Digital Signal Processing Reference
In-Depth Information
Exercise 11.5: Multirate is so useful!
Consider the following block dia-
gram:
l
g
r
,
y
i
d
.
,
©
,
L
s
x
[
n
]
M
↑
LP
{
π/
M
}
z
−L
M
↓
y
[
n
]
and show that this system implements a fractional delay (i.e. show that the
transfer function of the system is that of a pure delay, where the delay is not
necessarily an integer).
To see a practical use of this structure, consider now a data transmission
system over an analog channel. The transmitter builds a discrete-time sig-
nal
s
[
n
]
; this is converted to an analog signal
s
c
(
t
)
via an interpolator with
period
T
s
, and finally
s
c
is transmitted over the channel. The signal takes
a finite amount of time to travel all the way to the receiver; say that the
transmission time over the channel is
t
0
seconds: the received signal
(
t
)
ˆ
s
c
(
t
)
is
therefore just a delayed version of the transmitted signal,
ˆ
s
c
(
t
)=
s
c
(
t
−
t
0
)
ˆ
(
)
At the receiver,
is sampled with a sampler with period
T
s
so that no
aliasing occurs to obtain
s
c
t
s
ˆ
[
n
]
.
ˆ
(a) Write out the Fourier Transform of
s
c
(
t
)
as a function of
S
c
(
j
Ω)
.
ˆ
(b) Write out the DTFT of the received signal sampled with rate
T
s
,
s
[
n
]
.
(c) Now we want to use the above multirate structure to compensate for
the transmission delay. Assume
t
0
=
4.6
T
s
; determine the values for
M
and
L
in the above block diagram so that
s
ˆ
[
n
]=
s
[
n
−
D
]
,where
D
has the smallest possible value (assume an ideal lowpass filter
in the multirate structure).
∈
Exercise 11.6: Multirate filtering.
Assume
H
(
z
)
is an ideal lowpass filter
with cutoff frequency
π/
10. Consider the system described by the following
block diagram:
M
↑
M
↓
x
[
n
]
H
(
z
)
y
[
n
]
(a) Compute the transfer function of the system for
M
=
2.
(b) Compute the transfer function of the system for
M
=
5.
