Section 3 2 PerformanceDMoverFading

Introduction
AWGN channels
Fading Channels

Chapter 3: Physical-layer transmission techniques
Section 3.2: Performance analysis over fading channels

Mobile communications - Chapter 3: Physical-layer transmission techniques

Section 3.2: Performance analysis over fading channels

1


Introduction
AWGN channels
Fading Channels

Outline of the lecture notes
1

Introduction

2

AWGN channels
Signal-to-Noise power ratio and bit/symbol energy
Error probability for BPSK and QPSK
Approximate symbol and bit error probabilities for typical modulations

3

Fading Channels
Introduction
Outage probability
Average probability of error

Mobile communications - Chapter 3: Physical-layer transmission techniques

Section 3.2: Performance analysis over fading channels

2


Introduction
AWGN channels
Fading Channels

Introduction

We now consider the performance of the digital modulation
techniques discussed in the previous chapter when used over AWGN
channels and channels with flat-fading.
There are two performance criteria of interest: the probability of
error, defined relative to either symbol or bit errors, and the outage
probability, defined as the probability that the instantaneous
signal-to-noise ratio falls below a given threshold.
Wireless channels may also exhibit frequency selective fading and
Doppler shift. Frequency-selective fading gives rise to intersymbol
interference (ISI), which causes an irreducible error floor in the
received signal.

Mobile communications - Chapter 3: Physical-layer transmission techniques

Section 3.2: Performance analysis over fading channels

3


Introduction
AWGN channels
Fading Channels

Signal-to-Noise power ratio and bit/symbol energy
Error probability for BPSK and QPSK
Approximate symbol and bit error probabilities for typical modulations

Signal-to-Noise power ratio and bit/symbol energy

In this section we define the signal-to-noise power ratio (SNR) and
its relation to energy-per-bit (𝐸𝑏 ) and energy-per-symbol (𝐸𝑠 ).
We then examine the error probability on AWGN channels for
different modulation techniques as parameterized by these energy
metrics. Our analysis uses the signal space concepts of previous
section.
]
[
In an AWGN channel, the modulated signal 𝑠(𝑡) = Re 𝑢(𝑡)𝑒𝑗2𝜋𝑓𝑐 𝑡
has receiver noise 𝑛(𝑡) added to it prior to reception. The noise 𝑛(𝑡)
is a white Gaussian random process with zero-mean and power
spectral density 𝑁0 /2.

The received signal is thus 𝑟(𝑡) = 𝑠(𝑡) + 𝑛(𝑡).

Mobile communications - Chapter 3: Physical-layer transmission techniques

Section 3.2: Performance analysis over fading channels

4


Introduction
AWGN channels
Fading Channels

Signal-to-Noise power ratio and bit/symbol energy
Error probability for BPSK and QPSK
Approximate symbol and bit error probabilities for typical modulations

Signal-to-Noise power ratio and bit/symbol energy (cont.)

We define the received signal-to-noise power ratio (SNR) as the
ratio of the received signal power 𝑃𝑟 to the power of the noise
within the bandwidth of the transmitted signal 𝑠(𝑡).
The received power 𝑃𝑟 is determined by the transmitted power and
the path loss and multipath fading.
The noise power is determined by the bandwidth of the transmitted
signal and the spectral properties of 𝑛(𝑡). Specifically, if the
bandwidth of the complex envelope 𝑢(𝑡) of 𝑠(𝑡) is 𝐵 then the
bandwidth of the transmitted signal 𝑠(𝑡) is 2𝐵.

Mobile communications - Chapter 3: Physical-layer transmission techniques

Section 3.2: Performance analysis over fading channels

5


Introduction
AWGN channels
Fading Channels

Signal-to-Noise power ratio and bit/symbol energy
Error probability for BPSK and QPSK
Approximate symbol and bit error probabilities for typical modulations

Signal-to-Noise ratio and bit/symbol energy (cont.)
Since the noise 𝑛(𝑡) has uniform power spectral density 𝑁0 /2, the
total noise power within the bandwidth 2𝐵 is
𝑃𝑛 = 𝑁0 /2 × 2𝐵 = 𝑁0 𝐵. So, the received SNR is given by
𝑆𝑁 𝑅 =

𝑃𝑟
.
𝑁0 𝐵

(1)

