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- Digital Communications I:
Modulation and Coding Course
Period 3 - 2007
Catharina Logothetis
Lecture 8
- Last time we talked about:
Some bandpass modulation schemes
M-PAM, M-PSK, M-FSK, M-QAM
How to perform coherent and non-
coherent detection
Lecture 8 2
- Example of two dim. modulation
ψ 2 (t )
16QAM 8PSK s3 “011”
“010” “001”
ψ 2 (t ) s4 s2
“0000” “0001” “0011” “0010”
s1 s2 3
s3 s4 Es
“000”
“110” s1
s5 ψ 1 (t )
“1000” “1001” “1011” “1010”
s5 s6 s7 s8
1 “111” “100”
-3 -1 1 3
ψ 1 (t )
s6 s8
s9 s10 -1
s11 12 s “101” s 7
ψ 2 (t )
“1100” “1101” “1111” “1110”
QPSK “00”
s 2“01” s1
s13 s14 -3
s15 16s
“0100” “0101” “0111” “0110” Es
ψ 1 (t )
Lecture 8
s3 “11” “10”
s4 3
- Today, we are going to talk about:
How to calculate the average probability
of symbol error for different modulation
schemes that we studied?
How to compare different modulation
schemes based on their error
performances?
Lecture 8 4
- Error probability of bandpass modulation
Before evaluating the error probability, it is
important to remember that:
Type of modulation and detection ( coherent or non-
coherent), determines the structure of the decision circuits
and hence the decision variable, denoted by z.
The decision variable, z, is compared with M-1 thresholds,
corresponding to M decision regions for detection purposes.
ψ 1 (t )
T r1
∫
0
⎡ r1 ⎤ Decision
r (t ) ⎢M⎥ =r
r Circuits
ˆ
m
ψ N (t ) ⎢ ⎥ Compare z
T ⎢rN ⎥
⎣ ⎦
with threshold.
∫
0 rN
Lecture 8 5
- Error probability …
The matched filters output (observation vector= r ) is
the detector input and the decision variable is a z = f (r)
function of r , i.e.
For MPAM, MQAM and MFSK with coherent detection z = r
For MPSK with coherent detection z = ∠r
For non-coherent detection (M-FSK and DPSK), z =| r |
We know that for calculating the average probability of
symbol error, we need to determine
Pr(r liesinside Zi | si sent)≡ Pr(z satisfies
conditionCi | si sent)
Hence, we need to know the statistics of z,
which depends on the modulation scheme
and the detection type.
Lecture 8 6
- Error probability …
AWGN channel model: r = s i + n
Signal vector s i = (ai1 , ai 2 ,..., aiN ) is deterministic.
Elements of noise vector n = (n1 , n2 ,..., nN ) are i.i.d
Gaussian random variables with zero-mean and
variance N 0 / 2 . The noise vector pdf is
1 ⎛ n2⎞
pn (n) = exp⎜ − ⎟
(πN 0 )N /2
⎜ N0 ⎟
⎝ ⎠
The elements of observed vector r = (r1 , r2 ,..., rN ) are
independent Gaussian random variables. Its pdf is
1 ⎛ r − si 2 ⎞
pr (r | s i ) = exp⎜ − ⎟
(πN 0 )N / 2 ⎜ N 0 ⎟
⎝ ⎠
Lecture 8 7
- Error probability …
BPSK and BFSK with coherent detection:
⎛ s1 − s 2 / 2 ⎞
PB = Q⎜ ⎟
⎜ N /2 ⎟
ψ 2 (t ) ⎝ 0 ⎠ ψ 1 (t )
“0” “1”
s1 s2 “0” s1 − s 2 = 2 Eb
BPSK BFSK s1 Eb
− Eb Eb ψ 1 (t )
s 2“1”
ψ 2 (t )
s1 − s 2 = 2 Eb Eb
⎛ 2 Eb ⎞ ⎛ Eb ⎞
PB = Q⎜ ⎟ PB = Q⎜ ⎟
⎜ N ⎟ ⎜ N ⎟
⎝ 0 ⎠ ⎝ 0 ⎠
Lecture 8 8
- Error probability …
Non-coherent detection of BFSK Decision variable:
2 / T cos( ω 1 t )
Difference of envelopes
r11
( )2 z = z1 − z2
T
∫
0
z1 = r11 + r12
2 2
2 / T sin( ω 1 t )
T r12
r (t )
∫
0
( )2 +
z
Decision rule:
ˆ
m
2 / T cos( ω 2 t ) if z (T ) > 0, m = 1
ˆ
if z (T ) < 0, m = 0
ˆ
T r21
( )2
∫
-
0
z 2 = r21 + r22
2 2
2 / T sin( ω 2 t )
T r22
∫
0
( )2
Lecture 8 9
- Error probability – cont’d
Non-coherent detection of BFSK …
1 1
PB = Pr( z1 > z 2 | s 2 ) + Pr( z 2 > z1 | s1 )
2 2
= Pr( z1 > z 2 | s 2 ) = E [Pr( z1 > z 2 | s 2 , z 2 )]
∞
= ∫ Pr( z1 > z 2 | s 2 , z 2 ) p ( z 2 | s 2 )dz 2 = ∫
∞
⎡ ∞ p ( z | s )dz ⎤ p( z | s )dz
0 0 ⎢ ∫z2
⎣ 1 2 1⎥
⎦ 2 2 2
1 ⎛ Eb ⎞
PB = exp⎜ −
⎜ 2N ⎟⎟
Rayleigh pdf Rician pdf
2 ⎝ 0 ⎠
Similarly, non-coherent detection of DBPSK
1 ⎛ Eb ⎞
PB = exp⎜ −
⎜ N ⎟ ⎟
2 ⎝ 0 ⎠
Lecture 8 10
- Error probability ….
Coherent detection of M-PAM
Decision variable:
z = r1
“00” “01” “11” “10”
s1 s2 s3 s4
4-PAM ψ 1 (t )
− 3 Eg − Eg 0 Eg 3 Eg
ψ 1 (t )
T r1
∫
ML detector
r (t ) (Compare with M-1 thresholds) ˆ
m
0
Lecture 8 11
- Error probability ….
Coherent detection of M-PAM ….
Error happens if the noise, n1 = r1 − s m , exceeds in amplitude
one-half of the distance between adjacent symbols. For symbols
on the border, error can happen only in one direction. Hence:
(
Pe (s m ) = Pr | n1 |=| r1 − s m |> E g ) for 1 < m < M ;
P (s ) = Pr (n = r − s
e 1 1 1 1 > Eg ) (
and Pe (s M ) = Pr n1 = r1 − s M < − E g )
M −2
( ) ( ) ( )
M
1 1 1
PE ( M ) =
M
∑ Pe (s m ) =
m =1 M
Pr | n1 |> E g + Pr n1 > E g + Pr n1 < − E g
M M
2( M − 1) ⎛ 2 E g ⎞
=
2( M − 1)
M
(
Pr n1 > E g =
M
)
2( M − 1) ∞
∫ Eg pn1 (n)dn = M Q⎜ N 0 ⎜
⎟
⎟
⎝ ⎠
( M 2 − 1)
Es = (log 2 M ) Eb = Eg
3
Gaussian pdf with
2( M − 1) ⎛ 6 log 2 M Eb ⎞
⎟ zero mean and variance
Q⎜
N0 / 2
PE ( M ) = ⎜ M 2 −1 N ⎟
M ⎝ 0 ⎠
Lecture 8 12
- Error probability …
Coherent detection ψ 2 (t )
of M-QAM
“0000” “0001”
s1 s2 s 3“0011”4 “0010”
s
“1000”
s “1001”
s s 7“1011” 8 “1010”
s
5 6
16-QAM ψ 1 (t )
s9 s10 s11 s12
“1100” “1101” “1111” “1110”
ψ 1 (t ) s13 s14 s15 s16
T r1 ML detector “0100” “0101” “0111” “0110”
∫
0 (Compare with M − 1 thresholds)
r (t ) Parallel-to-serial
ˆ
m
converter
ψ 2 (t )
T r2 ML detector
∫
0 (Compare with M − 1 thresholds)
Lecture 8 13
- Error probability …
Coherent detection of M-QAM …
M-QAM can be viewed as the combination of two M − PAM
modulations on I and Q branches, respectively.
