Comm-04-Phase and Frequency Modulation

Chapter 4Chapter 4Chapter 4Chapter 4Phase and Frequency ModulationPhase and Frequency Modulation

Wireless Information Transmission System Lab.Wireless Information Transmission System Lab.Institute of Communications EngineeringInstitute of Communications Engineeringg gg gNational Sun National Sun YatYat--sensen UniversityUniversity

OutlineOutline

◊ 4.1 Introduction

◊ 4.2 Basic Definitions

4 3 F M d l i◊ 4.3 Frequency Modulation

◊ 4.4 Phase-locked Loopp

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Chapter 4.1Chapter 4.1Chapter 4.1Chapter 4.1IntroductionIntroduction


4.14.1 IntroductionIntroduction◊ In this chapter, we study a second family of continuous-wave(CW)

modulation systems namely angle modulation in which the anglemodulation systems, namely, angle modulation, in which the angle of the carrier wave is varied according to the baseband signals.

In this method of mod lation the amplit de of the carrier a e is◊ In this method of modulation, the amplitude of the carrier wave is maintained constant.

◊ There are two common forms of angle modulation, namely, phase modulation and frequency modulation.

◊ An important feature of angle modulation is that it can provide better discrimination against noise and interference than amplitude

d l imodulation.

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4.14.1 IntroductionIntroduction◊ However, this improvement in performance is achieved at the

expense of increased transmission bandwidthexpense of increased transmission bandwidth.

◊ Moreover the improvement in the noise performance with angle◊ Moreover, the improvement in the noise performance with angle modulation is achieved at the expense of increased system complexity in both the transmitter and receiver.p y

◊ Such a trade-off is not possible with amplitude modulation.p p

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Chapter 4.2Chapter 4.2Chapter 4.2Chapter 4.2Basic DefinitionsBasic Definitions


4.2 Basic Definitions4.2 Basic Definitions◊ Let θi(t) denote the angle of a modulated sinusoidal carrier at time

t; it is assumed to be a function of the information bearing signalt; it is assumed to be a function of the information–bearing signal or message signal.

◊ We express the resulting angle modulated wave as◊ We express the resulting angle-modulated wave as (4.1)

where A is the carrier amplitude( ) ( )cosc is t A tθ⎡ ⎤= ⎣ ⎦

where Ac is the carrier amplitude.◊ The average frequency in Hertz over an interval from t to t+Δt is

given by ( ) ( )t t tθ θ+ Δgiven by(4.2)

◊ The instantaneous frequency of the angle-modulated signal s(t):

( ) ( ) ( )2

i it

t t tf t

tθ θ

πΔ

+ Δ −=

Δ◊ The instantaneous frequency of the angle-modulated signal s(t):

( ) ( ) ( ) ( ) ( )1lim lim i i it t t d tf t f t

θ θ θ⎡ ⎤+ Δ −= = =⎢ ⎥

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( ) ( )0 0

lim lim2 2i tt t

f t f tt dtπ πΔΔ → Δ →

= = =⎢ ⎥Δ⎣ ⎦

4.2 Basic Definitions4.2 Basic Definitions◊ For an unmodulated carrier, the angle θi(t) is given by

( ) 2i c ct f tθ π φ= +

and corresponding phasor rotates with a constant angular velocity equal to 2πfc. The constant is the value of θi(t) at t=0.cφ

◊ There are an infinite number of ways in which the angle θi(t) may be varied in some manner with the message (baseband) signalbe varied in some manner with the message (baseband) signal.

W h ll id l t l d th d h◊ We shall consider only two commonly used methods, phase modulation and frequency modulation.

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4.2 Basic Definitions4.2 Basic Definitions◊ Phase modulation (PM) is that form of angle modulation in which

the instantaneous angle θi(t) is varied linearly with the messagethe instantaneous angle θi(t) is varied linearly with the message signal as shown by

(4.4)( ) ( )2i c pt f t k m tθ π= + ( )The term 2πfct represents the angle of the unmodulated carrier; kprepresents the phase sensitivity of the modulator, expressed in p p y pradians per volt on the assumption that m(t) is a voltage waveform.

For convenience, we have assumed in Eq. (4.4) that the angle of the unmodulated carrier is zero at t=0. The phase-modulated signal s(t)i h d ib d i h i d i bis thus described in the time domain by

(4.5)( ) ( )cos 2s t A f t k m tπ⎡ ⎤= +⎣ ⎦

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( ) ( )cos 2c c ps t A f t k m tπ⎡ ⎤+⎣ ⎦

4.2 Basic Definitions4.2 Basic Definitions◊ Frequency modulation (FM) is that form of angle modulation in

which the instantaneous frequency fi(t) is varied linearly with the imessage signal m(t), as shown by

(4.6)( ) ( )i c ff t f k m t= +fc : The frequency of the unmodulated carrierkf : The frequency sensitivity of the modulator (Hertz per volt)f

Integrating Eq. (4.6) with respect to time and multiplying the result by 2π, we get

( ) ( )2 2t

t f t k m dθ π π τ τ+ ∫ (4.7)where, for convenience, we have assumed that the angle of the

( ) ( )0

2 2i c ft f t k m dθ π π τ τ= + ∫

t⎡ ⎤

unmodulated carrier wave is zero at t=0. The frequency-modulated signal is therefore described in the time domain by

(4 8)

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( ) ( )0

cos 2 2t

c c fs t A f t k m dπ π τ τ⎡ ⎤= +⎢ ⎥⎣ ⎦∫ (4.8)

4.2 Basic Definitions4.2 Basic Definitionsa) Carrier wave

b) Sinusoidal modulating signal

c) Amplitude-modulated signal

d) Phase-modulated signal

) F d l t d i l

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e) Frequency-modulated signal

Properties of AngleProperties of Angle--Modulated WavesModulated Waves

◊ Property 1: Constancy of Transmitted Power:◊ From both Eqs. (4.4) and (4.7), we readily see that the

amplitude of PM and FM waves is maintained at a constant l l t th i lit d A f ll ti tvalue equal to the carrier amplitude Ac for all time t,

irrespective of the sensitivity factors kp and kf . ◊ Consequently the average transmitted power of angle◊ Consequently, the average transmitted power of angle-

modulated waves is a constant, as shown by 1 2

(4.9)where it is assumed that the load resistance is 1 ohm

1 22

P Aav c=

where it is assumed that the load resistance is 1 ohm.2VP

R⎛ ⎞

=⎜ ⎟⎝ ⎠

12

R⎝ ⎠


◊ Property 2: Nonlinearity of the Modulation ProcessB th PM d FM i l t th i i l f iti◊ Both PM and FM waves violate the principle of superposition.

