1. Geometric Representation of Modulation Signals
Digital Modulation involves
Choosing a particular signal waveform for transmission for a particular symbol
For M possible symbols, the set of all signal waveforms are:
For binary modulation, each bit is mapped to a signal from a signal set S that has two signals.
We can view the elements of S as points in vector space.

2. Geometric Representation of Modulation Signals
Vector space
We can represent the elements of S as linear combination of basis signals i (t).
The number of basis signals is the dimension of the vector space.
Basis signals are orthogonal to each-other.
Each basis is normalized to have unit energy.

3. Geometric Representation of Modulation Signals
Let {j(t)| j = 1,2,…,N} represent a basis of S such that
(1) Any symbol, si(t) 
si(t)=
(2) Basis signals are orthogonal to each other in time
(3) Each basis signal is normalized to have unit energy
E =
Basis signals  Coordinate system for S
Gram-Schmidt process  systematic way to obtain basis for S

4. Example Two signal waveforms to be used for transmission
The basis signal Q I
One dimensional
Constellation Diagram

5. QPSK Constellation Diagram
Rotation by /4 obtain new QPSK signal set 
3/2
/2 Q I
M1 = Q I
/4
54
7/4
3/4
M2 =
Es = 2Eb

6. Signal Space Characterization of QPSK Signal Constellations
π/4 00
3π/4 01
5π/4 11
7π/4 10
si2
si1
grey coded
QPSK signal
binary symbol 00
π/2 01 π 11
3π/2 10
si2
si1
grey coded
QPSK signal
binary symbol
ith QPSK signal, based on message points (si1, si2) defined in tables
si(t) = si1,1(t) + si22(t)
for i = 1,2 and 0 ≤ t ≤ Ts

7. /4 QPSK modulation Q = possible states for k for k-1 = n/4
modulated signal selected from 2 QPSK constellations shifted by /4
for each symbol  switch between constellations –total of 8 symbols
states 4 used alternately
phase shift between each symbol = nk = /4 , n = 1,2,3
- ensures minimal phase shift, k = /4 between successive symbols
- enables timing recovery & synchronization Q I
possible signal transitions
= possible states for k for k-1 = n/4
= possible states for k for k-1 = n/2

8. Constellation Diagram
Properties of Modulation Scheme can be inferred from the Constellation Diagram:
Bandwidth occupied by the modulation increases as the dimension of the modulated signal increases.
Bandwidth occupied by the modulation decreases as the signal_points per dimension increases (getting more dense).
Probability of bit error is proportional to the distance between the closest points in the constellation.
Euclidean Distance
Bit error decreases as the distance increases (sparse).

9. Linear Modulation Techniques
Digital modulation techniques classified as:
Linear
The amplitude of the transmitted signal varies linearly with the modulating digital signal, m(t).
They usually do not have constant envelope.
More spectrally efficient.
Poor power efficiency
Example: QPSK.
Non-linear / Constant Envelope

10. Constant Envelope Modulation
constant carrier amplitude - regardless of variations in m(t)
Better immunity to fluctuations due to fading.
Better random noise immunity.
improved power efficiency without degrading occupied spectrum
- use power efficient class C amplifiers (non-linear)
low out of band radiation (-60dB to -70dB)
use limiter-discriminator detection
- simplified receiver design
high immunity against random FM noise & fluctuations
from Rayleigh Fading
larger occupied bandwidth than linear modulation

11. Constant Envelope Modulation
Frequency Shift Keying
Minimum Shift Keying
Gaussian Minimum Shift Keying

12. Frequency Shift Keying (FSK)
Binary FSK
Frequency of the constant amplitude carrier is changed according to the message state  high (1) or low (0)
Discontinuous / Continuous Phase

13. Discontinuous Phase FSK
Switching between 2 independent oscillators for binary 1 & 0
switch
cos w2t
cos w1t
input data
phase jumps
sBFSK(t)= vH(t)
binary 1 =
binary 0
sBFSK(t)= vL(t)
results in phase discontinuities
discontinuities causes spectral spreading & spurious transmission
not suited for tightly designed systems

14. Continuous Phase FSK sBFSK(t) =
single carrier that is frequency modulated using m(t)
sBFSK(t) = =
where (t) =
m(t) = discontinuous bit stream
(t) = continuous phase function proportional to integral of m(t)

15. FSK Example 0 1 1 x a0 VCO modulated composite a1 signal 1 cos wct
Data
FSK
Signal
VCO
cos wct x 1 a0 a1
modulated composite
signal

16. Spectrum & Bandwidth of BFSK Signals
complex envelope of BFSK is nonlinear function of m(t)
spectrum evaluation - difficult - performed using actual time
averaged measurements
PSD of BFSK consists of discrete frequency components at fc
fc  nf , n is an integer
PSD decay rate (inversely proportional to spectrum)
PSD decay rate for CP-BFSK 
PSD decay rate for non CP-BFSK 
f = frequency offset from fc

