Digital Signal Processing 9.1 Discrete-Time Systems 205
Part A | 9.1
Given the above, one can proceed to impulse response
truncation with accuracy ". This procedure consists of
approximating the original IIR system with transfer
function H • .z
1
/ D
P C1
nD0 h • .n/z
n by the following
FIR one with memory M
O
H FIR .z
1
/ D
M
X
nD0
h • .n/z
n
D
M
X
nD0
b n z
n
:
(9.42)
Approximating a continuous-time system with a discrete-time one, which on top might be FIR, comes with
a number of advantages, especially for real-time applications.
Invariance of step response: the unit step signal in
discrete time is defined as follows
u step .n/ D
(
1; n 0
0; n < 0
,
U step .z/ D
C1 X
nD0
z
n
D
1
1 z 1 :
(9.43)
The Z-transform of the unit step signal is derived by
applying the lemma for the infinite sum of a geometric
progression with a ratio that in magnitude is less than 1.
Consider now an LTI system with impulse response
as follows
y step .t/ D L
1
fH.s/U step .s/g D L
1
H.s/
s
:
(9.44)
We will now determine a discrete-time transfer function
H.z/ such that the latter’s unit step response coincides
with that of the continuous-time system above. Mathematically, this is defined as follows
Z
1
fH.z/U step .z/g D y step .n/ D y step .nT s / : (9.45)
Then, by employing (9.43), one obtains that
H.z/ D .1 z
1
/Zfy step .nT s /g :
(9.46)
Since
z
1
D exp.sT s / and Y step .s/ D
H.s/
s
:
We can finally obtain that
H.z/ D Zfy ZOH .nT s /g :
(9.47)
In the above, it holds that
y ZOH .t/ D h ZOH .t/ ˝ h.t/ ,
Y ZOH .s/ D H ZOH .s/H.s/ D
1 e
sTs
s
H.s/ : (9.48)
Pole and Zero Matching
of a Transfer Function
This method derives from equation
z
1
D exp.sT s /
connecting variable z with complex frequency s. Combining the above with the following factorized form of
a continuous-time scalar transfer function
H.s/ D K s
.s C z 1 /.s C z 2 / : : : .s C z m /
.s C p 1 /.s C p 2 / : : : .s C p n /
; m Ä n :
(9.49)
One can map every pole and every zero of the continuous-time transfer function to a corresponding pole and
zero of an equivalent discrete-time transfer function
i D D exp.T s z i /; i D 1; : : : ; m ;
i D D exp.T s p i /; i D 1; : : : ; n :
(9.50)
In effect, the following discrete-time transfer function
is derived in its factorized from
H.z/ D k z
.z C 1/
nm
.z C 1 /.z C 2 / : : : .z C m /
.z C 1 /.z C 2 / : : : .z C n /
:
(9.51)
Factor .z C 1/
nm is introduced in the discrete-time
transfer function above in correspondence to factor
..s C 1/
nm with ! ˙j0, which is hidden in the continuous-time transfer function (9.49). The intent is to
equate the numerator degree to that of the denominator.
This is achieved by introducing an extra zero at imaginary infinity .˙j1/ with multiplicity .n m/ using
factor ..s C 1/
nm . In the discrete-time transfer function, factor ..s C 1/
nm is mapped to .z C 1/
nm , since
point ˙j1 on the s-plane is mapped to 1 on the zplane.
Finally, gain k in the discrete-time transfer function is determined by equating its value at a certain z
with that of the continuous-time transfer function at the
equivalent point s. For example, for z D 1, the corresponding point is s D 0, and, therefore, k is calculated
as follows
H.z/j zD1 D H.s/j sD0
, k z
2
nm
.1 C 1 /.1 C 2 / : : : .1 C m /
.1 C 1 /.1 C 2 / : : : .1 C n /
;
D K s
z 1 z 2 : : : z m
p 1 p 2 : : : p n
:
(9.52)