◊ For example, the message signal m(t) is made up of two different components m (t) and m (t): ( ) ( ) ( )m t m t m t+different components, m1(t) and m2(t):

◊ Let s(t), s1(t), and s2(t) denote the PM waves produced by m(t), m (t) and m (t) in accordance with Eq (4 4) respectively We

( ) ( ) ( )1 2m t m t m t= +

m1(t), and m2(t) in accordance with Eq. (4.4), respectively. We may express these PM waves as follows:

( )⎡ ⎤

( ) ( ) ( )2 4.4i c pt f t k m tθ π= +

( ) ( ) ( )( )1 2cos 2c c ps t A f t k m t m tπ⎡ ⎤= + +⎣ ⎦

( ) ( )1 1cos 2c c ps t A f t k m tπ⎡ ⎤= +⎣ ⎦ ( ) ( ) ( )s t s t s t≠ +

( ) ( ) ( )1 2m t m t m t= +

d l i ff i i f

( ) ( )2 2cos 2c c ps t A f t k m tπ⎡ ⎤= +⎣ ⎦( ) ( ) ( )1 2s t s t s t≠ +

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◊ Frequency modulation offers superior noise performance compare to amplitude modulation,

Properties of AngleProperties of Angle--Modulated WavesModulated Waves◊ Property 3: Irregularity of Zero-Crossings

◊ Zero-crossing are defined as the instants of time at which a gwaveform changes its amplitude from positive to negative value or the other way around.

◊ The zero-crossings of a PM or FM wave no longer have a perfect regularity in their spacing across the time-scale.

◊ The irregularity of zero-crossings in angle-modulated waves is attributed to the nonlinear character of the modulation process.

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◊ Property 4: Visualization Difficulty of Message Waveform◊ In AM, we see the message waveform as the envelope of the

modulated wave, provided the percentage modulation is less than 100 t100 percent.(AM: The percentage modulation over 100 percent→phase reversal→distortion)reversal→distortion)

◊ This is not so in angle modulation as illustrated by the◊ This is not so in angle modulation, as illustrated by the corresponding waveform of Figures 4.1d and 4.1e for PM and FM, respectively., p y

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◊ Property 5-Trade-OFF of Increased Transmission B d id h f I d N i P fBandwidth for Improved Noise Performance◊ An important advantage of angle modulation over amplitude

d l i i h li i f i d i fmodulation is the realization of improved noise performance.

◊ This advantage is attributed to the fact that the transmission of a◊ This advantage is attributed to the fact that the transmission of a message signal by modulating the angle of a sinusoidal carrier wave is less sensitive to the presence of additive noise than transmission by modulating the amplitude of the carrier.

Th i t i i f i hi d t th◊ The improvement in noise performance is achieved at the expense of a corresponding increase in the transmission bandwidth requirement of angle modulation

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bandwidth requirement of angle modulation.


◊ Property 5-Trade-OFF of Increased Transmission d id h f d i fBandwidth for Improved Noise Performance

◊ The use of angle modulation offers the possibility of exchanging i i h i i b d id h f i ian increase in the transmission bandwidth for an improvement in

noise performance.

◊ Such a trade-off is not possible with amplitude modulation since the transmission bandwidth of an amplitude modulated wave isthe transmission bandwidth of an amplitude-modulated wave is fixed somewhere between the message bandwidth W and 2W, depending on the type of modulation employed.p g yp p y

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Example 4.1 ZeroExample 4.1 Zero--CrossingsCrossings

◊ Consider a modulating wave m(t) that increases linearly with time t, starting at t=0 as shown bystarting at t 0, as shown by

( ), 0at t

m t≥⎧

= ⎨

where a is the slope parameter (see Figure 4 2a) In what

( )0, 0

m tt

= ⎨ <⎩

where a is the slope parameter (see Figure 4.2a). In whatfollows, we study the zero-crossings of the PM and FM wavesproduced by m(t) for the following set of parameters:produced by m(t) for the following set of parameters:

11 Hz4

1 volt/s

cf

a

=

=

18

1 volt/sa =


Fig 4 2 Starting at time t = 0 the figure displays (a) linearly increasing message signal m(t)

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Fig. 4.2 Starting at time t = 0, the figure displays (a) linearly increasing message signal m(t), (b)phase-modulated wave, and (c) frequency-modulated wave.

Example 4.1 ZeroExample 4.1 Zero--CrossingsCrossings◊ Phase Modulation:

◊ Phase-sensitivity factor k =π/2 radians/volt Applying Eq (4 5)◊ Phase sensitivity factor kp π/2 radians/volt. Applying Eq. (4.5) to the given m(t) yields the PM wave

( ) ( )cos 2 , 0c c pA f t k at tπ⎧ + ≥⎪

( ) ( ) ( )cos 2 4.5c c ps t A f t k m tπ⎡ ⎤= +⎣ ⎦

which is plotted in Figure 4.2b for Ac=1 volt.

( ) ( )( )

cos ,

cos 2 , 0c c p

c c

fs t

A f t t

π

π⎪= ⎨

<⎪⎩which is plotted in Figure 4.2b for Ac 1 volt.

◊ Let tn denote the instant of time at which the PM wave experiences a zero crossing; this occurs whenever the angle of p g; gthe PM wave is an odd multiple of π/2:

2 2 0 1 2pk af t k at f t n nππ π π

⎛ ⎞+ = + = + =⎜ ⎟2 2 , 0,1,2,

2c n p n c nf t k at f t n nπ π ππ

+ + +⎜ ⎟⎝ ⎠

…

12

nt

+=

1 0 1 2t n n= + =

20

2n

pc

t kf a

π

=+

, 0,1,2,2nt n n= + = …

Example 4.1 ZeroExample 4.1 Zero--CrossingsCrossings◊ Frequency Modulation:

◊ Frequency-sensitivity factor kf =1 Hz/volt Applying Eq (4 8)◊ Frequency sensitivity factor, kf 1 Hz/volt. Applying Eq. (4.8) yields the FM wave

( )2cos 2 0A f t k at tπ π⎧ + ≥⎪

( ) ( ) ( )0

cos 2 2 4.8t

c c fs t A f t k m dπ π τ τ⎡ ⎤= +⎢ ⎥⎣ ⎦∫

( ) ( )( )

cos 2 , 0

cos 2 , 0c c f

c c

A f t k at ts t

A f t t

π π

π

⎧ + ≥⎪= ⎨<⎪⎩

which is plotted in Figure 4.2c.◊ Invoking the definition of a zero-crossing, we can obtain:

