Angle Modulation
Lesson 15
BME 333 Biomedical Signals and Systems 2 - J.Schesser Angle Modulation • Two types: Frequency and Phase both less susceptible to noise but more difficult to construct and reconstruct.
x(t)=Aocos[ωot + ϕ(t)] • Let m(t) be a BL signal and fM<<ωo/2π • If ϕ (t) = kφm(t) then we have phase modulation (PM): x(t)=Aocos[ωot + kφm(t)]
t with modulation index ϕ m=| kφm(t)|peak
• If φ ( t ) = k ω m ( τ ) d τ then we have frequency modulation (FM): ∫ t −∞ x(t) = A cos[ω t + k m(τ )dτ ] 0 0 ∫ ω −∞ with instantaneous radian frequency ω(t)=dϕ(t)/dt= ωo+ kω m(t), with maximum frequency deviation Δ ω =| kωm(t)|peak, t φ = k m(τ )dτ ] and with modulation index m ω ∫ −∞ peak BME 333 Biomedical Signals and Systems 3 - J.Schesser Spectrum of Angle Modulation
• Unlike AM there is no single theory that can handle the general PM and FM cases • Let’s look at a simple example where
m(t)=mocos ωmt
PM è Ao cos(ωot+kφ mocosωmt)
FM è Ao cos[ωot+(kω mo/ωm)sin ωmt] • Let’s look at the following case:
v(t)=Ao cos[ωot+ϕmsin ωmt]
BME 333 Biomedical Signals and Systems 4 - J.Schesser Spectrum Continued
v(t)=+ Aoomm cos[ωtφ sin ωt ]
= Aoommomm [cos(ωt )cos(φφ sin ωt- ) sin( ωt )sin( sin ωt )]
jφmmsin ωt Note: e(=+ cosφφmm sin ωt) j sin ( mm sin ωt) ∞ aejkωm t = ∑ k −∞ 1 π and ae= j(φωmmm sin ωt− k t ) dt (ω ) k ∫ m 2π −π The solution of this integral is called a Bessel function of the 1st kind and order k and is denoted as: 1 π Jedt()φω= j(φωmmm sin ωt− k t ) ( ) km∫ m 2π −π k and JJkm (φφ )= (− 1)− km ( )
BME 333 Biomedical Signals and Systems 5 - J.Schesser Bessel Functions
1.2
1
0.8
0.6 0 1 0.4 2 3 0.2 4 5 0 10 -1 1 3 5 7 9 11 13 15 -0.2
-0.4
-0.6
BME 333 Biomedical Signals and Systems 6 - J.Schesser Spectrum of an Angle Modulated Signal
∞ eJejφωmmsin ωt() jk m t = ∑ kmφ −∞ It can be shown that
cos[ωomt+=φφω sin ω mt ] J00 ( m )cos t
++Jtt10(φωωωωmm )[cos( )−− cos( 0 m )
+++Jtt20(φωωωωmm )[cos( 2 ) cos( 0− 2 m )
++Jtt30(φωωωωmm )[cos( 3 )−− cos( 0 3 m ) +K • When look at AM, there are only 2 bands; however, for PM/FM there are an infinite number of frequencies. • We say that PM/FM disperses the spectrum of modulating signal, m(t).
• Note that if ϕm<0.20, then J0 (ϕm) ≈ 0.99 and Jn (ϕm) BME 333 Biomedical Signals and Systems 7 - J.Schesser Construction of FM signal and reconstruction of m(t) • Construction of FM signal can be achieved using a voltage controlled oscillator. L V2 V1 R C 0 • Reconstruction of m(t) can be achieved using an FM discriminator followed by an envelope detector 1.2 L V2 1 V1 0.8 R C 0.6 0.4 0 0.2 0 wo BME 333 Biomedical Signals and Systems 8 - J.Schesser Homework • TBD BME 333 Biomedical Signals and Systems 9 - J.Schesser