For ease of analog or digital information transmission and reception, modulation is the foremost important technique. In the present project, we’ll discuss about different modulation scheme in digital mode done by operating a switch/ key by the digital data. As we know, by modifying basic three parameters of the carrier signal, three basic modulation schemes can be obtained; generation and detection of these three modulations are discussed and compared with respect to probability of error or bit error rate (BER).

1. 1
天津职业技术师范大学
DEPARTMENT: ELECTRONICS ENGINEERING
SUBJECT: ADVANCED DIGITAL COMMUNICATION
PROJECT TITLE: DIGITAL MODULATION
STUDENT NAME: 唐德宁

2. 2
Abstract
For ease of analog or digital information transmission and reception, modulation is
the foremost important technique. In the present project, we’ll discuss about different
modulation scheme in digital mode done by operating a switch/ key by the digital data.
As we know, by modifying basic three parameters of the carrier signal, three basic
modulation schemes can be obtained; generation and detection of these three
modulations are discussed and compared with respect to probability of error or bit
error rate (BER).

3. 3
Contents
Abstract……………………………………………………………………………..…1
1 Introduction………………………………………………………………………..3
2 Digital Modulation…………………………………………………………….......3
2.1 Representation of Band-Pass Signals……………………………………….…4
3 The Challenge of Digital Modulation…………………………………………..….4
3.1 Bandwidth……………………………………………………………………5
3.2 Shannon Bandwidth…………………………………………………...……..6
3.3 Signal-to-Noise Ratio……………………………………………………..….6
3.4 Error Probability………………………………………………………………7
4 Types of modulation techniques…………………………………………...………8
4.1 Amplitude-Shift Keying (ASK)…………………………………….………..8
4.1.1 Advantages and disadvantages of ASK………………………..………10
4.2 Frequency-shift keying (FSK)……………………………………………...10
4.3 Phase-shift keying (PSK)………………………………………..…………10
4.4 Quadrature Amplitude Modulation (QAM)……………………..………….13
5 Performance of digital modulation techniques in presence of Noise…………….16
6 BER equations for the different modulation techniques………………...……….18
7 Comparison of Digital Modulation Schemes…………………………...………..19
8 Applications of digital modulation techniques…………………………………...20
9 Conclusion………………………………………………………………………...21
References………………………………………………………………...…………..22

4. 4
1 Introduction
Wireless communications is one of the most active areas of technology development
of our time and has become an ever-more important and prominent part of everyday
life. Modulation, by which data is transmitted by varying low-powered radio waves,
plays a key role in wireless communication systems. The goal of a modulation
technique is to provide high speed data transmission with good quality in the presence
of mobile channel impairments while occupying minimum bandwidth and requiring
the least amount of signal power. Most first generation systems were introduced in the
mid 1980s, and are characterized by the use of analog transmission techniques. The
primary disadvantages of analog transmission are its poor noise immunity and low
data rates. Second generation systems were introduced in the early 1990s, and all use
digital technology. Digital modulation offers many advantages over analog modulation
and greatly improves the performance of the communication systems. Many types of
digital modulation schemes are possible, and the choice of which one to use depends
on spectral efficiency, power efficiency, and bit error rate performance. A tradeoff
between power and spectral efficiency always exists in the design of a modulation
scheme. Furthermore, better bit error rate performance can be achieved by assigning
more bandwidth and a larger amount of signal power.
In this project, I will focus on some of the digital modulation techniques such as ASK,
FSK, PSK etc.
2 Digital Modulation
Modulation is the process of varying a sinusoidal carrier signal with a message
bearing signal in order to achieve a long distance transmission. A device that performs
modulation is known as a modulator while a device that performs the inverse
operation of modulation is known as a demodulator. Message information can be
embedded in the amplitude, frequency, or phase of the carrier, or any combination of
these. Modulation is generally performed to overcome signal transmission issues to
allow easy (low loss, low dispersion) propagation. Modulation techniques are expected
to have three positive properties:
Good Bit Error Rate Performance
Modulation schemes should achieve low bit error rate in the presence of fading,
Doppler spread, interference, and thermal noise.
Power Efficiency
Power limitation is one of the critical design challenges in portable and mobile
applications. Nonlinear amplifiers (Class C or Class D) are usually used to increase
power efficiency; however, a nonlinearity may degrade the bit error rate performance
of some modulation schemes. Constant envelope modulation techniques are used to
prevent the regrowth of spectral side-lobes during nonlinear amplification.
Spectral Efficiency
The modulated signals power spectral density should have a narrow main lobe and fast
roll-off of side lobes. Spectral efficiency is measured in units of bit/sec/Hz.
In analog modulation, the carrier signals are varied continuously in response to the

