CHAPTER 2

REVIEW OF DIGITAL MODULATION

SCHEMES

2.1 Introduction

The main distinction between pulse modulation and CW modulation is that, in CW

modulation some parameters of the modulated wave vary continuously with the mes-
sage. In pulse modulation some parameters of each pulse are modulated by a partic-
ular sample value of the message. Usually the pulses are quite short compared to the
time between them. So a pulse modulated wave is off most of the time. Because of
this property, pulse modulation offers two potential advantages over CW

1. The transmitted power can be concentrated into short bursts rather than being
delivered continuously.

2. The time interval between pulses can be filled with sample values from other
messages, thereby permitting the transmission of many messages on one com-
munication system.

The other difference between pulse and and CW modulation is the pulse wave
may contain appreciable DC and low frequency content. Thus pulse modulation
is a message processing technique rather than modulation in the usual sense since
Communication Systems I, First Edition. 9
By Osama A. Alkishriwo Copyright c 2019 John Wiley & Sons, Inc.
10 REVIEW OF DIGITAL MODULATION SCHEMES

efficient transmission entails a CW modulation to provide complete frequency trans-

lation. There are two basic types of pulse modulation
1. Analog pulse modulation.
2. Digital or coded pulse modulation.
Regardless of pulse modulation type, the key operation for pulsed communication
is extracting sample values from the message waveform. Thus, sampling theorem
plays important role in these systems. The sampling theorem states that
If a bandlimited message signal is sampled instantaneously at regular intervals
and at a rate of at least twice the highest significant message frequency, then the
samples contain all of the information of the original message.

2.2 Analog Pulse Modulation

If a message is adequately described by its sample values, it can be transmitted via

analog pulse modulation in which the sample values modulate a periodic pulse train
with one pulse for each sample. There are three types of analog pulse modulation
1. Pulse amplitude modulation PAM.
2. Pulse width (duration or length) modulation PWM (PDM or PLM).
3. Pulse position modulation PPM.
PWM and PPM are called pulse time modulation PTM. Figure 2.1 shows typical
message and its corresponding pulse modulated waveforms.

Figure 2.1 Analog pulse modulation signals.

In general pulse modulated waves have appreciable DC and low frequency con-
tent. Direct transmission may therefore be difficult. Hence ,most pulse systems have
a carrier modulation step in which the pulses are converted to radio frequency pulses.
Figure 2.2 shows a complete pulse transmission system.
 DIGITAL PULSE MODULATION 11

(a)

(b)

Figure 2.2 Pulse modulation system: (a) Transmitter, (b) Receiver.

The transmitted signal is mathematically represented as

v(t) = Ac mp (t) cos(ωc t) (2.1)

where fc >> fs and mp (t) is pulse modulated signal, v(t) is a DSB and therefore
envelope detection can be employed at the receiver if mp (t) ≥ 0 and no carrier phase
reversal.
For pulse resolution, the required baseband bandwidth is at least 1/2τ , where τ
is the nominal pulse duration. The practical advantage of pulse modulation depends
on the pulse duration being small compared to the time interval between pulses, Ts ,
i.e. τ << Ts ≤ 1/2W , where W is the maximum frequency content of the message
signal. Thus the baseband transmission bandwidth B is given as
1
B≥ >> W (2.2)
2τ
and the transmission bandwidth for carrier modulation is
1
BT = 2B ≥ >> 2W (2.3)
τ

2.3 Digital Pulse Modulation

In digital pulse modulation, the message signal is represented using a coded group
of digital (discrete amplitude) pulses. The techniques used to generate such digital
pulses are: Pulse code modulation (PCM), delta modulation (DM), and differential
pulse code modulation (DPCM).

2.3.1 Pulse Code Modulation (PCM)

The elements of PCM generation are shown in Fig. 2.3.
The parameters of the encoder are n and µ where n is the number of digital pulses
in the code word, and µ is the number of possible amplitude levels of each pulse.
12 REVIEW OF DIGITAL MODULATION SCHEMES

Figure 2.3 PCM generation system.

They are related to the number of quantizer levels Q as given

Q = µn or n = logµ (Q) (2.4)

The most common form of PCM is binary PCM for which µ = 2, and the number
of quantizer levels is some powers of 2, that is Q = 2n . Because several digits are
required for each message sample, the PCM bandwidth will be much greater than
the message bandwidth. The PCM bandwidth can written as
nfs
B≥ = nW = W logµ (Q) (2.5)
2
This means that the baseband PCM bandwidth is at least n times the message band-
width W . PCM is not susceptible to noise as CW systems. But the quantization
noise is the basic limitation of PCM systems which can be reduced by increasing the
number of quantizer levels Q. Thus, the signal–to–quantization noise power for the
case of uniform quantization is given by

SNR = 20 n log10 (µ) dB (2.6)

for the case of binary PCM (µ = 2), Eq. 2.6 is written as

SNR = 6 n dB (2.7)

Thus the SNR increases by increasing n, at the same time the bandwidth, will
also increase. However, the SNR increases more rapidly than the bandwidth, giving
trade–off relationship between them. Of course if the bandwidth is too large, the
error rate at the demodulator output will be increased because of channel noise for a
specified signal level.