22 , 0,1,2,2c n f nf t k at n nππ π π+ = + = …

1 1⎛ ⎞⎛ ⎞21 1= , 0,1,2, 2n c c f

f

t f f ak n nak

⎛ ⎞⎛ ⎞− + + + =⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠…

1

21

( )1= 1 9 16 , 0,1,2, 4nt n n− + + = …


◊ Comparing the zero-crossing results derived for PM and FM waves, we may make the following observations once the linear modulatingwe may make the following observations once the linear modulating wave begins to act on the sinusoidal carrier wave:

1. For PM, regularity of the zero-crossings is maintained; the , g y g ;instantaneous frequency changes from the unmodulated value of fc=1/4 Hz to the new constant value of ( )/ 2 0.5Hzc pf k a π+ =

2. For FM, the zero-crossings assume an irregular form; as expected, the instantaneous frequency increases linearly with time t.

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4.2 Basic Definitions4.2 Basic Definitions◊ Comparing Eq. (4.5 ) with (4.8) reveals that an FM signal may be

regarded as a PM signal in which the modulating wave is ( )0

tm dτ τ∫g g g

in place of m(t). ( )

0∫

( ) ( )cos 2 (4.5)c c ps t A f t k m tπ⎡ ⎤= +⎣ ⎦

( ) ( )0

cos 2 2 (4.8)t

c c fs t A f t k m dπ π τ τ⎡ ⎤= +⎢ ⎥⎣ ⎦∫◊ The FM signal can be generated by first integrating m(t) and then

using the result as the input to a phase modulator, as in Figure 4.3a.◊ Conversely, a PM signal can be generated by first differentiating

m(t) and then using the result as the input to a frequency modulator, as in Figure 4 3bas in Figure 4.3b.

◊ We may thus deduce all the properties of PM signals from those of FM signals and vice versa Henceforth we concentrate our

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FM signals and vice versa. Henceforth, we concentrate our attention on FM signals.

4.2 Basic Definitions4.2 Basic Definitions

Figure 4.3 Illustrating the relationship between frequency modulation and phase modulation. (a) Scheme for generating an FM wave by using a phase modulator, (b) scheme for generating a PM wave by using a frequency modulatorgenerating a PM wave by using a frequency modulator.

( )θ ( )f

Unmodulatedsignal

( )i tθ ( )if t

2 cf tπ cfsignal

PM signal ( )2 c pf t k m tπ + ( )2

pc

k dm tf

dtπ+

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FM signal ( )0

2 2t

c ff t k m dπ π τ τ+ ∫ ( )c ff k m t+

Chapter 4.3 Chapter 4.3 Chapter 4.3 Chapter 4.3 Frequency ModulationFrequency Modulation


4.3 Frequency Modulation4.3 Frequency Modulation

◊ The FM signal s(t) define by Eq. (4.8) is a nonlinear function of the modulating signal m(t) which makes frequency modulation amodulating signal m(t), which makes frequency modulation a nonlinear modulation process.

Ho then can e tackle the spectral anal sis of FM signal? We◊ How then can we tackle the spectral analysis of FM signal? We propose to provide an empirical answer to this important question by proceeding in the same manner as with AM modulation, that is, weproceeding in the same manner as with AM modulation, that is, we consider the simplest case possible, namely, single-tone modulation.

◊ Consider then a sinusoidal modulating signal define by◊ Consider then a sinusoidal modulating signal define by

(4 10)( ) ( )2t A f t (4.10)( ) ( )cos 2m mm t A f tπ=

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4.3 Frequency Modulation4.3 Frequency Modulation◊ The instantaneous frequency of the resulting FM signal is

( ) ( ) ( )cos 2 cos 2f t f k A f t f f f tπ π+ + Δ (4.11)(4.12)

( ) ( ) ( )cos 2 cos 2i c f m m c mf t f k A f t f f f tπ π= + = + Δ

f mf k AΔ =◊ The quantity Δf is called the frequency deviation, representing the

maximum departure of the instantaneous frequency of the FM signal form the carrier frequency fc.q y fc

◊ A fundamental characteristic of an FM signal is that the frequency deviation Δf is proportional to the amplitude of the modulating signal and is independent of the modulating frequencyindependent of the modulating frequency.

◊ Using Eq. (4.11), the angle θi(t) of the FM signal is obtained as

( ) ( ) ( )2 2 i 2t ff d f fθ Δ∫

◊ The ratio of the frequency deviation Δf to the modulation

( ) ( ) ( )0

2 2 sin 2i i c mm

ft f t dt f t f tf

θ π π π= = +∫

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frequency fm is commonly called the modulation index of the FM signal.


◊ The modulation index is denoted by β:m

ff

β Δ=

mf

( ) ( )2 sin 2i c mt f t f tθ π β π= +

◊ The parameter β represents the phase deviation of the FM signal, i.e. the maximum departure of the angle θi(t) from the angle 2πfct of the unmodulated carrier. β is measured in radians.

◊ The FM signal itself is given by

(4.16)( ) ( )cos 2 sin 2c c ms t A f t f tπ β π⎡ ⎤= +⎣ ⎦

Depending on the value of the modulation index β, we may distinguish two cases of frequency modulation:

N b d FM f hi h β i ll d di

28

◊ Narrow-band FM, for which β is small compared to one radian.◊ Wide-band FM, for which β is large compared to one radian.


◊ Narrow-band frequency modulation◊ Consider Eq (4 16) which defines an FM signals resulting◊ Consider Eq. (4.16), which defines an FM signals resulting

form the use of sinusoidal modulating signal. Expanding this relation, we get, g

A i th t th d l ti i d β i ll d t( ) ( ) ( ) ( ) ( ) ( )cos 2 cos sin 2 sin 2 sin sin 2 4.17c c m c c ms t A f t f t A f t f tπ β π π β π⎡ ⎤ ⎡ ⎤= −⎣ ⎦ ⎣ ⎦

◊ Assuming that the modulation index β is small compared to one radian, we may use the following two approximations:

⎡ ⎤⎡ ⎤ ( ) ( )sin sin 2 sin 2m mf t f tβ π β π⎡ ⎤⎣ ⎦( )cos sin 2 1mf tβ π⎡ ⎤⎣ ⎦

( ) ( ) ( ) ( ) ( )cos 2 sin 2 sin 2 4 18s t A f t A f t f tπ β π π( ) ( ) ( ) ( ) ( )cos 2 sin 2 sin 2 4.18c c c c ms t A f t A f t f tπ β π π−

