Part A | 9.1
204 Part A Fundamentals
2. Time-domain response matching at sampling instants:
Impulse response matching
Step response matching.
3. Dynamic matching, i. e., matching of all poles and
zeroes appearing in the transfer function.
Approximate Differentiation or Integration
Consider a continuous-time signal x.t/ being sampled
with sampling interval T s and the discrete-time signal
x.n/ is obtained. Then, the following approximations
can be derived.
Approximate derivative with respect to time using
backward difference
P
x.t/ '
x.n/ x.n 1/
T s
) s Á
1 z
1
T s
:
(9.34)
Approximate derivative with respect to time using forward difference
P
x.t/ '
x.n C 1/ x.n/
T s
) s Á
z 1
T s
D
1 z
1
T s z 1 :
(9.35)
Approximate integral with respect to time using the
trapezoidal rule
x 1 .nT s / D
nTs
Z
0
x../ d D
.n1/Ts Z
0
x../ d C
nTs
Z
.n1/Ts
x../ d
, x 1 ..n 1/T s / C x 1 .n/ ;
x 1 .n/ ' T s
x.n/ C x.n 1/
2
:
(9.36)
However,
X 1 .s/ D
1
s
X.s/ and
X 1 .z/ D z
1 X 1 .z/ C T s
1 C z
1
2
X.z/ :
(9.37)
The above in effect yields
1
s
Á
T s
2
1 C z
1
1 z 1 , s Á
2
T s
1 z
1
1 C z 1 :
(9.38)
Equations (9.34), (9.35), and (9.38) mean that when
presented with a transfer function H.s/ in the complex
frequency (s) domain, describing a continuous-time LTI
system, it is possible to derive a discrete-time transfer
function H.z/, or H.z
1
/, defining a discrete-time LTI
system by substituting variable s with the approximation desired.
Impulse Response Matching
The methods presented previously were based on approximating the integration or differentiation operator
appearing in the differential (dynamic) equation of an
LTI system. However, an LTI system can, also, be described in full by its response to an arbitrary input
signal. Common choices include using either Dirac’s
delta (impulse) or the unit step signal (Heaviside signal).
In effect, if the impulse or step response of
a continuous-time system is sampled, then a discretetime description can be derived as follows.
Invariance of impulse response: the impulse delta is
defined as follows in discrete time
•.n/ D
(
0; n ¤ 0
1; n D 0
, .z/ D 1 :
(9.39)
Given the transfer function, H.z/ of a discrete-time LTI
system, it is easy to prove that, just like in the continuous time case, H.z/ is the Z-transform of the system’s
impulse response.
Indeed, the output y.n/ of a discrete-time LTI system is given by the following convolutional sum
y.n/ D u.k/ ˝ h.k/ ,
C1 X
kDD1
u.k/ h.n k/ ) Y.z/
D H.z/ U.z/ :
(9.40)
In effect,
u.n/ D •.n/ ) U.z/ D 1 ) Y.z/ D H.z/ :
Consider a continuous-time LTI system with impulse
response h.t/ D L
1
fH.s/g, which is being sampled at
rate f s . This means that continuous-time signal h • .t/ D
P C1
nDD1 h.nT s /•.t nT s / corresponds to discrete-time
signal h • .n/ Á h.nT s /, i. e., a sample set of h.t/.
The Z-transform of h • .n/,
H • .z
1
/ D
C1 X
nD0
h • .n/z
n
;
defines a discrete-time LTI system, the impulse response of which, by definition, coincides with that of
continuous-time transfer function H.s/, at all sampling
instants.
Assume, now, that for every positive M there is
a positive integer " such that
C1 X
nDMC1
h
2
• .n/ < " :
(9.41)