2.3.2 Delta Modulation (DM)

Delta modulation is a technique where an analog signal can be encoded directly into
binary digits. It is the simplest method for converting an analog signal to a digital
form. In exchange of equipment saving, DM generally requires a larger transmission
bandwidth than PCM. Figure 2.4 shows the block diagram of DM transmitter and
receiver.
The limitation of DM is slope overload which occurs when the signal changes too
rapidly such that the approximation m̃(t) cannot follow the signal m(t). That is the
 DIGITAL PULSE MODULATION 13

(a)

(b)

Figure 2.4 Delta modulation system: (a) Modulator, (b) Demodulator.

slope of m(t) is more than the slope of m̃(t). The sufficient condition for no slope
overload is
2πW A
∆fs ≥ 2πW A or fs > (2.8)
∆
where W is the message bandwidth and A is the maximum amplitude of the signal
m(t). We can find the transmission bandwidth for DM system as

fs
B≥ (2.9)
2
The mean square value of the quantization noise in DM is ∆2 /3. It is often as-
sumed that the power spectral density of the quantization noise is that of bandlimited
white noise, i.e. it is constant up to frequency fs and zero beyond that. Then, the
signal–to–quantization noise is given by:
 3
3 fs
SNR = (2.10)
8π 2 W

2.3.3 Differential Pulse Code Modulation (DPCM)

In this system the difference between m(t) and m̃(t) shown in DM system is quan-
tized and encoded. m̃(t) is generated by decoding the signal and applied to an inte-
grator. The m̃(t) has a variable step size ranging from ±∆ ± Q∆/2, where Q is the
14 REVIEW OF DIGITAL MODULATION SCHEMES

(a)

(b)

Figure 2.5 DPCM system: (a) Modulator, (b) Demodulator.

number of quantizer levels, so it more accurately follows m(t). If Q = 2, the DPCM

reduces to DM. The modulator and demodulators are shown in Figs. 2.5(a) and (b).
Clearly DPCM with Q > 2 requires equipments just as complex as conventional
PCM. In return, it offers potential transmission bandwidth reduction. This follows
since the difference signal m(t) − m̃(t) is adequately represented with fewer quan-
tizer levels, if m(t) does not change drastically from sample to sample.

2.3.4 Comparison between PCM and DM

1. Signal–to–Noise Ratio
If the channel signal–to–noise ratio is high, then the performance of PCM and
DM is limited by the quantization noise. For the same bandwidth, the perfor-
mance of DM is always worse than PCM. However the performance of DM
can be considerably improved by using a variable step size. For speech trans-
mission, it has been found that there is little difference in the performance of
adaptive DM and PCM systems operating at a bit rate of about 64 kbps. The
overall SNR of a DM system is also lower than the overall SNR of PCM system
using the same bandwidth.

2. Bandwidth requirements
With the use of PCM, speech transmission is found to be of good quality when
f s = 8 kHz and n = 8. The corresponding bit rate is 64 kbps. To obtain
comparable quality using DM, the sampling rate has been shown to be about
100 kHz. However, it has been later shown that with continuous variable slope
DM it is possible to achieve good signal quality at about 32 kbps.
 DIGITAL PULSE MODULATION 15

3. Equipment complexity
The hardware required to implement DM is much simpler than that required
for implementing PCM. Single integrated circuit chips (continuously-variable
step DM) called CODECS are rapidly becoming available. In comparison PCM
coder/decoders require two chips for implementation: one for processing the
analog signal and the second for encoding the sampled analog signal. Thus the
PCM hardware is more expensive than DM hardware.

2.3.5 Error probability of binary digital pulses

The basic elements of a digital baseband receiver is shown in Fig. 2.6.

Figure 2.6 Baseband digital receiver.

The additive noise is white Gaussian with two sided power spectral density η/2
and polar binary signal is assumed, i.e. the transmitted 10 s and 00 s are represented by
+A and −A respectively. The error probability is given by
 
A
Pe = Q (2.11)
σ
√ R∞
where Q(x) = 1/ 2π x exp(−y 2 /2) dy.
The same expression of the error probability is obtained for the case of unipolar
binary system in which a 10 and 00 are represented by 2A and zero volt respectively.
The difference between the polar and unipolar binary systems is in the required av-
erage transmitted power.
The above expression of the error probability can be written in terms of the signal–
to–noise ratio at the filter output or in terms of the average energy per bit.

S Sav A2
= = 2 for polar signaling (2.12)
N ηB σ

and

S Sav 2A2
= = 2 for unipolar signaling (2.13)
N ηB σ

Thus,
r !
Sav
Pe = Q , for polar signaling (2.14)
N
16 REVIEW OF DIGITAL MODULATION SCHEMES

and
r !
Sav
Pe = Q , for unipolar signaling (2.15)
2N

Now if the filter bandwidth is taken B = rb = 1/Tb , where Tb is the bit duration,
then N = ηB = ηrb . Therefore the error probability can be rewritten as

s !
Eb
Pe = Q , for polar signaling (2.16)
η

and
s !
Eb
Pe = Q , for unipolar signaling (2.17)
2η

where Eb = Sav Tb .

2.3.6 M-Ary Signaling Scheme

In M-ary scheme, the output of the pulse generator takes one of M possible levels
(M > 2). Each level corresponds to a distinct input symbol, where M is the available
distinct input symbols. If the symbols in the input sequence are equiprobable and
statistically independent, and rs is the symbol signaling rate, then the information
rate at the output of the pulse generator is rs log2 (M ) bits/sec. Thus, each pulse
contains log2 (M ) bits of information. Figure 2.7 shows a block diagram of the
transmitter and receiver of M-ary signaling scheme.
The minimum required bandwidth for the M-ary scheme is
1 rs
B≥ = (2.18)
2Ts 2

2.3.7 Error Probability in M-ary Signaling

Assuming polar M-ary with M even, and 2A is the spacing between adjacent pulse
amplitude levels, i.e. the pulse amplitude is Ak = ±A, ± 3A, · · · , ± (M − 1)A.
Assuming equiprobable symbols so p(Ak ) = 1/M , and assuming the added noise is
white Gaussian noise of power spectral density of η/2 watt/Hz, the error probability
is
   
M −1 A
Pe = 2 Q (2.19)
M σ
But the signal–to–noise ratio at the input to the A/D converter is
 2
S Sav M 2 − 1 A2 M2 − 1 A
= = = (2.20)
N ηB 3 ηB 3 σ
 DIGITAL PULSE MODULATION 17

(a)

(b)

Figure 2.7 M-ary signaling scheme: (a) Transmitter, (b) Receiver.