In systems with interference, we often use the received
signal-to-interference-plus-noise power ratio (SINR) in place of SNR
for calculating error probability. If the interference statistics
approximate those of Gaussian noise then this is a reasonable

approximation.
The received SINR is given by
𝑆𝑁 𝑅 =

𝑃𝑟
.
𝑁0 𝐵 + 𝑃𝐼

(2)

where 𝑃𝐼 is the average power of the interference.
Mobile communications - Chapter 3: Physical-layer transmission techniques

Section 3.2: Performance analysis over fading channels

6


Introduction
AWGN channels
Fading Channels

Signal-to-Noise power ratio and bit/symbol energy
Error probability for BPSK and QPSK
Approximate symbol and bit error probabilities for typical modulations

Signal-to-Noise ratio and bit/symbol energy (cont.)
The SNR is often expressed in terms of the signal energy per bit 𝐸𝑏
or per symbol 𝐸𝑠 as
𝑆𝑁 𝑅 =

𝑃𝑟
𝐸𝑠
𝐸𝑏
=
=
.
𝑁0 𝐵
𝑁0 𝐵𝑇𝑠
𝑁0 𝐵𝑇𝑏

(3)

where 𝑇𝑠 and 𝑇𝑏 are the symbol and bit durations, respectively. For
binary modulation (e.g., BPSK), 𝑇𝑠 = 𝑇𝑏 and 𝐸𝑠 = 𝐸𝑏 .
For data shaping pulses with 𝑇𝑠 = 1/𝐵 (e.g., raised cosine pulses
with 𝛽 = 1), one will have SNR = 𝐸𝑠 /𝑁0 for multilevel signaling
and SNR = 𝐸𝑏 /𝑁0 for binary signaling. For general pulses,
𝑇𝑠 = 𝑘/𝐵 for some constant 𝑘, we have 𝑘 × SNR = 𝐸𝑠 /𝑁0 .

The quantities 𝛾𝑠 = 𝐸𝑠 /𝑁0 and 𝛾𝑏 = 𝐸𝑏 /𝑁0 are sometimes called
the SNR per symbol and the SNR per bit, respectively.

Mobile communications - Chapter 3: Physical-layer transmission techniques

Section 3.2: Performance analysis over fading channels

7

Introduction
AWGN channels
Fading Channels

Signal-to-Noise power ratio and bit/symbol energy
Error probability for BPSK and QPSK
Approximate symbol and bit error probabilities for typical modulations

Signal-to-Noise ratio and bit/symbol energy (cont.)
For performance specification, we are interested in the bit error
probability 𝑃𝑏 as a function of 𝛾𝑏 .
However, for M-array signaling (e.g., MPAM and MPSK), the bit
error probability depends on both the symbol error probability and
the mapping of bits to symbols. Thus, we typically compute the
symbol error probability 𝑃𝑠 as a function of 𝛾𝑠 based on the signal
space concepts of previous section and then obtain 𝑃𝑏 as a function
of 𝛾𝑏 using an exact or approximate conversion.
The approximate conversion typically assumes that the symbol
energy is divided equally among all bits, and that Gray encoding is
used so that at reasonable SNRs, one symbol error corresponds to
exactly one bit error.
These assumptions for M-array signaling lead to the approximations:
𝛾𝑏 ≈

𝛾𝑠
𝑃𝑠
and 𝑃𝑏 ≈
.
log2 𝑀
log2 𝑀

Mobile communications - Chapter 3: Physical-layer transmission techniques

Section 3.2: Performance analysis over fading channels

(4)
8


Introduction
AWGN channels
Fading Channels

Signal-to-Noise power ratio and bit/symbol energy
Error probability for BPSK and QPSK
Approximate symbol and bit error probabilities for typical modulations

Error probability for BPSK and QPSK
Consider BPSK modulation with coherent detection and perfect
recovery of the carrier frequency and phase. With binary modulation
each symbol corresponds to one bit, so the symbol and bit error rates
are the same. The transmitted signal is 𝑠1 (𝑡) = 𝐴𝑔(𝑡)𝑐𝑜𝑠(2𝜋𝑓𝑐 𝑡) to
send a 0 bit and 𝑠2 (𝑡) = −𝐴𝑔(𝑡)𝑐𝑜𝑠(2𝜋𝑓𝑐 𝑡) to send a 1 bit. Note
that for binary modulation where 𝑀 = 2, there is only one way to
make an error and 𝑑𝑚𝑖𝑛 is the distance between the two signal
constellation points, so the probability of error is also the bound:
)
(
𝑑𝑚𝑖𝑛
.