5. 5
input data. In contrast, in digital modulation, the changes in the signal are determined
by a fixed list, the modulation alphabet. Each entry of the alphabet represents a symbol
which consists of one or more bits and it is convenient to represent that alphabet on a
constellation diagram.
2.1 Representation of Band-Pass Signals
We can express the modulated signals in the complex envelope form
𝑠(𝑡) = 𝑅𝑒[𝑠̃( 𝑡)𝑒 𝑗2𝜋𝑓𝑐 𝑡
]
(2.1)
where
𝑠̃( 𝑡) = 𝑠𝐼̃(𝑡) + 𝑗𝑠̃ 𝑄(𝑡)
is the complex envelope, 𝑓𝑐 is the carrier frequency, and 𝑠𝐼̃(𝑡)) and 𝑠̃ 𝑄(𝑡) are the
in-phase and quadrature components of s(t). The band-pass waveform can also be
expressed in the quadrature form
𝑠(𝑡) = 𝑠𝐼̃(𝑡)𝑐𝑜𝑠2𝜋𝑓𝑐 𝑡 − 𝑠̃ 𝑄(𝑡)𝑠𝑖𝑛2𝜋𝑓𝑐 𝑡
(2.2)
Finally, the envelope-phase form of 𝑠(𝑡) is
𝑠(𝑡) = 𝑎(𝑡) 𝑐𝑜𝑠(2𝜋𝑓𝑐 𝑡 + 𝜙(𝑡))
(2.3)
where
𝑎(𝑡) = √𝑠̃ 𝐼
2
(𝑡) + 𝑠̃ 𝑄
2(𝑡)
𝜙(𝑡) = 𝑡𝑎𝑛−1
[
𝑠̃ 𝑄(𝑡)
𝑠𝐼̃(𝑡)
]
Here, 𝑎(𝑡) is the amplitude of the modulated signal and 𝜙(𝑡) is the phase of the
modulated signal. The complex envelope of any digital scheme can be written in a
standard form
𝑠𝐼̃(𝑡) = 𝐴 ∑ 𝑏(𝑡 − 𝑛𝑇, 𝑥 𝑛)𝑛 (2.4)
𝑥 𝑛 = (𝑥 𝑛, 𝑥 𝑛−1, … . 𝑥 𝑛−𝐾) (2.5)
where A is the amplitude, 𝑥 𝑛 is the sequence of complex data symbols, and 𝑏(𝑡, 𝑥𝑖)
is the shaping function. T is the symbol time and the baud rate is R = 1 T⁄
symbols/sec.
3 The Challenge of Digital Modulation
The selection of a digital modulation scheme should be done by making the best
possible use of the resources available for transmission, namely, bandwidth, power,

6. 6
and complexity, in order to achieve the reliability required.
3.1 Bandwidth
There is no unique definition of signal bandwidth. Actually, any signal s(t) strictly
limited to a time interval T would have an infinite bandwidth if the latter were defined
as the support of the Fourier transform of s(t). For example, consider the bandpass
linearly modulated signal
𝑣(𝑡) = ℜ[∑ 𝜉 𝑘 𝑠(𝑡 − 𝑛𝑇)𝑒 𝑗2𝜋𝑓0 𝑡∞
𝑛=−∞ ]
(3.1)
where ℜ denotes real part, fo is the carrier frequency, s(t) is a rectangular pulse with
duration T and amplitude 1, and (𝜉 𝑘) is a stationary sequence of complex uncorrelated
random variables with 𝐸(𝜉 𝑘) = 0 and 𝐸(|𝜉 𝑛|2) = 1Then the power density spectrum
of v(t) is given by
𝒢(𝑓) =
1
𝑓
[𝐺(−𝑓 − 𝑓0) + 𝐺(𝑓 − 𝑓0)] (3.2)
where
𝒢(𝑓) = 𝑇 [
𝑠𝑖𝑛𝜋𝑓𝑇
𝜋𝑓𝑇
]
2
(3.3)
The function 𝒢(𝑓) is shown figure 1.
The following are possible definitions of the bandwidth:
Half-power bandwidth: This is the interval between the two frequencies at which
the power spectrum is 3 dB below its peak value.
Equivalent noise bandwidth: This is given by
𝐵𝑒𝑞 =
1
2
∫ 𝒢(𝑓)𝑑𝑓
∞
−∞
𝑚𝑎𝑥 𝑓 𝒢(𝑓)
(3.4)
This measures the basis of a rectangle whose height is 𝑚𝑎𝑥𝑓 𝒢(𝑓) and whose area is
one-half of the power of the modulated signal.
Null-to-null bandwidth: This represents the width of the main spectral lobe.