Then the error probability is given by

  s !
M −1 3 Sav
Pe = 2 Q (2.21)
M 2(M 2 − 1) N

where Sav is the average power per symbol. The error probability can also be given
in terms of the average energy per symbol, Es = Sav Ts for B = rs as
  s !
M −1 3 Es
Pe = 2 Q (2.22)
M 2(M 2 − 1) η

2.3.8 Comparison of Bipolar Binary and M-ary Signaling Schemes

The binary and M-ary signaling are compared on the following bases
1. Both are required to give the same error probability.
2. The additive noise is white Gaussian noise with power spectral density η/2 and
the channel is ideal low pass.
3. Both schemes use the same pulse shape.
4. The input to both systems is assumed to be produced from an ergodic informa-
tion source which emits binary digits which are independent and equally likely
at a rate of rb bits/sec.
5. Each block of n binary digits is translated to one of M levels at the transmitter
(i.e. M = 2n ).
18 REVIEW OF DIGITAL MODULATION SCHEMES

The following observations can be made

1. Binary transmission has lower power requirements than M-ary scheme. For
M >> 2 and Pe << 1, the required transmitted power for the M-ary scheme
is M 2 log2 (M )/3 times that of binary scheme.
2. The M-ary schemes require less bandwidth than binary case and for M >> 2
and Pe << 1, the required bandwidth for the M-ary case is reduced by a factor
1/ log2 (M ) compared to that of the binary scheme.
3. M-ary schemes are more complex since the receiver has to decide on one of the
M levels using M − 1 comparators. In the binary case the decoding requires
only one comparator.

2.4 Digital Carrier Modulation Schemes

In this section, we will review the digital modulation schemes, namely amplitude
shift keying (ASK), frequency shift keying (FSK), and phase shift keying (PSK).
Figure 2.8 shows these different modulation waveforms for transmitting binary in-
formation over a bandpass channel,

Figure 2.8 Digital carrier modulation waveforms.

The transmitted waveform is either s1 (t) or s2 (t) depending on either a space

(zero) or a mark (one) is transmitted respectively. The waveforms s1 (t) and s2 (t)
have a duration of Tb and have a finite energy, i.e.
Z Tb
E1 = s21 (t) dt < ∞ (2.23)
0

Z Tb
E1 = s22 (t) dt < ∞ (2.24)
0
 DIGITAL CARRIER MODULATION SCHEMES 19

A block diagram of a binary data transmission scheme using digital modulation

is shown in Fig. 2.9.

Figure 2.9 Binary data transmission system.

In our analysis the following assumptions are made:

1. The transmission channel is assumed distortionless.
2. The channel noise is a white Gaussian noise.
3. s1 (t) and s2 (t) are equiprobable.
4. There is no intersymbol interference.

2.4.1 Detection of Binary Digital Modulation Schemes

The performance of the receiver is measured in terms of the probability of error. The
receiver that gives minimum error probability is said to be optimum.
The optimum receiver takes the form of a matched filter which can be imple-
mented as an integrate and dump correlation receiver. It is a coherent or synchronous
receiver which requires a local carrier reference having the same frequency and phase
as the transmitted carrier. Fig. 2.10(a) and (b) shows a matched filter and an equiv-
alent correlation receiver. These receivers will maximize the signal to noise ratio, γ,
and consequently result in a minimum error probability. The maximum, γ, is given
by:
Z ∞
|P (f )|2
Z
2
γm = df = Tb p2 (t) dt (2.25)
−∞ Gn (f ) η 0

where P (f ) = F {p(t)}, p(t) = s2 (t) − s1 (t), and Gn (f ) = η/2 is the P.S.D. of the
noise.

2.4.2 Amplitude Shift Keying (ASK)

The transmitted signal can be represented by

v(t) = A mp (t) cos(ωc t) (2.26)

20 REVIEW OF DIGITAL MODULATION SCHEMES

(a)

(b)

Figure 2.10 (a) Matched filter receiver, (b) Correlation receiver.

where mp (t) is the baseband binary signal which is equal to one or zero within a bit
duration Tb . The transmission bandwidth BT is given as

BT = 2B (2.27)

where B is the baseband bandwidth of mp (t) which depends on the binary sequence
of mp (t). In general

k
B= (2.28)
Tb
where k ≥ 1 and it depends on the pulse waveforms. k = 1 is usually used.
The error probability for coherent reception of ASK signal is
s ! s !
A2 Tb Eb
Pe = Q =Q (2.29)
4η η

where Eb = Sav Tb and Sav = A2 /4.

2.4.3 Frequency Shift Keying

In this scheme the transmitted waveforms are

s1 (t) = A cos[(ωc − ∆ω)t] (2.30)

s2 (t) = A cos[(ωc + ∆ω)t] (2.31)

 DIGITAL CARRIER MODULATION SCHEMES 21

within the bit duration Tb , representing space and mark respectively. The transmis-
sion bandwidth of the FSK signal is
BT = 2(∆f + B) (2.32)
where ∆f = ∆ω/2π, and B = krb is the baseband bandwidth as before.
(a) Coherent FSK
In this scheme a coherent receiver as shown in Fig. 2.10(b) is used. Local
carrier signals s1 (t) = A cos[(ωc − ∆ω)t] and s2 (t) = A cos[(ωc + ∆ω)t] are
generated at the receiver. The error probability is given by
s !
A2 Tb

sin(2∆ω Tb )
Pe = Q 1− (2.33)
2η 2∆ω Tb

The largest value is obtained if 2∆ω Tb = 3π/2. Thus, if ∆ω is selected to

satisfy this relation, then
s ! s !
0.61 A2 Tb 1.2 Eb
Pe = Q =Q (2.34)
η η

where Eb = Sav Tb and Sav = A2 /2.

