Digital Signal Processing Reference
In-Depth Information
9.7 Discrete-Time Processing of Analog Signals
Sampling and interpolation (or, more precisely, the A
Aconver-
sions strategies which we will see in the next Chapter) represent the entry
and exit points of the powerful processing paradigm for which discrete-time
signal processing is “famous”. Samplers and interpolators are the only inter-
faces with the physical world, while all the processing and the analysis are
performed in the abstract, dimensionless and timeless world of a general
purpose microprocessor.
The generic setup of a real-world processing device is as shown in Fig-
ure 9.13. In most cases, the sampler's and the interpolator's frequencies are
the same, and they are chosen as a function of the bandwidth of the class
of signals for which the device is conceived; let us assume that the input is
bandlimited to
/
DandD
/
l
g
r
,
y
i
d
.
,
©
,
L
s
2, if we reason in hertz). For the case
in which the processing block is a linear filter
H
Ω
=
π/
T
s
(or to
F
s
/
N
, the overall processing
chain implements an analog transfer function; from the relations:
(
z
)
T
s
X
c
j
T
s
1
e
j
ω
)=
X
(
(9.35)
e
j
ω
)=
e
j
ω
)
e
j
ω
)
Y
(
H
(
X
(
(9.36)
e
j
Ω
T
s
Y
c
(
j
Ω)=
T
s
Y
(
)
(9.37)
we have
e
j
Ω
T
s
Y
c
(
j
Ω)=
H
(
)
X
c
(
j
Ω)
(9.38)
So,forinstance,if
H
3, the pro-
cessing chain in Figure 9.13 implements the transfer function of an analog
lowpass filter with cutoff frequency
Ω
N
/
3(or,inhertz,
F
s
/
6).
(
z
)
is a lowpass filter with cutoff frequency
π/
y
[
n
]
x
[
n
]
x
c
(
t
)
H
(
z
)
I
(
t
)
y
c
(
t
)
1
/
T
s
1
/
T
s
Figure 9.13
Discrete-time processing of analog signals.
9.7.1 A Digital Differentiator
In Section 2.1.4 we introduced an approximation to the differentiator oper-
ator in discrete time as the first- order difference between neighboring sam-
ples. The processing paradigm that we have just introduced, will now allow
