frequency (fm) and phase (pm) modulations

The Hong Kong Polytechnic University EIE331: Communication Fundamentals

File name: eie331 04fm.pdf

ANGLE MODULATION:

FREQUENCY (FM) and PHASE (PM) MODULATIONS

• Basic definitions

• Narrow-band and wide-band frequency modulations

• Transmission bandwidth of angle modulated signals

• Phase-locked loop (PLL)

• Generation and demodulation of angle modulated signals

• FM stereo multiplexing

Geza KOLUMBAN — Dept. of Electronic and Information Engineering 1


References for angle modulation:Haykin: Section 3.10, pp. 154–193Lecture notesTutorial notes

Angle modulation: Carrier angle is varied according to the slowly-varyingmessage signal

An important feature of angle modulation:

• It can provide a better discrimination (robustness) against noise andinterference than AM

• This improvement is achieved at the expense of increasedtransmission bandwidth

• In case of angle modulation, channel bandwidth may be exchangedfor improved noise performance

• Such trade-off is not possible with AM



BASIC DEFINITIONS

Relationship between the angle and frequency of a sinusoidal signal

Sinusoidal carrierc(t) = Ac cos[θi(t)]

Angle of carrierθi(t) [rad]

Instantaneous frequency of carrier

fi(t) =12π

ωi(t) =12π

dθi(t)dt

=12π

θi(t) [Hz]

In the case of an unmodulated carrier, the angle becomes

θi(t) = 2πfct + φc



Phase modulation (PM)

Sinusoidal carrier: c(t) = a(t) cos[θi(t)]

PM: The angle θi(t) of carrier is varied linearly with the message signal m(t)

θi(t) = 2πfct + kpm(t)

Amplitude of carrier is constant: a(t) = Ac

Phase-modulated waveform

s(t) = Ac cos[2πfct + kpm(t)]

where • fc denotes the carrier frequency (i.e., frequency of unmodulated signal)

• kp is the phase sensitivity of the PM modulator expressed in radiansper volt

• It is assumed that the angle of unmodulated carrier is zero at t = 0



Frequency modulation (FM)

Sinusoidal carrier: c(t) = a(t) cos[θi(t)]

The instantaneous frequency fi(t) of carrier is varied linearly with the messagesignal m(t)

12π

dθi(t)dt

=12π

θi(t) = fc + kfm(t)

Angle of carrier

θi(t) = 2π

∫ t

0

fi(τ)dτ = 2πfct + 2πkf

∫ t

0

m(τ)dτ

Amplitude of carrier is constant:

a(t) = Ac



Frequency-modulated waveform

s(t) = Ac cos[2πfct + 2πkf

∫ t

0

m(τ)dτ

]

where • fc denotes the carrier frequency (i.e., frequency of unmodulated signal)

• kf is the frequency sensitivity of the FM modulator expressed in Hertzper volt

• It is assumed that the angle of unmodulated carrier is zero at t = 0



PM and FM signal in the time domain produced by a single tone message signal

Note: • The similarity between the angle modulated signals

• Amplitude of angle modulated signals is constant



MOST IMPORTANT PROPERTIES OF PM and FM SIGNALS – Part I

PM signal in the time domain:

sPM(t) = Ac cos[2πfct + kpm(t)]

FM signal in the time domain:

sFM(t) = Ac cos[2πfct + 2πkf

∫ t

0

m(τ)dτ

]

1. Amplitude of PM and FM signals is constant

2. Because the information is carried by the angle of carrier, a nonlinearoperation (also nonlinear distortion) that preserves the angle has no influenceon the angle modulation systems (i.e., it does not cause distortion).Consequently, even a hard limiter may be used to fix the amplitude of aPM or FM signal



MOST IMPORTANT PROPERTIES OF PM & FM SIGNALS – Part II

sPM(t) = Ac cos[2πfct + kpm(t)] sFM(t) = Ac cosh2πfct + 2πkf

R t

0m(τ)dτ

i3. A close relationship exists between the PM and FM signals:

FM wavePhase

modulator��

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PM wave

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kp = 2πkf kf = kp

2π

Conclusions:

• A PM/FM modulator may be used to generate an FM/PM waveform• FM is much more frequently used than PM• All the properties of a PM signal may be deduced from that of an FM signal• Henceforth, in the remaining part of our studies we deal only with FM

signals



FREQUENCY MODULATION

Goal: Determine the spectrum and transmission bandwidth of an FM signal

FM signal sFM(t) = Ac cosh2πfct + 2πkf

R t

0m(τ)dτ

iis a nonlinear function of

message signal m(t)