( ) ( ) ( ) ( ){ } ( )1cos 2 cos 2 cos 2 4.192c c c c m c ms t A f t A f f t f f tπ β π π⎡ ⎤ ⎡ ⎤+ + − −⎣ ⎦ ⎣ ⎦

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( ) ( ) ( ) ( ){ } ( )2 ⎣ ⎦ ⎣ ⎦

( ) ( )1sin sin cos cos2

α β α β α β⎡ ⎤= − − +⎣ ⎦


◊ This expression is somewhat similar to the corresponding one defining an AM signal (from Example 3 1):defining an AM signal (from Example 3.1):

( ) ( ) ( ) ( ){ } ( )1cos 2 cos 2 cos 2 4.202AM c c c c m c ms t A f t A f f t f f tπ μ π π⎡ ⎤ ⎡ ⎤= + + + −⎣ ⎦ ⎣ ⎦

where μ is the modulation factor of the AM signal.

{ }2 ⎣ ⎦ ⎣ ⎦

◊ Compare Eqs. (4.19) and (4.20), we see that the basic difference between an AM signal and a narrow-band FM gsignal is that the algebraic sign of the lower side frequency in the narrow-band FM is reversed.

◊ Thus, a narrow-band FM signal requires essentially the same transmission bandwidth (i e 2f ) as the AM signal

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transmission bandwidth (i.e. 2fm) as the AM signal.


◊ Example 4.2 Phase Noise◊ Phase noise is often introduced by oscillators in band-pass

communications and has a number of causes.◊ Some causes are the deterministic, such as those created by

changes in oscillator temperature, supply voltage, physical vibration magnetic field humidity or output load impedancevibration, magnetic field, humidity, or output load impedance.

◊ The phase noise due to these sources may be minimized by good designgood design.

◊ Other sources are categorized as random, which can be controlled but not eliminated by appropriate circuitry, such as y pp p y,phase-lock loops (PLL).

◊ The phase noise introduced by oscillators has a multiplicative

31

effect on an angle-modulated signal.


◊ Example 4.2 Phase Noise (cont.)F l if ( ) i l d l d i l d ( ) i h◊ For example, if s(t) is an angle-modulated signal, and c(t) is the receiver oscillator, having phase noise φn(t), then when translating the signal from f to f (see section 3 7) the output istranslating the signal from fc to fb (see section 3.7), the output is

( ) ( ) ( ) ( ) ( )cos 2 cos 2c c c b ns t c t A f t t f f t tπ φ π φ⎡ ⎤⎡ ⎤= + × − +⎣ ⎦ ⎣ ⎦

( ) ( )( ) ( ) ( ) ( )( )cos 2 cos 2 22

cb n c b n

A f t t f f t t

A

π φ φ π φ φ⎡ ⎤= + − + − + +⎣ ⎦

where the high frequency term has been removed by a band pass

( ) ( )cos 22

cb n

A f t tπ φ φ⎡ ⎤≈ + −⎣ ⎦

where the high frequency term has been removed by a band-pass filter centered around fb.

◊ Thus the phase noise of the oscillator directly affects the

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◊ Thus the phase noise of the oscillator directly affects the information component of the angle-modulated signal.


◊ Wide-band frequency modulationThe following studies the spectrum of the single tone FM signal◊ The following studies the spectrum of the single-tone FM signal of Eq. (4.16) for an arbitrary value of the modulation index β.

◊ By using the complex representation of band pass signals

( ) ( ) ( )cos 2 sin 2 4.16c c ms t A f t f tπ β π⎡ ⎤= +⎣ ⎦

◊ By using the complex representation of band-pass signals described in Chapter 2: (Carrier frequency fc compared to the bandwidth of the FM signal is large enough)g g g )

( ) ( )( ) ( )Re exp 2 sin 2 4.21c c ms t A j f t j f tπ β π⎡ ⎤= +⎣ ⎦

( ) ( )Re exp 2 cs t j f tπ⎡ ⎤= ⎣ ⎦

33

( ) ( ) periodic functiowhere ex np sin 2c ms t A j f tβ π →⎡ ⎤= ⎣ ⎦


◊ Wide-band frequency modulationWe may therefore expend in the form of complex Fourier( )t◊ We may therefore expend in the form of complex Fourier series as follows:

(4 23)

( )s t

( ) ( )exp 2n ms t c j nf tπ∞

= ∑ (4.23)( ) ( )n=−∞∑

( ) ( )1 2

1 2exp 2m

m

f

n m mfc f s t j nf t dtπ

−= −∫

(4.24)(4 26)

( )1 2

1 2exp sin 2 2m

m

f

m c m mff A j f t j nf t dtβ π π

−⎡ ⎤= −⎣ ⎦∫

A2 mx f tπ=

(4.26)

(4 28)

( )exp sin2

cn

Ac j x nx dxπ

πβ

π −⎡ ⎤= −⎣ ⎦∫

( ) ( )1 iJ j dπ

β β⎡ ⎤∫( )A J β (4.28)

(4 31)

( ) ( )exp sin2nJ j x nx dx

πβ β

π −⎡ ⎤= −⎣ ⎦∫∵( )n c nc A J β=

( ) ( ) ( )∞⎡ ⎤⎡ ⎤∑

nth order Bessel function of the first kind.

34

(4.31)( ) ( ) ( )Re exp 2c n c mn

s t A J j f nf tβ π=−∞

⎡ ⎤⎡ ⎤= ⋅ +⎣ ⎦⎢ ⎥⎣ ⎦∑

4.3 Frequency Modulation4.3 Frequency Modulation◊ Taking the Fourier transforms of both sides of Eq. (4.31)

(4.32)( ) ( ) ( ) ( )cAS f J f f nf f f nfβ δ δ∞

⎡ ⎤= + + +⎣ ⎦∑ ( )

◊ In Figure 4.6 we have plotted the Bessel function Jn(β) versus the d l ti i d β f diff t iti i t l f

( ) ( ) ( ) ( )2 n c m c m

n

S f J f f nf f f nfβ δ δ=−∞

⎡ ⎤= − − + + +⎣ ⎦∑

modulation index β for different positive integer values of n.

35FIGURE4.6 Plots of Bessel functions of the first kind.


