CHAPTER 7. DT LTI SYSTEMS WITH FINITE MEMORY
156
To conclude this section, we compute the z-transform of x[n + 1], the advance or shifting to the left of x[n] by one sample. Be definition, we have Z[x[n + 1]] =
∞ X
x[n + 1]z − n =
n=0 ∞ X
=
z
∞ X
x[n + 1]z − (n+1)+1
n=0
x[n + 1]z − (n+1)
n=0
which can be written as, defining l := n + 1, Z[x[n + 1]] = z
∞ X
" x[l]z − l = z
l=1
=
#
∞ X
x[l]z
−l
− x[0]z
−0
l=0
z[X (z) − x[0]]
(7.33)
Using the same procedure, we can obtain £ ¤ Z[x[n + 2]] = z 2 X (z) − x[0] − x[1]z − 1 (7.34) We give an explanation of (7.33). The sequence x[n + 1] shifts x[n] one sample to the left. Thus x[0] is shifted from n = 0 to n = − 1 and x[n + 1] becomes non-positive time even though x[n] is positive time. The z-transform of x[n + 1] is defined only for the positive-time part of x[n + 1] and, consequently, will not contain x[0]. Thus we subtract x[0] from X (z) and then multiply it by z to yield (7.33). Equation (7.34) can be similarly explained. We see that advancing a sequence by one sample is equivalent to the multiplication of z in its z-transform, denoted as Unit-sample advance ←→ z In other words, a system that carries out unit-sample time advance has transfer function z. Note that such a system is not causal and cannot be implemented in real time. Moreover because (7.33) and (7.34) involve x[0] and x[1], their uses are not as convenient as (7.30).
7.8
Composite systems: Transform domain or time domain?
We have introduced four mathematical equations to describe DT LTI systems. They are 1. Convolutions 2. High-order difference equations 3. Sets of first-order difference equations of special form, called state-space (ss) equations 4. DT rational transfer functions The first three are all expressed in terms of time index and are called time-domain descriptions. The last one is based on the z-transform and is called the transform-domain description. Note that for DT LTI memoryless systems, the first three descriptions reduce to y[n] = αu[n] and the last description reduces to Y (z) = αU (z), for some constant α. In the transform domain, we have Z [output] = transfer function × Z[input]
(7.35)
This equation is purely algebraic; it does not involve time advance or time delay. For example, consider the unit-delay element shown in Figure 7.2(c) with y[n] = u[n − 1]. Although the element is denoted by UD, the equation y[n] = UDu[n] is mathematically meaningless. In
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