(5)
𝑃𝑏 = 𝑄 √
2𝑁0
In previous chapter, we have 𝑑𝑚𝑖𝑛 =∥ s1 − s2 ∥=∥ 𝐴 − (−𝐴) ∥= 2𝐴.
The energy-per-bit can be determined by
∫ 𝑇𝑏
∫ 𝑇𝑏
∫ 𝑇𝑏
𝐸𝑏 =
𝑠21 (𝑡)𝑑𝑡 =
𝑠22 (𝑡)𝑑𝑡 =
𝐴2 𝑔 2 (𝑡) cos2 (2𝜋𝑓𝑐 𝑡)𝑑𝑡 = 𝐴2 .
0

0

Mobile communications - Chapter 3: Physical-layer transmission techniques

0

Section 3.2: Performance analysis over fading channels

(6)

9


Introduction
AWGN channels
Fading Channels

Signal-to-Noise power ratio and bit/symbol energy
Error probability for BPSK and QPSK
Approximate symbol and bit error probabilities for typical modulations

Error probability for BPSK and QPSK (cont.)
Thus, the signal
√ constellation for
√ BPSK in terms of energy-per-bit is
given by s0 = 𝐸𝑏 and s√1 = − 𝐸𝑏 . This yields the minimum
distance 𝑑𝑚𝑖𝑛 = 2𝐴 = 2 𝐸𝑏 . Substituting this into (5) yields
(√
)
( √ )
(√ )
2𝐸𝑏
2 𝐸𝑏
2𝛾𝑏 .
(7)
𝑃𝑏 = 𝑄 √
=𝑄
=𝑄
𝑁0
2𝑁0
QPSK modulation consists of BPSK modulation on both the
in-phase and quadrature components of the signal. With perfect
phase and carrier recovery, the received signal components
corresponding to each of these branches are orthogonal. Therefore,
the bit error probability
(√ ) on each branch is the same as for

BPSK:𝑃𝑏 = 𝑄 2𝛾𝑏 .
Mobile communications - Chapter 3: Physical-layer transmission techniques

Section 3.2: Performance analysis over fading channels

10


Introduction
AWGN channels
Fading Channels

Signal-to-Noise power ratio and bit/symbol energy
Error probability for BPSK and QPSK
Approximate symbol and bit error probabilities for typical modulations

Error probability for BPSK and QPSK (cont.)

The symbol error probability equals the probability that either
branch has a bit error:
[
(√ )]2
2𝛾𝑏
.
𝑃𝑠 = 1 − 1 − 𝑄

(8)

Example: Find the bit error probability 𝑃𝑏 and symbol error
probability 𝑃𝑠 of QPSK assuming 𝛾(𝑏√= 7 )dB. Solution: We have

𝛾𝑏 = 107/10 = 5.012, then 𝑃𝑏 = 𝑄 2𝛾𝑏 = 7.726 × 10−4 and
[
(√ )]2
𝑃𝑠 = 1 − 1 − 𝑄 2𝛾𝑏
= 1.545 × 10−3 .

Mobile communications - Chapter 3: Physical-layer transmission techniques

Section 3.2: Performance analysis over fading channels

11


Introduction
AWGN channels
Fading Channels

Signal-to-Noise power ratio and bit/symbol energy
Error probability for BPSK and QPSK
Approximate symbol and bit error probabilities for typical modulations

Approximate symbol and bit error probabilities for typical
modulations
Many of the approximations or exact values for 𝑃𝑠 derived above for
coherent modulation are in the following form:
(√
)
𝑃𝑠 (𝛾𝑠 ) ≈ 𝛼𝑀 𝑄
𝛽𝑀 𝛾𝑠
(9)

where 𝛼𝑀 and 𝛽𝑀 depend on the type of approximation and the
modulation type. In the below table, we summarize the specific values of
𝛼𝑀 and 𝛽𝑀 for common 𝑃𝑠 expressions for PSK, QAM, and FSK
modulations based on the derivations in the prior sections.
Modulation
BFSK:
BPSK:
QPSK,4QAM:
MPAM:
MPSK:
Rectangular MQAM:
Nonrectangular MQAM:

Ps (γs )
√

Ps ≈ 2 Q  γs

q
6γ s
Ps ≈ 2(MM−1) Q
M 2 −1


√
Ps ≈ 2Q 2γs sin(π/M )

q
√
3γ s
Ps ≈ 4( √MM−1) Q
M −1


q
3γ s
Ps ≈ 4Q
M −1

Mobile communications - Chapter 3: Physical-layer transmission techniques

Pb (γb )