7. 7
Figure 1. Power density spectrum of a linearly modulated signal with rectangular
waveforms.
Fractional power containment bandwidth: This bandwidth definition states that
the occupied bandwidth is the band that contains (1 − ε) of the total signal
power.
Bounded-power spectral density bandwidth: This states that everywhere outside
this bandwidth the power spectral density must fall at least a certain level (e.g., 35
or 50 dB) below its maximum value.
Although the actual value of the signal bandwidth depends on the definition that has
been accepted for the specific application, in general, we can say that
B =
α
T
(3.5)
where T is the duration of one of the waveforms used by the modulator, and α reflects
the definition of bandwidth and the selection of waveforms. For example, for Eq. (3.3)
the null-to-null bandwidth provides B = 2 T⁄ , that is, α = 2. For 3-dB bandwidth,
α = 0.88. For equivalent-noise bandwidth, we have α = 1.
3.2 Shannon Bandwidth
To make it possible to compare different modulation schemes in terms of their
bandwidth efficiency, it is useful to consider the following definition of bandwidth.
Consider a signal set and its geometric representation based on the orthonormal set of
signals {𝛹𝑖(𝑡)}𝑖=1
𝑁
defined over a time interval with duration T. The value of N is
called the dimensionality of the signal set. We say that a real signal x(t) with Fourier
transform X( f −T 2⁄ < 𝑡 < T 2⁄ at level ∈ if
∫ 𝑥2
|𝑡|>𝑇 2⁄
(𝑡)𝑑𝑡 < 𝜖 (3.6)
and is bandlimited with bandwidth B at level ∈ if
∫ |𝑋(𝑓)|2
|𝑓|>𝐵
𝑑𝑓 < 𝜖

8. 8
(3.7)
Then for large BT the space of signals that are time limited and bandlimited at level ϵ
has dimensionality N = 2BT. Consequently, the Shannon bandwidth of the signal set is
defined as
B =
N
2T
(3.8)
and is measured in dimensions per second.
3.3 Signal-to-Noise Ratio
Assume from now on that the information source emits independent, identically
distributed binary digits with rate Rs digits per second, and that the transmission
channel adds to the signal a realization of a white Gaussian noise process with power
spectral density N0/2.
The rate, in bits per second, that can be accepted by the modulator is
𝑅 𝑠 =
𝑙𝑜𝑔2 𝑀
𝑇
(3.9)
where M is the number of signals of duration T available at the modulator, and 1/T is
the signaling rate. The average signal power is
𝒫 =
ℰ
𝑇
= ℰ 𝑏 𝑅 𝑠 (3.10)
where ℰ is the average signal energy and ℰ 𝑏 = ℇ 𝑙𝑜𝑔2⁄ 𝑀 is the energy required to
transmit one binary
digit. As a consequence, if B denotes the bandwidth of the modulated signal, the ratio
between signal power and noise power is
𝒫
𝑁0 𝐵
=
ℰ 𝑏
𝑁0
𝑅 𝑠
𝐵
(3.11)
This shows that the signal-to-noise ratio is the product of two quantities, namely, the
ratio ℰ 𝑏 𝑁0⁄ , the energy per transmitted bit divided by twice the noise spectral density,
and the ratio 𝑅 𝑠 B⁄ representing the bandwidth efficiency of the modulation scheme.
In some instances the peak energy ℰ 𝑝 is of importance. This is the energy of the
signal with the maximum amplitude level.
3.4 Error Probability
The performance of a modulation scheme is measured by its symbol error
probability P(e), which is the probability that a waveform is detected incorrectly, and
by its bit error probability, or bit error rate (BER) Pb(e), the probability that a bit sent
is received incorrectly. A simple relationship between the two quantities can be
obtained by observing that, since each symbol carries𝑙𝑜𝑔2 𝑀 bits, one symbol error
causes at least one and at most 𝑙𝑜𝑔2 𝑀 bits to be in error,
𝑃(𝑒)
𝑙𝑜𝑔2 𝑀
≤ 𝑃𝑏(𝑒) ≤ 𝑃(𝑒)
(3.12)

