# Angle Modulation, II. Lecture topics FM bandwidth and Carson s rule. Spectral analysis of FM. Narrowband FM Modulation. Wideband FM Modulation

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1 Angle Modulation, II EE 179, Lecture 12, Handout #19 Lecture topics FM bandwidth and Carson s rule Spectral analysis of FM Narrowband FM Modulation Wideband FM Modulation EE 179, April 25, 2014 Lecture 12, Page 1

2 Bandwidth of Angle-Modulated Waves Angle modulation is nonlinear and complex to analyze. Early developers thought that bandwidth could be reduced to 0. They were wrong. FM has infinite bandwidth. Two approximations for FM: narrowband approximation (NBFM) wideband approximation (WBFM) NBFM: if a(t) = m(u)du and k f a(t) 1, then B s 2B m and ϕ FM (t) A ( cosω c t k f a(t)sinω c t ) WBFM: if f = max k f m(t) is peak frequency deviation, then B s = 2 f +2B m This is known as known as Carson s rule. J.R. Carson, Proc. IRE, EE 179, April 25, 2014 Lecture 12, Page 2

3 Narrowband FM Let a(t) = t m(u)du. Define complex FM signal: ˆϕ FM (t) = Ae j(ωct+k fa(t)) = Ae jk fa(t) e jωct By definition, ϕ FM (t) = Re (ˆϕ FM (t) ). Maclaurin power series expansion of ˆϕ FM (t): ( ) A 1+jk f a(t) k2 f 2! a2 (t)+ +j nkn f n! an (t)+ e jωct This expansion for ˆϕ FM (t) shows that the bandwidth is infinite. Since kf n /n! 0, all but a small amount of power is in a finite band. Using ϕ FM (t) = Re (ˆϕ FM (t) ), the FM signal is ( ) A cosω c t k f a(t)sinω c t k2 f 2! a2 (t)cosω c t+ k3 f 3! a3 (t)sinω c t+ If k f a(t) is small then the first two terms are sufficient. EE 179, April 25, 2014 Lecture 12, Page 3

4 Narrowband FM (cont.) If k f a(t) 1 then all but first two terms are negligible. The narrowband FM approximation is ϕ FM (t) A ( cosω c t k f a(t)sinω c t ) NBFM signal has bandwidth 2B, same as bandwidth of AM. NBFM has power 1 2 A2, which does not depend directly on m(t). Narrowband approximation for phase modulation is ϕ PM (t) A ( cosω c t k p m(t)sinω c t ) SNR of NBFM will be discussed later. The narrowband approximation is a special case of linearization, finding the linear approximation to a function at a given point. EE 179, April 25, 2014 Lecture 12, Page 4

5 Tone Frequency Modulation, f c = 20, f m = 1 ϕ FM (t) = cos ( 2πf c +k a a(t) ), k f a(t) = 0.2,0.8, k a = k a = k a = 3.2 EE 179, April 25, 2014 Lecture 12, Page 5

6 Fourier Transforms of Tone FM ϕ FM (t) = cos ( 2πf c +k a a(t) ), k f a(t) = 0.2,0.8, k a = k a = k a = 3.2 EE 179, April 25, 2014 Lecture 12, Page 6

7 Wideband FM (WBFM) Bandwidth Analysis Sampling theorem: any signal with bandwidth B Hz can be reconstructed from samples taken at any rate f s > 2B, so that t k = k/f s = kt s. The reconstruction process interpolates using a low-pass filter: m(t) = m(t k )sinc(π(t t k )) k= If m(t) is approximated by a step signal, then the reconstruction filter is a shaped low-pass filter. EE 179, April 25, 2014 Lecture 12, Page 7

8 WBFM Bandwidth Analysis (cont.) Staircase approximation to m(t) is sum of pulses of width 1/2B: m(t) = k= m(t k )Π(2B(t t k )) Frequency of modulated signal is constant over each cell : instantaneous frequency is ω i (t) = ω c t+k f m(t k ) ϕ FM,k (t) = Π(2B(t t k ))cos(ω i t) By the modulation theorem, the contribution to the spectrum of ϕ FM (t) from each cell is ( ) 1 ω 4B sinc +ωc +k f m(t k ) + 1 ( ) ω 2B 4B sinc ωc k f m(t k ) 2B (We ignore phase shift factors e ±j2πω, i.e., the above is the magnitude of the transform.) EE 179, April 25, 2014 Lecture 12, Page 8