(b) Noncoherent FSK
In this scheme, a noncoherent receiver shown in Fig. 2.11 is used.

Figure 2.11 Noncoherent detection of FSK.

The error probability is given by

A2
 
1
Pe = exp − (2.35)
2 4No
where No is the noise power at the filter output No = η BT . If BT = 2rb , then
No = 2η rb and
A2 Tb
   
1 1 Eb
Pe = exp − = exp − (2.36)
2 8η 2 4η
where Eb = Sav Tb = A2 Tb /2.
22 REVIEW OF DIGITAL MODULATION SCHEMES

2.4.4 Phase Shift Keying (PSK)

In PSK scheme the transmitted waveforms are
s1 (t) = −A cos(ωc t) (2.37)

s2 (t) = A cos(ωc t) (2.38)

The transmission bandwidth of the PSK signal is BT = 2B, where B = krb is the
baseband bandwidth as before.
A correlator receiver shown in Fig. 2.12 is used to recover the transmitted bit se-
quence. A local reference signal s2 (t) − s1 (t) = 2A cos(ωc t) which is synchronized
in frequency and phase with the incoming signal.

Figure 2.12 Correlator receiver.

The error probability is given by

s ! s !
A2 Tb 2Eb
Pe = Q =Q (2.39)
η η

where Eb = Sav Tb and Sav = A2 /2.

2.4.5 Differentially Coherent PSK (DPSK)

The DPSK scheme can be thought of as a noncoherent version of the PSK scheme.
Fig. 2.13(a) and (b) are the modulator and demodulator of DPSK signal. The differ-
ential encoding operation performed by the modulator is explained in Fig. 2.14. The
encoding process starts with an arbitrary first bit, say 1, and thereafter the encoded
bit stream dk is generated by

dk = dk−1 bk ⊕ d¯k−1 b̄k (2.40)

The transmission bandwidth is the same as PSK. The probability of error can be
shown to be equal to
A2 Tb
   
1 1 Eb
Pe = exp − = exp − (2.41)
2 2η 2 η
 DIGITAL CARRIER MODULATION SCHEMES 23

(a)

(b)

Figure 2.13 (a) DPSK modulator, (b) DPSK demodulator.

Figure 2.14 Differential encoding and decoding.

2.4.6 Minimum Shift Keying (MSK)

Minimum shift keying is a special case of continuous phase frequency shift keying
where the peak frequency deviation ∆f = rb /4 = 1/4Tb . This means the two sig-
naling frequencies are fc + 1/4Tb and fc − 1/4Tb . This corresponds to the minimum
frequency spacing that allows two FSK signals to be coherently orthogonal. The
name minimum shift keying implies the minimum frequency spacing.
MSK is sometimes called fast FSK, as the frequency spacing used is only half of
that used in conventional noncoherent FSK. MSK is also equivalent to QPSK. It is
represented as
     
2πt 2πt
sM SK (t) = A be (t) sin cos(ωo t) + A bo (t) cos sin(ωo t)(2.42)
4Tb 4Tb
where be (t) is the even bit stream of the baseband data stream b(t). It consists of
alternate bits b2 , b4 , b6 , · · · and bo (t) is the odd bit stream of b(t) consisting of al-
ternate bits b1 , b3 , b5 , · · · where each bit in both streams is held for two bit inter-
vals Ts = 2Tb . The waveforms sin(2πt/4Tb ) and cos(2πt/4Tb ) are chosen to pass
through zero precisely at the end of the signal time in be (t) and bo (t), respectively.
24 REVIEW OF DIGITAL MODULATION SCHEMES

The spectrum of MSK has a main lobe which is 1.5 times as wide as the main
lobe of QPSK, while the side lobes in MSK are relatively much smaller compared to
the main lobe making filtering much easier.
A block diagram of the MSK transmitter and receiver is shown in Fig. 2.15

(a)

(b)

Figure 2.15 (a) MSK transmitter, (b) MSK receiver.

2.4.7 Gaussian Minimum Shift Keying (GMSK)

GMSK can be viewed as a derivative of minimum shift keying (MSK). It is a binary
modulation scheme which used in digital mobile cellular systems such as global
system for mobile (GSM).
The simplest way to generate GMSK is to pass a NRZ message bit stream through
a Gaussian baseband filter followed by an FM modulator or MSK transmitter as
shown in Fig. 2.16.
The baseband Gaussian pulse shaping filter smoothes the phase trajectory of the
MSK signal and hence stabilizes the instantaneous frequency variation over time.
The GMSK premodulation filter has an impulse response given by
√
π 2 t2
 
π
h(t) = exp − 2 (2.43)
α α
 DIGITAL CARRIER MODULATION SCHEMES 25

Figure 2.16 GMSK transmitter using direct FM generation.

and the transfer function given by

H(f ) = exp(−α2 f 2 ) (2.44)

The parameter α is related to the 3 dB baseband bandwidth B of H(f ) as

p
ln(2) 0.5887
α= √ = (2.45)
2B B

The GMSK filter is defined from B and the baseband symbol duration T . Thus, it
is customary to define GMSK by its BT product. As the BT product decreases the
side lobe levels of the power spectrum for GMSK signal fall of rapidly as shown in
Fig. 2.17. The BT product of infinity is equivalent to MSK spectrum.

Figure 2.17 Normalised frequency from carrier, (f − fc )T .