Angle modulations (including FM and PM) are nonlinear modulation processes

Consequently, spectrum of FM signal may not be determined in the frequencydomain using Fourier transform

Empirical approach is required to determine the spectrum and transmissionbandwidth of FM signal where the following single-tone sinusoidal messagesignal is considered

m(t) = Am cos(2πfmt)



Definition of frequency deviation and modulation index

Let m(t) = Am cos(2πfmt) denote the single-tone message (modulating) signal

Then the instantaneous frequency of FM signal becomes

fi(t) = fc + kfAm cos(2πfmt) = fc + ∆f cos(2πfmt) (1)

In (1), ∆f = kfAm is the frequency deviation, representing the maximumdeparture of instantaneous frequency of FM signal from the carrier frequency fc

Angle of FM signal is

θi(t) = 2π

∫ t

0

fi(τ)dτ = 2πfct +∆f

fmsin(2πfmt) = 2πfct + β sin(2πfmt) (2)

In (2), β = ∆f/fm is the modulation index, representing the maximumdeparture of angle of FM signal from angle 2πfct of unmodulated carrier



Units of frequency deviation and modulation index

∆f = kfAm [Hz] β =∆f

fm[rad]

Recall the relationship between FM and PM signals

FM: s = (t) = sFM(t) = Ac cos[2πfct + β sin(2πfmt)]

PM: sPM(t) = Ac cos[2πfct+kpAm cos(2πfmt)] =⇒ !! β ⇔ kpAm

Two cases are distinguished:

• Narrow-band FM, for which β = kfAm

fm<< 1 rad

• Wide-band FM, for which β = kfAm

fm>> 1 rad

The spectrum of narrow- and wide-band FM signals are completely different



Spectrum of narrow-band FM signal

Narrow-band FM signal in the time domain is

s(t) = Ac cos[2πfct + β sin(2πfmt)] where β << 1 rad

Using a trigonometric identity cos(α + β) = cos α cos β − sin α sin β we get

s(t) = Ac cos(2πfct) cos[β sin(2πfmt)]− Ac sin(2πfct) sin[β sin(2πfmt)] (3)

If the modulation index is much smaller than 1 radian then

Since limα→0

{cos α} = 1, =⇒ cos[β sin(2πfmt)] ≈ 1

Since limα→0

{sin α} = α, =⇒ sin[β sin(2πfmt)] ≈ β sin(2πfmt)

Hence, (3) simplifies to

s(t) ≈ Ac cos(2πfct)− βAc sin(2πfct) sin(2πfmt)



From the equation developed on the previous transparency

s(t) ≈ Ac cos(2πfct)− βAc sin(2πfct) sin(2πfmt)

we get using the trigonometric identity sin α sin β = 12[cos(α−β)−cos(α+β)]

s(t) = sFM(t) ≈ Ac cos(2πfct)+12βAc {cos[2π(fc + fm)t]− cos[2π(fc − fm)t]}

Recall, the single-tone modulated AM signal in the time domain is

sAM(t) ≈ Ac cos(2πfct) +12µAc {cos[2π(fc + fm)t] + cos[2π(fc − fm)t]}

Comparing sFM(t) and sAM(t) we conclude:

• If β = µ then the only difference between AM and narrow-bandFM signals is that in the latter the sign of the lower side frequencycomponent is reversed



Spectrum and transmission bandwidth of narrow-band FM signal

Spectrum of narrow-band FM and AM signals

Narrow-band FM:SFM(f) = F{sFM(t)}

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AM:SAM(f) = F{sAM(t)}

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Transmission bandwidth of narrow-band FM is equal to that of AM:

BT = 2W

where W is the bandwidth of the low-pass message signal



Spectrum of wide-band FM signal

Equation giving the FM signal in the time domain was developed on page 13,see (3) on page 13

For convenience, that equation is repeated here:

s(t) = Ac cos(2πfct) cos[β sin(2πfmt)]−Ac sin(2πfct) sin[β sin(2πfmt)] (4)

Recall, only periodic signals may be expanded as a trigonometric Fourier series.Fourier series and, consequently, periodic signals have a discrete spectrum

Even though s(t) is not necessarily periodic, the terms cos[β sin(2πfmt)] andsin[β sin(2πfmt)] are periodic and each can be expanded as a Fourier series



