Phân tích đặc điểm của các loại kênh phading trong truyền dẫn

Fading Channels: Capacity, BER and Diversity
Master Universitario en Ingenier´ıa de Telecomunicaci´on
I. Santamar´ıa
Universidad de Cantabria


Introduction

Capacity

BER

Diversity

Conclusions

Contents

Introduction
Capacity
BER
Diversity
Conclusions

Fading Channels: Capacity, BER and Diversity

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Introduction

Capacity

BER

Diversity

Conclusions

Introduction

We have seen that the randomness of signal attenuation
(fading) is the main challenge of wireless communication
systems
In this lecture, we will discuss how fading affects
1. The capacity of the channel
2. The Bit Error Rate (BER)

We will also study how this channel randomness can be used
or exploited to improve performance → diversity

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Introduction

Capacity

BER

Diversity

Conclusions

General communication system model
Source

Channel
Encoder

b[n]

sˆ[n]

s[n]
Modulator

Channel

⊕

Demod.

Channel
Decoder

bˆ[n]

Noise

The channel encoder (FEC, convolutional, turbo, LDPC ...)
adds redundancy to protect the source against errors
introduced by the channel
The capacity depends on the fading model of the channel
(constant channel, ergodic/block fading), as well as on the
channel state information (CSI) available at the Tx/Rx
Let us start reviewing the Additive White Gaussian Noise (AWGN)
channel: no fading

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Introduction

Capacity

BER

Diversity

Conclusions

AWGN Channel
Let us consider a discrete-time AWGN channel
y [n] = hs[n] + r [n]
where r [n] is the additive white Gaussian noise, s[n] is the
transmitted signal and h = |h|e jθ is the complex channel

The channel is assumed constant during the reception of the
whole transmitted sequence
The channel is known to the Rx (coherent detector)
The noise is white and Gaussian with power spectral density
N0 /2 (with units W/Hz or dBm/Hz, for instance)
We will mainly consider passband modulations (BPSK, QPSK,
M-PSK, M-QAM), for which if the bandwith of the lowpass
signal is W the bandwidth of the passband signal is 2W
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Introduction

Capacity

BER

Diversity

Conclusions

Signal-to-Noise-Ratio
Transmitted signal power: P
Received signal power: P|h|2
Total noise power: N = 2WN0 /2 = WN0
The received SNR is
SNR = γ =

P|h|2
WN0

In terms of the energy per symbol P = Es /Ts
We assume Nyquist pulses with β = 1 (roll-off factor) so
W = 1/Ts
Under these conditions
SNR =
Fading Channels: Capacity, BER and Diversity

Es Ts |h|2
Es |h|2
=
Ts N0
N0
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Introduction

Capacity

BER

Diversity

Conclusions

For M-ary modulations, the energy per bit is

Eb =

Es
log(M)

⇒

SNR =

Eb |h|2
N0 log(M)

To model the noise we generate zero-mean circular complex
Gaussian random variables with σ 2 = N0
r [n] ∼ CN(0, σ 2 )

or

r [n] ∼ CN(0, N0 )

The real and imaginary parts have variance
2
σI2 = σQ
=

Fading Channels: Capacity, BER and Diversity

σ2
N0
=

2
2

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Introduction

Capacity

BER

Diversity

Conclusions

Average SNR and Instantaneous SNR
Sometimes, we will find useful to distinguish between the average
and the instantaneous signal-to-noise ratio
Average ⇒ SNR = γ¯ =

P
WN0

Instantaneous ⇒ SNR = γ¯ |h|2 =

P
|h|2
WN0

AWGN channel: N0 and h are assumed to be known
(coherent detection), therefore we typically take (w.l.o.g)
P
h = 1 ⇒ SNR = γ¯ = WN
, and the SNR at the receiver is
0
constant
Fading channels: The instantaneous signal-to-noise-ratio
SNR = γ = γ¯ |h|2 is a random variable
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Introduction

Capacity

BER

Diversity

Conclusions

Capacity AWGN Channel
Let us consider a discrete-time AWGN channel
y [n] = hs[n] + r [n]
where r [n] is the additive white Gaussian noise, s[n] is the
transmitted symbol and h = |h|e jθ is the complex channel
The channel is assumed constant during the reception of the

whole transmitted sequence
The channel is known to the Rx (coherent detector)
The capacity (in bits/seg or bps) is given by the well-known
Shannon’s formula
C = W log (1 + SNR) = W log (1 + γ)
2

where W is the channel bandwidth, and SNR = P|h|
WN0 ; with P
2
being the transmit power, |h| the power channel gain and N0 /2
the power spectral density (PSD) of the noise
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Introduction