For a given BT product, the occupied RF bandwidth containing a given percent-

age of power in a GMSK signal can be obtained. Table 2.1 shows occupied RF
bandwidth as a fraction of channel data rate rb containing a given percentage of
power in GMSK and MSK for different values of BT product.
GMSK signals can be detected using orthogonal coherent detector as shown in
Fig. 2.18(a) or with simple noncoherent detector such as standard FM discriminator
as shown in Fig. 2.18 (b).
26 REVIEW OF DIGITAL MODULATION SCHEMES

Table 2.1 Occupied RF bandwidth as a fraction of rb for different BT product and percentage
of power included.
BT % of power included

90% 99% 99.9% 99.99%

0.2 GMSK 0.52 0.79 0.99 1.22
0.25 GMSK 0.57 0.86 1.09 1.37
0.5 GMSK 0.69 1.04 1.33 2.08
MSK 0.78 1.2 2.76 6

LPF
Modulated
RF input Demodulated
IF local Decision binary signal
signal
𝜋 2 oscillator device

LPF

(a)

Modulated IF Sampler Demodulated Binary

input FM Decision signal
BPF
discriminator device

(b)

Figure 2.18 (a) Coherent detector, (b) Noncoherent detector.

The bit error rate for GMSK for a white Gaussian noise channel is given as
s !
2δEb
Pe = Q (2.46)
η

where
(
0.68, for GMSK with BT = 0.25
δ= (2.47)
0.85, for MSK (BT = ∞)

2.5 Comparison of Digital Modulation Systems

The choice of a modulation method depends on the specific application. It may de-
pend on the simplicity of equipment and compatibility with other equipment already
in use, or on the relative immunity to noise and channel impairments.
 M-ARY SIGNALING SCHEMES 27

(a) Bandwidth requirements

The bandwidth of PSK and ASK is of the order of 2rb whereas for FSK it is
somewhat larger than 2rb . Thus if the bandwidth is of primary concern, the
FSK is generally not considered.

(b) Power requirements

The probability of error versus the peak power is given in Fig. 2.19. The error
probability in most practical systems is in the range of 10−4 to 10−7 . It is
clear that PSK scheme requires the least amount of power followed by DPSK,
coherent FSK, coherent ASK, noncoherent FSK, and noncoherent ASK. The
power requirement of DPSK is about 1 dB more than PSK, and NFSK requires
about 7 dB more power than PSK. Of course the saving of power by 1 dB in a
large communication network results in saving of millions of dollars a year.

Figure 2.19 Probability of error for binary digital modulation schemes. (Note that the
average signal power for ASK schemes is A2 /4, whereas it is A2 /2 for other schemes).

(c) Equipment Complexity

The coherent systems are more complex than the noncoherent systems. NFSK
is less complex than the DPSK system but practically both are chosen from the
simplicity point of view.

(d) Immunity to channel impairments

If the channel has fading then noncoherent systems are preferred compared to
coherent systems which require local reference signal. If the transmitter has
serious power limitation then PSK or DPSK is preferred because it requires less
power for a given error probability.

2.6 M-ary Signaling Schemes

M-ary signaling schemes can be used in conjuction with digital carrier modulation
techniques. That is one of M signals s1 (t), s2 (t), · · ·, sM (t) is sent during each
28 REVIEW OF DIGITAL MODULATION SCHEMES

signaling interval Ts . These signals are generated by changing the amplitude, fre-
quency, or phase of a carrier in M discrete steps. Thus resulting in M-ary ASK,
M-ary FSK, M-ary APK, and M-ary PSK digital modulation schemes. In general
M-ary signaling schemes are preferred over binary schemes for transmitting digital
information over bandpass channel, when conservation of bandwidth (at the expense
of increasing power) is required or when conservation of power (at the expense of
increasing bandwidth) is required. M-ary PSK are widely used because of its con-
servation of bandwidth.

2.6.1 M-ary Phase Shift Keying (MPSK)

We assume that the input to the modulator is an independent sequence of equiproba-
ble binary digits, and the modulator takes blocks of k binary digits and assign one of
M possible waveforms to each block (M = 2k ).
The phase of the carrier is allowed to take one of M possible values φm =
2πm/M , (m = 0, 1, · · · , M − 1). Thus during each interval Ts , the M possible
signals that can be transmitted are
 
2πm
sm (t) = A cos ωc t + , m = 0, 1, · · · , M − 1 and 0 ≤ t ≤ Ts (2.48)
M
If the binary sequence to be transmitted has a bit rate of rb then the bandwidth re-
quired using binary PSK schemes is about 2rb . Now if a block of k bits are taken and
using M-ary signaling scheme with M = 2k and Ts = kTb or rs = rb /k, the band-
width required will be of the order of BT = 2rs = 2rb /k . Thus the M-ary PSK
signaling scheme offers a reduction in bandwidth by a factor of k over the binary
PSK scheme.
The coherent demodulation schemes are used to demodulate the M-ary PSK. The
probability of error in an optimum M-ary PSK signaling scheme is approximated by
s !
2Es 2
π
Pe = 2 Q sin , M ≥4 (2.49)
η M

where the signal to noise ratio at the receiver input is large.

For the case M = 4 is called four phase PSK or quadrature PSK (QPSK). The
transmitted wave forms during each interval Ts are one of the following
s1 (t) = A cos(ωc t)
s2 (t) = A cos(ωc t + π/2) = −A sin(ωc t)
s3 (t) = A cos(ωc t + π) = −A cos(ωc t), for 0 ≤ t ≤ Ts (2.50)
s4 (t) = A cos(ωc t + 3π/2) = A sin(ωc t)
The error probability is given by
s !
A2 Ts
Pe = 2 Q (2.51)
2η
 M-ARY SIGNALING SCHEMES 29

In QPSK two binary bits are represented by one of the four signals. That is s1 (t)
represents 00, and s2 (t) represents 01, and so on.

