Bandpass modulation schemes transmit data by varying the amplitude, phase, or frequency of a carrier signal. Common schemes include M-PAM, M-PSK, M-FSK, and M-QAM. At the receiver, data is detected through either coherent or non-coherent detection. Coherent detection uses carrier phase estimation while non-coherent does not. The average probability of symbol error is calculated for each modulation scheme and used to compare their performance.

1. TRUY N THÔNG SỀ Ố
DIGITAL COMMUNICATION
Tu n 5ầ
1

2. Reference
• “Digital communications: Fundamentals and
Applications” by Bernard Sklar
• Catharina Logothetis’s Lecture, Uppsala
University
2

3. Last time
• Signal space used for detection
• Signal detection in AWGN channels
– Correlator/Matched-filter Demodulator
– Maximum likelihood
• The techniques to reduce ISI
– Pulse shaping
– Equalization
3

4. Today: BANDPASS MODULATION
• Các cách đi u ch d i qua (bandpass modulationề ế ả
schemes)
– M-PAM, M-PSK, M-FSK, M-QAM
• Tách sóng t i đ u thu (detect the transmittedạ ầ
information at the receiver)
– Coherent detection
– Non-coherent detection
• Tính xác su t l i trung bình (the average probability ofấ ỗ
symbol error) c a các d ng đi u chủ ạ ề ế
• So sánh các d ng đi u chạ ề ế
4

5. 5

6. Block diagram of a DCS
Format
Source
encode
Format
Source
decode
Channel
encode
Pulse
modulate
Bandpass
modulate
Channel
decode
Demod.
SampleDetect
Channel
Digital modulation
Digital demodulation
6

7. Bandpass modulation (đi u ch d i qua)ề ế ả
• Bandpass modulation:
– Là quá trình bi n tín hi u d li u thành d ng sóng sin cóế ệ ữ ệ ạ
biên đ / pha / t n s ho c k t h p thay đ i theo tín hi uộ ầ ố ặ ế ợ ổ ệ
đó.
– The process of converting data signal to a sinusoidal
waveform where its amplitude, phase or frequency, or a
combination of them, is varied in accordance with the
transmitting data.
7

8. Bandpass modulation (đi u ch d i qua)ề ế ả
• Bandpass signal:
là xung d i g c (baseband pulse shape), có năng l ngả ố ượ
• Gi đ nh:ả ị
– là d ng xung ch nh t, năng l ng đ n v (rectangular pulseạ ữ ậ ượ ơ ị
shape with unit energy).
– Gray coding is used for mapping bits to symbols.
– denotes average symbol energy given by
( ) Ttttit
T
E
tgts ic
i
Ti ≤≤+∆−+= 0)()1(cos
2
)()( φωω
)(tgT
)(tgT gE
sE ∑=
=
M
i is E
M
E 1
1
8

9. Remind: Gray code
Dec Gray Binary
0 000 000
1 001 001
2 011 010
3 010 011
4 110 100
5 111 101
6 101 110
7 100 111
9

10. Demodulation and detection
(gi i đi u ch và dò tìm/tách sóng)ả ề ế
• Demodulation (gi i đi u ch ):ả ề ế Tín hi u nh n đ c sệ ậ ượ ẽ
đ c chuy n sang tín hi u d i g c, l c và l y m u.ượ ể ệ ả ố ọ ấ ẫ
• Detection (dò tìm/tách sóng): Các m u này đ c dùng đẫ ượ ể
dò tìm các giá tr đã g i đi theo các quy lu t, ví d MLị ở ậ ụ
detection rule.










Nz
z

1
z=
∫
T
0
)(1 tψ
∫
T
0
)(tNψ
)(tr
1z
Nz
z Decision
circuits
(ML detector)
mˆ
10

11. Coherent and nonconherent detections
• Coherent detection (s tách tín hi u đ ng b /nh t quán)ự ệ ồ ộ ấ
– dùng pha c a sóng mang (carrier’s phase) đ tách tín hi u +ủ ể ệ
dùng c l ng pha (phase estimation) đ u thu.ướ ượ ở ầ
• Noncoherent detection (s tách tín hi u không đ ng b ).ự ệ ồ ộ
– không dùng pha c a tín hi u nên không c n c l ng phaủ ệ ầ ướ ượ
(phase estimation) đ u thu.ở ầ
– Ưu đi m: gi m đ ph c t p so v i coherent detectionể ả ộ ứ ạ ớ
– Khuy t: Xác su t l i l n h n coherent detection.ế ấ ỗ ớ ơ
11

