9.1 • THE Z-TRANSFOR.M 355y ( 11 ]
(^5).. .......
4
3
2
Figure 9.1 The solution to y[n + 1] = ~y[n] + 1 with Yo = 0.
-------.-EXERCISES FOR SECTION 9.1
1. Use t he definition of the z-transform t o find X (z) = 3 (x.,] = 3[x[n]].
(a) For t he sequence Xn = x(n] = ( ~ r.
(b) For t he sequence x,. = x(n] = e"".
(c) For t he sequence x., = x(n] = n.(^2) • U se 3[ e ion] = z -z ef.O an d 3[ e -ion] = %- e - •O z t 0 prove t' 1la t 3{ $10 · ( an )] = zi-2cos•i•( • ) z (o)z-+l ·
- Show that the z-transfor m of the delayed uni t-step sequence
{1 for n > mXn = u(n -m) = 0 for n :( m
, JS X( Z ) = z l - m • - l.- Find a.nd simplify over a common denominator the foUowing z-tra nsforrns.
(a) X (z) = 3(2" + 4"]
(b) X(z) = 3 [3" + 3]
(c) X (z) = 3(2" + 2n)[ ( ) ]( ..^1 +•-1)b- Show t hat 3-^1 <•~l) (l-:• 1 > = .,_ 1 and supply all the details.
6. Show that t he convolution sequences x,. = 1 and Yn = n is Wn = x., * y,. = n(~+l),
and that 3(wn) = 3[x,.]3[y.,].- Prove the following propert ies of z-transforms.
(a) Linearity: 3(cx,. + dy,..] = cX(z) + dY(z).