This document provides an overview of digital modulation techniques including PAM, PSK, QAM, and FSK. PAM encodes information in signal amplitude, PSK in phase, and QAM combines both. FSK is a nonlinear technique that encodes in carrier frequency. Key parameters like energy, minimum distance, and bit rate are defined for each scheme. Equations show how to represent, expand, and calculate metrics for the different modulation types.

1. TELE4653 Digital Modulation &
Coding
Digital Modulation
Wei Zhang
w.zhang@unsw.edu.au
School of Electrical Engineering and Telecommunications
The University of New South Wales

2. Outline
PAM (ASK)
PSK
QAM
FSK
TELE4653 - Digital Modulation & Coding - Lecture 2. March 8, 2010. – p.1/2

3. Modulation
Source information to be transmitted is usually in the form of
a binary data stream.
The transmission medium, i.e., communication channel
suffers from noise, attenuation, distortion, fading, and
interference.
Digital Modulation - To generate a signal that represents the
binary data stream and matches the characteristics of the
channel
Modulation with Memoryless or with Memory
TELE4653 - Digital Modulation & Coding - Lecture 2. March 8, 2010. – p.2/2

4. Definitions
1
Signaling Interval: Ts . Signaling (Symbol) Rate: Rs = Ts .
Ts
Bit Interval: Tb = k for a signal carrying k bits of
information.
Bit Rate: R = kRs = Rs logM .
2
M
Average signal energy: Eavg = m=1 pm Em with pm being
the probability of the mth signal.
Eavg Eavg
Average energy per bit: Ebavg = k = logM
.
2
TELE4653 - Digital Modulation & Coding - Lecture 2. March 8, 2010. – p.3/2

5. PAM
The signal waveform may be represented as
sm (t) = Am p(t), 1 ≤ m ≤ M (1)
where p(t) is a pulse of duration T and Am denotes the
amplitude with the mth value, given by
Am = 2m − 1 − M, 1 ≤ m ≤ M (2)
i.e., the amplitudes are ±1, ±3, ±5, · · · , ±(M − 1).
Digital amplitude modulation is usually called amplitude-shift
keying (ASK).
TELE4653 - Digital Modulation & Coding - Lecture 2. March 8, 2010. – p.4/2

7. PAM
The energy of signal sm (t) is given by
∞
Em = A2 p2 (t)dt = A2 Ep
m m (3)
−∞
The average signal energy is
M M
Ep
Eavg = p m Em = A2
m
M
m=1 m=1
(M 2 − 1)Ep
= . (4)
3
and the average energy per bit is
(M 2 − 1)Ep
Ebavg = M
. (5)
3 log2
TELE4653 - Digital Modulation & Coding - Lecture 2. March 8, 2010. – p.6/2

8. Bandpass PAM
The bandpass PAM signals are carrier-modulated bandpass
signals with lowpass equivalents of the form s ml (t) = Am g(t),
where Am and g(t) are real. The signal waveform is
sm (t) = sml (t)ej2πfc t = Am g(t) cos(2πfc t) (6)
The energy of signal sm (t) is given by
A2
Em = m Eg . (7)
2
Moreover,
(M 2 −1)Eg (M 2 −1)Eg
Eavg = 6 , Ebavg = 6 logM
. (8)
2
TELE4653 - Digital Modulation & Coding - Lecture 2. March 8, 2010. – p.7/2

10. Bandpass PAM - Expansion
For basedband PAM, the expansion of sm (t) = Am p(t) is
sm (t) = Am Ep φ(t) (9)
where p(t)
φ(t) = . (10)
Ep
For bandpass PAM, the expansion of sm (t) = sml (t)ej2πfc t
is [Tutorial 1]
Eg
sm (t) = Am φ(t) (11)
2
where
2
φ(t) = g(t) cos(2πfc t). (12)
Eg
TELE4653 - Digital Modulation & Coding - Lecture 2. March 8, 2010. – p.9/2

11. Bandpass PAM - dmin
The bandpass PAM can be represented as the one-dimensional
E
vector: sm = Am 2g , where Am = ±1, ±3, · · · , ±(M − 1).
The Euclidean distance between any pair of signal points is
2
Eg
dmn = s m − sn = |Am − An | (13)
2
For adjacent signal points |Am − An | = 2, it has
12 log2 M
dmin = 2Eg = E
2 − 1 bavg
(14)
M
where in the last equality Eq. (8) is used.
TELE4653 - Digital Modulation & Coding - Lecture 2. March 8, 2010. – p.10/2

12. Phase Modulation
The bandpass PM signal waveform may be represented as
2π(m−1)
j
sm (t) = g(t)e M ej2πfc t , 1 ≤ m ≤ M
2π(m − 1)
= g(t) cos + 2πfc t (15)
M
2π(m−1)
Let θm = M , m = 1, 2, · · · , M . Then,
sm (t) = g(t) cos θm cos(2πfc t) − g(t) sin θm sin(2πfc t). (16)
Digital phase modulation is usually called phase-shift keying
(PSK).
The PSK signals have equal energy, Eavg = Em = 1 Eg .
2
TELE4653 - Digital Modulation & Coding - Lecture 2. March 8, 2010. – p.11/2

14. Phase Modulation - Expansion
2π(m−1)
j
The expansion of PM signal s(t) = g(t)e M ej2πfc t is
Eg Eg ˜
sm (t) = cos(θm )φ(t) + sin(θm )φ(t) (17)
2 2
where [see Tutorial 1]
2
φ(t) = g(t) cos(2πfc t) (18)
Eg
˜ 2
φ(t) = − g(t) sin(2πfc t) (19)
Eg
TELE4653 - Digital Modulation & Coding - Lecture 2. March 8, 2010. – p.13/2

