Performance Evaluation of M-ary Frequency Shift Keying Radio Modems via Measurements and Simulations

Performance Evaluation of M-ary Frequency Shift Keying Radio Modems via Measurements and Simulations

Submitted by

ERIC NII OTORKUNOR SACKEY

Department of Electrical Engineering Blekinge Institute of Technology

Karlskrona, Sweden September 2006

This thesis is presented as part of the Degree of Master of Science in Electrical Engineering with emphasis on Telecommunications/Signal Processing.

Abstract

M-ary Frequency Shift Keying is a power efficient modulation scheme that is currently used by manufacturers of low power low data rate data transmission equipment. The power efficiency of this modulation increases as the signal alphabet increases at the expense of increased complexity and reduced bandwidth efficiency. There is, however, a gap between the performance of real world systems employing Frequency Shift Keying (FSK) and that of theoretical FSK systems. To investigate the nature of this gap, a comparison is needed between the performance of real world systems using FSK and that of theoretical FSK systems.

This thesis investigates the nature of this gap by simulating 2, 4 and 8-level FSK systems in additive white Gaussian noise channel using MATLAB, measuring of the performance of commercially available data transmission equipment manufactured by RACOM s.r.o of the Czech Republic, and comparison of their performances.

Some important results have been illustrated and also, it is observed that the gap between the performance of theoretical and real world systems using FSK is about 1 dB at a bit error rate (BER) of 10^-3 and widens as BER decreases.

Acknowledgements

The completion of my studies at Blekinge Institute of Technology, Sweden would have been impossible without the generous help of others. Foremost, I give credit to the almighty God. I also extend my gratitude to my supervisors, Dr. Abbas Mohammed of the School of Engineering and Mr. Jiri Hruska of RACOM s.r.o of the Czech Republic, for their encouragement and support.

I also thank the staff of RACOM s.r.o., especially Karel and Marek, who have helped me during the period of measurements by answering questions and providing a conducive environment for my work.

Finally, I wish to express my sincere appreciation to Mr. Kwaku Boadu for his support, guidance and encouragement throughout my studies at Blekinge Institute of Technology, Sweden.

Table of Contents

Abstract 2

Acknowledgements 3

Table of Contents 4

List of Figures 6

List of Tables 8

1 Introduction 9

2 Modulation 10

2.1 Modulation Format 10

2.2 Digital Modulation – an Overview 10

2.3 Factors that Influence the Choice of Digital Modulation 11 2.4 Additive White Gaussian Noise Channel 12

3 FSK Background 15

3.1 Constant Envelope Modulation 15

3.2 Binary Frequency Shift Keying 15

3.2.1 Binary FSK signal Modulator 15 3.2.2 Coherent Demodulation and Error Performance 19 3.2.3 Noncoherent Demodulation and Error Performance 21

3.2.4 Power Spectral Density 24

3.3 M-ary FSK 27

3.3.1 Modulator, Demodulator and Error Performance 27 3.3.2 Coherent Versus Noncoherent 36

3.3.3 Power Spectral Density 38

3.3.4 Bandwidth Efficiency 42

4 Simulations and Results 44

4.1 Simulation of AWGN Channel 44

4.2 Simulation of Binary FSK System in AWGN Channel 45

4.2.1 Coherent System 45

4.2.2 Noncoherent System 46

4.3 Simulation of 4 and 8-level FSK Systems in AWGN Channel 48 5 Measuring of Performance of Commercially Available Radio Modems 52

5.1 RACOM s.r.o 52

5.2 Measurements 52

5.2.1 Extraction of BER from measured PER 54

5.2.2 Calculation of Noise Power 55

5.2.3 Processing of Measurement Data on MR400 Radio Modem 56 5.2.4 Processing of Measurement Data on MX160 Radio Modem 58

6 Comparison of Performance of 2, 4 and 8-level FSK Systems 62

6.1 Bounds on Communication 62

6.2 Comparison of Performance of Simulated FSK Systems 64

6.3 Theory versus Reality 66

6.4 Comparison of Performance using Shannon’s Capacity Curve 67

7 Conclusions 69

8 References 70

A Acronyms 71

List of Figures

3.1 Noncoherent BFSK modulator 16

3.2 Coherent BFSK modulator 17

3.3 Coherent BFSK demodulator: correlator implementation 19 3.4 Coherent BFSK demodulator: matched filter implementation 20 3.5 Probability of error of coherently demodulated BFSK signal 21 3.6 BFSK noncoherent demodulator: correlator-squarer implementation 22 3.7 BFSK noncoherent demodulator: matched filter implementation 23 3.8 Probability of error of noncoherently demodulated BFSK signal 24

3.9 Coherent M-ary FSK modulator 27

3.10 Coherent M-ary FSK demodulator: correlator implementation 28 3.11 Coherent M-ary FSK demodulator: matched filter implementation 28 3.12 Bit error probability of coherently demodulated M-ary FSK signals 30 3.13 Symbol error probability of coherently demodulated M-ary FSK signals 30