◊ We can develop further insight into the behavior of the Bessel function J (β) by making use of the following properties:function Jn(β) by making use of the following properties:

1. For n even, we have Jn(β)=J-n(β); on the other hand, for n odd, we ha e J (β) J (β) That ishave Jn(β)=-J-n(β). That is

(4.33)2 F ll l f th d l ti i d β h

( ) ( ) ( )1 for all nn nJ J nβ β−= −

2. For small values of the modulation index β, we have

( )0 1J β ⎫⎪

(4.34)( )1 2J ββ

⎪⎪⎬⎪

3. ( ) 0, 2nJ nβ

⎪> ⎪⎭

( )2 1J β∞

∑36

(4.35)( )2 1nn

J β=−∞

=∑


◊ Thus, using Eqs. (4.32) through (4.35) and the curves of Figure 4.6, we may make the following observations:we may make the following observations:

1. The spectrum of an FM signal contains a carrier component (n=0) and an infinite set of side frequencies located symmetrically on q y yeither side of the carrier at frequency separations of fm, 2fm, 3fm, ….(An AM system gives rise to only one pair of side frequencies.)( y g y p q )

2. For the special case of β small compared with unity, only the Bessel coefficients J0(β) and J1(β) have significant values (see 4 34) so thatcoefficients J0(β) and J1(β) have significant values (see 4.34), so that the FM signal is effectively composed of a carrier and a single pair of side frequencies at fc ± fm.(This situation corresponds to the special case of narrowband FM that was considered previously)

37


3. The amplitude of the carrier component of an FM signal is dependent on the modulation index β The physical explanation for thison the modulation index β. The physical explanation for this property is that the envelope of an FM signal is constant, so that the average power of such a signal developed across a 1–ohm resistor is also constant, as shown by

(4.36)21 21 (Using (4.31) and (4.35))

2 cP A=

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EXAMPLE 4.3 Spectra of FM SignalsEXAMPLE 4.3 Spectra of FM Signals

◊ In this example, we wish to investigate the ways in which variations in the amplitude and frequency of a sinusoidal modulating signalin the amplitude and frequency of a sinusoidal modulating signal affect the spectrum of the FM signal.

◊ Consider first the case when the frequency of the modulating signal is fixed, but its amplitude is varied, producing a corresponding p p g p gvariation in the frequency deviation Δf.

◊ Consider next the case when the amplitude of the modulating signal is fixed; that is, the frequency deviation Δf is maintained constant,

d h d l i f f i i dand the modulation frequency fm is varied.

39


FIGURE4 7 Di t lit d tFIGURE4.7 Discrete amplitude spectra of an FM signal, normalized with respect to the carrier amplitude, for the case of sinusoidal modulation of fixedcase of sinusoidal modulation of fixed frequency and varying amplitude. Only the spectra for positive frequencies are shown

40

shown.


◊ We have an increasing number of spectral lines crowding into the fixed frequency interval f -Δf<| f |<f + Δffixed frequency interval fc-Δf<| f |<fc+ Δf .

◊ When β approaches infinity, the bandwidth of the FM wave approaches the limiting value of 2Δf, which is an important pointapproaches the limiting value of 2Δf, which is an important point to keep in mind.

FIGURE 4.8 Discrete amplitude spectra of FM i l li d ith t t than FM signal, normalized with respect to the

carrier amplitude, for the case of sinusoidal modulation of varying frequency and fixed amplitude Only the spectra for positive

41

amplitude. Only the spectra for positive frequencies are shown.

Transmission Bandwidth of FM SignalsTransmission Bandwidth of FM Signals

◊ In theory, an FM signal contains an infinite number of side frequencies so that the bandwidth required to transmit such a signalfrequencies so that the bandwidth required to transmit such a signal is similarly infinite in extent.

◊ In practice, however, we find that the FM signal is effectively p , , g ylimited to a finite number of significant side frequencies compatible with a specified amount of distortion.

◊ Consider the case of an FM signal generated by a single-tonemodulating wave of frequency fm.◊ In such an FM signal, the side frequencies that are separated

from the carrier frequency fc by an amount greater than the frequency deviation Δf decrease rapidly toward zero so that thefrequency deviation Δf decrease rapidly toward zero, so that the bandwidth always exceeds the total frequency excursion, but nevertheless is limited.

42

nevertheless is limited.

Transmission Bandwidth of FM SignalsTransmission Bandwidth of FM Signals

◊ We may thus define an approximate rule for the transmission bandwidth of an FM signal generated by a single-tone modulating signal of frequency fm as follows:

1⎛ ⎞ L 2B fβ Δ

(4.38)12 2 2 1T mB f f fβ

⎛ ⎞Δ + = Δ +⎜ ⎟

⎝ ⎠

Large 2Small 2

T

T m

B fB f

ββ→ Δ→

This empirical relation is known as Carson’s rule.

◊ For a more accurate assessment of the bandwidth requirement of an FM signal, we may thus define the transmission bandwidth of an FM wave as the separation between the two frequenciesan FM wave as the separation between the two frequencies beyond which none of the side frequencies is greater than 1% of the carrier amplitude obtained when the modulation is removed.

43

the carrier amplitude obtained when the modulation is removed.

Chapter 4 4 Chapter 4 4 Chapter 4.4 Chapter 4.4 PhasePhase--locked Looplocked Loop


4.4 Phase4.4 Phase--Locked LoopLocked Loop

◊ The phase-locked loop (PLL) is a negative feedback system, the operation of which is closely linked to frequency modulationoperation of which is closely linked to frequency modulation.

◊ It can be used for synchronization frequency division/multiplication◊ It can be used for synchronization, frequency division/multiplication, frequency modulation, and indirect frequency demodulation.

◊ Basically, the phase-locked loop consists of three major components: a multiplier, a loop filter, and a voltage-controlled p p , p f , goscillator (VCO) connected together in the form of a feedback loop, as in Figure 4.16.

◊ The VCO is a sinusoidal generator whose frequency is determined

45

by a voltage applied to it from an external source.


FIGURE 4.16 Phase-locked loop.

◊ We assume that initially we have adjusted the VCO so that when the◊ We assume that initially we have adjusted the VCO so that when the control voltage is zero, two conditions are satisfied:

1 The frequency of the VCO in precisely set at the unmodulated carrier1. The frequency of the VCO in precisely set at the unmodulated carrier frequency fc.

2. The VCO output has a 90-degree phase-shift with respect to the

46

2. The VCO output has a 90 degree phase shift with respect to the unmodulated carrier wave.


◊ Suppose then that the input signal applied to the phase-locked loop is an FM signal defined byis an FM signal defined by

(4.59)( ) ( )1sin 2c cs t A f t tπ φ⎡ ⎤= +⎣ ⎦t

∫where Ac is the carrier amplitude and .

