# Mobile Communications - Performance of Digital Modulation over

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```					                                            Performance of Digital Modulation over Wireless Channels

Mobile Communications
Performance of Digital Modulation over Wireless Channels

Wen-Shen Wuen

Trans. Wireless Technology Laboratory
National Chiao Tung University

WS Wuen      Mobile Communications                                     1

Outline   Performance of Digital Modulation over Wireless Channels

Outline

1   Modulation Performance in Fading Channels

WS Wuen      Mobile Communications                                     2
Modulation Performance in Fading Channels   Performance of Digital Modulation over Wireless Channels

Modulation Performance in Slow Flat-Fading Channels

Slow Flat-Fading Channel: the attenuation and phase shift of
the signal is constant over at least one symbol interval.

r(t) = α(t)s(t)e−jθ(t) + n(t),                  0≤t ≤T

α(t): gain of the channel
θ(t): phase shift of the channel

WS Wuen        Mobile Communications                                     4

Modulation Performance in Fading Channels   Performance of Digital Modulation over Wireless Channels

Evaulate Pe in a Slow Flat-Fading Channel

Average the probability of error of the particular modulation in
AWGN channel over the possible ranges of signal strength due
The probability of error in AWGN channels is viewed as a
conditional error probability where the condition is that α is
ﬁxed.
Probability of error in slow ﬂat-fading channel = averaging the
error in AWGN channels over the fading probability density
function:                     ∞
Pe =            Pe (X )p(X )dX
0
Pe (X ): the probability of error for an arbitrary modulation at a
speciﬁc value of signal to noise ratio X .
X : signal to noise ratio, X = α2 Eb /N0
p(X ): the probability density function of X due to the fading
channel.
WS Wuen        Mobile Communications                                     5
Modulation Performance in Fading Channels      Performance of Digital Modulation over Wireless Channels

Pe in Slow Flat Rayleigh Fading Channels
distribution ⇒ fading power α2 and X have a chi-square
distribution with two degrees of freedom.
1 −X
p(X ) =         e Γ ,X ≥ 0
Γ
E
where Γ = Nb α2 is the average signal to noise ratio. For α2 = 1, Γ
0
corresponds to the average Eb /N0 for the fading channel.
Probability of error of coherent BPSK in slow ﬂat-fading channel
∞          1 X
Pe,BPSK =                 Q( 2X ) e− Γ dX
0              Γ

1                Γ
Pe,BPSK =            1−
2               1+Γ
Probability of error of coherent BFSK in slow ﬂat-fading channel

1                Γ
Pe,BFSK =            1−
2               2+Γ

WS Wuen           Mobile Communications                                     6

Modulation Performance in Fading Channels      Performance of Digital Modulation over Wireless Channels

Pe in Slow Flat Rayleigh Fading Channels, cont’d

Probability of error of DPSK in slow ﬂat-fading channel

1
Pe,DPSK =
2(1 + Γ)

Probability of error of noncoherent orthogonal BFSK in slow
1
Pe,NCBFSK =
2+Γ
For large Eb /N0
1
Pe,BPSK ≈
4Γ
1
Pe,BFSK ≈
2Γ
1
Pe,DPSK ≈
2Γ
1
Pe,NCBFSK ≈
Γ
WS Wuen           Mobile Communications                                     7
Modulation Performance in Fading Channels   Performance of Digital Modulation over Wireless Channels

Pe in Slow Flat Rayleigh Fading Channels, cont’d

Probability of error of
coherent GMSK in slow
             
1          δΓ                     1
Pe,GMSK = 1 −
2        δΓ + 1                   4δΓ

where

0.68       for BT = 0.25
δ
0.85       for BT = ∞

WS Wuen        Mobile Communications                                     8

Modulation Performance in Fading Channels   Performance of Digital Modulation over Wireless Channels

Modulation in Frequency Selective Mobile Channels

intersymbol interference ⇒ irreducible BER ﬂoor.
Motion ⇒ time-varying Doppler spread ⇒ irreducible BER ﬂoor
Simulation is the major tool for analyzing frequency selective
Irreducible error ﬂoor in a frequency selective channel is
caused by ISI when
the undelayed signal component is removed through multipath
cancellation
a non-zero value of normalized rms delay spread d = στ /Ts
the sampling time of a receiver is shift as a result of delay spread

error bursts
Large delay spreads ⇒ timing errors and ISI are the dominant
error mechanisms.

WS Wuen        Mobile Communications                                     9
Modulation Performance in Fading Channels     Performance of Digital Modulation over Wireless Channels

Irreducible Bit Error Rate Due to Multipath

WS Wuen          Mobile Communications                                     10

Modulation Performance in Fading Channels     Performance of Digital Modulation over Wireless Channels

Irreducible Bit Error Rate Floor
Irreducible bit error rate ﬂoor Pﬂoor

Pﬂoor ≤ d2 for 0.02 ≤ d ≤ 0.1

For example, the rms delay spread in a typical urban
environment is approximately στ = 2.5µs. To keep στ < 0.1Ts
requires that the data rate not exceed 40 kbps.

Example 1
Using the approximation Pﬂoor ≤ (στ /Ts )2 , ﬁnd the maximum data rate that can be transmitted
through a channel with delay spread στ = 3µs, using either BPSK or QPSK modulation, such that
the probability of bit error Pe is less than 10−3 .
Solution:
For BPSK
στ 2            στ                      1
Pﬂoor ≤         ⇒ Tb ≥          = 94.87µs ⇒ Rb =    = 10.54 kbps
Tb              P                      Tb
ﬂoor

For QPSK
στ
Ts ≥             = 94.87µs
Pﬂoor

2
Since there are two bits per symbol in QPSK, the data rate is Rb = 2Rs = T = 21.01 kbps
s

WS Wuen          Mobile Communications                                     11

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