ECE 5670 : Digital Communications
Lecture 22: Time, Frequency and Antenna Diversity
1
4/7/2011 and 4/12/2011
Instructor: Salman Avestimehr
Introduction
We have seen that the communication (even with coding) over a slow fading flat wireless
channel has very poor reliability. This is because there is a significant probability that the
channel is in outage and this event dominates the total error event (which also include the
effect of additive noise). The key to improving the reliability of reception (and this is really
required for wireless systems to work as desired in practice) is to reduce the chance that the
channel is in outage. This is done by communicating over “different” channels and the name
of the game is to harness the
diversity
available in the wireless channel.
There are three
main types: temporal, frequency and antennas. We will see each one in this lecture, starting
time diversity.
Time Diversity Channel
The temporal diversity occurs in a wireless channel due to mobility. The idea is to code and
communicate over the order of time over which the channel changes (called the
coherence
time). This means that we need enough delay tolerance by the data being communicated.
A simple singletap model that captures time diversity is the following:
y
ℓ
=
h
ℓ
x
ℓ
+
w
ℓ
,
ℓ
= 1
, . . . , L.
(1)
Here the index
ℓ
represents a single time sample over
different
coherence time intervals. By
interleaving across different coherence time intervals, we can get to the time diversity channel
in Equation (1). A good statistical model for the channel coefficients
h
1
, . . . , h
L
is that they
are all independent and identically distributed. Further more, in a environment with lots of
multipath we can suppose that they are complex Gaussian random variables (the socalled
Rayleigh fading).
1
Based on lecture notes of Professor Pramod Viswanath at UIUC.
1
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A Single Bit Over a Time Diversity Channel
For ease of notation, we start out with
L
= 2 and just a single bit to transmit. The simplest
way to do this is to
repeat
the same symbol: i.e., we set
x
1
=
x
2
=
x
±
√
E.
(2)
At the receiver, we have
four
(real) voltages:
ℜ
[
y
1
]
=
ℜ
[
h
1
]
x
+
ℜ
[
w
1
]
,
(3)
ℑ
[
y
1
]
=
ℑ
[
h
1
]
x
+
ℑ
[
w
1
]
,
(4)
ℜ
[
y
2
]
=
ℜ
[
h
2
]
x
+
ℜ
[
w
2
]
,
(5)
ℑ
[
y
2
]
=
ℑ
[
h
2
]
x
+
ℑ
[
w
2
]
.
(6)
As usual, we suppose
coherent
reception, i.e., the receiver has full knowledge of the exact
channel coefficients
h
1
, h
2
. By now, it is quite clear that the receiver might as well take the
appropriate weighted linear combination to generate a single real voltage and then make the
decision with respect to the single transmit voltage
x
. This is the matched filter operation:
y
MF
=
ℜ
[
y
1
]
ℜ
[
h
1
] +
ℑ
[
y
1
]
ℑ
[
h
1
] +
ℜ
[
y
2
]
ℜ
[
h
2
] +
ℑ
[
y
2
]
ℑ
[
h
2
]
.
(7)
Using the complex number notation,
y
MF
=
ℜ
[
h
*
1
y
1
] +
ℜ
[
h
*
2
y
2
]
,
(8)
=
(

h
1

2
+

h
2

2
)
x
+ ˜
w.
(9)
Here ˜
w
is a real Gaussian random variable because it is the sum of four independent and
identically distributed Gaussians, but they get scaled by the real and imaginary parts of
h
1
and
h
2
. Therefore ˜
w
is zero mean and has variance:
Var( ˜
w
) =
σ
2
2
(

h
1

2
+

h
2

2
)
.
(10)
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 Spring '11
 SCAGLIONE
 Normal Distribution, Frequency, Radio resource management, Diversity scheme

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