Lab10: FM Spectra and VCO

Transcription

1 Lab10: FM Spectra and VCO Prepared by: Keyur Desai Dept. of Electrical Engineering Michigan State University ECE458 Lab 10

2 What is FM? A type of analog modulation Remember a common strategy in analog modulation? Vary some aspect of a Carrier signal w.r.t message A simple case: When message is sinusoid (single tone) x(t) = A c cos[ω c t + φ c + A m sin(ω m t + φ m )], (1) where (A c, ω c, φ c ) describe the carrier sinusoid where (A m, ω m, φ m ) specify the modulator sinusoid

3 What is FM? x(t) = A c cos[ω c t + φ c + A m sin(ω m t + φ m )], (2) Strictly speaking, it is not the frequency of the carrier that is modulated sinusoidally, but rather the instantaneous phase of the carrier Phase modulation is a better term

4 What is FM? Let us discuss what we understand by Phase of a sinusoid s(t) = A cos[φ(t)] (3) where, φ(t) is instantaneous phase: φ(t) = ωt + θ d dt φ(t) =? d dt φ(t) = ω In general the phase may not increase / change linearly and hence d dt φ(t) = ω(t) We call ω(t) instantaneous frequency

5 What is FM? FM signal is: ( t ) x c (t) = A cos ω c t + β m(τ) dτ 0 Instantaneous phase of this signal is: ( t ) φ(t) = ω c t + β m(τ) dτ 0 Then Instantaneous frequency is: d dt φ(t) = ω c + β m(t), where β is the frequency sensitivity of the FM modulator

6 What is FM? Figure: An audio signal (top) may be carried by an AM or FM radio wave

7 What is FM? Figure: An example of frequency modulation. The top diagram shows the modulating signal superimposed on the carrier wave. The bottom diagram shows the resulting frequency-modulated signal

8 Why FM? Historical motivation was: Constant amplitude signals are easier to process What we mean by process? Analog Circuit operations, such as Amplify, Multiply Amplification of AM signal is (maybe was) difficult Bandwidth-Fidelity tradeoff: FM allows to allocate more bandwidth to improve SNR How do I spend more bandwidth? Increase the frequency sensitivity β of FM modulator

9 Spectra of FM x(t) = A c cos[ω c t + φ c + A m sin(ω m t + φ m )], (4) What is the spectrum (Fourier domain representation) of the above signal? Answer is non-trivial and require us to go through an elegant mathematical derivation that involves Bessel functions Remember the first rule of Fourier Club: Series for periodic functions and transform for aperiodic functions

10 Bessel function We will need Bessel functions. The Bessel function is defined as (Hold your breathe): J α (x) = 1 2π π π e i(ατ x sin τ) dτ (5) From where such a madness arises? For a while focus only on the question: Can we compute the integral Remember exponential series: e x = n=0 x n n! = 1 + x + x 2 2! + x 3 3! + x 4 4! + (6)

11 Bessel function If we plug the exponential series into Bessel function definition we end up with J α (x) = m=0 Where did the integral go? ( 1) m x ) 2m+α (7) m!γ(m + α + 1)( 2 It is consumed by a simpler concept called Gamma function: Γ(z) = 0 t z 1 e t dt (8)

12 Spectra of FM What good is the concept of Bessel function for? e jβ sin(ωmt) = k= J k (β)e jkωmt. (9) What does it mean? J k (β) is the amplitude of the kth harmonic in the Fourier-series expansion of the periodic signal exp[jβ sin(ω m t)]

13 Bessel function Figure: Plot of Bessel function of the first kind, J α (x), for integer orders α = 0, 1, 2

14 Spectra of FM How does this signal look like in Fourier domain? x(t) = cos[ω c t + β sin(ω m t)], (10) Is x(t) periodic? Then we will have to work with Fourier series

15 Spectra of FM = re = re = cos[ω c t + β sin(ω m t)] (11) { = re e j[ωct+β sin(ωmt)]} (12) { = re e jωct e jβ sin(ωmt)} (13) { e jωct { k= k= k= J k (β)e jkωmt } J k (β)e j(ωc+kωm)t } (14) (15) J k (β) cos[(ω c + kω m )t] (16)

16 Remember the fourier series? where Fourier Series of e jβ sin(ω mt) f (t) = + n= c ne inωot, (17) c n = ω π o ωo f (t)e inωot dt. (18) 2π π ωo Let us derive fourier series of e jβ sin(ωmt). c n = ω π m ωm e jβ sin(ωmt) e jn ωm t dt (19) 2π c n = ω m 2π π ωm π ωm π ωm J α (x) = 1 2π e j(n ωm t β sin(ωmt)) dt (20) π π e j(ατ x sin τ) dτ (21)