12. Coherent and nonconherent detections
12

13. Coherent detections
• Các v n đ đ u thu gây ra b i:ấ ề ở ầ ở (Source of carrier-phase
mismatch at the receiver):
– S tr / trì hoãn (Propagation delay).ự ễ
– Các dao đ ng n i c a đ u thu (The oscillators at the receiverộ ộ ủ ầ
which generate the carrier signal, are not usually phased locked
to the transmitted carrier).
13

14. Coherent detection ..
– Circuits such as Phase-Locked-Loop (PLL) are
implemented at the receiver for carrier phase
estimation ( ).
PLL
Oscillator 90 deg.
( ) )()(cos
2
)()( tntt
T
E
tgtr ii
i
T +++= αφω ( )αω ˆcos
2
+t
T
c
( )αω ˆsin
2
+t
T
c
Used by
correlators
αα ˆ≈
I branch
Q branch
14

15. Bandpass Modulation Schemes
• One dimensional waveforms
– Amplitude Shift Keying (ASK)
– M-ary Pulse Amplitude Modulation (M-PAM)
• Two dimensional waveforms
– M-ary Phase Shift Keying (M-PSK)
– M-ary Quadrature Amplitude Modulation (M-
QAM)
• Multidimensional waveforms
– M-ary Frequency Shift Keying (M-FSK)
15

16. One dimensional modulation, demodulation
and detection
• Amplitude Shift Keying (ASK) modulation:
( )φω += t
T
E
ts c
i
i cos
2
)(
( )cos
2
)(
,,1)()(
1
1
ii
c
ii
Ea
t
T
t
Mitats
=
+=
==
φωψ
ψ 
)(1 tψ
1s2s
0 1E
“0” “1”
On-off keying (M=2):
16

17. One dimensional mod.,…
• M-ary Pulse Amplitude modulation (M-PAM)
( )t
T
ats cii ωcos
2
)( =
( )
( )
gs
gii
gi
c
ii
E
M
E
MiEE
EMia
t
T
t
Mitats
3
)1(
12
)12(
cos
2
)(
,,1)()(
2
22
1
1
−
=
−−==
−−=
=
==
s
ωψ
ψ 
4-PAM:
)(1 tψ
2s1s
0gE3−
“00” “01”
4s3s
“11” “10”
gE− gE gE3
17

18. • Coherent detection of M-PAM
∫
T
0
)(1 tψ
ML detector
(Compare with M-1 thresholds)
)(tr
1z
mˆ
One dimensional mod.,...–cont’d
18

19. Two dimensional modulation, demodulation and
detection (M-PSK)
• M-ary Phase Shift Keying (M-PSK)






+=
M
i
t
T
E
ts c
s
i
π
ω
2
cos
2
)(
( ) ( )
2
21
21
2211
2
sin
2
cos
sin
2
)(cos
2
)(
,,1)()()(
iis
sisi
cc
iii
EE
M
i
Ea
M
i
Ea
t
T
tt
T
t
Mitatats
s==






=





=
−==
=+=
ππ
ωψωψ
ψψ 
19

20. Two dimensional mod.,… (MPSK)
)(1 tψ
2s1s
bE
“0” “1”
bE−
)(2 tψ
3s
7s
“110”
)(1 tψ
4s 2s
sE
“000”
)(2 tψ
6s 8s
1s
5s
“001”
“011”
“010”
“101”
“111” “100”
)(1 tψ
2s 1s
sE
“00”
“11”
)(2 tψ
3s 4s“10”
“01”
QPSK (M=4)
BPSK (M=2)
8PSK (M=8)
20

21. Two dimensional mod.,…(MPSK)
• Coherent detection of MPSK
Compute Choose
smallest2
1
arctan
z
z φˆ
|ˆ| φφ −i
∫
T
0
)(1 tψ
∫
T
0
)(2 tψ
)(tr
1z
2z
mˆ
21

22. Two dimensional mod.,… (M-QAM)
• M-ary Quadrature Amplitude Mod. (M-QAM)
( )ic
i
i t
T
E
ts ϕω += cos
2
)(
( ) ( )
3
)1(2
andsymbolsPAMareandwhere
sin
2
)(cos
2
)(
,,1)()()(
21
21
2211
−
=
==
=+=
M
Eaa
t
T
tt
T
t
Mitatats
sii
cc
iii
ωψωψ
ψψ 
( )