3.14 Noncoherent M-ary FSK modulator 31

3.15 Noncoherent M-ary FSK demodulator: correlator-squarer implementation 32 3.16 Noncoherent M-ary FSK demodulator: matched filter implementation 33 3.17 Noncoherent M-ary FSK demodulator: envelope detector implementation 34 3.18 Bit error probability of noncoherently demodulated M-ary FSK signals 35 3.19 Symbol error probability of noncoherently demodulated M-ary FSK signals 35 3.20 Comparison of BER for coherent and noncoherent BFSK 36 3.21 Comparison of BER for coherent and noncoherent 4-FSK 37 3.22 Comparison of BER for coherent and noncoherent 8-FSK 37 3.23 Power-density spectrum of BFSK signal (h = 0.5, 0.6, 0.7) 39 3.24 Power-density spectrum of BFSK signal (h = 0.8, 0.9, 0.95) 39 3.25 Power-density spectrum of 4-FSK signal (h = 0.2, 0.3, 0.4) 40 3.26 Power-density spectrum of 4-FSK signal (h= 0.5, 0.6, 0.7) 40 3.27 Power-density spectrum of 8-FSK signal (h = 0.125, 0.2, 0.3) 41 3.28 Power-density spectrum of 8-FSK signal (h = 0.4, 0.5, 0.6) 41 3.29 Power-density spectra of M-ary FSK signals for M = 2, 4 and 8 (h = 0.5) 42 4.1 Simulation model for coherent BFSK system 45 4.2 Performance of simulated coherent BFSK system 46 4.3 Simulation model for noncoherent BFSK system 47 4.4 Performance of simulated noncoherent BFSK system 48 4.5 Simulation model for coherent 4-level FSK system 49 4.6 Performance of simulated coherent 4-level FSK system 50 4.7 Performance of simulated noncoherent 4-level FSK system 50 4.8 Performance of simulated coherent 8-level FSK system 51 4.9 Performance of simulated noncoherent 8-level FSK system 51

5.1 MR400 radio modem 53

5.2 MX160 radio modem 53

5.3 PER versus packet length at -108 dBm for MR400 radio modem 56

5.4 Performance curve of MR400 radio modem 58 5.5 PER versus packet length at -104 dBm for MX160 radio modem 59

5.6 Performance curve of MX160 radio modem 60

5.7 Comparison of performance of MR400 and MX160 radio modems 61 6.1 Energy versus spectral efficiency of an optimum system 63 6.2 Comparison of simulated symbol error probabilities for coherent M-ary FSK 64 6.3 Comparison of simulated bit error probabilities for coherent M-ary FSK 65 6.4 Comparison of simulated symbol error prob for noncoherent M-ary FSK 65 6.5 Comparison of simulated bit error probabilities for noncoherent M-ary FSK 66 6.6 Comparison of theoretical and practical BER for 2 and 4-level FSK systems 67 6.7 Comparison of theoretical and practical orthogonal modulation techniques

at BER of 10^-5 68

List of Tables

3.1 Bandwidth efficiency of coherent M-ary FSK signals 43 5.1 Technical data for MR400 and MX160 radio modems 53 5.2 Raw data from measurements on MR400 radio modem 54 5.3 Raw data from measurements on MX160 radio modem 54 5.4 Processed data from measurements on MR400 radio modem 56 5.5 Processed data from measurements on MX160 radio modem 58 6.1 Spectral efficiency and S/N required to achieve BER of 10^-5 67

Chapter 1 Introduction

M-ary frequency shift keying (FSK) is a power efficient modulation scheme whose efficiency improves as the number of frequencies employed (M) increases at the expense of additional complexity and smaller bandwidth efficiency. This scheme has been found advantageous in low rate low power applications. There is a difference, however, between the performance of theoretical M-ary FSK systems and that of real world systems employing M-ary FSK modulation schemes.

The primary objective of this thesis is to simulate 2, 4 and 8-level FSK systems in additive white Gaussian noise (AWGN) channel using MATLAB, compare their performance to that of real world systems employing Gaussian minimum shift keying (GMSK) and 4-level FSK modulation techniques and explain the difference. Basic simulations of both coherent and noncoherent 2, 4 and 8-level FSK systems are considered and measurements of performance of real world systems are focused on commercially available data transmission equipment, manufactured by RACOM s.r.o of the Czech Republic, using GMSK and 4-level FSK modulation schemes. In order to establish the difference between the performance of theoretical and practical FSK systems, only basic simulations are considered since their performance is close to that of theory.

In the following chapters, we discuss general modulation in brief, factors that influence the choice of a particular digital modulation scheme, how the comparison of performance of different digital modulation types are made and a model of AWGN channel. Next we follow with the theoretical background of binary and M-ary frequency shift keying (FSK). The next chapter would be simulations of 2, 4 and 8-level FSK systems, for both coherent and noncoherent demodulation, followed by measurements of performance of commercially available data transmission equipment (radio transceivers), by RACOM s.r.o. of Czech Republic, using GMSK and 4-level FSK. Finally we compare the performance of the commercially available data transmission equipment to theory and end with the conclusions.

Chapter 2 Modulation

This chapter describes what modulation is, and an overview of digital modulation. It also describes the factors that influence the choice of a particular digital modulation scheme, and a model for additive white Gaussian noise (AWGN) channel.

2.1 Modulation Format

A modulation format is the means by which information is encoded unto a signal.

Information, or data, can be carried in the amplitude, frequency, or phase of a signal. A modulation scheme can be analog (where data is contained in a set of continuous values) or digital (where the data is contained in a set of discrete values). The information signal is called modulating waveform. When the information is encoded on a signal, the signal is called a modulated waveform. The process of bringing the bandpass signal down to baseband is denoted as demodulation. We distinguish detection from demodulation by denoting detection as the process of extracting the information from the baseband demodulated signal. A receiver consists of a demodulator and a detector. Noncoherent techniques (where the reference phase is unknown; discussed in chapter 3.2.4) can often be implemented in demodulation or detection, such that the two terms can be used interchangeably.

