This preview shows pages 1–2. Sign up to view the full content.
Lecture 9: Capacity of the Continuous time AWGN Channel
Introduction
In the penultimate lecture we saw the culmination of our study of reliable communication
on the discrete time AWGN channel. We concluded that there is a threshold called capac
ity below which we are guaranteed arbitrarily reliable communication and above which all
communication is hopelessly unreliable. But the real world is analog and in the last lecture
we saw in detail the engineering way to connect the
continuous time
AWGN channel to the
discrete time one. In this lecture we will connect these two story lines into a ﬁnal statement:
we will derive a formula for the capacity of the continuous time AWGN channel. This is the
largest rate of reliable communication (as measured in bits/second) and depends only on the
two key physical resources: bandwidth and power. We will see the utility of this formula by
getting a feel for how the capacity changes as a function of the two physical has more impact
on the capacity
The Continuous Time AWGN Channel
The channel is, naturally enough,
y
(
t
) =
x
(
t
) +
w
(
t
)
,
t >
0
.
(1)
The power constraint of
¯
P
Watts on the transmit signal says that
lim
N
→∞
1
NT
Z
NT
0
(
x
(
t
))
2
dt
≤
¯
P.
(2)
The (twosided) bandwidth constraint of
W
says that much of the energy in the transmit
signal is contained within the spectral band
£

W
2
,
W
2
/
.
We would like to connect this to the discrete time AWGN channel:
y
[
m
] =
x
[
m
] +
w
[
m
]
,
m
≥
1
.
(3)
This channel came with the discretetime power constraint:
lim
N
→∞
1
N
N
X
m
=1
(
x
(
m
])
2
≤
P
∀
N.
(4)
We have already seen that there are
W
channel uses per second in the continuous time
channel if we constrain the bandwidth of the analog transmit voltage waveform to
W
Hz.
So, this ﬁxes the sampling rate to be
W
and thus unit time in the discrete time channel
corresponds to
1
W
seconds.
To complete the connection we need to:
1. connect the two power constraints
¯
P
and
P
;
2. ﬁnd an appropriate model for the continuous time noise
w
(
t
) and connect it to the
variance of the additive noise
w
[
m
].
We do these two steps next.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '08
 PRAMODVISWANATH

Click to edit the document details