ECE 5670 : Digital Communications
Lecture 9: Capacity of the Continuous time AWGN Channel
1
2/22/2011
Instructor: Salman Avestimehr
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 resources 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
integraldisplay
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
bracketleftbig

W
2
,
W
2
bracketrightbig
.
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
summationdisplay
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.
1
Based on lecture notes of Professor Pramod Viswanath at UIUC.
1
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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
].
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 Spring '11
 SCAGLIONE
 Signal Processing, lim, Additive white Gaussian noise, time awgn channel

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