2.2 Digital Modulation – an Overview

Modern communication systems use digital modulation techniques. Advancement in very large-scale integration (VLSI) and digital signal processing (DSP) technology have made digital modulation more cost effective than analog transmission systems. Digital modulation offers many advantages over analog modulation. Some advantages include greater noise immunity and robustness to channel impairments, easier multiplexing of various forms of information (for example, voice, data, and video), and greater security.

Further more, digital transmissions accommodate digital error-control codes which detect and/or correct transmission errors, and support complex signal conditioning and processing techniques such as source coding, encryption, and equalization to improve the performance of the overall communication link. New multipurpose programmable digital signal processors have made it possible to implement digital modulators and demodulators completely in software. Instead of having a particular modem design permanently frozen as hardware, embedded software implementations now allow alterations and improvements without having to redesign or replace the modem.

In digital wireless communication systems, the modulating signal (e.g., the message) may be represented as a time sequence of symbols or pulses, where each symbol has m finite states. Each symbol represent n bits of information, where n = log2m

communication systems, and many more are sure to be introduced. Some of these techniques have subtle differences between one another, and each technique belongs to a family of related modulation methods. For example, Frequency Shift Keying (FSK) may be either coherently or noncoherently detected; and may have two, four, eight or more possible levels per symbol, depending on the manner in which information is transmitted within a single symbol [1].

2.3 Factors That Influence the Choice of Digital Modulation

Several factors influence the choice of a digital modulation scheme. A desirable modulation scheme provides low bit error rates at low received signal-to-noise ratios, performs well in multipath and fading conditions, occupies a minimum bandwidth, and is easy and cost effective to implement. Existing modulation schemes do not simultaneously satisfy all of these requirements. Some modulation schemes are better in terms of bit error rate performance, while others are better in terms of bandwidth efficiency. Depending on the demands of a particular application, tradeoffs are made when selecting a digital modulation.

The performance of a modulation scheme is often measured in terms of its power efficiency and bandwidth efficiency. Power efficiency describes the ability of a modulation technique to preserve the fidelity of the digital message at low power levels.

In a digital communication system, in order to increase noise immunity, it is necessary to increase the signal power. However, the amount by which the signal power should be increased to obtain a certain level of fidelity (i.e., an acceptable bit error probability) depends on the particular type of modulation employed. The power efficiency (sometimes called energy efficiency) of a digital modulation scheme is a measure of how favorable this tradeoff between fidelity and signal power is made, and is often expressed as the ratio of the signal energy per bit to noise power spectral density (Eb/No) required at the input of the receiver for a certain probability of error (say 10^-5).

Bandwidth efficiency describes the ability of a modulation scheme to accommodate data within a limited bandwidth. In general, increasing the data rate implies decreasing the pulse width of a digital symbol, which increases the bandwidth of the signal. Thus, there is an unavoidable relationship between data rate and bandwidth occupancy. However, some modulation schemes perform better than others in making this tradeoff. Bandwidth efficiency reflects how efficiently the allocated bandwidth is utilized and is defined as the ratio of the throughput data rate per Hertz in a given bandwidth. If R is the data rate in bits per second, and B is the bandwidth occupied by the modulated radio frequency signal, then bandwidth efficiency ηB is expressed as

ηB =

B

R bps/Hz (2.1)

The system capacity of a digital communication system is directly related to the bandwidth efficiency of the modulation scheme, since a modulation with a greater value of ηB will transmit more data in a given spectrum allocation.

There is a fundamental upper bound on achievable bandwidth efficiency.

Shannon’s channel coding theorem states that for an arbitrary small probability or error,

the maximum possible bandwidth efficiency is limited by the noise in the channel, and is given by channel capacity formula. The Shannon’s bound for additive white Gaussian noise (AWGN) non-fading channel is given by;

ηBmax = B

C = log2(1 + N

S ) (2.2)

where C is the channel capacity in bits per second, B is the radio frequency (RF) bandwidth, and S/N is the signal-to-noise ratio.

In design of a digital communication system, very often there is a tradeoff between bandwidth efficiency and power efficiency. For example, adding error control coding to a message increases the bandwidth occupancy (and this, in turn, reduces the bandwidth efficiency), but at the same time reduces the required power for a particular bit error rate, and hence trades bandwidth efficiency for power efficiency. On the other hand, higher level modulation schemes (M-ary keying), except M-ary FSK, decrease bandwidth occupancy but increase the required received power, and hence trades power efficiency for bandwidth efficiency.

While power and bandwidth considerations are very important, other factors also affect the choice of a digital modulation scheme. For example, for all personal communication systems which serve a large user community, the cost and complexity of the subscriber receiver must be minimized, and a modulation which is simple to detect is most attractive. The performance of a modulation scheme under various types of channel impairments such as Rayleigh and Ricean fading and multipath time dispersion, given a particular demodulator implementation, is another key factor in selecting a modulation.

In wireless systems where interference is a major issue, the performance of a modulation scheme in an interference environment is extremely important. Sensitivity to detection of time jitter, caused by time-varying channels, is also an important consideration in choosing a particular modulation scheme. In general, the modulation, interference, and implementation of the time-varying effects on a channel as well as the performance of the specific demodulator are analyzed as a complete system using simulation to determine relative performance and ultimate selection [1].

2.4 Additive White Gaussian Noise Channel

Additive white Gaussian noise (AWGN) channel is a universal channel model for analyzing modulation schemes. In this model, the channel does nothing but add a white Gaussian noise to the signal passing through it. This implies that the channel’s amplitude frequency response is flat (thus with unlimited or infinite bandwidth) and phase response is linear for all frequencies so that the modulated signal pass through it without any amplitude or phase loss or distortion of frequency components. Fading does not exist.