√
Pb = Q γb

√
Pb = Q 2γb

√
Pb ≈ Q 2γb
q

6γ b log2 M
2(M −1)
Pb ≈ M
log2 M Q
(M 2 −1)
p


Pb ≈ log2 M Q
2γb log2 M sin(π/M )
2
√

q

3γ b log2 M
−1)
Pb ≈ √4(M M
Q
(M −1)
log2 M

q
3γ b log2 M
4
Pb ≈ log M Q
(M −1)
2

Section 3.2: Performance analysis over fading channels

12


Introduction
AWGN channels
Fading Channels

Introduction
Outage probability
Average probability of error

Introduction

In AWGN the probability of symbol error depends on the received

SNR 𝛾𝑠 . In a fading channel, the received signal power varies
randomly over distance or time due to shadowing and/or multipath
fading. Thus, in fading 𝛾𝑠 is a random variables with distribution
𝑝𝛾𝑠 (𝛾), and therefore 𝑃𝑠 (𝛾𝑠 ) is also random.
The performance metric when 𝛾𝑠 is random depends on the rate of
change of the fading. There are three different performance criteria
that can be used to characterize the random variable 𝑃𝑠 :
The outage probability, 𝑃𝑜𝑢𝑡 , defined as the probability that 𝛾𝑠 falls
below a given value corresponding to the maximum allowable 𝑃𝑠 .
The average error probability, 𝑃𝑠 , averaged over the distribution of
𝛾𝑠 .

Mobile communications - Chapter 3: Physical-layer transmission techniques

Section 3.2: Performance analysis over fading channels

13


Introduction
AWGN channels
Fading Channels

Introduction
Outage probability
Average probability of error

Introduction (cont.)

If the power of the received signal (with fading) is changing slowly

(slow-fading), then a deep fade will affect many simultaneous
symbols. Thus, fading may lead to large error bursts, which cannot
be corrected for with coding of reasonable complexity. Therefore,
these error bursts can seriously degrade end-to-end performance.
In this case acceptable performance cannot be guaranteed over all
time or, equivalently, throughout a cell, without drastically
increasing transmit power. Under these circumstances, an outage
probability is specified so that the channel is deemed unusable for
some fraction of time or space.

Mobile communications - Chapter 3: Physical-layer transmission techniques

Section 3.2: Performance analysis over fading channels

14


Introduction
AWGN channels
Fading Channels

Introduction
Outage probability
Average probability of error

Outage probability
The outage probability relative to 𝑃𝑜𝑢𝑡 is defined as
∫ 𝛾0
𝑃𝑜𝑢𝑡 = 𝑝 (𝛾𝑠 < 𝛾0 ) =
𝑝𝛾𝑠 (𝛾)𝑑𝛾.

(10)

0

where 𝛾0 typically specifies the minimum SNR required for
acceptable performance. For example, if we consider digitized voice,
𝑃𝑏 = 10−3 is an acceptable error rate since it generally cannot be
detected by the human ear. Thus, for a BPSK signal in Rayleigh
fading, 𝛾𝑏 < 7 dB would be declared an outage, so we set 𝛾0 = 7 dB.
In Rayleigh fading with 𝑝𝛾𝑠 (𝛾) = 𝛾1 𝑒−𝛾𝑠 /𝛾 𝑠 , one will have
𝑠

𝑃𝑜𝑢𝑡 =

∫

0

𝛾0

1 −𝛾𝑠 /𝛾 𝑠
𝑒
𝑑𝛾 = 1 − 𝑒−𝛾0 /𝛾 𝑠 .
𝛾𝑠

Mobile communications - Chapter 3: Physical-layer transmission techniques

Section 3.2: Performance analysis over fading channels

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15


Introduction
AWGN channels
Fading Channels

Introduction
Outage probability
Average probability of error

Outage probability (cont.)

Example: Determine the required 𝛾 𝑏 for BPSK modulation in slow
Rayleigh fading such as 50% of the time (or in space),
𝑃𝑏 (𝛾𝑏 ) < 10−4 .
Solution: For BPSK modulation in AWGN,
√ the target BER is
obtained
at
8.5
dB
(i.e.,
for
𝑃
(𝛾
)
=

𝑄(
2𝛾𝑏 ), one have
𝑏
𝑏
(
)
𝑃𝑏 100.85 = 10−4 ). Thus, 𝛾0 = 8.5 dB, since we want
𝑃𝑜𝑢𝑡 = 𝑝(𝛾𝑏 < 𝛾0 ), we have
𝛾𝑏 =

