Digital Signal Processing Reference
In-Depth Information
Exercise 11.4: Digital processing of continuous-time signals.
In
your grandmother's attic you just found a treasure: a collection of super-
rare 78 rpm vinyl jazz records. The first thing you want to do is to transfer
the recordings to compact discs, so you can listen to them without wearing
out the originals. Your idea is obviously to play the record on a turntable and
use an A
l
g
r
,
y
i
d
.
,
©
,
L
s
D converter to convert the line-out signal into a discrete-time se-
quence, which you can then burnonto a CD. The problem is, you only have a
“modern” turntable, which plays records at 33 rpm. Since you're a DSP wiz-
ard, you know you can just go ahead, play the 78 rpm record at 33 rpm and
sample the output of the turntable at 44.1 KHz. You can then manipulate
the signal in the discrete-time domain so that, when the signal is recorded
on a CD and played back, it will sound right.
Design a system which performs the above conversion. If you need to get
on the right track, consider the following:
/
•
Call
s
the continuous-time signal encoded on the 78 rpm vinyl (the
jazz music).
(
t
)
•
Call
x
the continuous-time signal you obtain when you play the
record at 33 rpm on the modern turntable.
(
t
)
•
Let
x
[
n
]=
x
(
nT
s
)
,with
T
s
=
1
/
44, 100.
Answer the following questions:
(a) Express
x
(
t
)
in terms of
s
(
t
)
.
(b) Sketch the Fourier transform
X
(
j
Ω)
when
S
(
j
Ω)
is as in the follow-
ing figure. The highest nonzero frequency of
S
π
)
·
16,000 Hz (old records have a smaller bandwidth than modern ones).
(
j
Ω)
is
Ω
max
=(
2
|
S
(
j
Ω)
|
−
Ω
max
Ω
max
(c) Design a system to convert
x
so that, when
you interpolate
y
[
n
]
to a continuous-time signal
y
(
t
)
with interpola-
tion period
T
s
, you obtain
Y
[
n
]
into a sequence
y
[
n
]
(
j
Ω)=
S
(
j
Ω)
.
(d) What if you had a turntable which plays records at 45 rpm? Would
your system be different? Would it be better?
