ECE 5670 : Digital Communications
Lecture 16: Orthogonal Frequency Division Modulation (OFDM) and
Capacity of the Wireline Channel
1
3/10/2011
Instructor: Salman Avestimehr
Introduction
In this lecture we will see in detail the OFDM method to convert the wireline channel into a
parallel AWGN channel. We also see that this achieves the
capacity
of the wireline channel
– in other words, the largest possible data rate of communication is achieved by the OFDM
method.
OFDM
Consider the frequency selective model that we have been working with as a good approxi
mation of the wireline channel:
y
[
m
] =
L

1
summationdisplay
ℓ
=0
h
ℓ
x
[
m
−
ℓ
] +
w
[
m
]
,
m
≥
1
.
(1)
We will convert the ISI channel in Equation (1) into a
collection
of AWGN channels, each of
different noise energy level:
ˆ
y
[
N
c
k
+
n
] =
ˆ
h
n
ˆ
x
[
N
c
k
+
n
] + ˆ
w
[
N
c
k
+
n
]
,
k
≥
0
,
n
= 0
. . . N
c
−
1
.
(2)
We will be able to make this transition by some very simple signal processing techniques.
Interestingly, these signal processing techniques are
universally
applicable to every wireline
channel, i.e., they do not depend on the exact values of channel coefficients
h
0
, . . . , h
L

1
. This
makes OFDM a very
robust
communication scheme over the frequencyselective channel.
Cyclic Prefix
Suppose we have mapped our information bits into
N
c
voltages. We will revisit the issue of
how these voltages were created from the information bits at a slightly later point in this
lecture. For now, we write them as a vector:
d
= [
d
[0]
, d
[1]
, . . . , d
[
N
c
−
1]]
t
.
1
Based on lecture notes of Professor Pramod Viswanath at UIUC.
1
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We use these
N
c
voltages to create an
N
c
+
L
−
1 block of
transmit
voltages as:
x
= [
d
[
N
c
−
L
+ 1]
, d
[
N
c
−
L
+ 2]
, . . . , d
[
N
c
−
1]
, d
[0]
, d
[1]
, . . ., d
[
N
c
−
1]]
t
,
(3)
i.e., we add a
prefix
of length
L
−
1 consisting of data symbols rotated cyclically (Figure 1).
The first
L
−
1 transmitted symbols contain the “data” symbols
d
[
N
c
−
(
L
−
1)]
, . . . , d
[
N
c
−
1].
The next
N
c
transmitted voltages or symbols contain the “data” symbols
d
[0]
, d
[1]
, . . ., d
[
N
c
−
1]. In particular, for a 2tap frequencyselective channel we have the following result of cyclic
precoding:
x
[1]
=
d
[
N
c
−
1]
x
[2]
=
d
[0]
x
[3]
=
d
[1]
.
.
.
x
[
N
c
+ 1]
=
d
[
N
c
−
1]
With this input to the channel (1), consider the output
y
[
m
] =
L

1
summationdisplay
ℓ
=0
h
ℓ
x
[
m
−
ℓ
] +
w
[
m
]
,
m
= 1
, . . . , N
c
+ 2(
L
−
1)
.
The first
L
−
1 elements of the transmitted vector
x
were constructed from circularly wrapped
elements of the vector
d
, which are included in the last
N
c
−
1 elements of
x
. The receiver
hence ignores the first
L
−
1 received symbols
y
[1]
, . . . , y
[
L
−
1]. The ISI extends over the
first
L
−
1 symbols and the receiver ignores it by considering only the output over the time
interval
m
∈
[
L, N
c
+
L
−
1].
Let us take a careful look at how the
N
receive voltages
(received at times
L
through
N
c
+
L
−
1) depend on the transmit voltages
d
[0]
, . . . , d
[
N
c
−
1]:
y
[
m
] =
L

1
summationdisplay
ℓ
=0
h
ℓ
d
[(
m
−
L
−
ℓ
) modulo
N
c
] +
w
[
m
]
.
(4)
See Figure (1).
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 Spring '11
 SCAGLIONE
 Frequency, NC

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