L5
Sampling, reconstruction, and
software radio
Until this point, your study of signals and systems has concerned only the continuoustime case
1
,
which dominated the early history of signal processing. About 50 years ago, however, the devel
opment of the modern computer generated research interest in digital signal processing (DSP), a
type of discretetime signal processing. Although hardware limitations made most realtime DSP
impractical at the time, the continuing maturation of the computer has been matched with a
continuing expansion of DSP. Much of that expansion has been into areas previously dominated
by continuoustime systems: our telephone network, medical imaging, music recordings, wireless
communications, and many more.
You do not need to worry whether the time and e
f
ort you have invested in studying continuous
time systems will be wasted because of the growth of DSP — digital systems are practically always
hybrids of analog and digital subsystems. Furthermore, many DSP systems are linear and time
invariant, meaning that the same analysis techniques apply, although with some modiﬁcations.
In this lab, you will explore some of the parallels between continuoustime systems and DSP
with a “software radio” designed to the same speciﬁcations as the receiver circuit you developed
on your protoboard.
1
Prelab
Our software radio is typical of many DSP systems in that both the available input and re
quired output are continuoustime signals. The conversion of a continuoustime input signal to
a discretetime signal is called
sampling
(or A/D conversion), and the conversion of a discrete
time signal to a continuoustime output signal is called
reconstruction
(or D/A conversion).
As discussed in class,
samples
f
(
nT
)
of a
bandlimited
analog signal
f
(
t
)
can be used to re
construct
f
(
t
)
exactly when the
sampling interval
T
and
signal bandwidth
Ω
= 2
π
B
satisfy
the
Nyquist criterion
T <
1
2
B
.
This is illustrated by the hypothetical system shown in Figure L5.1, where the analog signal
f
(
t
)
deﬁned at the output stage of a lowpass ﬁlter
H
1
(
ω
)
has a bandwidth
Ω
= 2
π
B
limited by
the bandwidth
Ω
1
= 2
π
B
1
of the ﬁlter.
A/D converter extracts the samples
f
(
nT
)
from
f
(
t
)
with a sampling interval of
T
, and D/A conversion of samples
f
(
nT
)
into an analog signal
y
(
t
)
can be envisioned as lowpass ﬁltering of a hypothetical signal
f
T
(
t
) =
∑
n
f
(
nT
)
δ
(
t

nT
)
using
the ﬁlter
H
2
(
ω
)
.
With an appropriate choice of
H
2
(
ω
)
, system output
y
(
t
)
will be identical to
f
(
t
)
in all its details as long as
T
<
1
2
B
1
.
The reason for that can be easily appreciated after
comparing the Fourier transforms
F
(
ω
)
and
F
T
(
ω
)
of signals
f
(
t
)
and
f
T
(
t
)
with the help of
Figure L5.2.
1
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 Spring '08
 Staff
 Digital Signal Processing, Signal Processing, LPF, software radio

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