The only distortion is introduced by the AWGN. The received signal is then equal to

r(t) = s(t) + n(t) (2.3)

where n(t) is the additive white Gaussian noise, and s(t) is the modulated signal.

The whiteness of n(t) implies that it is a stationary random process with a flat power spectral density (PSD) for all frequencies. It can be observed, however, that if Sn(f) = C for all frequencies, where C is a constant, then

( )

S_n

∫

∞

−

=

∫

∞

−

Cdf

so that the total power is infinite. Obviously, no real physical process can have infinite power and, therefore, a white process may not be a meaningful physical process.

However, quantum mechanical analysis of thermal noise shows that it has a power- spectral density given by [2]

S(f) = ²

(

)

in which h denotes Planck’s constant (equal to 6.6 x 10^-34 Js), k is Boltzmann’s constant (equal to 1.38 x 10^-23 J/K), and T denotes temperature in Kelvin.

This spectrum achieves it maximum at f = 0, and the value of this maximum is kT/2. The spectrum goes to zero as f goes to infinity, but the range of convergence to zero is very slow. For instance, at room temperature (T = 300 K), S(f) drops to 90 % of its maximum at about f = 2.0 x 10¹² Hz, which is beyond the frequencies employed in conventional communication systems. From this we conclude that thermal noise, although not precisely white, can be modeled for all practical purposes as a white process with the power spectrum equaling kT/2. The value kT/2 is usually denoted by N0; therefore, the power spectral density of additive white Gaussian noise is usually given as Sn(f) = N0/2 and is sometimes referred to as the two-sided power spectral density, emphasizing that this spectrum extends to both positive and negative frequencies.

According to the Wiener-Khinchine theorem, the autocorrelation function of the AWGN is

R(τ) = E{n(t)n(t-τ)} =

∫

( )

∞

−

= N₀e_j2πfτdf

∫

∞

∞

−

= δ

( )

(2.5)

where δ(τ) is the Dirac delta function. This shows that noise samples are uncorrelated no matter how close they are in time. The samples are also independent since the process is Gaussian.

At any time instance, the amplitude of n(t) obeys a Gaussian probability density function given by

p(η) = ⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ −

2 2

2 exp 2

2 1

σ η

πσ (2.6)

where η is used to represent the values of the random process n(t) and σ² is the variance of the random process. It is interesting to not that σ² = ∞ for AWGN process since σ² is the power of the noise, which is infinite due to its “whiteness”.

However, when r(t) is correlated with an orthonormal function φ (t), the noise in the output has a finite variance. In fact,

r =

∫

∞

−

where s =

∫

∞

−

dt t t s( )φ( )

and n =

∫

∞

−

dt t t n( )φ( ) The variance of n is

E{n²} = E{[

∫

∞

−

dt t t

n( )φ( ) ²}

= E{ ^∞

∫ ∫

∞

−

∞

∞

−

τ τ φ τ

φ t n dtd

t

n( ) ( ) ( ) ( )

=

∫ ∫

∞

−

∞

∞

−

τ τ φ φ

τ t dtd

n t n

E{ ( ) ( )} ( ) ( )

= N δ t τ φ t φ τ dtdτ ) ( ) ( ) 2 (

0 −

∫ ∫

∞

−

∞

∞

−

= N t dt ) 2 (

0

∫

∞

−

φ

= 2 N0

Then the probability density function of AWGN can be written as

p(n) = ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ −

0 2 0

1 exp

N n πN

Strictly speaking, the AWGN channel does not exist since no channel can have an infinite bandwidth. However, when the signal bandwidth is smaller than the channel bandwidth, many practical channels are approximately an AWGN channel. For example, the line-of-sight (LOS) radio channels, including fixed terrestrial microwave links and fixed satellite links, are approximately AWGN channels when the weather is good.

Wideband coaxial cables are also approximately AWGN channels since there is no other interference except Gaussian noise [3].

Chapter 3

FSK Background

This chapter talks briefly about constant envelope modulation; its advantages and disadvantages, and then discusses binary and M-ary FSK in detail.

3.1 Constant Envelope Modulation

Many practical radio communication systems use nonlinear modulation methods, where the amplitude of the carrier is constant, regardless of the variation in the modulating signal. The constant envelope family of modulations has the advantage of satisfying a number of conditions [1], some of which are:

• Power efficient class C amplifiers can be used without introducing degradation in the spectrum occupancy of the transmitted signal.

• Low out-of-band radiation of the order of -60 dB to -70 dB can be achieved.

• Limiter-discriminator detector can be used, which simplifies receiver design and provides high immunity against random frequency modulation noise and signal fluctuations due to Rayleigh fading.

While constant envelope modulations have many advantages, they occupy a larger bandwidth than linear modulation schemes. In situations where bandwidth efficiency is more important than power efficiency, constant envelope modulation is not well-suited.

3.2 Binary Frequency Shift Keying

3.2.1 Binary FSK Signal and Modulator

In binary frequency shift keying (BFSK), the frequency of a constant amplitude carrier signal is switched between two values according to the two possible message states, corresponding to a binary 1 or 0. Depending on how the frequency variations are imparted into the transmitted waveform, the FSK signal will have either a discontinuous phase or a continuous phase at bit transmissions. In general, a BFSK signal may be represented as

s1(t) = 2 cos(2 )

1

1 φ

πft+ T

E

b

b , 0 ≤ t ≤ Tb , for binary 1

s2(t) = 2 cos(2 )

2

2 φ

πf t+ T

E

b

b , 0 ≤ t ≤ Tb , for binary 0 (3.1)

where φ₁ and φ₂ are initial phases at t = 0, Tb is the bit period of the binary data, and Eb is the transmitted signal energy per bit.

