This document discusses bandwidth and spectral analysis of frequency modulation (FM). It covers Carson's rule for calculating FM bandwidth, the narrowband and wideband approximations of FM, and spectral analysis of FM signals. Key topics include FM bandwidth being infinite but having most power within a finite band, narrowband FM approximating to a sinusoid with bandwidth 2Bm, and wideband FM bandwidth given by Carson's rule as 2∆f + 2Bm, where ∆f is the peak frequency deviation and Bm is the modulation bandwidth.

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 |kf a(t)| ≪ 1, then Bs ≈ 2Bm and
ϕFM(t) ≈ A cos ωct − kf a(t) sin ωct
WBFM: if ∆f = max |kf m(t)| is peak frequency deviation, then
Bs = 2∆f + 2Bm
This is known as known as Carson’s rule.
J.R. Carson, Proc. IRE, 1922.
EE 179, April 25, 2014 Lecture 12, Page 2

3. Narrowband FM
Let a(t) =
t
−∞ m(u) du. Define complex FM signal:
ˆϕFM(t) = Aej(ωct+kf a(t))
= Aejkf a(t)
· ejωct
By definition, ϕFM(t) = Re ˆϕFM(t) .
Maclaurin power series expansion of ˆϕFM(t):
A 1 + jkf a(t) −
k2
f
2!
a2
(t) + · · · + jn
kn
f
n!
an
(t) + · · · ejωct
This expansion for ˆϕFM(t) shows that the bandwidth is infinite.
Since kn
f /n! → 0, all but a small amount of power is in a finite band.
Using ϕFM(t) = Re ˆϕFM(t) , the FM signal is
A cos ωct − kf a(t) sin ωct −
k2
f
2!
a2
(t) cos ωct +
k3
f
3!
a3
(t) sin ωct + · · ·
If kf 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 |kf a(t)| ≪ 1 then all but first two terms are negligible.
The narrowband FM approximation is
ϕFM(t) ≈ A cos ωct − kf a(t) sin ωct
NBFM signal has bandwidth 2B, same as bandwidth of AM.
NBFM has power 1
2A2, which does not depend directly on m(t).
Narrowband approximation for phase modulation is
ϕPM(t) ≈ A cos ωct − kpm(t) sin ωct
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, fc = 20, fm = 1
ϕFM(t) = cos 2πfc + kaa(t) , |kf a(t)| = 0.2, 0.8, 3.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
−1
−0.5
0
0.5
1
ka
= 0.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
−1
−0.5
0
0.5
1
ka
= 0.8
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
−1
−0.5
0
0.5
1
ka
= 3.2
EE 179, April 25, 2014 Lecture 12, Page 5

6. Fourier Transforms of Tone FM
ϕFM(t) = cos 2πfc + kaa(t) , |kf a(t)| = 0.2, 0.8, 3.2
−40 −30 −20 −10 0 10 20 30 40
0
2
4
6
ka
= 0.2
−40 −30 −20 −10 0 10 20 30 40
0
2
4
6
ka
= 0.8
−40 −30 −20 −10 0 10 20 30 40
0
1
2
3
ka
= 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 fs > 2B, so that tk = k/fs = kTs.
The reconstruction process interpolates using a low-pass filter:
m(t) =
∞
k=−∞
m(tk) sinc(π(t − tk))
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(tk)Π(2B(t − tk))
Frequency of modulated signal is constant over each “cell”:
◮ instantaneous frequency is ωi(t) = ωct + kf m(tk)
◮ ϕFM,k(t) = Π(2B(t − tk)) cos(ωit)
By the modulation theorem, the contribution to the spectrum of ϕFM(t)
from each cell is
1
4B
sinc
ω + ωc + kf m(tk)
2B
+
1
4B
sinc
ω − ωc − kf m(tk)
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 sinc f contains 90% of energy/power, since
1
−1
sinc2
f df = 0.902823 ,
Thus we need include spectrum only from
ωc − kf mp − 2πB to ωc + kf mp + 2πB
Phase modulation bandwidth analysis is similar. PM bandwidth depends
on ˙m(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) = cos ωmt ⇒ Bm = fm = 2πωm.
a(t) =
t
−∞
m(u) du =
1
ωm
sin ωmt (assuming a(−∞) = 0)
ˆϕFM = A exp jωct +
kf
ωm
sin ωmt = Aejωct
ejβ sin ωmt
where β = kf / ωm is frequency deviation ratio (also called FM modulation
index).
Since ejβ sin ωmt is periodic with frequency ωm,
Jn(β) =
1
2π
π
−π
ej(β sin t−nt)
dt ⇒ ejβ sin ωmt
=
n
Jn(β)ejnωmt
Jn is a Bessel function. Jn(β) is negligible if n > β + 1.
EE 179, April 25, 2014 Lecture 12, Page 11

12. Bessel Function Jn(β)
EE 179, April 25, 2014 Lecture 12, Page 12

13. US Broadcast FM
◮ Frequency range: 88.0 − −108.0 MHz
◮ Channel width: 200 KHz (100 channels)
◮ Channel center frequencies: 88.1, 88.3, . . . , 107.9
◮ Frequency deviation: ±75 KHz
◮ US FM is narrowband, since 75 × 103 ≪ 88 × 106
◮ 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 42–50 MHz band of the spectrum in 1940. In 1945, at the behest of
RCA (David Sarnoff CEO), the FCC moved FM to 88–108 MHz, obsoleting all existing receivers.
EE 179, April 25, 2014 Lecture 12, Page 13

14. NBFM Modulation
For narrowband signals, |kf a(t)| ≪ 1 and |kpm(t)| ≪ 1,
ˆϕNBFM ≈ A(cos ωc − kf a(t) sin ωct)
ˆϕNBPM ≈ A(cos ωc − kf m(t) sin ωct)
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
vo(t) =
+1 vi(t) > 0
−1 vi(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
vi(t) = A(t) cos θ(t) , where θ(t) = ωct + kf
t
−∞
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
vo(θ) =
+1 cos θ > 1
−1 cos θ < 1
which is periodic in θ with period 2π. Its Fourier series is
vo(θ) =
4
π
cos θ − 1
3 cos 3θ + 1
5 cos 5θ + · · ·
=
4
π
cos ωct + kf m(u) du − 1
3 cos 3 ωct + kf 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):
ωi(t) = ωc + kf m(t)
The signal with frequency ωi(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