Combined Linear & Constant Envelope Modulation
M-ary modulation: digital baseband data sent by varying RF carrier’s
(i) envelope ( eg. MASK)
(ii) phase /frequency ( eg. MPSK, MFSK)
(iii) envelope & phase  offer 2 degrees of freedom ( eg. MQAM)
(i) n bits encoded into 1 of M symbols, M  2n
(ii) each symbol mapped to signal si(t), M possible signals:
s1(t),…,sM(t)
(iii) a signal, si(t) , sent during each symbol period, Ts = n.Tb
1
Combined Linear & Constant Envelope Modulation
M-ary modulation is useful in bandlimited channels
• greater B  log2M
• significantly higher BER
- smaller distances in constellation
- sensitive to timing jitter
MPSK
MQAM
MFSK
OFDM
2
Mary Phase Shift Keying
Carrier phase takes 1 of M possible values – amplitude constant
i = 2(i-1)/M,
i = 1,2,…M
Modulated waveform:
si(t) =
2 Es
2


cos 2f c t 
(i  1) 
Ts
M


Es = log2MEb
Ts = log2MTb
2
Ts
2

Es cos (i  1)
M

for i = 1,2,…M
i = 1,2,…M
energy per symbol
symbol period
written in quadrature form as:
si(t) =
0  t Ts,
Basis Signal ?
2


 cos2f c t   Es sin  (i  1)
M



 sin 2f c t 

3
Mary Phase Shift Keying
Orthogonal basis signals
1(t) =
2(t) =
2
cos2f ct 
Ts
2
sin 2f ct 
Ts
defined over 0  t  Ts
MPSK signal can be expressed as
sMPSK(t) =






 Es cos (i  1) 1 (t )  Es sin  (i  1) 2 (t ) 
2
2




i = 1,2,…M
4
Mary Phase Shift Keying
• MPSK basis has 2 signals  2 dimensional constellation
• M-ary message points equally spaced on circle with radius Es
• MPSK is constant envelope when no pulse shaping is used
2(t)
MPSK signal can be
• coherently detected
• = Arctan(Y/X)
•Minimum | I -  |
• non-coherent detected
with differential encoding
1(t)
Es
 
2 Es sin 

M 
5
Mary Phase Shift Keying
Probability of symbol error in AWGN channel – using
distance between adjacent symbols as
 
2 Es sin  
M 
Pe = average symbol error probability in AWGN channel
Pe 
 2 Eb log 2 M
  
2Q
sin   
N0
 M 

When differentially encoded & non-coherently detected,
Pe estimated for M  4 as:
Pe 
2Q

4 Es
sin
2M
No
6
Power Spectrum of MPSK
Ts = Tblog2M
- Ts = symbol duration
- Tb = bit duration
Ps(f) = ¼ { Pg(f-fc) + Pg( -f-fc) }
PMPSK(f) =
PMPSK(f) =
Es
2
 sin  ( f  f )T
c s

