082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

698 Chapter 22 Difference equations and the z transform
Solutions
1 0, 7, −5, 14
4
2
7 , 63
4 , 64
63
3 (a) 5, 29
2
(b)
1
3 , 7 18
(c)
29
3 , 52
9
4 (a) 3,5,7, 9,11
(b) 5,7,2, 9, −3
(c) 1,1,1, 1,1
TechnicalComputingExercises22.6
1 Writeacomputer program to implementthe
difference equations given in Engineeringapplication
22.5. Verifythat your resultsagreepreciselywith
those given in Table 22.1.
2 Modifyyour program sothat the input sequence is
x[n] = [0 1 0 0 0 0 0 0 0 0 0 0].Theoutput
is known astheimpulseresponse ofthe system.
Observe the shape ofthe plot andcompareitto the
output sequence in the Example.
3 Now change your program sothatitwill generate a
total of100 samples ofoutputandchange the input
sequence to:
x[n] = sin(2πn/100)
You should observe asinewave on the outputy(n).
Makeanoteofits amplitude.
4 Investigatewhat happensto the amplitudeofthe
outputwhenyou increasethe frequency ofthe input
sinewave. The frequency can be increased by
reducing the value ofthe denominatorin the sine
function, forexample,
x[n] = sin(2πn/10)
5 Does the result agreewith the description ofthe
operation ofthe discrete-time filterdescribedin the
example?
22.7 DEFINITIONOFTHEzTRANSFORM
Supposewehaveasequence f[k],k ∈ N.Suchasequencemayhavearisenbysampling
a continuoussignal. We define itsztransformtobe
∞∑
F(z) = Z{f[k]} = f[k]z −k (22.7)
k=0
Weseefromthedefinitionthattheztransformisaninfiniteseriesformedfromtheterms
ofthe sequence.Explicitly, wehave
Z{f[k]}=f[0]+ f[1]
z
+ f[2]
z 2 + f[3]
z 3 +···
Inmostengineeringapplicationswedonotactuallyneedtoworkwiththeinfiniteseries
since it is often possible to express this in a closed form. The closed form is generally
valid for values ofzwithin a region known as the radius of absolute convergence as
will become apparent from the following examples.