M-ary PSK and QAM

 Practically Used Digital Carrier Schemes 

Combinations:

(a)Multi-level Phase Shift Keying: M-ary PSK: Digital TV

Transmission (DTH, Set top boxes etc.) and Satellite
communication
(b)M-ary Phase Shift Keying (M-PSK) + Amplitude Shift Keying
(ASK) : Digital Quadrature Amplitude Modulation (QAM): ADSL
Modems, Cable modems, Digital TV Transmission and satellite
communication
(c)Frequency Shift Keying (FSK) with special restrictions: Gaussian
Minimum Shift Keying (GMSK): GSM voice communication in
cellular mobile telephony
 M-ary Phase Shift Keying (M-PSK):
Multilevel PSK: BW Efficient PSK
 Select M (No. of phases or symbols) = 2n
 n=1, M=2: Binary PSK (BPSK) or simply PSK : Only two
different phases of carrier (0 and π) used to modulate 1 and 0
respectively (single bit).  One bit/symbol
 n = 2, M=4 is called Quadrature PSK (QPSK): Four different
Phases of carrier used to modulate two successive bits. 
Two bits/symbol
 n=3, M=8 is called 8-PSK where 8 different phases of carrier
used to modulate three successive bits (Three bits/symbol)
and so on.
 In M-ary PSK successive phase angles differ by 2π/M  n-
bits/symbol and M is number of symbols (phases)
 Bandwidth requirement can be improved by using M-ary
coding techniques by a factor of n.
 Quadrature PSK (QPSK) Modulator
 Needs 2-bits of information signal simultaneously to modulate it
with sinusoidal carrier with one of the four possible phase angles.

s (t ) = I (t ) cos(ωc t ) + Q(t ) sin(ωc t )

 I(t) and Q(t) =+1V if bit is at logic ‘1’, I(t) and Q(t) =-1V if bit is at
logic ‘0’.
 QPSK Phasor or Constellation Diagram:
QPSK signals ( symbols ) :
3π 3π  5π 
0 0 → −1 − 1 → s (t ) = 2 cos(ω c t + )→  or − 
4 4  4 
5π 5π  3π  
0 1 → −1 1 → s (t ) = 2 cos(ω c t + )→  or −  
4 4  4  
 Four Symbols; differs in phase( separation π / 2)
π π 7π  
1 0 → 1 − 1 → s (t ) = 2 cos(ω c t + )→
 or − 
4 4 4  

7π 7π  π 
1 1 → 1 1 → s (t ) = 2 cos(ω c t + )→  or − 
4 4  4 


 BW required for QPSK = 2Rb /2 =Rb

= Half the bandwidth required for
BPSK
 However detection error
performance degrades compared
to BPSK
 QPSK Demodulator:

 Analysis: output of upper LPF = I(t)/2

 output of lower LPF = Q(t)/2
 Increase in M will improve BW requirement further but detection becomes
more difficult and error in detection increases as the successive phase
differences become narrower.
 General M-ary PSK equation
∗ M= 2n; M =No. of point (symbols) in constellation, n= no. of bits/symbol, Transmission BW
required BT = 2Rb/n = 2Rsym; Symbol rate Rsym = Rb/n; Symbol period Tsym = nTb
s (t ) = a mψ 1 (t ) + bmψ 2 (t );
a m = A cos(θ m ), bm = − A sin(θ m ); a m + bm = A 2 = const.
2 2

2π
θm = θ0 + ( m − 1); m = 1,2,..., M ; θ 0 = Re f . phase
M
ψ 1 (t ) and ψ 2 (t ) are orthogonal carrier signal at same carrier frequency f c
They are also orthonormal over [0, Tb ] ( Energy in ψ 1 (t ) and ψ 2 (t ) is unity over period Tb )
2 2
ψ 1 (t ) = cos( 2πf c t ), ψ 2 (t ) = sin( 2πf c t )
Tb Tb
2 2
s (t ) = A cos(θ m ) cos( 2πf c t ) − A sin(θ m ) sin( 2πf c t ); kTs ym ≤ t < ( k + 1)Ts ym
Tb Tb
2
= A cos( 2πf c t + θ m ); kTs ym ≤ t < ( k + 1)Ts ym ; k int eger
Tb
Further Improvement in Transmission
Bandwidth using M-ary Techniques
∗ Each of the three digital modulation techniques
(ASK,FSK or PSK) can be refined (M-ary) to send more
than one bit at a time. It is possible to send two bits on
one wave by defining four different phases(QPSK).
∗ This technique could be further refined to send three
bits at the same time by defining 8 different phases (8-
PSK) or four bits by defining 16 phases (16-PSK) etc.
The same approach can be used for amplitude and
frequency shift keying.
∗ In practice, the maximum number of bits that can be
sent with any one of these techniques is about five bits
(limited due to detection error). For further
improvement the solution is to combine modulation
techniques.
 Combination of M-ary PSK with ASK:
M-ary QAM
∗ Popular technique called Quadrature Amplitude Modulation
(QAM) involves splitting the signal into different phases, and
each phase with different amplitudes.
∗ If in QPSK we use two different amplitudes for each phase
modulated carrier than it results in 8 different combinations of
phase and amplitudes (8 different symbols): called 8-QAM.
Here 3 bits are modulated simultaneously (3 bits/symbol). Each
symbol differs with the other in either amplitude or phase or
both.
∗ A variety of forms of QAM are available and some of the more
common forms include 16 QAM, 32 QAM, 64 QAM, 128 QAM,
and 256 QAM.
∗ M-ary QAM is generalization of M-ary PSK with amplitude not
same for all symbols.
∗ M-ary PSK and M-ary QAM share a common property: Both are
examples of linear modulation (like DSBSC), BW requirement
improved by a factor of n, M number of points (symbols) in
constellation diagram.
 8-QAM Modulator