One obvious way to generate a FSK signal is to switch between two independent oscillators according to whether the data bit is a 0 or a 1, as shown in Figure 3.1.

Normally this form of FSK generation results in a waveform that is discontinuous at the switching times, and for this reason this type of FSK is called discontinuous of noncoherent FSK. Equation (3.1) represents a discontinuous FSK signal since φ₁ and φ₂ are not the same in general.

Oscillator 1

s1(t) =

Figure 3.1 Noncoherent BFSK modulator.

ince the phase discontinuities pose several problems, such as spectral spreading and

The second type of FSK is the coherent one where the two signals have the same S

spurious transmissions, discontinuous FSK is generally not used in highly regulated wireless systems.

initial phase φ at t = 0;

s1(t) = 2 cos(2 )

1 φ

πft+ T

E

b

b , 0 ≤ t ≤ Tb , for binary 1

2(t) =

s 2 cos(2 )

2 φ

πf t+ T

E

b

b , 0 ≤ t ≤ Tb , for binary 0 (3.2) Control line

Binary data input, a_k )

2 2 cos(

1

1 φ

πft+ T

E

b b

Oscillator 2

s2(t) = 2 cos(2 )

2

2 φ

πf t+ T

E

b b

Multiplexer f1, Ø1

f

f2, Ø2

i, Øi

This type of FSK can be generated by the modulator as shown in Figure 3.2. The

or coherent demodulation of the coherent FSK signal, the two frequencies are so chosen

= 0 That is

) 2

cos(

) 2

cos( ₂

0

1 φ π φ

π + +

∫

=

frequency synthesizer generates two frequencies, f and f1 2, which are synchronized. The binary input data controls the multiplexer. The bit timing must be synchronized with the carrier frequencies. If a 1 is present, s (t) will pass and if a 0 is present, s1 2(t) will pass.

s1(t) and s2(t) are always there regardless of the data input.

Figure 3.2 Coherent BFSK modulator.

F

that the two signals are orthogonal:

dt t s t s

Tb

) ( ) ( ₂

0

∫

dt t f t

f

Tb

[ ] [

{

}

T

∫

0

2 1 2

1 ) 2 cos2 ( )

( 2 2 cos

1 π φ π

]

=

{ [

] }

f

f_{1 2} sin 2 ( ^{1 2}) 2 ⁰ )

( 4

1 π φ

π + ^{+ +} + f f t ^Tb

f

f_{1 2} sin2 ( ^{1 2}) ⁰ )

( 4

1 −

− π

π

s₁(t) =

Control line

Binary data input, ak

) 2

2 cos(

1 φ

πft+ T

E

b b

Frequency Synthesizer

s₂(t) = 2 cos(2 )

2 φ

πf t+ T

E

b b

Multiplexer

φ

1, f

,φ fi

φ

2, f

[

]

f

f_{1 2} cos2 sin2 ( ^{1 2}) sin2 cos2 ( ^{1 2}) ⁰ )

( 4

1 + + +

+ φ π φ π

π ⁺

=

Tb

t f f f

f_{1 2} sin2 ( ^{1 2}) ⁰ )

( 4

1 −

− π

π

{

}

π( ) ^{cos2 sin2 ( ) sin2 cos( ) sin2} 4

1

2 1 2

1 2

1

− +

+

+ f + f T^b f f T^b

f

f +

=

Tb

f f f

f sin2 ( )

) (

4 1

2 1 2

1

− π −

π

= 0 This implies that

Tb

f

f )

(

2π ₁+ ₂ = 2πn (3.3)

and

Tb

f

f )

(

2π ₁− ₂ mπ

= (3.4)

where and are integers. Solving equations (3.3) and (3.4) simultaneously leads to m n

Tb

m n

4 2 +

Tb

m n

4 2 − f1 = and f₂ =

Δf

2 =

Tb

m

2 2

1 f

f − =

Tb

4 Thus we conclude that for orthogonality and f₁ f₂must be an integer multiple of 1

and their difference must be integer multiple of Tb

2

1 . Using Δf we can rewrite the two frequencies as

Tb

n 2 2

2

1 f

f +

f1 = f_c+Δf and f₂= f_c−Δf , which leads to = f_c = ,

Tb

2 where is the nominal carrier frequency which must be an integer multiple of f_c 1 for orthogonality.

Tb

When the separation is chosen as 1 , then the phase continuity will be maintained at bit transitions, and the FSK is called Sunde’s FSK [3]. As a matter of fact, if the separation is

Tk , where k is an integer, the phase of the coherent FSK of equation (3.2) is b

Tb

2 always continuous. The minimum separation for orthogonality between and f₁ f₂is 1 . However, this separation cannot guarantee continuous phase. A particular form of FSK called minimum shift keying (MSK) not only has the minimum separation but also has continuous phase. However, MSK is much more than ordinary FSK, it has properties that ordinary FSK doesn’t have. It must be generated by methods other than the one described in Figure 3.2. MSK is an important modulation method scheme but not included in the scope of this thesis.

3.2.2 Coherent Demodulation and Error Performance in AWGN Channel

Coherent demodulation requires knowledge of the reference phase or exact phase recovery, meaning local oscillators, phase-lock-loops, and carrier recovery circuits may be required, adding to the complexity of the receiver. The demodulator can be implemented with two correlators as shown in Figure 3.3, where the two reference signals are cos(2πf₁t) andcos(2πf₂t). They must be synchronized with the received signal.