17. Spectrum & Bandwidth of BFSK Signals
Transmission Bandwidth of BFSK Signals (from Carson’s Rule)
B = bandwidth of digital baseband signal
BT = transmission bandwidth of BFSK signal
BT = 2f +2B
assume 1st null bandwidth used for digital signal, B
- bandwidth for rectangular pulses is given by B = Rb
- bandwidth of BFSK using rectangular pulse becomes
BT = 2(f + Rb)
if RC pulse shaping used, bandwidth reduced to:
BT = 2f +(1+) Rb

18. ? ? General FSK signal and orthogonality
Two FSK signals, VH(t) and VL(t) are orthogonal if ?
interference between VH(t) and VL(t) will average to 0 during
demodulation and integration of received symbol
received signal will contain VH(t) and VL(t)
demodulation of VH(t) results in (VH(t) + VL(t))VH(t) ?

19. vH(t) = vL(t) = = vH(t) vL(t) = then An FSK signal for 0 ≤ t ≤ Tb and
vH(t) vL(t) are orthogonal if Δf sin(4πfcTb) = -fc(sin(4πΔf Tb)

20. CPFSK Modulation
elimination of phase discontinuity improves
spectral efficiency & noise performance
consider binary CPFSK signal defined over the interval 0 ≤ t ≤ T
s(t) =
θ(t) = phase of CPFSK signal
θ(t) is continuous  s(t) is continuous at bit switching times
θ(t) increases/decreases linearly with t during T
θ(t) = θ(0) ±
‘+’ corresponds to ‘1’ symbol
‘-’ corresponds to ‘0’ symbol
h = deviation ratio of CPFSK
0 ≤ t ≤ T

21. To determine fc and h by substitution
2πfct + θ(0) +
= 2πf2 t+ θ(0)
2πfct + θ(0) -
= 2πf1t+ θ(0)
f1 =
f2 =
yields
and
thus
fc=
h = T(f2 – f1)
nominal fc = mean of f1 and f2
h ≡ f2 – f1 normalized by T

22. FSK modulation index = kFSK (similar to FM modulation index)
symbol ‘1’  θ(T) - θ(0) = πh
symbol ‘0’  θ(T) - θ(0) = -πh
θ(T) = θ(0) ± πh
At t = T 
‘1’ sent  increases phase of s(t) by πh
‘0’ sent  decreases phase of s(t) by πh
variation of θ(t) with t follows a path consisting of straight lines
slope of lines represent changes in frequency
FSK modulation index = kFSK (similar to FM modulation index)
kFSK =
peak frequency deviation F = |fc-fi | =

23. Phase Tree  depicted from t = 0 phase transitions across
interval boundaries of
incoming bit sequence
θ(t) - θ(0) = phase of CPFSK signal is even or odd multiple of πh at even or odd multiples of T
θ(t) - (0) rads
3πh
2πh πh
-πh
-2πh
-3πh
0 T 2T 3T 4T 5T 6T t

24.  thus change in phase over T is either π or -π
Phase Tree is a manifestation of phase continuity – an inherent characteristic of CPFSK
0 ≤ t ≤ T
θ(t) = θ(0) ±
θ(t) - (0) 3π 2π π -π
-2π
-3π
0 T 2T 3T 4T 5T 6T t
 thus change in phase over T
is either π or -π
change in phase of π = change in phase of -π
e.g. knowing value of bit i doesn’t help to find the value of bit i+1

25. CPFSK = continuous phase FSK
phase continuity during inter-bit switching times
si(t) =
0 ≤ t ≤ T
= 0 otherwise
for i = 1, 2
assume fi given by as
fi =
nc = fixed integer
si(t) =
0 ≤ t ≤ T
for i = 1, 2
= 0 otherwise

26. BFSK constellation: define two coordinates as
for i = 1, 2
i(t) =
0 ≤ t ≤ T
= 0 otherwise
let nc = 2 and T = 1s (1Mbps) then f1 = 3MHz, f2 = 4MHz
1(t) =
0 ≤ t ≤ T
= 0 otherwise
2(t) =

27. BFSK Constellation s1(t) = 0 ≤ t ≤ T = = 0 otherwise 0 ≤ t ≤ T s2(t) =
BFSK Constellation
s1(t) =
0 ≤ t ≤ T
= 0 otherwise =
0 ≤ t ≤ T
s2(t) =
= 0 otherwise =

28. Coherent BFSK Detector
2 correlators fed with local coherent reference signals
difference in correlator outputs compared with threshold to
determine binary value
output + -
r(t)
Decision
Circuit
sin wct
cos wct 
Pe,BFSK =
Probability of error in coherent FSK receiver given as:

29. Non-coherent Detection of BFSK
operates in noisy channel without coherent carrier reference
pair of matched filters followed by envelope detector
- upper path filter matched to fH (binary 1)
- lower path filter matched to fL (binary 0)
envelope detector output sampled at kTb  compared to threshold
r(t)
output
Decision
Circuit + - 
Envelope
Detector
Matched Filter fL Tb fH
Average probability of error in non-coherent FSK receiver:
Pe,BFSK, NC =