2.6.2 Quadrature Amplitude Modulation (QAM)

QAM sometimes is called amplitude phase keying (MAPK). QAM is multilevel
quadrature carrier system similar to that described for QPSK except that different
amplitudes are included in the signal constellation as shown in Fig. 2.20.

Figure 2.20 QAM constellations: (a) 4-QAM, (b) 16-QAM, and (c) 64-QAM.

The general form of an M-ary QAM signal can be written as

r r
2Emin 2Emin
si (t) = ai cos(2πfc t) + bi sin(2πfc t) (2.52)
Ts Ts

where i = 1, 2, · · · , M − 1, 0 ≤ t ≤ Ts , Emin is the energy of the signal with the

lowest amplitude, and {ai , bi } are a pair of independent integers chosen according
the location of a particular signal point. It is important to note that QAM doesn’t
have constant energy per symbol, nor constant distances between possible symbol
states. √ √
The coordinates of the ith message point are at ai Emin and bi Emin where
{ai , bi } is an element of L × L matrix given by

 
(−L + 1, L − 1) (−L + 3, L − 1) ··· (L − 1, L − 1)
 (−L + 1, L − 3) (−L + 3, L − 3) ··· (L − 1, L − 3)
 

{ai , bi } =  .. .. .. .. (2.53)
.
 
 . . . 
(−L + 1, −L + 1) (−L + 3, −L + 1) ··· (L − 1, −L + 1)
√
where L = M .
For example, the L × L of 16-QAM is given as
30 REVIEW OF DIGITAL MODULATION SCHEMES

 
(−3, 3) (−1, 3) (1, 3) (3, 3)
 
 (−3, 1) (−1, 1) (1, 1)
(3, 1) 
{ai , bi } = 
(−3, −1)
 (2.54)
 (−1, −1) (1, −1) (3, −1) 

(−3, −3) (−1, −3) (1, −3) (3, −3)

A generalized block diagram of the QAM modulator is shown in Fig. 2.21. The
2 − L level converter (VI,i = ai and VQ,i =√bi ) generates L level signals having a
symbol rate Rs = R/ log2 (M ), where L = M and M is the number of levels of
the QAM.

(a)

(b)

Figure 2.21 M-QAM: (a) Modulator, (b) Demodulator.

The transmission bandwidth is given by

R
BT = 2B = 2k (2.55)
log2 (M )
 SPREAD SPECTRUM MODULATION 31

where B is the LPF bandwidth and k ≥ 1/2.

A simple approximation for the probability of symbol error for MQAM (M even)
signalling in Gaussian noise is
√ ! s ! √ ! r !
M −1 3 Es M −1 6 Sav
Ps = 4 √ Q =4 √ Q (2.56)
M M −1 η M M −1 N

where (Es = Sav Ts ) is the average energy per QAM symbol and N = ηBT is the
noise power in the RF bandwidth.

2.7 Spread Spectrum Modulation

Spread spectrum (SS) is a technique in which a modulated signal is modulated a

second time such that the resulting waveform produce negligible interference on
other signal operating on the same frequency band. The resulting waveform is a
very wide band signal.
The main two spread spectrum techniques are
1. Direct sequence spread spectrum (DSSS)
2. Frequency hopping spread spectrum (FHSS)
A brief explanation of each technique is given below.

2.7.1 Direct Sequence Spread Spectrum

1. Basic Principles
A direct sequence spread spectrum signal is a technique in which the amplitude
of an already modulated signal (e.g. binary PSK or QPSK) is amplitude mod-
ulated by a very high rate NRZ binary stream of digits. Thus, if the original
signal is a binary PSK signal s(t) given by
p
s(t) = 2Ps d(t) cos(ωo t) (2.57)

where d(t) is the data sequence bit rate rb and Ps is the average power of s(t).
Thus, the DSSS signal is
p
v(t) = g(t) s(t) = 2Ps g(t) d(t) cos(ωo t) (2.58)

where g(t) is a pseudo-random noise (PN) binary sequence having values ±1. It
is generated in a deterministic repetitive manner. However, the sequence length
before repetition is usually extremely long and it is assumed to be random which
implies there is no correlation between the value of a given bit and the value of
any other bits. Also, the bit rate rc of g(t) is much greater than the bit rate rb of
d(t). The bit rate of g(t) is called chip rate rc to distinguish it from the data bit
rate.
32 REVIEW OF DIGITAL MODULATION SCHEMES

Since the bandwidth of BPSK signal s(t) is 2rb , then the bandwidth of the
BPSK spread spectrum signal v(t) is 2rc and the spectrum has been spread by
the ratio rc /rb .
If the power transmitted by s(t) and v(t) is the same and equal to Ps , then the
power spectral density Gs (f ) is reduced by the factor rb /rc . Figure 2.22 shows
BPSK system transmitter and receiver with spread spectrum technique.

𝑑(𝑡) 𝑣(𝑡) 𝑣1 (𝑡) 𝑣𝑜′ (𝑡) 𝑣𝑜 (𝑡)

× × Channel × × ∫

2𝑃𝑠 cos(𝜔𝑜 𝑡) 𝑔(𝑡) 𝑔(𝑡) 2 cos(𝜔𝑜 𝑡)

Transmitter Receiver

Figure 2.22 BPSK system with spread spectrum technique.

0
At the receiver, vo (t) is given by

0 p √
vo (t) = 2Ps g(t) g(t) d(t) 2 cos2 (ωo t)
p
= Ps d(t) [1 + cos(2ωo t)]

where we use g 2 (t) = 1.