Capacity

BER

Diversity

Conclusions

Sometimes, we will find useful to express C in bps/Hz (or
bits/channel use)
C = log (1 + SNR)

A few things to recall
1. Shannon’s coding theorem proves that a code exists that
achieves data rates arbitrarily close to capacity with
vanishingly small probability of bit error
The codewords might be very long (delay)
Practical (delay-constrained) codes only approach capacity

2. Shannon’s coding theorem assumes Gaussian codewords, but
digital communication systems use discrete modulations
(PSK,16-QAM)

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Introduction

Capacity

BER

Diversity

Conclusions

10
Shaping gain (1.53 dB)
Gaussian inputs
log(1+SNR)

Capacity (bps/Hz)

8

64-QAM
6
16-QAM

4

4-QAM (QPSK)

2

Pe (symbol) = 10

-6

(uncoded)

0
5

10

15

20

25

30

SNR dB

For discrete modulations, with increasing SNR we should increase
the constellation size M (or use a less powerful channel encoder)
to increase the rate and hence approach capacity: Adaptive
modulation and coding (more on this later)
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Introduction

Capacity

BER

Diversity

Conclusions

Capacity fading channels
Let us consider the following example1
Fading channel
AWGN
C = 1 bps/Hz

Channel
Encoder

Random
Switch
AWGN
C = 2 bps/Hz

The channel encoder is fixed
The switch takes both positions with equal probability
We wish to transmit a codeword formed by a long sequence of
coded bits, which are then mapped to symbols
What is the capacity for this channel?
1

Taken from E. Biglieri, Coding for Wireless Communications, Springer,
2005
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Introduction

Capacity

BER

Diversity

Conclusions

The answer depends on the rate of change of the fading
1. If the switch changes position every symbol period, the
codeword experiences both channels with equal probabilities
so we could use a fixed channel encoder, and transmit at a
maximum rate
1
1
C = C1 + C2 = 1.5 bps/Hz
2
2
2. If the switch remains fixed at the same (unknown) position
during the transmission of the whole codeword, then we should
use a fixed channel encoder, and transmit at a maximum rate
C = C1 = 1 bps/Hz
or otherwise half of the codewords would be lost
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Introduction

Capacity

BER

Diversity

Conclusions

What would be the capacity if the switch chooses from a
2
continuum of AWGN channels whose SNR, γ = P|h|
WN0 , 0 ≤ γ < ∞,
follows an exponential distribution (Rayleigh channel)


C1 = log(1 + γ 1 )
Channel
Encoder

C2 = log(1 + γ 2 )
Random
Switch

C3 = log(1 + γ 3 )


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Introduction

Capacity

BER

Diversity

Conclusions

Fast fading (ergodic) channel
If the switch changes position every symbol period, and the
codeword is long enough so that the transmitted symbols
experience all states of the channel (fast fading or ergodic
channel)
∞

C = E [log(1 + γ)] =

log(1 + γ)f (γ)dγ

(1)

0

where f (γ) = γ1 e −γ/γ , with γ being the average SNR
Eq. (1) is the ergodic capacity of the fading channel
For Rayleigh channels the integral in (1) is given by
C=
where E1 (x) =
2

1
exp
2 ln(2)

∞ e−t
t dt
x

1
γ

E1

1
γ

is the Exponential integral function2

y=expint(x) in Matlab

Fading Channels: Capacity, BER and Diversity

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Introduction

Capacity

BER

Diversity

Conclusions

The ergodic capacity is in general hard to compute in closed
form (we can always resort to numerical integration)
We can apply Jensen’s inequality to gain some qualitative
insight into the effect of fast fading on capacity

Jensen’s inequality
If f (x) is a convex function and X is a random variable
E [f (X )] ≥ f (E [X ])
If f (x) is a concave function and X is a random variable
E [f (X )] ≤ f (E [X ])
log(·) is a concave function, therefore
C = E [log(1 + γ)] ≤ log (1 + E [γ]) = log (1 + γ)
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Capacity Channel:
BER ErgodicDiversity
SISO Fading
Capacity

Introduction

Conclusions

The
capacity fading

of a fading
with2 ]receiver
CSIσis2 =
less1.than the
ume iid
Rayleigh
withchannel
σh2 = E[|h|
= 1, and
z

✫

capacity of an AWGN channel with the same average SNR
7

6

Capacity (bps/Hz)

