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

´ — Dept. of Electronic and Information Engineering G´ eza KOLUMBAN

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The Hong Kong Polytechnic University

EIE331: Communication Fundamentals

References for angle modulation: Haykin: Section 3.10, pp. 154–193 Lecture notes Tutorial notes Angle modulation: Carrier angle is varied according to the slowly-varying message signal An important feature of angle modulation: • It can provide a better discrimination (robustness) against noise and interference than AM • This improvement is achieved at the expense of increased transmission bandwidth • In case of angle modulation, channel bandwidth may be exchanged for improved noise performance • Such trade-off is not possible with AM ´ — Dept. of Electronic and Information Engineering G´ eza KOLUMBAN

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The Hong Kong Polytechnic University

EIE331: Communication Fundamentals

BASIC DEFINITIONS Relationship between the angle and frequency of a sinusoidal signal Sinusoidal carrier c(t) = Ac cos[θi(t)] Angle of carrier θi(t) [rad] Instantaneous frequency of carrier fi(t) =

1 dθi(t) 1 ˙ 1 ωi(t) = = θi(t) [Hz] 2π 2π dt 2π

In the case of an unmodulated carrier, the angle becomes θi(t) = 2πfct + φc

´ — Dept. of Electronic and Information Engineering G´ eza KOLUMBAN

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The Hong Kong Polytechnic University

EIE331: Communication Fundamentals

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 radians per volt • It is assumed that the angle of unmodulated carrier is zero at t = 0 ´ — Dept. of Electronic and Information Engineering G´ eza KOLUMBAN

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The Hong Kong Polytechnic University

EIE331: Communication Fundamentals

Frequency modulation (FM) Sinusoidal carrier:

c(t) = a(t) cos[θi(t)]

The instantaneous frequency fi(t) of carrier is varied linearly with the message signal m(t) 1 dθi(t) 1 ˙ = θi(t) = fc + kf m(t) 2π dt 2π Angle of carrier Z

Z

t

θi(t) = 2π

fi(τ )dτ = 2πfct + 2πkf 0

t

m(τ )dτ 0

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

´ — Dept. of Electronic and Information Engineering G´ eza KOLUMBAN

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The Hong Kong Polytechnic University

EIE331: Communication Fundamentals

Frequency-modulated waveform · ¸ Z t s(t) = Ac cos 2πfct + 2πkf m(τ )dτ 0

where • fc denotes the carrier frequency (i.e., frequency of unmodulated signal) • kf is the frequency sensitivity of the FM modulator expressed in Hertz per volt • It is assumed that the angle of unmodulated carrier is zero at t = 0

´ — Dept. of Electronic and Information Engineering G´ eza KOLUMBAN

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The Hong Kong Polytechnic University

EIE331: Communication Fundamentals

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 ´ — Dept. of Electronic and Information Engineering G´ eza KOLUMBAN

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The Hong Kong Polytechnic University

EIE331: Communication Fundamentals

MOST IMPORTANT PROPERTIES OF PM and FM SIGNALS – Part I PM signal in the time domain: sP M (t) = Ac cos[2πfct + kpm(t)] FM signal in the time domain: ·

Z

¸

t

sF M (t) = Ac cos 2πfct + 2πkf

m(τ )dτ 0

1. Amplitude of PM and FM signals is constant 2. Because the information is carried by the angle of carrier, a nonlinear operation (also nonlinear distortion) that preserves the angle has no influence on the angle modulation systems (i.e., it does not cause distortion). Consequently, even a hard limiter may be used to fix the amplitude of a PM or FM signal ´ — Dept. of Electronic and Information Engineering G´ eza KOLUMBAN

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The Hong Kong Polytechnic University

EIE331: Communication Fundamentals

MOST IMPORTANT PROPERTIES OF PM & FM SIGNALS – Part II sP M (t) = Ac cos[2πfct + kpm(t)]

h

sF M (t) = Ac cos 2πfct + 2πkf

Rt 0

m(τ )dτ

i

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





Phase modulator

FM wave

 

  
  

kp = 2πkf

  



Frequency modulator

PM wave


  


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 ´ — Dept. of Electronic and Information Engineering G´ eza KOLUMBAN

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The Hong Kong Polytechnic University

EIE331: Communication Fundamentals

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

FM signal sF M (t) = Ac cos 2πfct + 2πkf message signal m(t)

