1. 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
(iii) a signal, si(t) , sent during each symbol period, Ts = n.Tb
(ii) each symbol mapped to signal si(t), M possible signals:
s1(t),…,sM(t)

2. 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

3. 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) =
0  t Ts, i = 1,2,…M
Es = log2MEb
energy per symbol
Ts = log2MTb
symbol period
written in quadrature form as:
si(t) =
for i = 1,2,…M
Basis Signal ? 2 Ts

4. Mary Phase Shift Keying
defined over 0  t  Ts
2(t) =
Orthogonal basis signals
sMPSK(t) =
i = 1,2,…M
MPSK signal can be expressed as

5. Mary Phase Shift Keying
MPSK basis has 2 signals  2 dimensional constellation
M-ary message points equally spaced on circle with radius
MPSK is constant envelope when no pulse shaping is used
2(t)
1(t)
MPSK signal can be
coherently detected
 = Arctan(Y/X)
Minimum | I -  |
non-coherent detected
with differential encoding

6. Mary Phase Shift Keying
Probability of symbol error in AWGN channel – using
distance between adjacent symbols as
Pe = average symbol error probability in AWGN channel
Pe 
When differentially encoded & non-coherently detected,
Pe estimated for M  4 as:
2 Q
4 Es No
sin  2M
Pe 

7. Power Spectrum of MPSK Ps(f) = ¼ { Pg(f-fc) + Pg( -f-fc) }
Ts = Tblog2M
- Ts = symbol duration
- Tb = bit duration
Ps(f) = ¼ { Pg(f-fc) + Pg( -f-fc) }
PMPSK(f) =
PMPSK(f) =

8. PSD for M = 8 & M = 16 rect pulses RCF Increase in M with
normalized PSD (dB)
fc-½Rb fc-¼Rb fc fc+¼Rb fc+½Rb
-10
-20
-30
-40
-50
-60
fc-⅔Rb fc-⅓Rb fc+⅓Rb fc+⅔Rb
PSD for M = 8 & M = 16
rect pulses
RCF
Increase in M with
Rb held constant
Bnull decreases
 B increases
denser constellation
 higher BER

9. MPSK Bandwidth Efficiency vs Power Efficiency2 4 8 16 32 64
B = Rb/Bnull
0.5
1.0
1.5
2.5 3
Eb/N0 (dB)
10.5
14.0
18.5
23.4
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

10. 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

11. Mary- Quadrature Amplitude Modulation
allows amplitude & phase to vary
general form of M-ary QAM signal given by
0  t  Ts i = 1,2,…M
si(t) =
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

12. Mary- Quadrature Amplitude Modulation
Assuming rectangular pulses - basis functions given by
1(t) =
0  t  Ts
2(t) =
QAM signal given by:
ai1(t) bi2(t)
si(t) =
0  t  Ts i = 1,2,…M
coordinates of ith message point =
and
(ai, bi) = element in L2 matrix, where L =

13. Mary- Quadrature Amplitude Modulation
e.g. let M = 16, then {ai,bi} given based on
ai1(t) bi2(t)
{ai,bi} =
1(t) 2(t)
s11(t) =
0  t  Ts
1(t) 2(t)
s21(t) = -3
0  t  Ts

14. 16 ary- Quadrature Amplitude Modulation
2(t)
1(t)
1.5
0.5
-0.5
-1.5
1011
1010
0001
0011
1001
1000
0000
0010
1110
1100
0100
0101
1111
1101
0110
0111
QAM: modulated signal is hybrid of phase & amplitude modulation
each message point
corresponds to a quadbit
Es is not constant – requires linear channel

15. Mary- Quadrature Amplitude Modulation
In general, for any M = L2
ai1(t) bi2(t)
{ai,bi} =

16. Mary- Quadrature Amplitude Modulation
The average error probability, Pe for M-ary QAM is approximated by
assuming coherent detection
AWGN channel
no fading, timing jitter
Pe 
In terms of average energy, Eav
Pe 
Power Spectrum & Bandwidth Efficiency of QAM = MPSK
Power Efficiency of QAM is better than MPSK

17. Mary- Quadrature Amplitude Modulation
M-ary QAM - Bandwidth Efficiency & Power Efficiency
Assume Optimum RC filters in AWGN
Does not consider fading, jitter, - overly optimistic 28 5
1024
33.5 24
18.5 15
10.5
Eb/N0 (BER = 10-6) 6 4 3 2 1
B = Rb/Bnull
4096
256 64 16 M

18. Mary Frequency Shift Keying
MFSK - transmitted signals defined as
si(t) =
0  t Ts, i = 1,2,…M
fc = nc/2Ts
nc = fixed integer
si(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

19. 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 

20. Mary Frequency Shift Keying
MFSK non-coherent detection
using matched filters followed by envelope detectors
average probability of error, Pe
Pe =
bound Pe  use leading terms of binomial expansion
Pe 

21. MFSK Channel Bandwidth
Coherent detection
B =
Non-coherent detection
B =
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

22. M-ary QAM - Bandwidth Efficiency & Power Efficiency28 5
1024
33.5 24
18.5 15
10.5
Eb/N0 (BER = 10-6) 6 4 3 2 1
B = Rb/Bnull
4096
256 64 16 M
Coherent M-ary FSK - Bandwidth Efficiency & Power Efficiency
7.5
0.29 32
6.9
8.2
9.3
10.8
13.5
Eb/N0 (BER = 10-6)
0.18
0.42
0.55
0.57
0.4
B = Rb/Bnull 64 16 8 4 2 M

23. Summary of M-ary modulation in AWGN ChannelB 3
2.5 2
1.5 1
0.5
MPSK/QAM
Coherent MFSK M 30 25 20 15 10 5
MPSK
QAM
Coherent MFSK
(BER = 10-6)
EB/N0

24. log10 Eb/ N0 16 PSK 15 12 16 QAM BFSK 9 6 4 FSK 4 PSK/QAM 3 BPSK
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
log2 C/B 15 12 9 6 3
Error Free Region
16 PSK
16 QAM
4 PSK/QAM
BPSK
4 FSK
16 FSK
BFSK
-1.6dB

25. Power & BW Efficiency Bandwidth Efficiency B =
Power Efficiency Eb/N0 = energy used by a bit for detection
B =
Bandwidth Efficiency

26. 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 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