As it is usual practice to have bit duration to be multiplies of the half periods of
the carrier period 1/fo , hence the integrator output vo (t) is
p
vo (t) = Ps d(t) (2.59)

It can be shown that the statistical properties of the noise are not affected by
the spread spectrum technique, so the overall performance of the system is not
affected. Thus, the probability of error is the same as that of BPSK without
spread spectrum i.e.
s !
2Eb
Pe = Q (2.60)
η

where Eb is bit energy and η/2 is the two sided power spectral density of the
noise.
2. Signal Tone Interference p
Assume a sinusoidal signal of power Pj and of carrier frequency fo , 2Pj cos(ωo t+
 SPREAD SPECTRUM MODULATION 33

θ) interferes with DSSS signal. It can be shown that the output of the receiver
vo (t) for the case of no receiver noise n(t) is given by
p p
vo (t) = Ps d(t) + Pj g(t) cos(θ) (2.61)

The power spectral density of the interfering signal is

2
Pj E{cos2 (θ)}

sin(πf /rc )
Gj (f ) = (2.62)
2rc πf /rc

Since it can be verified that an integrator with integration period Tb is approxi-

mately equivalent to a low pass filter with cutoff frequency rb = 1/Tb and since
rb << rc , then in the frequency range ±rb the power spectral density can be
approximated by

Pj E{cos2 (θ)}
Gj (f ) = , |f | ≤ rb (2.63)
2rc

Now since we consider the interfering tone signal is applied to the channel
instead of the noise n(t), the noise power spectral density η/2 at the output of
the integrate and dump filter is to be replaced by Gj (f ) obtained in Eq. (2.63).
Thus, the probability of error is given by
s !
2Eb rc
Pe = Q
Pj E{cos2 (θ)}
s !
Ps rc 2
= Q (2.64)
Pj rb E{cos2 (θ)}

The angle θ is the phase of Jamming sinusoidal waveform with respect to carrier
information signal and it is random variable so E{cos2 (θ)} = 1/2, then
q  q 
Pe = Q 4Ps rc
Pj rb = Q 2Ps
Pje (2.65)

where Pje = Pj /2(rc /rb ) and is called effective Jamming power. The ratio
rc /rb measures the extent to which the effect of the mean jamming power Pj /2
is reduced by the chipping. It is called processing gain

Gp = rc /rb (2.66)

3. Spread Spectrum Applications in Code Division Multiple Access (CDMA)

In CDMA each user is given an individual and distinctive pseudo noise (PN)
code. These codes are assumed uncorrelated with one another. To illustrate the
principle of CDMA, assume that at a given time, each of k users is transmit-
ting data at the same carrier frequency fo , using DS spread spectrum and his
34 REVIEW OF DIGITAL MODULATION SCHEMES

particular code is gi (t). Then, each receiver is presented with the same input
waveform.
k p
X
v(t) = 2Ps gi (t) di (t) cos(ωo t + θi ) (2.67)
i=1

where each signal is assumed to present the same power Ps to the receiver. This
can be achieved using power control technique. Each pseudo random sequence
gi (t) has the same chip rate rc , and di (t) is data transmitted by user i, and data
rate for each user is assumed the same rb , θi is a random phase and statistically
independent for different users. Thermal noise is omitted for simplification and
at the end its power spectral density η/2 can be added to the interference power
spectral density.
Figures 2.23 (a) and (b) show the transmitter and the receiver block diagrams,
respectively.

𝑑1 (𝑡) × ×
2𝑃𝑠 cos(𝜔𝑜 𝑡 + 𝜃1 ) 𝑔1 (𝑡)

𝑑2 (𝑡) × ×

2𝑃𝑠 cos(𝜔𝑜 𝑡 + 𝜃2 ) 𝑔2 (𝑡) + 𝑣(𝑡)

⋮ ⋮ ⋮

𝑑𝑘 (𝑡) × ×

2𝑃𝑠 cos(𝜔𝑜 𝑡 + 𝜃𝑘 ) 𝑔𝑘 (𝑡)

(a)

× × 𝑣𝑜1 (𝑡)
𝑔1 (𝑡) 2 cos(𝜔𝑜 𝑡 + 𝜃1 )
× × 𝑣𝑜2 (𝑡)
𝑣(𝑡)
𝑔2 (𝑡) 2 cos(𝜔𝑜 𝑡 + 𝜃2 )

⋮ ⋮ ⋮ ⋮ ⋮

× × 𝑣𝑜𝑘 (𝑡)

𝑔𝑘 (𝑡) 2 cos(𝜔𝑜 𝑡 + 𝜃𝑘 )

(b)

Figure 2.23 CDMA: (a) Transmitter, (b) Receiver.

 SPREAD SPECTRUM MODULATION 35

Each branch of the receiver side in Fig. 2.23 (b) represent a receiver for each
user. At√receiver 1, the signal v(t) of Eq. (2.67) is multiplied by g1 (t) and
also by 2 cos(ωo t + θ1 ) and applied to the integrate and dump filter. It is not
difficult to show that the output vo1 (t) is given by
k p
X
vo1 (t) = Ps g1 (t) gi (t) di (t) cos(θi − θ1 ) (2.68)
i=1

p k p
X
= Ps d1 (t) + Ps g1 (t) gi (t) di (t) cos(θi − θ1 )
i=2

Assume that all gi ’s make transition at the same time, that is the product g1 (t) gi (t)
has the same chip rate fc of each of them. For simplicity, we write g1 (t) gi (t) =
g1i (t) and cos(θi − θ1 ) = cos(θ1i ), then Eq. (2.69) becomes

p k p
X
vo1 (Tb ) = Ps d1 (Tb ) + Ps g1i (Tb ) di (Tb ) cos(θ1i ) (2.69)
i=2

Comparing Eq. (2.69) with Eq. (2.61) we see they are similar except that Eq.
(2.69) has k − 1 independent interfering signals whereas Eq. (2.61) has one
interfering signal. Thus, if we let Pj = Ps and cos2 (θ) = 1/2, then the total
power spectral density Gj (f ) of (k − 1) independent interferers is the sum of
the power spectral density.
Ps
Gj (f ) = (k − 1) , |f | ≤ rb (2.70)
4rc
The bit error probability is found using Eq. (2.65) by letting Pj = (k − 1)Ps to
obtain
q 
4 rc
Pe = Q k−1 rb (2.71)