15. Phase Modulation - dmin
The bandpass PM can be represented as the two-dimensional
Eg Eg
vector: sm = 2 cos θm , 2 sin θm , m = 1, 2, · · · , M . Note
θm = 2π(m−1) for m = 1, 2, · · · , M .
M
The Euclidean distance between any pair of signal points is
2
Eg
dmn = sm − sn = | cos θm − cos θn |2 + | sin θm − sin θn |2
2
= Eg [1 − cos(θm − θn )]. (20)
TELE4653 - Digital Modulation & Coding - Lecture 2. March 8, 2010. – p.14/2

16. Phase Modulation - dmin
For adjacent signal points |m − n| = 1, it has
2π
dmin = Eg 1 − cos( ) (21)
M
2 π
= 2Eg sin (22)
M
2 π
= 2 Ebavg log2 M × sin (23)
M
π π
For large values of M , we have sin M ≈ M, and then
π2
dmin ≈ 2 Ebavg log2 M × 2 (24)
M
TELE4653 - Digital Modulation & Coding - Lecture 2. March 8, 2010. – p.15/2

17. QAM
The quadrature amplitude modulation (QAM) signal waveform
may be expressed as
sm (t) = (Ami + jAmq )g(t)ej2πfc t , m = 1, 2, · · · , M
= Ami g(t) cos(2πfc t) − Amq g(t) sin(2πfc t), (25)
where Ami and Amq are the information-bearing signal
amplitudes of the quadrature carriers and g(t) is the signal pulse.
TELE4653 - Digital Modulation & Coding - Lecture 2. March 8, 2010. – p.16/2

18. QAM
Alternatively, the QAM signal may be expressed as
sm (t) = rm g(t)ejθm ej2πfc t
= rm g(t) cos(θm + 2πfc t), (26)
where rm = A2 + A2 and θm = tan−1 (Amq /Ami ).
mi mq
It is apparent that QAM signal can be viewed as combined
amplitude rm and phase θm modulation.
TELE4653 - Digital Modulation & Coding - Lecture 2. March 8, 2010. – p.17/2

20. QAM - Expansion
˜
Similar to PSK case, φ(t) in (18) and φ(t) in (19) can be used as
orthonormal basis for expansion of QAM signals [Tutorial 1]
Eg Eg ˜
sm (t) = Ami φ(t) + Amq φ(t) (27)
2 2
which results in vector representation of the form
Eg Eg
sm = (sm1 , sm2 ) = Ami , Amq (28)
2 2
and
2 Eg
Em = s m = A2 + A 2 .
mi mq (29)
2
TELE4653 - Digital Modulation & Coding - Lecture 2. March 8, 2010. – p.19/2

21. QAM - dmin
The Euclidean distance between any pair of signal vectors in
QAM is
dmn = s m − sn 2
Eg
= [(Ami − Ani )2 + (Amq − Anq )2 ]. (30)
2
In the case when the signal amplitude take values of
√
±1, ±3, · · · , ±( M − 1) on both Ami and Amq , the signal space
diagram is rectangular, as shown in Fig. on next page. In this
case,
dmin = 2Eg . (31)
TELE4653 - Digital Modulation & Coding - Lecture 2. March 8, 2010. – p.20/2

23. Square QAM
In the case of square QAM (i,e., M = 4, 16, 64, 256, · · · ) with
√
amplitudes of ±1, ±3, · · · , ±( M − 1)
√ √
M M
1 Eg
Eavg = (A2 + A2 )
m n
M 2
m=1 n=1
Eg 2M (M − 1) M −1
= × = Eg (32)
2M 3 3
and
M −1
Ebavg = Eg (33)
3 log2 M
6 log2 M
dmin = Ebavg (34)
M −1
TELE4653 - Digital Modulation & Coding - Lecture 2. March 8, 2010. – p.22/2

24. PAM, PSK, QAM
For bandpass PAM, PSK, and QAM, the signaling schemes are
of the general form
sm (t) = Am g(t)ej2πfc t , m = 1, 2, · · · , M (35)
For PAM, Am is real, equal to ±1, ±3, · · · , ±(M − 1)
j 2π (m−1)
For PSK, Am is complex and equal to e M
For QAM, Am = Ami + jAmq .
Therefore, PAM and PSK can be considered as special cases of
QAM.
TELE4653 - Digital Modulation & Coding - Lecture 2. March 8, 2010. – p.23/2

25. FSK
The FSK signal waveform is
sm (t) = sml (t)ej2πfc t , 1 ≤ m ≤ M, 0 ≤ t ≤ T
2E
= cos (2πfc t + 2πm∆f t) (36)
T
2E j2πm∆f t
where sml (t) = T e , 1 ≤ m ≤ M and 0 ≤ t ≤ T .
FSK signaling is a nonlinear modulation scheme, whereas ASK,
PSK, and QAM are linear modulation schemes.
TELE4653 - Digital Modulation & Coding - Lecture 2. March 8, 2010. – p.24/2

26. FSK
FSK is an orthogonal signaling if [ sml (t), snl (t) ] = 0, m = n.
T
2E
sml (t), snl (t) = ej2π(m−n)∆f t dt (37)
T 0
and
[ sml (t), snl (t) ] = 2Esinc(2T (m − n)∆f ) (38)
FSK is an orthogonal signaling when ∆f = k/2T . The minimum
frequency separation ∆f that guarantees orthogonality is
1
∆f = 2T .
TELE4653 - Digital Modulation & Coding - Lecture 2. March 8, 2010. – p.25/2