9. 9
When the transmission takes place over a channel affected by additive white Gaussian
noise, and the modulation scheme is memoryless, the symbol error probability is upper
bounded as follows:
𝑃(𝑒) ≤
1
2𝑀
∑ ∑ 𝑒𝑟𝑓𝑐 (
𝑑𝑖𝑗
2√𝑁0
)𝑀
𝑗=1
𝑗≠1
𝑀
𝑖=1
(3.13)
where dij is the Euclidean distance between signals si(t) and sj(t),
𝑑𝑖𝑗
2
= ∫ [𝑠𝑖(𝑡) − 𝑠𝑗(𝑡)]
2𝑇
0
𝑑𝑡
(3.14)
and erfc(.) denotes the Gaussian integral function
𝑒𝑟𝑓𝑐(𝑥) =
2
√ 𝜋
∫ 𝑒−𝑧2
𝑑𝑧
∞
𝑥
(3.15)
Another function, denoted 𝑄(𝑥), is often used in lieu of erfc(.). This is defined as
𝑄(𝑥) =
1
2
𝑒𝑟𝑓𝑐 (
𝑥
√2
)
(3.16)
A simpler upper bound on error probability is given by
|𝑃(𝑒)| ≤
𝑀−1
2
𝑒𝑟𝑓𝑐 (
𝑑 𝑚𝑖𝑛
2√𝑁0
)
(3.17)
where 𝑑 𝑚𝑖𝑛 = 𝑚𝑖𝑛𝑖≠𝑗 𝑑𝑖𝑗
A simple lower bound on symbol error probability is given by
𝑃(𝑒) ≥
1
𝑀
𝑒𝑟𝑓𝑐 (
𝑑 𝑚𝑖𝑛
2√𝑁0
)
(3.18)
By comparing the upper and the lower bound we can see that the symbol error
probability depends exponentially on the term dmin, the minimum Euclidean distance
among signals of the constellation. In fact, upper and lower bounds coalesce
asymptotically as the signal-to-noise ratio increases. For intermediate signal-to-noise
ratios, a fair comparison among constellations should take into account the error
coefficient as well as the minimum distance. This is the average number v of nearest
neighbors [i.e., the average number of signals at distance dmin from a signal in the
constellation; for example, this is equal to 2 for M-ary phase-shift keying (PSK),
M > 2]. A good approximation to P(e) is given by
𝑃(𝑒) ≈
𝑣
2
𝑒𝑟𝑓𝑐 (
𝑑 𝑚𝑖𝑛
2√𝑁0
) (3.19)
Roughly, at 𝑃(𝑒) = 10−6
, doubling v accounts for a loss of 0.2 dB in the
signal-to-noise ratio.

10. 10
4 Types of modulation techniques
There are four major modulation techniques used by communication systems
nowadays to transport baseband digital data onto a carrier. These modulation
techniques are:
Amplitude-Shift Keying (ASK)
Frequency-Shift Keying (FSK)
Phase-Shift Keying (PSK)
Quadrature Amplitude Modulation (QAM)
4.1 Amplitude-Shift Keying (ASK)
ASK represents digital data as variations in the amplitude of a carrier signal. For
example the transmitter could send the carrier 2𝐴 𝑐𝑜𝑠 𝑤𝑐 𝑡 to represent a logic 1, while
using the carrier 𝐴 𝑐𝑜𝑠 𝑤𝑐 𝑡 to represent a logic 0. This is shown in the diagram
below. The receiver detects the amplitude of the carrier to recover the original bit
stream.
A special case of ASK is when a logic 1 is represented by 𝐴 𝑐𝑜𝑠 𝑤𝑐 𝑡 (i.e., the
presence of a carrier) and a logic 0 is represented by a zero voltage (i.e., the absence of
a carrier). This special case is called On-Off Keying (OOK) and is shown below.
Notice that you can visualize ASK as the process of Amplitude Modulation (AM)
using a “Polar NRZ” digital baseband message signal. In other words, we say that
ASK is the result of multiplying a binary Polar NRZ signal 𝑚 𝑡 (with appropriate DC
shift) times a sinusoidal carrier. This is shown in the diagram below:

11. 11
The above diagram shows that a general ASK signal is simply an AM signal with a
modulation index m < 1, while an OOK is an AM signal with a modulation index m =
1. Hence, an envelope detector can be used at the receiver to demodulate the ASK
signal. In addition, since ASK is a special case of AM modulation, the bandwidth of
ASK is 2B centered around the carrier frequency, where B is the bandwidth of the
Polar NRZ signal. Since the bandwidth of Polar NRZ is equal to the data bit rate (𝑓0)
of the bit stream to be sent, the bandwidth of ASK is 𝟐𝒇 𝟎 (Hz). The following is a
sketch of the PSD for an ASK signal. It consists of two replicas of the PSD for a Polar
NRZ signal with additional carrier impulses. You can see that the bandwidth of this
ASK signal is approximately 2𝑓0 (Hz).
4.1.1 Advantages and disadvantages of ASK
Advantages Disadvantages
- ASK is the simplest kind of
modulation to generate and
detect.
- It can be used only when the
signal-ti-noise ratio (SNR) is
very high.
- Its bandwidth is too big (equals
2𝑓0).
4.2 Frequency-shift keying (FSK)
In FSK the instantaneous frequency of the carrier signal is shifted between two
possible frequency values termed the mark frequency (representing a logic 1) and the
space frequency (representing a logic 0). This is shown in the diagram below.