∗ 4-Phases* 2 Amplitudes = 8 combinations

 8-QAM: Truth Table and Phasor
(Constellation) Diagram
I Q C 8-QAM Symbol: s(t) Ampl Phas
itude e
0 0 0 0.765cos(ωct+3π/4) 0.765 3π/4
0 0 1 1.848cos(ωct+3π/4) 1.848 3π/4
0 1 0 0.765cos(ωct+π/4) 0.765 π/4
0 1 1 1.848cos(ωct+π/4) 1.848 π/4
1 0 0 0.765cos(ωct+5π/4) 0.765 5π/4
1 0 1 1.848cos(ωct+5π/4) 1.848 5π/4
1 1 0 0.765cos(ωct+7π/4) 0.765 7π/4
1 1 1 1.848cos(ωct+7π/4) 1.848 7π/4
 8 QAM Demodulator:

∗ 2-bit ADC: Reverse of 2-bit DAC at the receiver

 Constellation Diagram of 16-QAM, 32-
QAM and 64-QAM

MODULATION BITS PER SYMBOL MIN. BW

BPSK 1 2Rb
QPSK 2 Rb
8PSK 3 2Rb/3
8QAM 3 2Rb/3
16QAM 4 Rb/2
32QAM 5 2Rb/5
64QAM 6 Rb/3
 General M-ary QAM equation
∗ It is modification of M-ary PSK equation where amplitudes
are not same for all symbols and also select M = 22n to get
complete square constellation diagram.

s(t ) = a mψ 1 (t ) + bmψ 2 (t ); m = 1,2,..., M

a m = rm cos(θ m ), bm = −rm sin(θ m )
a m + bm = rm .
2 2 2

bm
θ m = − tan −1 ( )
am
ψ 1 (t ) and ψ 2 (t ) are orthogonal carrier signal at same carrier frequency f c
They are also orthonormal over [0, Tb ]
2 2
ψ 1 (t ) = cos(2πf c t ), ψ 2 (t ) = sin(2πf c t )
Tb Tb
2 2
s(t ) = a m cos(2πf c t ) + bm sin(2πf c t ); kTsym ≤ t < (k + 1)Tsym
Tb Tb
2
= rm cos(2πf c t + θ m ); kTsym ≤ t < (k + 1)Tsym
Tb
General M-ary QAM generation and
detection for M = 22n
 Performance Analysis of Digital
Communication systems in presence
of additive noise
 Performance Metrics (Measures)
∗ Digital Communication Systems
∗ Metrics are data rate (Rb bps) and probability of bit error
∗ Error is introduced when symbol ‘0’ is detected as ‘1’ and vice-versa.
∗ This error occurs when the channel noise is higher than decision
threshold.
∗ Pb is a theoretical expectation of Bit Error Rate (BER) for a given
system
∗ BER is a measured (historical) record of a system’s bit error
performance.
∗ BER is measured and then compared with Pb to evaluate system’s
performance
∗ Pb is a function of ratio of Energy per bit (Eb) to the noise power
spectral density (N0) i.e. Eb/N0
∗ Eb/N0 : SNR per bit
∗ Without channel noise/distortion/sync. problem, we will never make bit
errors.
Overview: The Gaussian Random Variable (R.V.)
∗ Let X is R.V. having Gaussian Probability distribution function
pX(x) with zero mean and unity variance.
∗ X is called standard Gaussian or Normal R.V.
 Results of Noise Analysis of Baseband
Systems (No carrier modulation):
∗ Assuming channel contributes additive white Gaussian Noise
(AWGN) of zero mean and with uniform noise PSD N0 /2
∗ The expression for the bit error probability of a binary data
communication system is given by:

Polar Signaling

Unipolar (ON-OFF)
Signaling

For M-ary PAM: Symbol Error Rate:

Polar Signaling
Results of Noise Analysis of Digital Carrier
Modulation Schemes:
∗ Binary ASK (coherent detection):

∗ Binary FSK (coherent detection):

∗ Binary PSK (BPSK):

∗ Binary MSK (coherent detection):

∗ Binary ASK (Non- coherent detection):

∗ Binary FSK (Non- coherent detection):

∗ Binary DPSK:
 Comparison of Noise Performance of M-ary PSK
Systems
• The bit error probability for all the systems
decrease monotonically with increasing values
of Eb/No.
• For high value of Eb/No BPSK and QPSK, produce
a smaller bit error probability than the other
systems.
• QPSK and BPSK provides almost same error
performance because the reduction in error
distance for QPSK is offset by the decrease in its
bandwidth.
• Both systems provide optimum performance..
• The QPSK system transmits, in a given
bandwidth, twice as many bits of information as
a conventional coherent BPSK system with the
same error rate performance.
• However, a QPSK system requires a more
sophisticated carrier recovery circuit than a
BPSK system.
• For higher level PSKs (≥8), the noise
performance degrades.
 Comparison of Noise Performance of QAM Systems
∗ For a large number of signal
points (i.e., M>4), QAM
outperforms PSK.
∗ This is because the distance
between signaling points in a
PSK system is smaller than the
distance between points in a
comparable QAM system.