No error occurs if no error is detected on either I and Q
branches. Hence:
Considering the symmetry of the signal space and orthogonality
of I and Q branches:
PE ( M ) = 1 − PC ( M ) = 1 − Pr(no error detected on I and Q branches)
Pr(no error detected on I and Q branches) = Pr(no error on I)Pr(no error on Q)
(
= Pr(no error on I)2 = 1 − PE ( M ))
2
⎛ 1 ⎞ ⎛ 3 log 2 M Eb ⎞
PE ( M ) = 4⎜1 − ⎟Q⎜
⎜ M −1 N ⎟
⎟ Average probability of
⎝ M ⎠ ⎝ 0 ⎠
symbol error for M − PAM
Lecture 8 14
- Error probability …
Coherent detection
of MPSK
ψ 2 (t )
s 3 “011”
“010”
s4 “001”
s2
Es
“110”
s“000”
1
8-PSK s5 ψ 1 (t )
ψ 1 (t ) “111”
s6 s8“100”
T r1 s
“101” 7
∫
r1 φ
0 ˆ
r (t ) ˆ
m
arctan Compute Choose
ψ 2 (t ) r2 | φi − φ |
ˆ smallest
T
∫
0
r2 Decision variable
z = φ = ∠r
ˆ
Lecture 8 15
- Error probability …
Coherent detection of MPSK …
The detector compares the phase of observation vector to M-1
thresholds.
Due to the circular symmetry of the signal space, we have:
M π /M
1
PE ( M ) = 1 − PC ( M ) = 1 −
M
∑ Pc (s m ) = 1 − Pc (s1 ) = 1 − ∫
m =1
−π / M
pφˆ (φ )dφ
where
2 Es ⎛ E ⎞ π
pφˆ (φ ) ≈ cos(φ ) exp⎜ − s sin 2 φ ⎟;
⎜ N ⎟ | φ |≤
π N0 ⎝ 0 ⎠ 2
It can be shown that
⎛ 2 Es ⎛ π ⎞⎞ ⎛ 2(log 2 M )Eb ⎛ π ⎞⎞
⎜
PE ( M ) ≈ 2Q⎜ sin ⎜ ⎟ ⎟ or ⎜
PE ( M ) ≈ 2Q⎜ sin ⎜ ⎟ ⎟
⎝ N0 ⎝ M ⎠⎟
⎠ ⎝ N0 ⎝ M ⎠⎟
⎠
Lecture 8 16
- Error probability …
Coherent detection of M-FSK
ψ 1 (t )
T r1
∫
0
⎡ r1 ⎤ ML detector:
r (t ) ⎢M⎥ =r
r Choose
the largest element ˆ
m
ψ M (t ) ⎢ ⎥ in the observed vector
T ⎢rM ⎥
⎣ ⎦
∫
0 rM
Lecture 8 17
- Error probability …
Coherent detection of M-FSK …
The dimensionality of signal space is M. An upper
bound for average symbol error probability can be
obtained by using union bound. Hence
⎛ Es ⎞
PE ( M ) ≤ (M − 1)Q⎜
⎜ N ⎟
⎟
⎝ 0 ⎠
or, equivalently
⎛
PE ( M ) ≤ (M − 1)Q⎜
(log 2 M )Eb ⎞
⎟
⎜ N0 ⎟
⎝ ⎠
Lecture 8 18
- Bit error probability versus symbol error
probability
Number of bits per symbol k = log 2 M
For orthogonal M-ary signaling (M-FSK)
PB 2 k −1 M /2
= k =
PE 2 − 1 M − 1
PB 1
lim =
k →∞ P 2
E
For M-PSK, M-PAM and M-QAM
PE
PB ≈ for PE
- Probability of symbol error for binary
modulation
Note!
• “The same average symbol
energy for different sizes of
PE signal space”
Eb / N 0 dB
Lecture 8 20
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