◊ Let the VCO output in the phase locked loop be defined by

( ) ( )1 02

t

ft k m dφ π τ τ= ∫◊ Let the VCO output in the phase-locked loop be defined by

(4.61)( ) ( )2cos 2π φ⎡ ⎤= +⎣ ⎦v cr t A f t t

where Av is the amplitude. With a control voltage v(t) applied to a VCO input, the angle is related to v(t) by the integral( )2φ t

(4.62)where k is the frequency sensitivity of the VCO measured in Hertz

( ) ( )2 02φ π υ= ∫

t

vt k t dt

47

where kv is the frequency sensitivity of the VCO, measured in Hertz per volt.


◊ The object of the phase-locked loop is to generate a VCO output r(t) that has the same phase angle (except for the fixed difference of 90that has the same phase angle (except for the fixed difference of 90 degrees) as the input FM signal s(t).

Th ti i h l ψ ( ) h t i i ( ) b d◊ The time-varying phase angle ψ1(t) characterizing s(t) may be due to modulation by a message signal m(t) as in Eq. (4.60), in which case we wish to recover ψ1(t) in order to estimate m(t)case we wish to recover ψ1(t) in order to estimate m(t).

◊ In other applications of the phase-locked loop, the time-varying h l ψ ( ) f th i i i l ( ) b t dphase angle ψ1(t) of the incoming signal s(t) may be an unwanted

phase shift caused by fluctuations in the communication channel; in this latter case we wish to track ψ1(t) so as to produce a signal withthis latter case, we wish to track ψ1(t) so as to produce a signal with the same phase angle for the purpose of coherent detection (synchronous demodulation).

48


◊ To develop an understanding of the phase-locked loop, it is desirable to have a model of the loopto have a model of the loop.

◊ In what follows, we first develop a nonlinear model, which is b tl li i d t i lif th l isubsequently linearized to simplify the analysis.

49

Nonlinear Model of the PLLNonlinear Model of the PLL◊ According to Figure 4.16, the incoming FM signal s(t) and the VCO

output r(t) are applied to the multiplier producing two components:output r(t) are applied to the multiplier, producing two components:1. A high- frequency component, represented by the double- frequency

term ( ) ( )i 4 φ φ⎡ ⎤⎣ ⎦k A A f t t t

2. A low- frequency component, represented by the difference-

( ) ( )1 2sin 4π φ φ⎡ ⎤+ +⎣ ⎦m c v ck A A f t t t

2. A low frequency component, represented by the differencefrequency term

( ) ( )1 2sin φ φ⎡ ⎤−⎣ ⎦m c vk A A t t

where km is the multiplier gain, measured in volt-1.◊ The loop filter in the phase-locked loop is a low-pass filter, and its p f p p p f ,

response to the high- frequency component will be negligible.

50

Nonlinear Model of the PLLNonlinear Model of the PLL◊ Therefore, discarding the high-frequency component (i.e., the

double- frequency term) the input to the loop filter is reduced todouble frequency term), the input to the loop filter is reduced to(4.63)

where ψ (t) is the phase error defined by( ) ( )sinm c ee t k A A tυ φ⎡ ⎤= ⎣ ⎦

where ψe(t) is the phase error defined by( ) ( ) ( )1 2e

t

t t tφ φ φ= −

∫ (4.64)

◊ The loop filter operates on the input e (t) to produce an output v(t)

( ) ( )1 02

tt k dυφ π υ τ τ= − ∫

◊ The loop filter operates on the input e (t) to produce an output v(t) defined by the convolution integral

(4 65)( ) ( ) ( )υ τ τ τ∞

= ∫t e h t d (4.65)where h(t) is the impulse response of the loop filter.

( ) ( ) ( )υ τ τ τ−∞

= −∫t e h t d

51

Nonlinear Model of the PLLNonlinear Model of the PLL◊ Using Eqs. (4.62) to (4.64) to relate ψe(t) and ψ1(t), we obtain the

following nonlinear integro-differential equation as descriptor of thefollowing nonlinear integro differential equation as descriptor of the dynamic behavior of the phase-locked loop:

( ) ( )d t d tφ φ ∞

∫ (4.66)where K0 is a loop-gain parameter defined by

( ) ( ) ( ) ( )102 sine

e

d t d tK h t d

dt dtφ φ

π φ τ τ τ∞

−∞⎡ ⎤= − −⎣ ⎦∫

where K0 is a loop gain parameter defined by(4.67)

◊ Equation (4 66) suggest the model shown in Figure 4 17 for a phase-0 m cK k k A Aυ υ=

◊ Equation (4.66) suggest the model shown in Figure 4.17 for a phaselocked loop.

I thi d l h l i l d d th l ti hi b t (t)◊ In this model we have also included the relationship between v(t) and e(t) as represented by Eqs. (4.63) and (4.65).

52

DerivatinDerivatin of Eq. 4.66of Eq. 4.66

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )( )1 2

2 sin

e

t

t t t

t k d t e h t d e t k A A t

φ φ φ

φ π υ τ τ υ τ τ τ φ∞

= −

= − = − = ⎡ ⎤⎣ ⎦∫ ∫( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( )

( ) ( ) ( ) ( )

1 0

1 0

2 , sin

= 2 sin

2 sin =

m c e

t

m c e

t

t k d t e h t d e t k A A t

t k k A A k h k dkd

t K k h k dkd K k k A A

υ υ

υ υ

φ π υ τ τ υ τ τ τ φ

φ π φ τ τ

φ π φ τ τ

−∞

∞

−∞

∞

= = = ⎡ ⎤⎣ ⎦

− −⎡ ⎤⎣ ⎦

⎡ ⎤⎣ ⎦

∫ ∫

∫ ∫∫ ∫( ) ( ) ( ) ( )

( ) ( ) ( )

1 0 00

1 0 0

2 sin =

= 2 sin

e m c

t

e

t K k h k dkd K k k A A

t K k h k d dk

υ υφ π φ τ τ

φ π φ τ τ

−∞

∞

−∞

= − −⎡ ⎤⎣ ⎦

− −⎡ ⎤⎣ ⎦

∫ ∫∫ ∫

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )

1 2

01 02 sin

=

e

t

e

t t tt t t

K k h k d dkt

φ φ φ

π φ τ τφ∞

−∞

∂ ∂ ∂= −

∂ ∂ ∂

∂ −⎡ ⎤∂ ⎣ ⎦∫ ∫

( ) ( )

=

(by using the Leibniz integral rule)

( ) ( ) ( , )( ) ( ( ) ) ( ( ) ) )b b

t t

b a f xf x dx f b f a dxα αα α αα α α α α

−∂ ∂

∂ ∂ ∂ ∂= − +∫ ∫

( ) ( )( )

( ) ( )

1 00

( , ) ( ( ), ) ( ( ), ) )

2 sin

a a

e

f x dx f b f a dx

h ktK k

t

α α

α α α α αα α α α

τφπ φ

= − +∂ ∂ ∂ ∂

∂ −∂= − ⎡ ⎤⎣ ⎦∂

∫ ∫t

ddk

t

τ∞

−∞ ∂∫

∫

53

t ⎣ ⎦∂( ) ( ) ( )1

02 sin e

tt

K k h t k dkt

φπ φ

−∞

∞

−∞

∂∂

= − −⎡ ⎤⎣ ⎦∂

∫

∫

Nonlinear Model of the PLLNonlinear Model of the PLL

FIGURE 4.17 Nonlinear model of the phase-locked loop.