) 2 cos( πf₂t received

signal, r(t)

dt

Tb

∫

) 2 cos( πf₁t

dt

Tb

∫

∑

l₁

l2

l

Threshold detector

0 1

0

Figure 3.3 Coherent BFSK demodulator: correlator implementation.

The receiver is optimum in the sense that it minimizes the error probability for equally likely binary signals. When signal s1(t) is transmitted, the upper correlator yields a signal l1 with a positive signal component and a noise component. However, the lower correlator output l2, due to the signal’s orthogonality, has only a noise component. Thus the output of the summer is most likely above zero, and the threshold detector will most likely produce a 1. When signal s2(t) is transmitted, opposite things happen to the two correlators and the threshold detector will most likely produce a 0. However, due to the noise nature that it values range from -∞ to +∞, occasionally the noise amplitude might overpower the signal amplitude, and then detection errors happen.

An alternative to Figure 3.3 is to use matched filter implementation of the demodulator that matches cos(2πf₁t)- cos(2πf₂t)(Figure 3.4). Both correlator and matched filter implementation are equivalent in terms of error performance.

h(T_b - t) ¹

Sample at t = Tb

l

Threshold detector

0 0 received

signal, r(t)

h(t) = cos(2πf₁t)−cos(2πf₂t)

Figure 3.4 Coherent BFSK demodulator: matched filter implementation.

In the presence of AWGN channel, the received signal is (t) + n(t), i = 1, 2

r(t) = si

where n(t) is the additive white Gaussian noise with zero mean and a two-sided power spectral density N0/2. The bit error probability for an equally likely binary signal is given by

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛ + −

0

2 12 2 1

2 2 N

E E E

Q E ρ

= (3.5) [3]

Pb

where E1 and E are the energies of the binary signals, 2 ρ₁₂is the correlation co-efficient of the binary signals, and Q(z) is the Q-function defined as

dz z

z

) 2 / 2 exp(

1 ₂

∫

∞

Q(z) = π

For Sunde’s FSK signals, E1 = E2 = Eb, and ρ = 0 since the signals are orthogonal. ₁₂ Thus the error probability is

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ N0

Q E^b

= (3.6)

Pb

where Eb is the average transmitted bit energy of the FSK signal. A plot of equation (3.6) is shown in Figure 3.5.

0 5 10 15 10^-8

10^-7 10^-6 10^-5 10^-4 10^-3 10^-2 10^-1 10⁰

Eb/No (dB)

Bit error probability, Pb

Figure 3.5 Probability of error of coherently demodulated BFSK signal.

3.2.3 Noncoherent Demodulation and Error Performance in AWGN Channel Noncoherent demodulation techniques do not require knowledge of the reference phase, eliminating the need for phase-lock-loops, local oscillators, and carrier recovery circuits.

Non coherent demodulation techniques are generally less expensive and easier to build than coherent techniques (since coherent reference signals do not have to be generated), and are often preferred, though they can degrade performance under certain channel conditions.

Coherently generated FSK signals can be noncoherently demodulated to avoid the carrier recovery. Noncoherently generated FSK can only be noncoherently demodulated.

Both are referred to as noncoherent FSK and the demodulation problem becomes a problem of detecting signals with unknown phases. It can be shown that the optimum receiver for noncoherent demodulation is a quadrature receiver. It can be implemented using correlators or equivalently, matched filters. With the assumption that the binary noncoherent FSK signals are equally likely and of equal energies, the demodulator using correlators is shown in Figure 3.6.

r(t)

Comparator dt

Tb

∫

∑

dt

Tb

∫

0

Squarer

Squarer

dt

Tb

∫

∑

dt

Tb

∫

0

Squarer

Squarer )

2 cos( πf₁t

) 2 sin( πf₁t

) 2 cos( πf₂t

) 2 sin( πf₂t

2

l1

2

l2

If l₁²> l₂² choose 1 If l₁²< l₂² choose 0

Figure 3.6 BFSK noncoherent demodulator: correlator-squarer implementation.

The received signal (ignoring noise for the moment) with unknown phase can be written as

), 2

cos( πft+φ

A _i

(t) = i = 1, 2

si

) 2 sin(

sin )

2 cos(

cos ft A ft

A θ π_i − θ π_i

=

) 2 cos(

cos ft

A θ π_i

The signal consists of inphase and quadrature components and )

2 sin(

sin ft

A θ π_i respectively. Thus the signal is partially correlated with cos(2πf_it) and partially correlated tosin(2πf_it). The outputs of the inphase and quadrature correlators will be

2 cosθ ATb

2 sinθ ATb

and , respectively. Depending on the value of the unknown phase, these two outputs could be anything in ⎟

⎠

⎜ ⎞

⎝

⎛ − , 2 2

b

b AT

AT . Fortunately the squared sum of these two signals is not dependent on the unknown phase. That is

2 2

2 sin 2

cos ⎟

⎠

⎜ ⎞

⎝ +⎛

⎟⎠

⎜ ⎞

⎝

⎛AT_b θ AT_b θ

2

2 2

Tb

= A

This quantity is actually the mean value of the statistics when the signal sl_i² i(t) is transmitted and noise is taken into consideration. When si(t) is not transmitted the mean value of is zero. The comparator decides which signal was transmitted by checking these .

2

li 2

li

The matched filter equivalence of Figure 3.6, which has the same error performance, is shown in Figure 3.7.

r(t)

Envelope Detector

Envelop Detector

Comparator Sample at t = Tb

Sample at t = Tb

l1

l2

) ( 2

cos πf₁ T_b −t

) ( 2

cos πf₂ T_b −t

if l₁> l₂ choose 1 if l₁< l₂ choose 0

Figure 3.7 BFSK noncoherent demodulator: matched filter implementation.