9 WBFM Bandwidth Analysis (cont.) EE 179, April 25, 2014 Lecture 12, Page 9

10 WBFM Bandwidth Analysis (cont.) The first lobe of sincf contains 90% of energy/power, since 1 1 sinc 2 f df = , Thus we need include spectrum only from ω c k f m p 2πB to ω c +k f m p +2πB Phase modulation bandwidth analysis is similar. PM bandwidth depends on ṁ(t), how rapidly m(t) varies. EE 179, April 25, 2014 Lecture 12, Page 10

11 Frequency Modulation of Tone Spectral analysis of FM is difficult/impossible for general signals. Consider sinusoidal input m(t) = cosω m t B m = f m = 2πω m. t a(t) = m(u)du = 1 sinω m t (assuming a( ) = 0) ω m ( ˆϕ FM = Aexp jω c t+ k ) f sinω m t = Ae jωct e jβsinωmt ω m where β = k f /ω m is frequency deviation ratio (also called FM modulation index). Since e jβsinωmt is periodic with frequency ω m, J n (β) = 1 2π π π e j(βsint nt) dt e jβsinωmt = n J n is a Bessel function. J n (β) is negligible if n > β +1. J n (β)e jnωmt EE 179, April 25, 2014 Lecture 12, Page 11

12 Bessel Function J n (β) EE 179, April 25, 2014 Lecture 12, Page 12

13 US Broadcast FM Frequency range: MHz Channel width: 200 KHz (100 channels) Channel center frequencies: 88.1, 88.3,..., Frequency deviation: ±75 KHz US FM is narrowband, since Signal bandwidth: high-fidelity audio requires ±20 KHz, so bandwidth is available for other applications: Muzak (elevator music) (1936) Stock market quotations Interactive games Stereo uses sum and difference of L/R audio channels FM radio was assigned the MHz band of the spectrum in In 1945, at the behest of RCA (David Sarnoff CEO), the FCC moved FM to MHz, obsoleting all existing receivers. EE 179, April 25, 2014 Lecture 12, Page 13

14 NBFM Modulation For narrowband signals, k f a(t) 1 and k p m(t) 1, ˆϕ NBFM A(cosω c k f a(t)sinω c t) ˆϕ NBPM A(cosω c k f m(t)sinω c t) We can use a DSB-SC modulator with a phase shifter. Phase modulation Frequency modulation EE 179, April 25, 2014 Lecture 12, Page 14

15 NBFM: Bandpass Limiter The input-output diagram for an ideal hard limiter is { +1 vi (t) > 0 v o (t) = 1 v i (t) < 0 This is a signum function, the output of a comparison against 0. A hard limiter can be implemented by an op amp without feedback. EE 179, April 25, 2014 Lecture 12, Page 15

16 NBFM: Bandpass Limiter (cont.) A bandpass limiter is a hard limiter cascaded with a bandpass filter. EE 179, April 25, 2014 Lecture 12, Page 16

17 NBFM: Bandpass Limiter (cont.) Input to bandpass limiter is t v i (t) = A(t)cosθ(t), where θ(t) = ω c t+k f m(u) du Ideally, A(t) is constant, but it may vary slowly. We assume A(t) > 0. The input to the bandpass filter is { +1 cosθ > 1 v o (θ) = 1 cosθ < 1 which is periodic in θ with period 2π. Its Fourier series is v o (θ) = 4 ( cosθ 1 π 3 cos3θ+ 1 cos5θ+ ) 5 = 4 ( cos (ω c t+k f m(u) du ) 1 π 3 (ω cos3 c t+k f ) ) m(u) du + The bandpass filter eliminates all but the first term. EE 179, April 25, 2014 Lecture 12, Page 17

18 WBFM Modulation: Direct Generation Using VCO A voltage controlled oscillator generates a signal whose instantaneous frequency proportional to an input m(t): ω i (t) = ω c +k f m(t) The signal with frequency ω i (t) is bandpass filtered, then used in a modulator. VCO can be constructed by using input voltage to control one or more circuit parameters: R: transistor with controlled gate voltage L: saturable core reactor C: reverse-biased semiconductor diode In all cases, feedback is used to adjust the frequency. EE 179, April 25, 2014 Lecture 12, Page 18

19 VCO Using Varactor The inductor is part of an LRC subcircuit whose frequency is determined by the capacitance on D1, which depends on the input voltage. The frequency of the output should be compared against a reference. EE 179, April 25, 2014 Lecture 12, Page 19

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