aT }\cos (wT) + e^{-aT}} | Cosine wave with exponential decay/growth |
| e^{akT} \cos (bkT) | |||
| t.e^{at} | \frac{1}{(s-a)^{2}} | \frac{Te^{-aT} z}{(e^{-aT}z – 1)^2} | Ramp with exponential decay,growth |
| kt.e^{aKT} | |||
| e^{at} f(t) | F(s − a) | F(e^{-aT}z) | Frequency-Shift Theorem. |
| e^{aKt} f(Kt) | |||
| f(t-nT) | e^{−snT} F(s) | z^{-n}F(z) | Time-Shift Theorem (a delay) |
| f(Kt-nT) | |||
| \frac{df(t)}{dt} | sF(s) | (1-z^{-1} )F(z) | Differentiation Theorem/Difference Theorem |
| f(Kt) – f(KT-T) | |||
| \int_{0}^{t}{f(\tau )d\tau } | \frac{F(s)}{s} | (\frac{z}{z-1}) F(z) | Integration Theorem/ Accumulation Theorem |
| f(Kt)+y(KT-T) | |||
| f(\infty ) | \underset{s\rightarrow 0}{\lim} \ sF(s) | \underset{z\rightarrow 1}{\lim} \ (1-z^{-1} ) F(z) | Final Value Theorem |
| f(0) | \underset{s\rightarrow \infty }{\lim} \ s F(s) | \underset{z\rightarrow \infty }{\lim} F(z) | Initial Value Theorem |
| f(t)g(t) | F(s) \otimesG(s) | F(z) \otimesG(z) | Convolution Theorem* |
| f(kT)g(kT) | |||
| \int\limits_{0}^{t}{}\begin{matrix} f(t-\tau )g(\tau )d(\tau )= \\f(t)\otimes g(t) \end{matrix} | F(s) G(s) | F(z) G(z) | Convolution Theorem* |
| F(kt) \otimesg(kt) | |||
| *Note that the symbol \otimes denotes convolution. | |||