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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)
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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
Combined Linear & Constant Envelope Modulation
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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 = log2MEb
energy per symbol
Ts = log2MTb
symbol period
written in quadrature form as:
si(t) =
for i = 1,2,…M
2
Ts
Basis Signal ?
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1(t) =
defined over 0 t Ts
2(t) =
Orthogonal basis signals
sMPSK(t) =
i = 1,2,…M
MPSK signal can be expressed as
Mary Phase Shift Keying
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• 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
Mary Phase Shift Keying
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Probability of symbol error in AWGN channel – using
distance between adjacent symbols as
Pe
Pe
When differentially encoded & non-coherently detected,
Pe estimated for M 4 as:
Pe = average symbol error probability in AWGN channel
Mary Phase Shift Keying
2 Q
4 Es
No
sin
2M
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Power Spectrum of MPSK
Ts = Tblog2M
- Ts = symbol duration
- Tb = bit duration
PMPSK(f) =
PMPSK(f) =
Ps(f) = ¼ { Pg(f-fc) + Pg( -f-fc) }
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Increase in M with
Rb held constant
• Bnull decreases
B increases
• denser constellation
higher BER
normalized PSD (dB)
fc-½Rb fc-¼Rb fc fc+¼Rb fc+½Rb
0
-10
-20
-30
-40
-50
-60
fc-⅔Rb fc-⅓Rb fc+⅓Rb fc+⅔Rb
PSD for M = 8 & M = 16
rect pulses
RCF
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M
2
4
8
16
32
64
B = Rb/Bnull
0.5
1.0
1.5
2
2.5
3
Eb/N0 (dB)
10.5
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
MPSK Bandwidth Efficiency vs Power Efficiency
bandwidth efficiency & power efficiency assume
• Ideal Nyquist Pulse Shaping (RC filters)
• AWGN channel without timing jitter or fading
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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
Mary Phase Shift Keying
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• allows amplitude & phase to vary
• general form of M-ary QAM signal given by
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
0 t Ts i = 1,2,…M
si(t) =
Mary- Quadrature Amplitude Modulation
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Assuming rectangular pulses - basis functions given by
1(t) =
0 t Ts
2(t) =
0 t Ts
(ai, bi) = element in L2 matrix, where L =
coordinates of ith message point =
and
ai1(t) + bi2(t)
si(t) =
0 t Ts i = 1,2,…M
QAM signal given by:
Mary- Quadrature Amplitude Modulation
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{ai,bi} =
e.g. let M = 16, then {ai,bi} given based on
ai1(t) + bi2(t)
1(t) + 2(t)
s11(t) = -3 3
0 t Ts
1(t) + 2(t)
s21(t) = -3
0 t Ts
Mary- Quadrature Amplitude Modulation
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QAM: modulated signal ishybrid of phase & amplitudemodulation
• each message point
corresponds to a quadbit
•Es is not constant – requireslinear channel
-1.5 -0.5 0.5 1.5
2(t)
1(t)
1.5
0.5
0
-0.5
-1.5
1011
1010
0001
0011
1001
1000
0000
0010
1110
1100
0100
0101
1111
1101
0110
0111
16 ary- Quadrature Amplitude Modulation
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ai1(t) + bi2(t)
{ai,bi} =
In general, for any M = L2
Mary- Quadrature Amplitude Modulation
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Pe
In terms of average energy, Eav
Power Spectrum & Bandwidth Efficiency of QAM = MPSK
Power Efficiency of QAM is better than MPSK
The average error probability, Pe for M-ary QAM is approximated by
Pe
• assuming coherent detection
• AWGN channel
• no fading, timing jitter
Mary- Quadrature Amplitude Modulation
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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
4
M
M-ary QAM - Bandwidth Efficiency & Power Efficiency
• Assume Optimum RC filters in AWGN
• Does not consider fading, jitter, - overly optimistic
Mary- Quadrature Amplitude Modulation
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MFSK - transmitted signals defined as
0 t Ts, i = 1,2,…M
si(t) =
• fc = nc/2Ts
• nc = fixed integer
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
0 t Ts, i = 1,2,…M
si(t) =
Mary Frequency Shift Keying
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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
Mary Frequency Shift Keying
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MFSK non-coherent detection
• using matched filters followed by envelope detectors
• average probability of error, Pe
Pe =
Pe
bound Pe use leading terms of binomial expansion
Mary Frequency Shift Keying
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MFSK Channel Bandwidth
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
Non-coherent detection
B =
22
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
Coherent M-ary FSK - Bandwidth Efficiency & Power Efficiency
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
4
M
M-ary QAM - Bandwidth Efficiency & Power Efficiency
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Summary of M-ary modulation in AWGN Channel
0 4 8 16 32 64 M
B
3
2.5
2
1.5
1
0.5
0
MPSK/QAM
Coherent MFSK
0 4 8 16 32 64 M
30
25
20
15
10
5
0
MPSK
QAM
Coherent MFSK
EB/N0
(BER = 10-6)
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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
-3 -2 -1 0 1 2 3 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
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Power Efficiency Eb/N0 = energy used by a bit for detection
B =
Bandwidth Efficiency
Power & BW Efficiency
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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 = EbC is the average signal power (measured @ receiver)
N = BN0 is the average noise power
Eb = STb is the average received bit energy at receiver
N0 = kT (Watts Hz–1) is the noise power density (Watts/Hz),
- thermal noise in 1Hz bandwidth in any transmission line
Power & BW Efficiency