  ( f  f c )Ts

Eb log 2 M
2
2
  sin  ( f  f c )Ts 
  

   ( f  f c )Ts 
 sin 2 ( f  f )T log M
c b
2


 2 ( f  f c )Tb log 2 M

2
2



  sin 2 ( f  f c )Tb log 2 M
  
  2 ( f  f c )Tb log 2 M



2



7
PSD for M = 8 & M = 16
0
rect pulses
RCF
-10
normalized PSD (dB)
-20
Increase in M with
Rb held constant
-30
• Bnull decreases
 B increases
• denser constellation
 higher BER
-40
-50
-60
fc-½Rb
fc-⅔Rb
fc-¼Rb
fc-⅓Rb
fc
fc+¼Rb
fc+⅓Rb
fc+½Rb
fc+⅔Rb
8
MPSK Bandwidth Efficiency vs Power Efficiency
M
2
4
8
16
2
B = Rb/Bnull 0.5 1.0 1.5
Eb/N0 (dB) 10.5 10.5 14.0 18.5
32
2.5
23.4
64
3
28.5
• B = bandwidth efficiency
• Rb = bit rate
• Bnull = 1st null bandwidth
• Eb/N0 for BER = 10-6
bandwidth efficiency & power efficiency assume
• Ideal Nyquist Pulse Shaping (RC filters)
• AWGN channel without timing jitter or fading
9
Mary Phase Shift Keying
Advantages:
Bandwidth efficiency increases with M
Drawbacks:
Jitter & fading cause large increase in BER as M increases
EMI & multipath alter instantaneous phase of signal
– cause error at detector
Receiver design also impacts BER
Power efficiency reduces for higher M
MPSK in mobile channels require Pilot Symbols or Equalization
10
Mary- Quadrature Amplitude Modulation
• allows amplitude & phase to vary
• general form of M-ary QAM signal given by
si(t) =
2Emin
2Emin


ai cos 2f ct 
bi sin 2f ct 
Ts
Ts
0  t  Ts
i = 1,2,…M
Emin = energy of signal with lowest amplitude
ai, bi = independent integers related to location of signal point
Ts = symbol period
• energy per symbol / distance between adj. symbols isn’t constant 
probability of correct symbol detection is not same for all symbols
• Pilot tones used to estimate channel effects
11
Mary- Quadrature Amplitude Modulation
Assuming rectangular pulses - basis functions given by
1(t) =
2
cos2f ct 
Ts
2(t) =
2
sin 2f ct 
Ts
0  t  Ts
0  t  Ts
QAM signal given by:
si(t) = Emin ai1(t) +
Emin
bi2(t)
0  t  Ts
i = 1,2,…M
coordinates of ith message point = ai Emin and bi Emin
(ai, bi) = element in L2 matrix, where L = M
12
Mary- Quadrature Amplitude Modulation
e.g. let M = 16, then {ai,bi} given based on
Emin
{ai,bi} =
ai1(t) +
Emin
bi2(t)
( 1,3)
(1,3)
(3,3) 
 ( 3,3)
 ( 3,1)
( 1,1)
(1,1)
(3,1) 


( 3,1) ( 1,1) (1,1) (3,1) 


(

3
,

3
)
(

1
,

3
)
(
1
,

3
)
(

3
,

3
)


s11(t) = -3 Emin 1(t) + 3 Emin 2(t)
0  t  Ts
s21(t) = -3 Emin 1(t) + Emin
0  t  Ts
2(t)
13
16 ary- Quadrature Amplitude Modulation
QAM: modulated signal is
hybrid of phase & amplitude
modulation
• each message point
corresponds to a quadbit
2(t)
1011
1001
1110
1111
1010
1000 0.5 1100
1101
-1.5
0001
-0.5 0 0.5
0000
0100
-0.5
1.5 1(t)
0110
0011
0010 -1.5 0101
0111
1.5
•Es is not constant – requires
linear channel
14
Mary- Quadrature Amplitude Modulation
In general, for any M = L2
Emin ai1(t) +
{ai,bi} =
Eminbi2(t)
 (  L  1, L  1) ( L  3, L  1)
 ( L  1, L  3) ( L  3, L  3)


...
...

( L  1, L  1) ( L  3, L  1)
... ( L  1, L  1) 
... ( L  1, L  3) 


...
...

... ( L  1, L  1)
15
Mary- Quadrature Amplitude Modulation
The average error probability, Pe for M-ary QAM is approximated by

Pe  41 

1  
Q
M  
2 Emin
N0




• assuming coherent detection
• AWGN channel
• no fading, timing jitter
In terms of average energy, Eav
Pe 
3Eav
1  

41 
Q
M   ( M  1) N 0





Power Spectrum & Bandwidth Efficiency of QAM = MPSK
Power Efficiency of QAM is better than MPSK
16
Mary- Quadrature Amplitude Modulation
M-ary QAM - Bandwidth Efficiency & Power Efficiency
• Assume Optimum RC filters in AWGN
• Does not consider fading, jitter, - overly optimistic
M
4
16
64
256
1024
4096
B = Rb/Bnull
1
2
3
4
5
6
15
18.5
24
28
33.5
Eb/N0 (BER = 10-6) 10.5
17
Mary Frequency Shift Keying
MFSK - transmitted signals defined as
si(t) =