5

AWGN Channel Capacity
Fading Channel Ergodic Capacity

4

3

2

1

0
−10

−5

Fading Channels: Capacity, BER and Diversity

0

5
SNR (dB)

10

15

20

17

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Introduction

Capacity

BER

Diversity

Conclusions

Block fading channel
If the switch remains fixed at the same (unknown) position
during the transmission of the whole codeword, there is no
nonzero rate at which long codewords can be transmitted with
vanishingly small probability or error
Strictly, the capacity for this channel model would be zero
In this situation, it is more useful to define the outage
capacity
Cout = r ⇒ Pr (log(1 + γ) < r ) = Pr (C (h) < r ) = Pout
Suppose we transmit at a rate Cout with probability of outage
Pout = 0.01 this means that with probability 0.99 the
instantaneous capacity of the channel (a realization of a r.v.) will
be larger than rate and the transmission will be successful; whereas
with probability 0.01 the instantaneous capacity will be lower than
the rate and the all bits in the codeword will be decoded incorrectly
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Introduction

Capacity

BER

Diversity

Conclusions

Pout is the error probability since data is only correctly
received on 1 − Pout transmissions

What is the outage probability of a block fading Rayleigh
channel with average SNR = γ for a transmission rate r ?
Pout = Pr (log(1 + γ) < r )
The rate is a monotonic increasing function with the
SNR = γ; therefore, there is a γmin needed to achieve the rate
⇒

log(1 + γmin ) = r

γmin = 2r − 1

and Pout can be obtained as
γmin

Pout =Pr (γ < γmin ) =

γmin

f (γ)dγ =
0

=1 − e

−

γmin
γ

Fading Channels: Capacity, BER and Diversity

=1−e

0

1 −γ/γ
e
dγ =
γ

r −1

−2

γ

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Introduction

Capacity

BER

Diversity

Conclusions

Pout vs Cout (rate) for γ = 100 (10 dB)
1

Rate (bps/Hz)

0.8
0.6
0.4
0.2
0
10 -4

10 -3

10 -2

Pout
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Introduction

Capacity

BER

Diversity

Conclusions

Summary
AWGN
L

L

h1

L



h1

C = log(1 + γ )

h1

Ergodic (fast fading)
L

L

h

h1 h2

h1 h2

L

h

L

C = E[log(1 + γ )]

Block fading
L

h1

L

h2

Fading Channels: Capacity, BER and Diversity

L



hn

Coutage

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Introduction

Capacity

BER

Diversity

Conclusions

Final note

All capacity results seen so far correspond to the case of
perfect CSI at the receiver side (CSIR)
If the transmitter also knows the channel, we can perform
power adaptation and the results are different
We’ll see more on this later
Now, let’s move to analyze the impact of fading on the Bit Error
Rate (BER)

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Introduction

Capacity

BER

Diversity

Conclusions

BER analysis for the AWGN channel (a brief reminder)
BPSK s[n] ∈ {−1, +1}
Ps = Pb = Q
where Q(x) =
QPSK s[n] ∈

2Es
N0

=Q

∞ 1
√
exp −(x 2 /2) ≤ 21
x
2π
√ , −1+j

√ , 1−j
√ , 1+j
√ }
{ −1−j
2
2
2
2

Ps = 1 − 1 − Q

√

√
2SNR
exp −(x 2 /2)

2

SNR

Nearest neighbor approximation Ps ≈ 2Q

√

SNR

With Gray encoding Pe = Ps /2
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Introduction

Capacity

BER

Diversity

Conclusions

M-PAM constellation Ai = (2i − 1 − M)d/2, i = 1, . . . , M
Distance between neighbors: d
Average symbol energy
Es =

1 2
(M − 1)
3

d
2

2

=

(M 2 − 1)d 2

12

+

2
Q
M

Symbol Error Rate
Ps =
=

M −2
2Q
M

d
2σr

2(M − 1)
Q
M

d
2σr

6 SNR
M2 − 1

With Gray encoding Pe ≈ Ps /M

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Introduction

Capacity

BER

Diversity

Conclusions

M-QAM: the constellations for the I and Q branches are
Ai = (2i − 1 − M)d/2, i = 1, . . . , M
√
At the I and Q branches we have orthogonal M − PAM
signals, each with half the SNR
Symbol Error Rate
Ps = 1 −

√
2( M − 1)
√
Q
1−
M

3 SNR
M −1

2

Nearest neighbor approximation (4 nearest neighbors)
Ps ≈ 4Q

Fading Channels: Capacity, BER and Diversity

3 SNR
M −1

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