It has been shown in applied mathematics that

cos[β sin(2πfmt)] = J0(β) +∞∑

n even

2Jn(β) cos(2πnfmt) (5)

sin[β sin(2πfmt)] =∞∑

n odd

2Jn(β) sin(2πnfmt) (6)

where n is positive, and the Bessel functions of the first kind, of order n andargument β are defined as

Jn(β) =12π

∫ π

−π

exp[j(β sinλ− nλ)]dλ



Substituting (5) and (6) (see page 17) into (4) (given on page 16) and expandingproducts of sine and cosine functions using trigonometric identities finally yieldsthe FM signal for arbitrary modulation index β in the time domain

s(t) =AcJ0(β) cos(2πfct)

+

∞Xn odd

AcJn(β)[cos(2πfc + n2πfm)t− cos(2πfc − n2πfm)t]

+

∞Xn even

AcJn(β)[cos(2πfc + n2πfm)t + cos(2πfc − n2πfm)t]

This equation is valid for both the narrow-band and wide-band FM signals

Recall the Fourier transform of a cosine signal with infinite duration is

F {cos(2πfct)} =1

2[δ(f − fc) + δ(f + fc)]



To get the spectrum of an FM signal, the Fourier transform of

s(t) =AcJ0(β) cos(2πfct)

+

∞Xn odd

AcJn(β)[cos(2πfc + n2πfm)t− cos(2πfc − n2πfm)t]

+∞X

n even

AcJn(β)[cos(2πfc + n2πfm)t + cos(2πfc − n2πfm)t]

has to be calculated

Observe

• Single-tone modulated FM signal has a discrete spectrum

• Spectrum contains a carrier-frequency line plus an infinite number of sideband

lines =⇒ Distortion-less transmission requires infinite bandwidth!

• All lines are equally spaced by the modulating frequency

• Odd-order lower sideband lines are reversed in phase



Properties of Bessel functions of the first kind

1. For small values of modulation index β the following approximations may beused (Recall, this is the case of narrow-band FM)

J0(β) = 1 and J1(β) = β2 and Jn(β) = 0, n > 2

2. Bessel functions Jn(β) as a function of modulation index β

Note: At certain values of Jn(β), the carrier disappears



Spectrum of a single-tone FM signal

Conditions: • Single-tone sinusoidal modulation m(t)• Spectra are normalized with respect to the carrier amplitude• Magnitude of spectra is shown only for positive frequencies

m(t): Frequency fixed, amplitude increased

β = 1.0

β = 2.0

β = 5.0

m(t): Amplitude fixed, frequency decreased

β = 1.0

β = 2.0

β = 5.0



TRANSMISSION BANDWIDTH OF FM SIGNAL

In theory, an FM signal contains an infinite number of side frequencies =⇒Bandwidth required for distortion-free transmission is infinite in extent whetheror not the message is band-limited

But implemented FM systems using finite bandwidth do exist and perform well

Explanation: Amplitude of side frequencies decays if we move away fromthe carrier frequency and sufficiently far away from the carrier the spectralcomponents becomes negligible

Experiments showed that if the amplitude of side frequency components is1 % then a distortion may not be noticed, if the amplitude of side frequencycomponents is 10 % then a small but noticeable distortion exists

By definition: The transmission bandwidth of an FM signal is the separationbetween the two frequencies beyond which none of the side frequencies isgreater than 1 % of carrier amplitude obtained when the modulation is removed



Determination of transmission bandwidth of a single-tone FM signal

In practice, the frequency deviation ∆f is fixed

Carson’s rule

BT ≈ 2(∆f + fm) = 2∆f

(1 +

1β

)

Easy to use, but Carson’s rule somewhat underestimates the bandwidthrequirement of an FM system

Exact 1 % bandwidth ofan FM signal

BT

∆f(β)



Transmission bandwidth of an arbitrary modulating signal

Let W denote the highest frequency component of the spectrum of messagesignal m(t). In case of a low-pass modulating signal, W is equal to thebandwidth of m(t)

Let D denote the deviation ratio that is defined as the ratio of maximumpossible frequency deviation to W . Recall, in built FM systems the frequencydeviation ∆f is fixed

Then the bandwidth of FM signal may be estimated by the Carson’s rulechanged according to the parameters of the arbitrary message signal

BT = 2∆f

(1 +

1D

)


frequency (fm) and phase (pm) modulations

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