+−−+−+−+−+−
−−−+−−+−
−−−+−−+−
=
)1,1()1,3()1,1(
)3,1()3,3()3,1(
)1,1()1,3()1,1(
, 21
MMMMMM
MMMMMM
MMMMMM
aa ii




22

23. Two dimensional mod.,… (M-QAM)
)(1 tψ
)(2 tψ
2s1s 3s 4s
“0000” “0001” “0011” “0010”
6s5s 7s 8s
10s9s 11s 12s
14s13s 15s 16s
1 3-1-3
“1000” “1001” “1011” “1010”
“1100” “1101” “1111” “1110”
“0100” “0101” “0111” “0110”
1
3
-1
-3
16-QAM
23

24. Two dimensional mod.,… (M-QAM)
• Coherent detection of M-QAM
∫
T
0
)(1 tψ
ML detector1z
∫
T
0
)(2 tψ
ML detector
)(tr
2z
mˆParallel-to-serial
converter
s)threshold1with(Compare −M
s)threshold1with(Compare −M
24

25. Multi-dimentional modulation, demodulation &
detection
• M-ary Frequency Shift keying (M-FSK)
( ) ( )
T
f
tit
T
E
t
T
E
ts c
s
i
s
i
2
1
2
)1(cos
2
cos
2
)(
=
∆
=∆
∆−+==
π
ω
ωωω
( )
2
1
0
cos
2
)(
,,1)()(
iis
s
ijii
M
j
jiji
EE
ji
jiE
at
T
t
Mitats
s==



≠
=
==
== ∑=
ωψ
ψ 
)(1 tψ
2s
1s
3s
)(3 tψ
)(2 tψ
sE
sE
sE
25

26. Multi-dimensional mod.,…(M-FSK)










Mz
z

1
z=
∫
T
0
)(1 tψ
∫
T
0
)(tMψ
)(tr
1z
Mz
z
ML detector:
Choose
the largest element
in the observed vector
mˆ
26

27. 27

28. Today
• Các cách đi u ch d i qua (bandpass modulationề ế ả
schemes)
– M-PAM, M-PSK, M-FSK, M-QAM
• Tách sóng t i đ u thu (detect the transmittedạ ầ
information at the receiver)
– Coherent detection
– Non-coherent detection
• Tính xác su t l i trung bình (the average probability ofấ ỗ
symbol error) c a các d ng đi u chủ ạ ề ế
• So sánh các d ng đi u chạ ề ế
28

29. Eb/N0 figure of merit in digital communications
• SNR or S/N là công su t tín hi u trung bình trênấ ệ
công su t nhi u trung bình. SNR đ c tính b ngấ ễ ượ ằ
bit-energy:
b
bb
R
W
N
S
WN
ST
N
E
==
/0
bR
W
: Bit rate
: Bandwidth
29
S: công suất tín hiệu trung bình (average signal power)
N: công suất nhiễu trung bình (average noise power)
Eb: năng lượng bit
Tb = 1/Rb

30. Example of Symbol error prob. For PAM signals
)(1 tψ
0
1s2s
bEbE−
Binary PAM
)(1 tψ0
2s3s
5
2 bE
5
6 bE
5
6 bE
−
5
2 bE
−
4s 1s
4-ary PAM
T t
)(1 tψ
T
1
0
30

31. 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.










Nr
r

1
r=
∫
T
0
)(1 tψ
∫
T
0
)(tNψ
)(tr
1r
Nr
r
Decision
Circuits
Compare z
with threshold.
mˆ
31

32. Error probability …
• AWGN channel model:
– Signal vector is deterministic (xác đ nh).ị
– Noise vector có thành ph n là các bi n ng uầ ế ẫ
nhiên Gaussian, tr trung bình = 0 (zero-mean) và bi n trị ế ị
(variance) là . The noise vector pdf:
– Observed vector có thành ph n là các bi nầ ế
ng u nhiên không ph thu c hàm Gaussian (independentẫ ụ ộ
Gaussian random variables). pdf :
),...,,( 21 iNiii aaa=s
),...,,( 21 Nrrr=r
),...,,( 21 Nnnn=n
nsr += i
2/0N
( ) 







−=
0
2
2/
0
exp
1
)(
NN
p N
n
nn
π
( ) 






 −
−=
0
2
2/
0
exp
1
)|(
NN
p i
Ni
sr
srr
π
32

33. Error probability …
• BPSK and BFSK with coherent detection:
)(1 tψ
2s1s
bE
“0” “1”
bE−
)(2 tψ
bE221 =−ss
)(1 tψ
2s
1s
)(2 tψ
bE
bE
bE221 =−ss“0”
“1”
BPSK BFSK
2
0








