Lab10: FM Spectra and VCO

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

What is FM? A type of analog modulation Remember a common strategy in analog modulation?

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

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

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

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

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

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) =?

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

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

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:

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:

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

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

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

Why FM? Historical motivation was: Constant amplitude signals are easier to process

Why FM? Historical motivation was: Constant amplitude signals are easier to process What we mean by process?

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

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

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

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?

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

Bessel function We will need Bessel functions. The Bessel function is defined as (Hold your breathe):

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)

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

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)

Bessel function If we plug the exponential series into Bessel function definition we end up with J α (x) = m=0 ( 1) m x ) 2m+α (7) m!γ(m + α + 1)( 2

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

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)

Spectra of FM What good is the concept of Bessel function for?

Spectra of FM What good is the concept of Bessel function for? e jβ sin(ωmt) = J k (β)e jkωmt. (9) What does it mean? k=

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)]

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

Spectra of FM How does this signal look like in Fourier domain? x(t) = cos[ω c t + β sin(ω m t)], (10)

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?

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

Spectra of FM cos[ω c t + β sin(ω m t)] (11)

Spectra of FM cos[ω c t + β sin(ω m t)] (11) { = re e j[ωct+β sin(ωmt)]} (12)

Spectra of FM 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)

Spectra of FM = 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= J k (β)e jkωmt } (14)

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= J k (β)e jkωmt } J k (β)e j(ωc+kωm)t } (14) (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)

Fourier Series of e jβ sin(ω mt) Remember the fourier series?

Fourier Series of e jβ sin(ω mt) Remember the fourier series? where f (t) = + n= c ne inωot, (17)

Fourier Series of e jβ sin(ω mt) Remember the fourier series? f (t) = + n= c ne inωot, (17) where c n = ω π o ωo f (t)e inωot dt. (18) 2π π ωo

Fourier Series of e jβ sin(ω mt) Remember the fourier series? f (t) = + n= c ne inωot, (17) where 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).

Fourier Series of e jβ sin(ω mt) Remember the fourier series? f (t) = + n= c ne inωot, (17) where 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 π ωm

Fourier Series of e jβ sin(ω mt) Remember the fourier series? f (t) = + n= c ne inωot, (17) where 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 e j(n ωm t β sin(ωmt)) dt (20)

Fourier Series of e jβ sin(ω mt) Remember the fourier series? f (t) = + n= c ne inωot, (17) where 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)