12. 12
Notice that FSK can be thought of as Frequency Modulation (FM) using a “Polar
NRZ” digital baseband signal as the message, and hence FSK can be seen as a subset
of FM modulation. Since FSK is a special case of FM modulation, the bandwidth of
FSK is given by Carson’s rule which says that 𝑩 𝑭𝑴 ≈ 𝟐𝜟𝒇 + 𝟐𝑩, where B is the
bandwidth of the Polar NRZ signal (equal to 𝒇 𝟎 (the bit rate)). Hence, the
bandwidth of FSK is 𝟐∆𝒇 + 𝟐𝒇 𝟎 (𝑯𝒛). In addition, all modulator and demodulator
circuits for FM are still applicable for FSK.
FSK has several advantages over ASK due to the fact that the carrier has a constant
amplitude. These are the same advantages present in FM which include: immunity to
non-linearities, immunity to rapid fading, immunity to adjacent channel interference,
and the ability to exchange SNR for bandwidth. FSK was used in early slow dial-up
modems.
4.3 Phase-shift keying (PSK)
In PSK, the data is conveyed by changing the phase of the carrier wave. One possible
representation (called Binary Phase-Shift Keying or BPSK) is to send logic 1 as a
cosine signal with zero phase shift and a logic 0 as a cosine signal but with a 180°
phase shift. We say in this case that the BPSK signal can assume one of two possible
symbols: 0°and 180°. This case is shown in the following Figure.

13. 13
BPSK can be thought of as a special case of Phase Modulation (PM) using a “Polar
NRZ” digital baseband message1
. In the case of BPSK, we select the peak phase
deviation to be Δ𝜃 = 𝜋/2 (i.e., 2Δ𝜃 = 𝜃𝑚𝑎𝑥 − 𝜃𝑚𝑖𝑛 = 𝜋). This value maximizes
immunity to phase noise. Since BPSK is a special case of PM, the bandwidth of PSK
is 2B + 2Δf, where B is the bandwidth for the polar NRZ signal and Δf = 0 since the
sinusoidal carrier signal does not change its frequency. Hence, the bandwidth of
BPSK is 2𝒇 𝟎 (Hz). A convenient way to represent PSK modulation is using a
constellation diagram. A constellation diagram consists of a group of points
representing the different symbols the carrier in a PSK modulated signal can assume.
For example, for BPSK, in which each bit is represented by one symbol (i.e., either
A 𝑐𝑜𝑠 𝑤𝑐 𝑡 or 𝐴 (𝑐𝑜𝑠 𝑤𝑐 𝑡 – 180°)), the constellation diagram consists of two points
(see Figure below). These two points have the same amplitude A, but they are 180°
apart. This means that a logic 1 corresponds to A 𝑐𝑜𝑠 𝑤𝑐 𝑡 , while a logic 0
corresponds to 𝐴 (𝑐𝑜𝑠 𝑤𝑐 𝑡 – 180°).
Another common example of PSK is Quadrature (or Quaternary) Phase-Shift
Keying (QPSK). QPSK uses four possible phases for the carrier

14. 14
(45°, 135°, 225°, 315°) but with the same carrier amplitude, as shown in the
constellation diagram below.
With four phases, QPSK can encode two bits per one symbol (see Figure below).
You can imagine QPSK as a special case of Phase Modulation (PM) in which the
baseband message signal m(t) is a digital M-ary signal (with M = 4). In this case, the
bandwidth of the M-ary baseband signal is B = Baud Rate = 𝒇 𝟎 𝟐⁄ , which means that
the bandwidth of the QPSK signal is 𝟐𝑩 + 𝟐∆𝒇 = 𝒇 𝟎 instead of 2𝒇 𝟎 for BPSK.
Hence, QPSK can be used to double the data rate compared to a BPSK system while
maintaining the same bandwidth of the modulated signal. Notice that any number of
phases may be used to construct a PSK constellation. Usually, 8-PSK is the highest
order PSK constellation deployed in practice (see the figure below).
In this case, each carrier symbol represents three bits. With more than eight phases,

