ADC Digital Modulation

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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

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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).

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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

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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

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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[�̃�(𝑡)𝑒𝑗2𝜋𝑓𝑐𝑡]

(2.1)

where

�̃�(𝑡) = 𝑠�̃�(𝑡) + 𝑗�̃�𝑄(𝑡)

is the complex envelope, 𝑓𝑐 is the carrier frequency, and 𝑠�̃�(𝑡)) and �̃�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) = 𝑠�̃�(𝑡)𝑐𝑜𝑠2𝜋𝑓𝑐𝑡 − �̃�𝑄(𝑡)𝑠𝑖𝑛2𝜋𝑓𝑐𝑡

(2.2)

Finally, the envelope-phase form of 𝑠(𝑡) is

𝑠(𝑡) = 𝑎(𝑡) 𝑐𝑜𝑠(2𝜋𝑓𝑐𝑡 + 𝜙(𝑡))

(2.3)

where

𝑎(𝑡) = √�̃�𝐼2(𝑡) + �̃�𝑄

2(𝑡)

𝜙(𝑡) = 𝑡𝑎𝑛−1 [�̃�𝑄(𝑡)

𝑠�̃�(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

𝑠�̃�(𝑡) = 𝐴 ∑ 𝑏(𝑡 − 𝑛𝑇, 𝑥𝑛)𝑛 (2.4)

𝑥𝑛 = (𝑥𝑛, 𝑥𝑛−1, … . 𝑥𝑛−𝐾) (2.5)

where A is the amplitude, 𝑥𝑛 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,

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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[∑ 𝜉𝑘𝑠(𝑡 − 𝑛𝑇)𝑒𝑗2𝜋𝑓0𝑡∞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 (𝜉𝑘) is a stationary sequence of complex uncorrelated

random variables with 𝐸(𝜉𝑘) = 0 and 𝐸(|𝜉𝑛|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∞

−∞

𝑚𝑎𝑥𝑓𝒢(𝑓)

(3.4)

This measures the basis of a rectangle whose height is 𝑚𝑎𝑥𝑓𝒢(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.

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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=1𝑁 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

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(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

𝑅𝑠 =𝑙𝑜𝑔2𝑀

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= ℰ𝑏𝑅𝑠 (3.10)

where ℰ is the average signal energy and ℰ𝑏 = 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

𝑁0𝐵=

ℰ𝑏

𝑁0

𝑅𝑠

𝐵

(3.11)

This shows that the signal-to-noise ratio is the product of two quantities, namely, the

ratio ℰ𝑏 𝑁0⁄ , the energy per transmitted bit divided by twice the noise spectral density,

and the ratio 𝑅𝑠 B⁄ representing the bandwidth efficiency of the modulation scheme.

In some instances the peak energy ℰ𝑝 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 carries𝑙𝑜𝑔2𝑀 bits, one symbol error

causes at least one and at most 𝑙𝑜𝑔2𝑀 bits to be in error,

𝑃(𝑒)

𝑙𝑜𝑔2𝑀≤ 𝑃𝑏(𝑒) ≤ 𝑃(𝑒)

(3.12)

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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=1𝑗≠1

𝑀𝑖=1

(3.13)

where dij is the Euclidean distance between signals si(t) and sj(t),

𝑑𝑖𝑗2 = ∫ [𝑠𝑖(𝑡) − 𝑠𝑗(𝑡)]

2𝑇

0𝑑𝑡

(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

2𝑒𝑟𝑓𝑐 (

𝑥

√2)

(3.16)

A simpler upper bound on error probability is given by

|𝑃(𝑒)| ≤𝑀−1

2𝑒𝑟𝑓𝑐 (

𝑑𝑚𝑖𝑛

2√𝑁0)

(3.17)

where 𝑑𝑚𝑖𝑛 = 𝑚𝑖𝑛𝑖≠𝑗𝑑𝑖𝑗

A simple lower bound on symbol error probability is given by

𝑃(𝑒) ≥1

𝑀𝑒𝑟𝑓𝑐 (

𝑑𝑚𝑖𝑛

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

2𝑒𝑟𝑓𝑐 (

𝑑𝑚𝑖𝑛

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.

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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 𝑤𝑐𝑡 to represent a logic 1, while

using the carrier 𝐴 𝑐𝑜𝑠 𝑤𝑐𝑡 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 𝑤𝑐𝑡 (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:

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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 𝟐𝒇𝟎 (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.

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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 𝑩𝑭𝑴 ≈ 𝟐𝜟𝒇 + 𝟐𝑩, where B is the

bandwidth of the Polar NRZ signal (equal to 𝒇𝟎 (the bit rate)). Hence, the

bandwidth of FSK is 𝟐∆𝒇 + 𝟐𝒇𝟎 (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.

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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 2𝒇𝟎 (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 𝑤𝑐𝑡 or 𝐴 (𝑐𝑜𝑠 𝑤𝑐𝑡 – 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 𝑤𝑐𝑡 , while a logic 0

corresponds to 𝐴 (𝑐𝑜𝑠 𝑤𝑐𝑡 – 180°).

Another common example of PSK is Quadrature (or Quaternary) Phase-Shift

Keying (QPSK). QPSK uses four possible phases for the carrier

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(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 = 𝒇𝟎 𝟐⁄ , which means that

the bandwidth of the QPSK signal is 𝟐𝑩 + 𝟐∆𝒇 = 𝒇𝟎 instead of 2𝒇𝟎 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,

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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.

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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) = 𝜙𝑄𝐴𝑀(𝑡) × 𝑐𝑜𝑠(𝑤𝑐𝑡)

= [𝑚1(𝑡)𝑐𝑜𝑠(𝑤𝑐𝑡) + 𝑚2(𝑡)𝑠𝑖𝑛(𝑤𝑐𝑡)] × 𝑐𝑜𝑠(𝑤𝑐𝑡)

=1

2𝑚1(𝑡) +

1

2𝑚1(𝑡)𝑐𝑜𝑠(2𝑤𝑐𝑡) +

1

2𝑚2(𝑡)𝑠𝑖𝑛(2𝑤𝑐𝑡)

(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

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.

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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

𝑓 (𝑥) =1

√2𝜋𝑒

−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) 𝑑𝛼𝑥

−∞

= ∫ 𝑓( 𝛼) 𝑑𝛼 =1

√2𝜋

∞

𝑥∫ 𝑒

−𝛼2

2∞

𝑥𝑑𝛼 (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

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.

- 𝑺𝒏(𝝎) =𝑵𝟎

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

𝐸𝑏 = 𝐸𝑠 = (𝐴2

2𝑇𝑠𝑦𝑚𝑏) 𝑃𝑟[1] + (

𝐴2

2𝑇𝑠𝑦𝑚𝑏) 𝑃𝑟[0] =

𝐴2

2𝑇𝑠𝑦𝑚𝑏 =

𝐴2

2𝑇0 =

𝐴2

2

1

𝑓0

Hence

𝐵𝐸𝑅 = 𝑄 (√2𝐸𝑏

𝑁0) = 𝑄 (

𝐴2

𝑁0𝑓0) = 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

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).

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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

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