The probability of error of noncoherent, orthogonal binary FSK signals can be shown to be

P _⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ − 2 0

2exp 1

N E_b

= (3.7)

b

The plot of equation (3.7) is shown in Figure 3.8 below.

0 5 10 15 10^-8

10^-7 10^-6 10^-5 10^-4 10^-3 10^-2 10^-1 10⁰

Eb/No (dB)

Bit error probability, Pb

Figure 3.8 Probability of error of noncoherently demodulated BFSK signal.

It is worth noting that the demodulators in Figures 3.6 and 3.7 are good for equiprobable, equal-energy, noncoherent signals. They do not require the signals to be orthogonal.

However, equation (3.7) is only applicable for orthogonal, equiprobable, equal-energy, noncoherent signals.

It was shown in chapter 3.2.1 that the minimum frequency separation for coherent FSK signals is 1/2T (where T is the symbol period). It can similarly be shown that that the minimum separation for noncoherent FSK signals is 1/T instead of 1/2T. Thus the separations for noncoherent FSK is double that of coherent FSK. Hence more system bandwidth is required for noncoherent FSK for the same symbol rate.

3.2.4 Power Spectral Density of BFSK

The Sunde’s FSK signal, assuming the initial phase to be zero, can be written as

T t T f

E

b c b

b )

2 ( 1 2

2 cos π + ≤t ≤T_b

s1(t) = , 0 , for binary 1

T t T f

E

c

b )

2 ( 1 2

2 cos π − ≤t≤T_b

s2(t) = , 0 , for binary 0 (3.8)

It is obvious from equation (3.8) that Sunde’s FSK signal can be further simplified as

T t T f

E

b c b

b )

2 ( 1 2

2 cos π ±

s(t) =

) 2

2 cos(

b c

b T

t t T f

E π ±π

=

) 2

cos(

b k

c T

a t t f

A π + π

= (3.9)

where

b b

T 2E

A = , and a_k= ± 1 Expanding equation (3.9) leads to

) 2 sin(

) sin(

) 2 cos(

)

cos( f t

T a t A t T f

a t

A _c

b k c

b

kπ π π π

−

s(t) =

) 2 sin(

) sin(

) 2 cos(

)

cos( f t

T Aa t t T f

A t _c

b k c

b

π π π π

− , 0≤t≤T_b

=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎭⎬

⎫

⎩⎨

⎧ + _k ^{−j ft}

b

e c

Tb jAa t

T

A πt π ₂^π

) sin(

) cos(

Re =

= _⎥ (3.10)

⎦

⎢ ⎤

⎣

⎡ _{−j ft} e c

t s( ) ^2π Re ~

)

~s(t

where is the complex envelope of the bandpass signal s(t), and defined as

) sin(

) cos(

b k

b T

jAa t T

A πt π

+ )

~(t s =

It can be shown that the power spectral density (PSD) of a bandpass signal

[

]

Re ⁻

s(t) = is the shifted version of the equivalent baseband signal or the complex envelope ~s(t)’s PSD S_B( f).

[

]

4

1 − + +

) ( f

S_s = ,

where is the power spectral density of the bandpass signal s(t). Therefore it suffices to determine the PSD of the equivalent baseband signal

) ( f S_s

)

~s(t . Since the inphase and quadrature component of the FSK signal of equation (3.10) are independent of each other, the PSD for the complex envelope is the sum of the PSDs of these two components.

) ( f

S_B = S_I(f)+S_Q(f) )

( f

S_I can easily be found since the inphase component is independent of data. It is defined on the entire time axis. Thus

2

cos ⎭⎬⎫

⎩⎨

⎧ ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ Tb

A t

F π

= ) ( f S_I

= ⎥

⎦

⎢ ⎤

⎣

⎡

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ +

⎟⎟+

⎠

⎜⎜ ⎞

⎝

⎛ −

b

b f T

f T A

2 1 2

1

4 δ δ

where F stands for Fourier transform. It is seen that the spectrum of the inphase part of the Sunde’s FSK signal are two delta functions.

It can also be shown that the PSD of a binary, bipolar, equiprobable, stationary, and uncorrelated digital waveform is just equal to the energy spectral density of the symbol shaping pulse divided by the symbol period. The symbol shaping pulse of the quadrature component is sin( )

Tb

A πt

, and therefore

b b

b

T T t

A t

T F ≤ ≤

⎭⎬

⎫

⎩⎨

⎧ ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ ,0

1 sin π ²

) ( f

S_Q =

dt T e

A t ^{j f}

T

b

b π ₂π

0

sin ⁻

∫

⎜⎜ ⎞

⎝

⎛

⎭⎬

⎫

⎩⎨

⎧

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ Tb

A t

F π

sin =

= [1 (2 ) ] ) cos(

2

f 2

T fT AT

b b b

π −

π

Thus,

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

−(2 ) ] 1

[

) cos(

1 2

f 2

T fT AT

T _b

b b

b π

= π ) ( f

S_Q

The complete baseband PSD of the binary FSK signal is the sum of S_I( f) andS_Q( f);

2 2 2

] ) 2 ( 1 [

) cos(

2 2

1 2

1

4 ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ + −

⎥⎦

⎢ ⎤

⎣

⎡

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ +

⎟⎟+

⎠

⎜⎜ ⎞

⎝

⎛ −

f T

fT T A

f T f T

A

b b b

b

b π

δ π δ

) ( f

S_B = (3.11)

3.3 M-ary FSK

3.3.1 Modulator, Demodulator, and Error Performance in AWGN Channel

The coherent modulator of binary FSK in Figure 3.2 can easily be extended to coherent M-ary FSK (Figure 3.9). Here the frequency synthesizer generates M signals with the designed frequencies and coherent phase, and the multiplexer chooses one of the frequencies, according to the n =log₂M bits.

f1

Multiplexer Frequency

Synthesizer

f2

fi

. . .

fM

Figure 3.9 Coherent M-ary FSK modulator.