15. 15
the error-rate becomes too high and there are better, though more complex,
modulation schemes available (such as QAM). Notice that in PSK, the constellation
points are usually positioned with uniform angular spacing around a circle. This
gives maximum phase-separation between adjacent points and thus the best
immunity to noise. Points are positioned on a circle so that all the different phases can
be transmitted with the same carrier amplitude. The axes in a constellation diagram
are called the in-phase (I) and quadrature (Q) axes, respectively, due to their 90°
separation. The nice thing about a constellation diagram is that it lends itself to
straightforward and simple implementation of PSK modulation in hardware. This is
because the PSK modulated signal can be generated by individually DSB-SC
modulating both a sine wave and a cosine wave and then adding the resulting
modulated carriers to each other. In such case, the constellation diagram is extremely
helpful since the amplitude of each point along the in-phase axis is the one used to
modulate the cosine wave and the amplitude along the quadrature axis is the one used
to modulate the sine wave. This procedure will be much more obvious when we
discuss QAM modulation in the next section. It is worth mentioning that BPSK and
QPSK can be regarded special cases of the more general QAM modulation, where the
amplitude of the modulating signal is constant (see next section).
Example: Find the bandwidth of an 8-PSK modulated signal if the data bit rate is
100 kbit/s.
Solution: For 8-PSK, Bandwidth = 2B = 2×Baud Rate
= 2 ×
100𝑘𝑏𝑝𝑠
𝑙𝑜𝑔2(8)
= 𝟐 ×
𝟏𝟎𝟎𝒌𝒃𝒑𝒔
𝟑𝒃𝒊𝒕𝒔/𝒔𝒚𝒎𝒃
= 66.67kHz.
4.4 Quadrature Amplitude Modulation (QAM)
QAM is a modulation scheme which conveys data by modulating the amplitude of
two carrier waves. These two waves (a cosine and a sine) are out of phase with each
other by 90° and are thus called quadrature carriers — hence the name of the scheme.
Both analog and digital QAM are possible. Analog QAM was used in NTSC and
PAL television systems, where the I- and Q-signals carry the components of
chrominance (color) information.
Let us start by remembering analog QAM, which allowed us to transmit two message
signals using two orthogonal carriers of the same frequency. The following Figure
shows this scheme. Notice that both modulated signals will occupy the same
frequency band around 𝑤𝑐.

16. 16
The two baseband signals can be separated at the receiver by synchronous
detection using two local carriers in phase quadrature. This can be shown by
considering the multiplier output 𝑥1(𝑡) of the top branch (see Figure above):
𝑥1(𝑡) = 𝜙 𝑄𝐴𝑀(𝑡) × 𝑐𝑜𝑠(𝑤𝑐 𝑡)
= [𝑚1(𝑡)𝑐𝑜𝑠(𝑤𝑐 𝑡) + 𝑚2(𝑡)𝑠𝑖𝑛(𝑤𝑐 𝑡)] × 𝑐𝑜𝑠(𝑤𝑐 𝑡)
=
1
2
𝑚1(𝑡) +
1
2
𝑚1(𝑡)𝑐𝑜𝑠(2𝑤𝑐 𝑡) +
1
2
𝑚2(𝑡)𝑠𝑖𝑛(2𝑤𝑐 𝑡)
(4.1)
The last two terms are suppressed by the lowpass filter (LPF), yielding the desired
output 𝑚1(𝑡) 2⁄ . Thus, in QAM two signals can be transmitted simultaneously over a
bandwidth of 2B, and still get separated at the receiver.
Digital QAM, on the other hand, is constructed using two M-ary baseband signals
(called i(t) and q(t)) modulating the two quadrature carriers. For example, in 16-QAM
both i(t) and q(t) are 4-ary digital baseband signals, which means each one of them
can assume one of four possibilities. This results in 4 × 4 = 16 possible carrier
symbols as shown in the constellation diagram below. Hence, 16-QAM uses 16
symbols, with each symbol representing a specific four-bit pattern.
For example, to send the bit sequence 100101110000 using 16-QAM, the bit stream is

17. 17
split into 4-bit groups, with each 4-bit pattern affecting i(t) and q(t) as shown in the
figure below.
Notice that the baud rate (i.e., the symbol rate) of the resulting 16-QAM signal is one
fourth that of the data bit rate. This is why the bandwidth of 16-QAM is 2×Baud Rate
= 2 𝑓0/4 = 𝑓0/2. You can see that this is correct because the bandwidth of each one
of the 4-ary signals is B =𝑓0/4 (one symbol per four bits). Performing DSB-SC
modulation for each one of these signals (i.e., QAM) results in a total bandwidth of
2B = 2 (𝑓0/4) = 𝑓0/2.
Example: Find the bandwidth of an 16-QAM modulated signal if the data bit rate is
8 Mbit/s.
Solution: For 16-QAM, Bandwidth = 2 × Baud Rate
= 2 ×
8𝑀𝑏𝑝𝑠
𝑙𝑜𝑔2(16)
= 𝟐 ×
𝟖𝑴𝒃𝒑𝒔
𝟒𝒃𝒊𝒕𝒔/𝒔𝒚𝒎𝒃
= 4MHz
In QAM, the constellation points are usually arranged in a square grid with equal
vertical and horizontal spacing called rectangular QAM (see the above constellation
diagram). The number of points in the grid is usually a power of 2 (2, 4, 8...). The
most common forms of QAM are 16-QAM, 64-QAM, 128-QAM and 256-QAM. By
moving to higher-order constellations, it is possible to transmit more bits per symbol,
which reduces bandwidth. However, if the mean energy of the constellation is to
remain the same, the points must be closer together and are thus more susceptible to
noise; this results in a higher bit error rate (BER) and, hence, higher order QAM can
deliver more data less reliably than lower-order QAM unless, of course, the SNR is
increased. Rectangular QAM constellations are, in general, sub-optimal in the sense
that they do not maximally space the constellation points for a given energy. However,
they have the considerable advantage that they are easier to generate and demodulated
using simple hardware. Non-square constellations achieve marginally better
performance but are harder to modulate and demodulate.