Thus, to insure a low probability of error, the processing gain (rc /rb ) must be
adjusted so that
rc k−1
>> (2.72)
rb 2
If the interfering signals is approximated by Gaussian distribution, the bit error
probability due to interference only is given by
q 
3
Pe = Q k−1 Gp (2.73)

where Gp = rc /rb is the processing gain. If the effect of the thermal noise is
added, the bit error probability is
 
Pe = Q p k−11 η (2.74)
3Gp + 2E
b

where η is the noise power spectral density and Eb is the bit energy.
36 REVIEW OF DIGITAL MODULATION SCHEMES

2.7.2 Frequency Hopping Spread Spectrum (FHSS)

A frequency-hopped (FHSS) signal uses a g(t) that is of the FM type, where there are
M = 2n hop frequencies controlled by the spreading code, in which n chip words
are taken to determine each hop frequency. An FHSS transmitter is shown in Fig.
2.24(a). The source information is modulated onto a carrier using conventional FSK
or BPSK techniques to produce an FSK or a BPSK signal. The frequency hopping
is accomplished by using a mixer circuit wherein the LO signal is provided by the
output of a frequency synthesizer that is hopped by the PN spreading code. The
serial-to-parallel converter reads n serial chips of the spreading code and outputs a
n-chip parallel word to the programmable dividers in the frequency synthesizer. The
n-chip word specifies one of the possible M = 2n hop frequencies, ω1 , ω2 , · · ·, ωM .

(a)

(b)

Figure 2.24 CDMA: (a) Transmitter, (b) Receiver.

The FH signal is decoded as shown in Fig. 2.24(b). Here, the receiver has the
knowledge of the transmitter, c(t), so that the frequency synthesizer in the receiver
can be hopped in synchronism with that at the transmitter. This despreads the FH
 ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING (OFDM) 37

signal, and the source information is recovered from the dehopped signal with the
use of a conventional FSK or BPSK demodulator, as appropriate.
In FHSS system, several users independently hop their carrier frequencies while
using BFSK modulators. Two cases might exist:

Case 1:
All users hop their carrier frequencies synchronously and this called slotted
frequency hopping. In this case, the probability of error is given by
    
1 Eb k−1 1 k−1
Pe = exp − 1− + (2.75)
2 2η M 2 M

where Eb is the bit energy, η is the noise power spectral density, k is the number
of users, and M is the number of possible hopping channels (slots).

Case 2:
All users hop their carrier frequencies asynchronously and this is called asyn-
chronous FH. The probability of error in this case is given by
   k−1 "   k−1 #
1 Eb 1 1 1 1 1
Pe = exp − 1− 1+ + 1− 1− 1+ (2.76)
2 η M Nb 2 M Nb

where Eb , η, k, M are as given in Eq. (2.75) and Nb is the number of bits per
hop.

2.8 Orthogonal Frequency Division Multiplexing (OFDM)

OFDM is one of multicarrier modulation systems which based on dividing high bit
rate r data into N low bit rate r/N data and sending each of these bit streams on one
of the N subcarriers. These subcarriers are orthogonal. Their frequencies fi = i/T ,
where i = 1, 2, · · · , N . The frequency f1 = 1/T = fb is the lowest subcarrier fre-
quency and it is sometimes called base frequency fb . Thus, the subcarrier frequencies
are multiples of this frequency, i. e. fb , 2fb , 3fb , · · · , N fb . At the transmitter, these
subcarrier frequencies are transferred to the operating frequency fo .
The modulation used in this technique for each of the subcarriers is a M-ary mod-
ulation. The most widely used are QPSK and QAM modulation. The available
bandwidth BT is divided among all N subcarriers. That is if each subcarrier requires
bandwidth equal to fb , then BT = N fb centered at fo . This means that fb = BT /N .
The subcarriers in the frequency domain are shown in Fig. 2.25.
A typical block diagram representation of an OFDM system is shown in Fig.
2.26. The delayed version of a symbol overlaps with the adjacent symbol causes
inter symbol interference (ISI). One simple solution to avoid this is to introduce a
guard band. However, we do not know the exact delay spread. To solve the problem,
we introduce the cyclic prefix (CP) which is basically making the symbol period
longer by copying the tail and blue it in the front as shown in Fig. 2.27.
38 REVIEW OF DIGITAL MODULATION SCHEMES

Figure 2.25 Subcarriers in the frequency domain.

Figure 2.26 Block diagram representation of an OFDM system.

Figure 2.27 Cyclic prefix.

Let Z(k) be the modulated symbols, then the output of the IFFT block is given
by
N −1  
X j2πkn
z(n) = Z(k) exp , n = 0, 1, · · · , N − 1 (2.77)
N
k=0

Equation (2.77) is referred to as inverse discrete Fourier transform (IFFT) of Z(k).

In turn, the inverse operation that allows the frequency samples Z(k) to be obtained
 ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING (OFDM) 39

from the sequence z(n) is called the N-point discrete Fourier transform (DFT) and
is given by
N −1  
X j2πkn
Z(k) = z(n) exp − , k = 0, 1, · · · , N − 1 (2.78)
n=0
N