=
N
E
QP b
B
2/
2/
0
21







 −
=
N
QPB
ss
0








=
N
E
QP b
B
33

34. Remind
• Q(x) = complementary error function = co-
error function
34

35. • Coherent detection of M-PAM
– Decision variable:
∫
T
0
)(1 tψ
ML detector
(Compare with M-1 thresholds)
)(tr
1r
mˆ
Error probability ….
)(1 tψ
2s1s
0gE3−
“00” “01”
4s3s
“11” “10”
gE− gE gE3
4-PAM
1rz =
35

36. Error probability ….
• Coherent detection of M-PAM ….
• Error happens if the noise, , exceeds in amplitude one-half
of the distance between adjacent symbols. For symbols on the
border, error can happen only in one direction. Hence:
( )
( ) ( )gMMege
gmme
ErnPErnP
MmErnP
−<−==>−==
<<>−==
ssss
ss
111111
11
Pr)(andPr)(
;1for||||Pr)(








−
−
=
0
2
2
1
log6)1(2
)(
N
E
M
M
Q
M
M
MP b
E
gbs E
M
EME
3
)1(
)(log
2
2
−
==
mrn s−= 11
Gaussian pdf with
zero mean and variance 2/0N
( ) ( ) ( )
( ) 






−
=
−
=>
−
=
−<+>+>
−
==
∫
∑
∞
=
0
1
111
1
2)1(2
)(
)1(2
Pr
)1(2
Pr
M
1
Pr
M
1
||Pr
2
)(
1
)(
1
N
E
Q
M
M
dnnp
M
M
En
M
M
EnEnEn
M
M
P
M
MP
g
E
ng
ggg
M
m
meE
g
s
36

37. Error probability …
• Coherent detection
of M-QAM
∫
T
0
)(1 tψ
ML detector1r
∫
T
0
)(2 tψ
ML detector
)(tr
2r
mˆParallel-to-serial
converter
s)threshold1with(Compare −M
s)threshold1with(Compare −M
)(1 tψ
)(2 tψ
2s1s 3s 4s“0000” “0001” “0011” “0010”
6s5s 7s 8s
10s9s 11s 12s
14s13s 15s 16s
“1000” “1001” “1011” “1010”
“1100” “1101” “1111” “1110”
“0100” “0101” “0111” “0110”
16-QAM
37

38. Error probability …
• Coherent detection of M-QAM …
• M-QAM can be viewed as the combination of two
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:
PAM−M
branches)QandIondetectederrornoPr(1)(1)( −=−= MPMP CE
( )( )22
1I)onerrorPr(no
Q)onerrorI)Pr(noonerrorPr(nobranches)QandIondetectederrornoPr(
MPE−==
=








−






−=
0
2
1
log31
14)(
N
E
M
M
Q
M
MP b
E Average probability of
symbol error for PAM−M
38

39. Error probability …
• Coherent detection
of MPSK
Compute Choose
smallest2
1
arctan
r
r φˆ
|ˆ| φφ −i
∫
T
0
)(1 tψ
∫
T
0
)(2 tψ
)(tr
1r
2r
mˆ
3s
7s
“110” )(1 tψ
4s 2s
sE
“000”
)(2 tψ
6s 8s
1s
5s
“001”
“011”
“010”
“101”
“111” “100”
8-PSK
Decision variable
r∠== φˆz
39

40. 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:
where
• It can be shown that
φφ
π
π φ
dpPP
M
MPMP
M
M
c
M
m
mcCE )(1)(1)(
1
1)(1)(
/
/
ˆ1
1
∫∑ −
=
−=−=−=−= ss














≈
MN
E
QMP s
E
π
sin
2
2)(
0
( )














≈
MN
EM
QMP b
E
π
sin
log2
2)(
0
2
or
2
||;sinexp)cos(
2
)( 2
00
ˆ
π
φφφ
π
φφ
≤





−≈
N
E
N
E
p ss
40

41. Error probability …
• Coherent detection of M-FSK










Mr
r

1
r=
∫
T
0
)(1 tψ
∫
T
0
)(tMψ
)(tr
1r
Mr
r
ML detector:
Choose
the largest element
in the observed vector
mˆ
41

42. 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
or, equivalently
( ) 







−≤
0
1)(
N
E
QMMP s
E
( ) ( )