18. 18
For example, the diagram of circular 16-QAM constellation is shown above. The
constellation diagram shown below is the one used in the V.32bis dial-up modem.
This modem provides 14.4 kbit/s using only 2400 baud rate. Can you calculate the
number of constellation points from these numbers?2
Note: It is worth mentioning that in practical systems, M-ary signals are shaped using
a raised-cosine pulse before modulating the two quadrature carriers. In such case, the
bandwidth of QAM (or PSK) becomes 2 × 𝐵𝑎𝑢𝑑 × (1 + 𝛽 )/2 instead of just
2 × 𝐵𝑎𝑢𝑑
5 Performance of digital modulation techniques in presence of Noise
We measured the performance for analog modulation techniques in terms of signal
quality, which was related to output signal-to-noise ratio (SNRout). For digital
modulation techniques, the performance is measured in terms of output bit error
rate (BER), which represents the number of erroneous bits that the receiver expects
per second. For example, a BER = 10-4
means that we expect on average 1 bit error
out of every 10,000 transmitted bits. We say the system exhibits good performance if
the 𝐵𝐸𝑅 ≤ 10−6
. Remember that we are using the Additive White Gaussian Noise
(AWGN) mathematical model to describe the noise on a communication channel.
Hence, the noise n(t) is considered as a Gaussian random process with zero average
and a variance 𝜎2
. The variance of the noise 𝜎2
is its average power.
Recall that for a standard Gaussian random variable X with zero-mean and unity
variance, the probability density function (pdf) is:

19. 19
𝑓 (𝑥) =
1
√2𝜋
𝑒
−𝑥2
2
(5.1)
For the purpose of our performance analysis, we will define the Quantile function
Q(x) as the complement of the cumulative distribution function F(x) of the standard
Gaussian random variable, i.e.,
𝑄( 𝑥 ) = 1 – 𝐹( 𝑥 ) = 1 − ∫ 𝑓( 𝛼) 𝑑𝛼
𝑥
−∞
= ∫ 𝑓( 𝛼) 𝑑𝛼 =
1
√2𝜋
∞
𝑥
∫ 𝑒
−𝛼2
2
∞
𝑥
𝑑𝛼 (5.2)
The diagram below gives a visual representation for Q(x) which represents the shaded
area under the standard Gaussian density curve:
Usually we use a table (similar to the one shown below) to lookup Q(x) values for
specific x arguments since the above integral has no closed form solution.

20. 20
6 BER equations for the different modulation techniques
A summary of the BER equations for the different modulation techniques is given
following table below.
where
- M = Number of possible symbols that the modulated signal can assume.
- k = the number of bits sent per transmitted symbol = log2 (M).
- Es = Average energy-per-transmitted-symbol in the modulated signal (Joule).
- Eb = Average energy-per-transmitted-bit in the modulated signal (Joule) = Es/k.
- 𝑺 𝒏(𝝎) =
𝑵 𝟎
𝟐
= Double-sided noise power spectral density (in W/Hz = Joule).
- To = Bit duration.
- Tsymb = Symbol duration = k To
- BER = Probability of bit-error = bit error rate.
Example:
Find the BER for BPSK if we use an optimal detector (a matched filter). Assume the
amplitude of the carrier is 𝐴 = 0.5 V, data rate is 2 bps, and 𝑁0 = 2 × 10−2
W/Hz.
Solution:
In BPSK there is one symbol per bit (i.e., a total of two symbols that the modulated
signal can assume). The two symbols can be written as:
𝑠1 = 𝐴 𝑐𝑜𝑠 (𝑤𝑐 𝑡 ) 𝑠2 = −𝐴 𝑐𝑜𝑠(𝑤𝑐 𝑡 ) = 𝐴 (𝑤𝑐 𝑡 − 𝜋)
The energy-per-symbol here is the same as the energy-per-bit and is equal for both
possible symbols. Hence, its average is:

21. 21
𝐸 𝑏 = 𝐸𝑠 = (
𝐴2
2
𝑇𝑠𝑦𝑚𝑏) 𝑃𝑟[1] + (
𝐴2
2
𝑇𝑠𝑦𝑚𝑏) 𝑃𝑟[0] =
𝐴2
2
𝑇𝑠𝑦𝑚𝑏 =
𝐴2
2
𝑇0 =
𝐴2
2
1
𝑓0
Hence
𝐵𝐸𝑅 = 𝑄 (√
2𝐸 𝑏
𝑁0
) = 𝑄 (
𝐴2
𝑁0 𝑓0
) = 𝑄 (√
0.52
2 × 10−2 × 2
) = 𝑄(√6.25) = 𝑄(2.5)
= 6.21 × 10−3
7 Comparison of Digital Modulation Schemes
Below are the BER curves for the different digital modulation schemes:
Comparing BPSK and QPSK with ASK and FSK, we notice that BPSK and QPSK
provide smaller bit error rate for the same Eb/No. In other words, for the same bit error
rate, we need less signal-to-noise ratio (Eb/No) to send BPSK and QPSK. This means
that BPSK and QPSK have better immunity to noise than ASK and FSK. Notice also
that the performance of BPSK is the same as that for QPSK, while the performance of
8-PSK and 16-PSK are worse (i.e., they require more signal-to-noise ratio to achieve
the same bit error rate). This is an expected result because 8-PSK and 16-PSK have
more constellation diagram points (which are now closer and closer to each other).
Also notice how 16-QAM has a superior performance compared to 16-PSK, which is

22. 22
to be expected because the constellation points are further apart in 16-QAM compared
to 16-PSK.
The following table shows the bandwidth requirements and the necessary signal-to
noise ratio (Eb/No) to achieve near error free transmission (this is 𝐵𝐸𝑅 ≈ 10−6
). Notice
that for higher order modulation techniques, we require less bandwidth but we need
more signal-to-noise ratio (Eb/No) to maintain small bit error rate (i.e., to maintain
good performance).
8 Applications of digital modulation techniques
The following are some current-day communication systems that use digital
modulation:
IEEE 802.11 (Wi-Fi): A very important Wireless Local Area Networking
technology. Since Wi-Fi has many variants, it uses different modulation
techniques such as: BPSK, QPSK, 16-QAM, 64-QAM and CCK
(Complementary Code Keying) (CCK is an extension of QPSK).
IEEE 802.16 (Wi-MAX): A very important Wireless Metropolitan Area
Network, and currently competes with ADSL for Internet delivery. Wi-MAX
switches dynamically between different modulation schemes such as: BPSK,
QPSK, 16-QAM, and 64-QAM. It uses these modulation schemes in combination
with OFDM (Orthogonal Frequency division multiplexing) (OFDM is an
extension of FDM).
DVB (Digital Video Broadcasting): This is the European standard for digital
television broadcasting. There are many variants within the standard: DVB-S (for
satellite broadcasting) uses QPSK or 8-PSK; DVB-C (for cable) uses 16-QAM,
32-QAM, 64-QAM, 128-QAM or 256-QAM; and DVB-T (for terrestrial
television broadcasting) uses 16-QAM or 64-QAM.
DAB (Digital Audio Broadcasting): Future European standard for digital radio
broadcasting, which should replace AM and FM radio broadcasting. DAB use
DQPSK (Differential QPSK) (DQPSK is a variation of QPSK).
ADSL: Currently one of the main choices for connecting to the Internet. Uses
adaptive QAM in a scheme called DMT (Discrete Multi-Tone modulation).

23. 23
9 Conclusion
An analysis of the digital modulation technique carried out in this project reveals that
the selection of a digital modulation technique is solely dependent on the type of
application. This is because of the fact that some of the technique provide lesser
complexities in the design of the modulation and demodulation system and prove
economic like the BASK, BFSK, BPSK and DPSK techniques and can be visualized
for the systems which really does not require high amount of precisions or when
economy is the major aspect and the BER performances can be tolerated. On the other
hand when the system designer has a sole consideration for the techniques like BASK,
BFSK, BPSK and designer has to think in terms of better modulation techniques. But
the criterion for higher data rate communication is taking the lead in almost every area
of communication and thus the ISI and BER realization become very important and
crucial aspect for any future digital modulation technique. Taking the above facts into
consideration, the design of a digital communication system is very trivial and is very
much applications oriented, as one application may require higher precision in data
reception where as the other may compromise on this aspect but may be rigid on the
aspect of the available bandwidth or power, thus the parameters like the modulation
bandwidth, power, channel noise and the bit error rate become very important
parameters in the designing of digital/wireless communication system.

24. 24
References
1. http://etd.nd.edu/ETD-db/theses/available/etd-12102006-195114/unrestricted/ZhangC1220
06.pdf
2. http://www.ece.ucsb.edu/courses/courses/ECE146/146B_S10Madhow/digital_modulation_v
3b.pdf
3. http://radio-1.ee.dal.ca/~ilow/6590/readings/0967_ch20.pdf
4. http://etd.nd.edu/ETD-db/theses/available/etd-12102006-195114/unrestricted/ZhangC122
006.pdf
5. http://fetweb.ju.edu.jo/staff/ee/mhawa/421/Digital%20Modulation.pdf