Digital Signal Processing Systems, Basic Filtering Types, and Digital Filter Realizations - Digital Signal Processing: Fundamentals and Applications - page 174

FIGURE 6.7

Relationship between Laplace transform and z-transform.

As shown in Figure 6.7 , the sampled signal, which is not quantized, with a sampling period of T is
written as

N
n¼ 0 xðnTÞdðt nTÞ¼xð 0 ÞdðtÞþxðTÞdðt TÞþxð 2 TÞdðt 2 TÞþ/

x s ðtÞ¼ (6.12)

Taking the Laplace transform and using the Laplace shift property as
Lðdðt nTÞÞ ¼ e nTs

(6.13)

leads to

N
n¼ 0 xðnTÞe nTs ¼ xð 0 Þe 0 Ts þ xðTÞe Ts þ xð 2 TÞe 2 Ts þ/

X s ðsÞ¼ (6.14)

Compare Equation (6.14) with the definition of a one-sided z-transform of the data sequence xðnÞ from
analog-to-digital conversion (ADC):

N
n¼ 0 xðnÞz n ¼ xð 0 Þz 0

þ xð 1 Þz 1 þ xð 2 Þz 2 XðzÞ¼ZðxðnÞÞ ¼ þ/ (6.15)

Clearly, we see the relationship of the sampled system in the Laplace domain and its digital system in
the z-transform domain by the following mapping:

z ¼ e sT (6.16)
Substituting s ¼a ju into Equation (6.16) , it follows that z ¼ e aTjuT . In the polar form, we
have

z ¼ e aT : uT (6.17)

Equations (6.16) and (6.17) give the following important conclusions.