−≤
0
2log
1)(
N
EM
QMMP b
E
42

43. Bit error probability và symbol error probability
• S bit/symbolố
• For orthogonal M-ary signaling (M-FSK)
• For M-PSK, M-PAM and M-QAM
2
1
lim
1
2/
12
2 1
=
−
=
−
=
∞→
−
E
B
k
k
k
E
B
P
P
M
M
P
P
Mk 2log=
1for <<≈ E
E
B P
k
P
P
43

44. 44

45. Error probability …
• Non-coherent detection of BFSK
∫
T
0
)cos(/2 1tT ω
∫
T
0)(tr
11r
12r
∫
T
0
∫
T
0
21r
22r
Decision rule:
)cos(/2 2tT ω
)sin(/2 2tT ω
)sin(/2 1tT ω
( )2
( )2
( )2
( )2
+
-
z
0ˆ,0)(if
1ˆ,0)(if
=<
=>
mTz
mTz
mˆ
2
12
2
111 rrz +=
2
22
2
212 rrz +=
21 zzz −=
Decision variable:
Difference of envelopes
45

46. Error probability – cont’d
• Non-coherent detection of BFSK …
• Non-coherent detection of DBPSK
[ ]
∫ ∫∫
∞ ∞∞



=>=
>=>=
>+>=
0
222121
0
2222221
2221221
112221
)|()|()|(),|Pr(
),|Pr()|Pr(
)|Pr(
2
1
)|Pr(
2
1
2
dzzpdzzpdzzpzzz
zzzEzz
zzzzP
z
B
ssss
ss
ss






−=
02
exp
2
1
N
E
P b
B






−=
0
exp
2
1
N
E
P b
B
Rayleigh pdf Rician pdf
46

47. Summary
47

48. Probability of symbol error for binary modulation
EP
dB/ 0NEb
Note!
• “The same average symbol
energy for different sizes of
signal space”
48

49. Probability of symbol error for M-PSK
EP
dB/ 0NEb
Note!
• “The same average symbol
energy for different sizes of
signal space”
49

50. Probability of symbol error for M-FSK
EP
dB/ 0NEb
Note!
• “The same average symbol
energy for different sizes of
signal space”
50

51. Probability of symbol error for M-PAM
EP
dB/ 0NEb
Note!
• “The same average symbol
energy for different sizes of
signal space”
51

52. Probability of symbol error for M-QAM
EP
dB/ 0NEb
Note!
• “The same average symbol
energy for different sizes of
signal space”
52

53. Example of samples of matched filter output for
some bandpass modulation schemes
53

54. Bài t pậ
T c đ bit (data rate) = 5000 bit/s.ố ộ
Tìm xác su t l i bit trung bình Pấ ỗ B khi bi t đ u thu dùng coherentế ầ
BPSK đ tách sóngể
HzWNmVA
tAtstAts
/10;1
)cos()();cos()(
11
0
0201
−
==
−== ωωCho
54

55. Bài t pậ
Cho 1 h th ng coherent BPSK có t c đ l i trung bình làệ ố ố ộ ỗ
100 l i/ngày v i t c đ data rate = 1000 bits/s, côngỗ ớ ố ộ
su t nhi uấ ễ
a) Tìm xác su t l i bit trung bìnhấ ỗ
b) N u ch nh công su t tín hi u nh n đ c trung bình là Sế ỉ ấ ệ ậ ượ
= 10-6
W thì xác su t l i bit trung bình là bao nhiêu?ấ ỗ
HzWN /10 10
0
−
=
55

56. Bài t p n p cho GVậ ộ
• Cách 1: n p tr c ti p cho GV (sau m i bu i h c)ộ ự ế ỗ ổ ọ
• Cách 2: g i email t i:ử ớ truyenthongsodtvt@gmail.com
• Th i h n n p bài: th ba ngày 3 tháng 5 (bu i h cờ ạ ộ ứ ổ ọ
cu i)ố
• Trong email và file n p ghi rõ h tên và mã s SVộ ọ ố
• Đi m bài t p: 30% t ng đi mể ậ ổ ể
56

57. Bài t p n p cho GVậ ộ
Ch n 1 trong các bài sau:ọ
1. Tìm hi u v Non-coherent detection (DPSK,ể ề
BDPSK, FSK)
2. Dùng Matlab mô ph ng đ so sánh s khácỏ ể ự
nhau c a các ki u đi u ch (v SNR vs Pủ ể ề ế ẽ E
ho c SNR vs BER)ặ
57