𝛾0
10.85
=
= 21.4 dB.
− ln(1 − 𝑃𝑜𝑢𝑡 )
− ln(1 − .05)

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Section 3.2: Performance analysis over fading channels

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16


Introduction
AWGN channels
Fading Channels

Introduction
Outage probability
Average probability of error

Average probability of error
The average probability of error is used as a performance metric
when 𝛾𝑠 is roughly constant over a symbol time. Then the averaged
probability of error is computed by integrating the error probability
in AWGN over the fading distribution:
∫ ∞
𝑃𝑠 =
𝑃𝑠 (𝛾)𝑝𝛾𝑠 (𝛾)𝑑𝛾.
(13)
0

where 𝑃𝑠 (𝛾) is the probability of symbol error in AWGN channels
with SNR 𝛾, which can be approximated by the expressions in the
aforementioned table.
For a given distribution of the fading amplitude 𝑟 (i.e., Rayleigh,
Rician, log-normal, etc.), we compute 𝑝𝛾𝑠 (𝛾) by making the change
of variable
𝑝𝛾𝑠 (𝛾)𝑑𝛾 = 𝑝(𝑟)𝑑𝑟.
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Section 3.2: Performance analysis over fading channels

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17

Introduction
AWGN channels
Fading Channels

Introduction
Outage probability
Average probability of error

Average probability of error (cont.)
For instance, in Rayleigh fading (refer to Zheng’s paper on Modified
Jake channel model), the received signal amplitude r has the
Rayleigh distribution
𝑝(𝑟) =

𝑟 −𝑟2 /(2𝜎2 )
𝑒
, 𝑟 ≥ 0.
𝜎2

(15)

The SNR per symbol for a given amplitude 𝑟 is
𝛾=

𝑟 2 𝑇𝑠
.
2𝜎𝑛2

(16)

where 𝜎𝑛2 = 𝑁0 /2 is the PSD of the noise in the in-phase and
quadrature branches.
Differentiating both sides of this expression yields
𝑑𝛾 =
Mobile communications - Chapter 3: Physical-layer transmission techniques

𝑟𝑇𝑠
𝑑𝑟.
𝜎𝑛2
Section 3.2: Performance analysis over fading channels

(17)
18


Introduction
AWGN channels
Fading Channels

Introduction
Outage probability
Average probability of error

Average probability of error (cont.)
Substituting (16) and (17) into (15) and then (14) yields
𝑝𝛾𝑠 (𝛾) =

𝜎𝑛2 −𝛾𝜎𝑛2 /(𝜎2 𝑇𝑠 )
𝑒
.

𝜎 2 𝑇𝑠

(18)

Since the average SNR per symbol 𝛾 𝑠 is just 𝜎 2 𝑇𝑠 /𝜎𝑛2 , one can
rewrite (18) as
1 𝛾/𝛾𝑠
𝑝𝛾𝑠 (𝛾) =
𝑒
,
(19)
𝛾𝑠
which is exponential. For binary signaling, this reduces to
𝑝𝛾𝑏 (𝛾) =

Mobile communications - Chapter 3: Physical-layer transmission techniques

1 𝛾/𝛾𝑏
𝑒
.
𝛾𝑏

Section 3.2: Performance analysis over fading channels

(20)

19


Introduction

AWGN channels
Fading Channels

Introduction
Outage probability
Average probability of error

Average probability of error (cont.)
Integrating the error probability of BPSK in AWGN over the
distribution (20) yields the following average probability of error for
BPSK in Rayleigh fading:
√
(
)
1
𝛾𝑏
1
𝑃𝑏 =
1−
≈
.
(21)
2
1 + 𝛾𝑏
4𝛾 𝑏
where the approximation holds for large 𝛾 𝑏 .
(√
)
If we use the general approximation 𝑃𝑠 ≈ 𝛼𝑀 𝑄 𝛽𝑀 𝛾𝑠 then the
average probability of symbol error in Rayleigh fading can be

approximated as
√
(
)
∫ ∞
(√
) 1
.5𝛽𝑀 𝛾 𝑠
𝛼𝑚
−𝛾/𝛾 𝑠
𝑃𝑠 ≈
𝛼𝑀 𝑄
𝛽𝑀 𝛾
𝑒
𝑑𝛾𝑠 =
1−
𝛾𝑠
2
1 + .5𝛽𝑀 𝛾 𝑠
0
𝛼𝑀
≈
.
2𝛽𝑀 𝛾 𝑠
Mobile communications - Chapter 3: Physical-layer transmission techniques

Section 3.2: Performance analysis over fading channels

20