◊ We see that the model resembles the block diagram of Figure 4.17. The multiplier at the input of the phase-locked loop is replaced by a subtracter and a sinusoidal nonlinearity, and the VCO by an integrator.

◊ The sinusoidal nonlinearity in the model of Figure 4.17 greatly increases the difficulty of analyzing the behavior of the phase-locked loop It would be helpful to linearize this model to simplify the

54

loop. It would be helpful to linearize this model to simplify the analysis.

Linear Model of the PLLLinear Model of the PLL

◊ When the phase error ψe(t) is zero, the phase-locked loop is said to be in phase-lock When ψ (t) is at all times small compared withbe in phase lock. When ψe(t) is at all times small compared with one radian, we may use the approximation

(4.68)( ) ( )sin e et tφ φ⎡ ⎤⎣ ⎦ ( )which is accurate to within 4 percent for ψe(t) less than 0.5 radians.

◊ We may represent the phase-locked loop by the linearized model

( ) ( )e eφ φ⎣ ⎦

◊ We may represent the phase locked loop by the linearized model shown in Figure 4.18a.

55Figure 4.18 Models of the phase-locked loop. (a)Linearized model.


◊ According to this model, the phase error ψe(t) is related to the input phase ψ1(t) by the linear integro-differential equationphase ψ1(t) by the linear integro differential equation

(4 69)( ) ( ) ( ) ( )1

02ed t d tK h t d

dt dtφ φ

π φ τ τ τ∞

−∞+ − =∫ (4.69)

◊ Transforming Eq. (4.69) into the frequency domain and solving for Φe( f ), the Fourier transform of ψe( f ), in terms of Φ1( f ), the

dt dt∞∫

Φe( f ), the Fourier transform of ψe( f ), in terms of Φ1( f ), the Fourier transform of ψ1(t), we get

( ) ( )1f fΦ Φ(4.70)

The function L( f ) in Eq. (4.70) is defined by

( ) ( ) ( )11e f fL f

Φ = Φ+

( f ) q ( ) y

(4.71)( ) ( )0

H fL f K

jf=

56

(4.71)where H( f ) is the transfer function of the loop filter.

jf


◊ The quantity L( f ) is called the open-loop transfer function of the phase-locked loopphase-locked loop.

◊ Suppose that for all values of f inside the baseband we make the magnitude of L( f ) very large compared with unity. Then from Eq. 4.70 we find that Φe( f ) approaches zero. That is, the phase of the VCO becomes asymptotically equal to the phase of thethe VCO becomes asymptotically equal to the phase of the incoming signal. Under this condition, phase-lock is established, and the objective of the phase-locked loop is thereby satisfied.j p p y

◊ From Figure 4.18a we see that V( f ), the Fourier transform of the h l k d l t t (t) i l t d t Φ ( f ) bphase-locked loop output v(t), is related to Φe( f ) by

(4 72)( ) ( ) ( )0KV f H f f= Φ

57

(4.72)( ) ( ) ( )eV f H f fkυ

Φ


Equivalently, in light of Eq. (4.71), we may writejf ( ) ( )

0

H fL f K

jf=

(4.73)( ) ( ) ( )e

jfV f L f fkυ

= Φ

( ) ( )jf k L f

( ) 0fjf

(4.74)F | L( f ) | 1

( ) ( ) ( )( ) ( )11

jf k L fV f f

L fυ= Φ

+jf◊ For | L( f ) | >> 1:

(4.75)( ) ( )1

jfV f fkυ

Φ

( ) (4.76)( ) ( )112

d tt

k dtυ

φυ

πTime-Domain:

◊ Thus, provided that the magnitude of the open-loop transfer function L( f ) is very large for all frequencies of interest, the phase locked loop may be modeled as a differentiator with its

58

phase-locked loop may be modeled as a differentiator with its output scaled by the factor 1/2πkv, as in Figure 4.18b.


Figure 4.18 Models of the phase-locked loop. (b) Simplified model when the loop gain is very large compared to unity.

◊ Therefore, substituting Eq. (4.60) in (4.76), we find that the resulting output signal of the phase-locked loop is approximately

(4.77)E i (4 77) h h h l i i h

( ) ( )fkt m t

kυυ

◊ Equation (4.77) states that when the loop operates in its phase-locked mode, the output v(t) of the phase-locked loop is approximately the same except for the scale factor k / k as the

59

approximately the same, except for the scale factor kf / kv, as the original message signal m(t).


◊ A significant feature of the phase-locked loop acting as a demodulator is that the bandwidth of the incoming FM signal can bedemodulator is that the bandwidth of the incoming FM signal can be much wider than that of the loop filter characterized by H( f ). The transfer function H( f ) can and should be restricted to the baseband.

◊ The complexity of the phase-locked loop is determined by the transfer function H( f ) of the loop filtertransfer function H( f ) of the loop filter.

◊ The simplest form of a phase-locked loop is obtained when H( f ) 1 th t i th i l filt d th lti h l k d l=1; that is, there is no loop filter, and the resulting phase-locked loop

is referred to as a first-order phase-locked loop.

60


◊ The order of the phase-locked loop is determined by the order of denominator polynomial of the closed-loop transfer function whichdenominator polynomial of the closed loop transfer function, which defines the output transform V( f ) in terms of the input transform Φ1( f ), as shown in Eq. (4.74).

◊ A major limitation of a first-order phase-locked loop is that the loop gain parameter K0 controls both the loop bandwidth as well as thegain parameter K0 controls both the loop bandwidth as well as the hold-in frequency range of the loop.