2 Es

cos (nc  i)t 
Ts
 Ts

0  t Ts,
i = 1,2,…M
• fc = nc/2Ts
• nc = fixed integer
si(t) =


2Es
i
cos 2  f c 
Ts
2Ts


 
t 

 
0  t Ts,
i = 1,2,…M
Each of M signals have
• equal energy
• equal duration
• adjacent sub carrier frequencies separated by 1/2Ts Hz
• sub carriers are orthogonal to each other
18
Mary Frequency Shift Keying
MFSK coherent detection - optimum receiver
• receiver has bank of M correlators or matched filters
• each correlator tuned to 1 of M distinct carrier frequencies
• average probability of error, Pe (based on union bound)
Pe 

M  1Q

Eb log 2 M
N0




19
Mary Frequency Shift Keying
MFSK non-coherent detection
• using matched filters followed by envelope detectors
• average probability of error, Pe
Pe =
M 1  (1) k 1  M  1
 
k 1  k  1



k
  kEs 

 exp 

 (k  1) N 0 
bound Pe  use leading terms of binomial expansion
Pe 
  Es 
 M 1 


 exp 
 2 
 2 N0 
20
MFSK Channel Bandwidth
Coherent detection B =
Non-coherent detection
Rb ( M  3)
2 log 2 M
B=
Rb M
2 log 2 M
Impact of increasing M on MFSK performance
bandwidth efficiency (B) of MFSK decreases
• MFSK signals are bandwidth inefficient (unlike MPSK)
power efficiency (P) increases
• with M orthogonal signals  signal space is not crowded
• power efficient non-linear amplifiers can be used without
performance degradation
21
M-ary QAM - Bandwidth Efficiency & Power Efficiency
M
4
16
64
256
1024
4096
B = Rb/Bnull
1
2
3
4
5
6
15
18.5
24
28
33.5
Eb/N0 (BER = 10-6) 10.5
Coherent M-ary FSK - Bandwidth Efficiency & Power Efficiency
M
2
0.4
B = Rb/Bnull
Eb/N0 (BER = 10-6) 13.5
4
0.57
10.8
8
0.55
9.3
16
0.42
8.2
32
0.29
7.5
64
0.18
6.9
22
Summary of M-ary modulation in AWGN Channel
B
3
2.5
EB/N0
(BER = 10-6)
2
1.5
1
0.5
0
MPSK/QAM
Coherent MFSK
0 4 8
16
32
30
25
20
15
10
5
0
64
M
MPSK
QAM
Coherent MFSK
0 4 8
16
32
64 M
23
Shannon Limit:
• Most schemes are away from Eb/N0 of –1.6 dB by 4dB or more
• FEC helps to get closer to Shannon limit
• FSK allows exchange of BW efficiency for power efficiency
log10 Eb/ N0
15
9 BPSK
16 FSK
4 PSK/QAM
6
Error Free Region
-3
16 QAM
BFSK
12
4 FSK
-1.6dB
16 PSK
-2
3
-1
0
1
2
3
log2 C/B
24
Power & BW Efficiency
Bandwidth Efficiency
 EbC 
C

 log 2 1 
B =
B
 N0B 
Power Efficiency Eb/N0 = energy used by a bit for detection
25
S
Eb
B
( dB ) 
( dB )  ( dB )
N
N0
R
Power & BW Efficiency
if C ≤ B log2(1+ S/N)  error free communication is possible
if C > B log2(1+ S/N)  some errors will occur
• assumes only AWGN (ok if BW << channel center frequency)
• in practice < 3dB (50%) is feasible
S = EbC is the average signal power (measured @ receiver)
N = BN0 is the average noise power
Eb = STb is the average received bit energy at receiver
N0 = kT (Watts Hz–1) is the noise power density (Watts/Hz),
- thermal noise in 1Hz bandwidth in any transmission line
26