The coherent M-ary FSK demodulator falls in the general form of detector for M- ary equiprobable, equal-energy signals with known phases. The demodulator consists of a bank of M correlators or matched filters (Figure 3.10 and Figure 3.11). At sample times t = kT, the receiver makes decisions based on the largest output of the correlators or matched filters. It is worth noting that the coherent M-ary FSK receivers in Figures 3.10 and 3.11 only require that the M-ary FSK signals be equiprobable, equal energy, and do not require them to be orthogonal.

S/P Converter b1 b₂ . . .

bn

Control lines

Binary input data

Figure 3.10 Coherent M-ary FSK demodulator: correlator implementation.

Figure 3.11 Coherent M-ary FSK demodulator: matched filter implementation.

) ( 2

cos πf₁ T−t

) ( 2

cos πf₂ T−t

) ( 2

cos πf_M T −t

Sample t = T_b

Sample at t =T_b

Sample at t =T_b .

. .

If l_i>l_j i j≠

∀ choose m_i

mi

l1

l2

lM

. . . Received

signal, r(t)

) 2 cos( πf_Mt Received

signal, r(t)

dt

Tb

∫

dt

Tb

∫

dt

Tb

∫

) 2 cos( πf₁t

) 2 cos( πf₂t

. . .

. . .

l1

If l_i> l_j i j l2

mi

∀ ≠ choose m_i .

. . lM

The exact expression for the symbol error probability for symmetrical signal set, equal energy and equiprobable, is given as

P

( ) _{[ ( ) ]}

dx x

N Q E

x _{s 0} ² _{M 1}

2 1 / exp 2

2

1 1 ⁻

∞

∞

−

⎪⎭ −

⎪⎬

⎫

⎪⎩

⎪⎨

⎧ −

−

−

∫

= (3.12) [3]

s

This expression does not require the signal set to be orthogonal, and cannot be analytically evaluated. If the signal set is equal-energy and orthogonal (not necessarily equiprobable), all distances between any two signals are equal. The distance d = 2E_s , and the upper bound obtained from equation (3.12) is

⎟⎟

⎠

⎞

⎜⎜

⎝

− ⎛

≤

0

) 1

( N

Q E

M ^s

Ps (3.13)

where E_s=E_blog₂M , and Q(z) is the Q-function defined in chapter 3.2.2.

/N

For fixed M this bound becomes increasingly tight as Es 0 is increased. Infact, it becomes a good approximation for P ≤ 10s ^-3.

For equally likely orthogonal M-ary signals, all symbol errors are equiprobable. That is, the demodulator may choose any one of the (M −1)erroneous orthogonal signals with equal probability. Hence it can be shown that the average bit error probability is given by

n s n

1P 2

2 ¹

−

−

s n

M 1P 2 ¹

−

− b=

P = , where n = log M. 2

) and symbol error probability (P

Bit error probability (Pb s) for coherently demodulated, equal-energy, equiprobable, and orthogonal M-ary FSK signals are shown in Figures 3.12 and 3.13.

0 5 10 15 10^-6

10^-5 10^-4 10^-3 10^-2 10^-1

Eb/No (dB)

Bit-error probability, Pb

Bit-error Probability of Coherent M-ary FSK

BFSK 4-FSK 8-FSK 16-FSK 32-FSK 64-FSK

Figure 3.12 Bit error probability of coherently demodulated M-ary FSK.

0 5 10 15

10^-8 10^-7 10^-6 10^-5 10^-4 10^-3 10^-2 10^-1

Eb/No (dB)

Symbol-error probability, Ps

Symbol-error Probability of Coherent M-ary FSK

BFSK 4-FSK 8-FSK 16-FSK 32-FSK 64-FSK

Figure 3.13 Symbol error probability of coherently demodulated M-ary FSK.

/N

It can be seen from Figures 3.12 and 3.13 that for the same Eb 0, error probability reduces when M increases, or for the same error probability, the required E /Nb 0 decrease as M increases. However, the speed of decrease in E /Nb 0 slows down when M gets larger.

The noncoherent modulator for binary FSK in Figure 3.2 can also be easily extended to noncoherent M-ary FSK by simply increasing the number of independent oscillators to M (Figure 3.14).

Multiplexer Oscillator 1

) 2

cos( _{1 1} f¹,φ¹

2 2, πf t+φ

A

Figure 3.14 Noncoherent M-ary FSK modulator

The noncoherent demodulator for M-ary FSK falls in the general form of detector for M- ary equiprobable, equal-energy signals with unknown phases as described in many communication books. The demodulator can be implemented in correlator-squarer form, or matched filter-squarer or matched filter-envelope detector form (Figures 3.15, 3.16 and 3.17).

Oscillator 2 ) 2

cos( _{2 1} f φ

φ πf t+ A

Oscillator M ) 2

cos( f_Mt _M

A π +φ

. . .

. . .

M

fM,φ

S/P Converter b1 b₂ . . . b _n

Control lines

Binary input data

i

fi,φ