Th h ld i f f t th f f i f◊ The hold-in frequency range refers to the range of frequencies for which the loop remains phase-locked to the input signal.

◊ It is for this reason that a first-order phase-locked loop is seldom used in practice.

61

Supplementary Material: Supplementary Material: Analysis of PLL Using Laplace Analysis of PLL Using Laplace Analysis of PLL Using Laplace Analysis of PLL Using Laplace

TransformTransform


The PhaseThe Phase--Locked LoopLocked Loop◊ The PLL basically consists of a multiplier, a loop filter, and a

voltage-controlled oscillator (VCO):

pp

g ( )

A i th t th i t t th PLL i th i id (t)◊ Assuming that the input to the PLL is the sinusoid xc(t)= Accos(2πfct+φ) and the output of the VCO is e0(t)= -Avsin(2πfct+ ), where represents the estimate of φ, the product of two signals is:φ

φwhere represents the estimate of φ, the product of two signals is:φ

( ) ( ) ( ) ( ) ( )0 cos 2 sin 2d c c c v ce t x t e t A f t A f tπ φ π φ= = − + +

63

( ) ( )1 12 2 sin sin 4c v c v cA A A A f tφ φ π φ φ= − − + +

The PhaseThe Phase--Locked LoopLocked Loop

◊ The loop filter is a low-pass filter that responds only to the low-

pp

frequency component 0.5AcAvsin(φ - ) and removes the component at 2fc.

h f h l fil id h l l ( )

φ

◊ The output of the loop filter provides the control voltage ev(t) for the VCO.Th VCO i i id l i l t ith i t t◊ The VCO is a sinusoidal signal generator with an instantaneous phase given by

t

∫where K is a gain constant in rad/s/V

( ) ( )2 2t

c c v vf t t f t K e dπ φ π τ τ−∞

+ = + ∫where Kv is a gain constant in rad/s/V.

( ) ( ) ( )ˆ

ort dt K e d K e tφφ τ τ= =∫

64

( ) ( ) ( ) or v v vt K e d K e tdtυφ τ τ

−∞∫


◊ By neglecting the double-frequency term resulting from the

pp

y g g q y gmultiplication of the input signal with the output of the VCO, the phase detector output is:

( )where is the phase error and Kd is a proportionality

( ) sind de Kψ ψ=

ψ φ φ= −constant.

◊ In normal operation, when the loop is tracking the phase of the i i i th h i ll A ltφ φincoming carrier, the phase error is small. As a result,φ φ−

( )sin φ φ φ φ− ≈ −

◊ With the assumption that | ψ |<<1, the PLL becomes linear.

( )

65


◊ The equations describing loop operation is conveniently

pp

◊ The equations describing loop operation is conveniently obtained by using Laplace transform notation .

◊ A loop model using Laplace-transformed quantities and p g p qassuming linear operation is shown in the following figure:

66


◊ The Laplace-transformed loop equations are:

pp

( ) ( ) ( ) ( )( ) ( ) ( )

d d d

v d

E s K s s K s

E s F s E s

⎡ ⎤= Φ −Θ = Ψ⎣ ⎦=( ) ( ) ( )

( ) ( )v d

v vK E ss

sΘ =

◊ The closed-loop transfer function:s

( ) ( ) ( ) ( ) /ds K K F s KF s sΘ

◊ The phase error transfer function:

( ) ( )( )

( )( )

( )( )

/1 /

v d

v d

s K K F s KF s sH s

s s K K F s KF s sΘ

=Φ + +

p f f

( ) ( ) ( )( )

( )( )

( )( ) ( ) ( )

1 1e

s s s s sH s H sK K F

Φ −Θ Ψ Θ= = − = − =

Φ Φ Φ

67

( ) ( ) ( ) ( ) ( ) ( )ev ds s s s K K F sΦ Φ Φ +


◊ The VCO control-voltage/input-phase transfer function:( ) ( ) ( )

pp

i i i h l d l f f i i

( ) ( )( )

( ) ( )( )

v dv

v v d

E s sH s K sF sH s

s K s K K F s= = =Φ +

◊ It is convenient to write the closed-loop transfer function in terms of the open-loop transfer function, which is defined as:

( ) ( )G sK K F s

◊ K=KvKd is the open-loop dc gain.

( ) ( ) ( ) ( )( )

1

opv dop

op

G sK K F sG s H s

s G s⇒ =

+v d p p g

◊ By appropriate choice of F(s), any order closed-loop transfer function can be obtained.

◊ For second-order passive loops, the transfer function is:

( ) ( )2 21 1s sF s H sτ τ+ += ⇒ =

68

( ) ( ) ( ) ( ) 21 2 1

1 1 1

F s H ss K s K sτ τ τ

= ⇒ =− + + +


◊ Second-order phase-locked-loop filters

pp

◊ Second order phase locked loop filters

69

The PhaseThe Phase--Locked LoopLocked Loop◊ Transfer functions and parameters for first- and second-order

h l k d l

pp

phase-locked loops

70


◊ Hence, the closed-loop system for the linearized PLL is second-

pp

order.◊ It is customary to express the denominator of H(s) in the standard

form:( ) 2 22 n nD s s sζω ω= + +

where ξ: loop damping factorωn: natural frequency of the loop

◊ The closed-loop transfer function becomes:

( )1 2 and 1 2n nK Kω τ ξ ω τ= = +

◊ The closed-loop transfer function becomes:

( ) ( )2 2

2 2

22

n n nK sH s

s sζω ω ω

ζω ω

− +=

+ +

71

2 n ns sζω ω+ +


◊ The frequency response of a second-order loop (with τ1»1)

pp

◊ ξ = 1 ⇒ critically damped loop response.◊ ξ < 1 ⇒ underdamped response.

72

ξ p p◊ ξ > 1 ⇒ overdamped response.

The PhaseThe Phase--Locked LoopLocked Loop◊ In practice, the selection of the bandwidth of the PLL involves

a trade off between speed of response and noise in the phase

pp

a trade-off between speed of response and noise in the phase estimate.

◊ On the one hand it is desirable to select the bandwidth of the◊ On the one hand, it is desirable to select the bandwidth of the loop to be sufficiently wide to track any time variations in the phase of the received carrier.p

◊ On the other hand, a wideband PLL allows more noise to pass into the loop, which corrupts the phase estimate.

Reference: Introduction to Spread-Spectrum Communications, by Roger L. Peterson, Rodger E. Ziemer, and David E. Borth, Appendix A, pp. 615-619, 1995 Prentice Hall, Inc.

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Comm-04-Phase and Frequency Modulation