Rt 0

m(τ )dτ

i

is a nonlinear function of

Angle modulations (including FM and PM) are nonlinear modulation processes Consequently, spectrum of FM signal may not be determined in the frequency domain using Fourier transform Empirical approach is required to determine the spectrum and transmission bandwidth of FM signal where the following single-tone sinusoidal message signal is considered m(t) = Am cos(2πfmt)

´ — Dept. of Electronic and Information Engineering G´ eza KOLUMBAN

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The Hong Kong Polytechnic University

EIE331: Communication Fundamentals

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 + kf Am cos(2πfmt) = fc + ∆f cos(2πfmt)

(1)

In (1), ∆f = kf Am is the frequency deviation, representing the maximum departure of instantaneous frequency of FM signal from the carrier frequency fc Angle of FM signal is Z

t

θi(t) = 2π

fi(τ )dτ = 2πfct + 0

∆f sin(2πfmt) = 2πfct + β sin(2πfmt) (2) fm

In (2), β = ∆f /fm is the modulation index, representing the maximum departure of angle of FM signal from angle 2πfct of unmodulated carrier ´ — Dept. of Electronic and Information Engineering G´ eza KOLUMBAN

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The Hong Kong Polytechnic University

EIE331: Communication Fundamentals

Units of frequency deviation and modulation index ∆f = kf Am [Hz]

β=

∆f [rad] fm

Recall the relationship between FM and PM signals FM: s = (t) = sF M (t) = Ac cos[2πfct + β sin(2πfmt)] PM: sP M (t) = Ac cos[2πfct+kpAm cos(2πfmt)] =⇒ !! β ⇔ kpAm Two cases are distinguished: • Narrow-band FM, for which β = • Wide-band FM, for which β =

kf A m fm

kf Am fm

> 1 rad

The spectrum of narrow- and wide-band FM signals are completely different ´ — Dept. of Electronic and Information Engineering G´ eza KOLUMBAN

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The Hong Kong Polytechnic University

EIE331: Communication Fundamentals

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

Note: At certain values of Jn(β), the carrier disappears ´ — Dept. of Electronic and Information Engineering G´ eza KOLUMBAN

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The Hong Kong Polytechnic University

EIE331: Communication Fundamentals

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

m(t): Amplitude fixed, frequency decreased

β = 1.0

β = 1.0

β = 2.0

β = 2.0

β = 5.0

β = 5.0

´ — Dept. of Electronic and Information Engineering G´ eza KOLUMBAN

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The Hong Kong Polytechnic University

EIE331: Communication Fundamentals

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 whether or 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 from the carrier frequency and sufficiently far away from the carrier the spectral components becomes negligible Experiments showed that if the amplitude of side frequency components is 1 % then a distortion may not be noticed, if the amplitude of side frequency components is 10 % then a small but noticeable distortion exists By definition: The transmission bandwidth of an FM signal is the separation between the two frequencies beyond which none of the side frequencies is greater than 1 % of carrier amplitude obtained when the modulation is removed ´ — Dept. of Electronic and Information Engineering G´ eza KOLUMBAN

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The Hong Kong Polytechnic University

EIE331: Communication Fundamentals

Determination of transmission bandwidth of a single-tone FM signal In practice, the frequency deviation ∆f is fixed Carson’s rule

µ

¶

1 β Easy to use, but Carson’s rule somewhat underestimates the bandwidth requirement of an FM system BT ≈ 2(∆f + fm) = 2∆f

1+

Exact 1 % bandwidth of an FM signal BT (β) ∆f

´ — Dept. of Electronic and Information Engineering G´ eza KOLUMBAN

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The Hong Kong Polytechnic University

EIE331: Communication Fundamentals

Transmission bandwidth of an arbitrary modulating signal Let W denote the highest frequency component of the spectrum of message signal m(t). In case of a low-pass modulating signal, W is equal to the bandwidth of m(t) Let D denote the deviation ratio that is defined as the ratio of maximum possible frequency deviation to W . Recall, in built FM systems the frequency deviation ∆f is fixed Then the bandwidth of FM signal may be estimated by the Carson’s rule changed according to the parameters of the arbitrary message signal µ ¶ 1 BT = 2∆f 1 + D

´ — Dept. of Electronic and Information Engineering G´ eza KOLUMBAN

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