EE 179, Lecture 13, Handout #20
US Broadcast FM
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US FM bandwidth specifications:
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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:
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Muzak (elevator music) (1936)
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Stock market quotations
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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 28, 2014
Lecture 13, Page 1
NBFM Modulation
For narrowband signals, |kf a(t)| ≪ 1 and |kp m(t)| ≪ 1,
ϕ̂NBFM ≈ A(cos ωc − kf a(t) sin ωc t)
ϕ̂NBPM ≈ A(cos ωc − kf m(t) sin ωc t)
We can use a DSB-SC modulator with a phase shifter.
Phase modulation
EE 179, April 28, 2014
Frequency modulation
Lecture 13, Page 2
NBFM: Bandpass Limiter
The input-output diagram for an ideal hard limiter is
(
+1 vi (t) > 0
vo (t) =
−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 28, 2014
Lecture 13, Page 3
NBFM: Bandpass Limiter (cont.)
A bandpass limiter is a hard limiter cascaded with a bandpass filter.
EE 179, April 28, 2014
Lecture 13, Page 4
NBFM: Bandpass Limiter (cont.)
Input to bandpass limiter is
vi (t) = A(t) cos θ(t) , where θ(t) = ωc t + kf
Z
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
(
+1 cos θ > 1
vo (θ) =
−1 cos θ < 1
which is periodic in θ with period 2π. Its Fourier series is
4
vo (θ) =
cos θ − 13 cos 3θ + 51 cos 5θ + · · ·
π
Z
Z
4
cos ωc t + kf m(u) du − 13 cos 3 ωc t + kf m(u) du + · · ·
=
π
The bandpass filter eliminates all but the first term.
EE 179, April 28, 2014
Lecture 13, Page 5
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:
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R: transistor with controlled gate voltage
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L: saturable core reactor
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C: reverse-biased semiconductor diode
In all cases, feedback is used to adjust the frequency.
EE 179, April 28, 2014
Lecture 13, Page 6
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 28, 2014
Lecture 13, Page 7
FM Demodulation
◮ Frequency-selective filter
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RC high-pass filter
H(f ) =
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j2πRC
≈ j2πRC
1 + j2πRC
(2πRC ≪ 1)
√
RLC circuit with carrier frequency ωc < ω0 = 1/ LC
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Differentiator
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Slope detection
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Zero-crossing detectors
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Phase-locked loop (not discussed today)
EE 179, April 28, 2014
Lecture 13, Page 8
Derivative Theorem for Fourier Transform
If G(f ) is the Fourier transform of g(t), then
dg(t)
⇋ j2πf G(f )
dt
and
Z
t
−∞
g(τ ) dτ ⇋
G(f ) 1
+ 2 G(0)δ(f )
j2πf
“Proof”:
Z
d ∞
dg(t)
=
G(f )ej2πf t dt
dt
dt −∞
Z ∞
Z ∞
d
j2πf G(f ) ej2πf t dt
G(f )ej2πf t dt =
=
dt
−∞
−∞
By the Fourier inversion theorem, j2πf G(f ) is transform of g ′ (t).
EE 179, April 28, 2014
Lecture 13, Page 9
Slope-Detecting Filter
Information in an FM signal is contained in the instantaneous frequency,
ωi (t) = ωc + kf m(t)
We can extract ωi using a slope-detecting filter, where
|H(f )| = a2πf + b
EE 179, April 28, 2014
Lecture 13, Page 10
FM Demodulator and Differentiator
EE 179, April 28, 2014
Lecture 13, Page 11
Envelope Detection using Ideal Differentiator
If ∆ω = kf mp < ωc we can use envelope detection.
Z t
d
ϕ̇FM (t) =
m(u) du
A cos ωc t + kf
dt
−∞
Z t
m(u) du (ωc + kf m(t))
= −A sin ωc t + kf
−∞
Z t
m(u) du − π
= A(ωc + kf m(t)) sin ωc t + kf
−∞
Envelope of ϕ̇FM (t) is A(ωc + kf m(t)). Important that A is constant.
EE 179, April 28, 2014
Lecture 13, Page 12
FM Detection Circuits
RC high-pass filter.
The transfer function is
H(f ) =
j2πRCf
≈ j2πRCf
1 + j2πRCf
(2πRC ≪ 1)
The impulse response is
h(t) = δ(t) − et/RC u(t)
EE 179, April 28, 2014
Lecture 13, Page 13
FM Detection Circuits (cont.)
√
RLC circuit with carrier frequency ωc < ω0 = 1/ LC
1
.
LC
The resonant frequency is ω0 = √
Transfer function H(f ) of RLC is quadratic with maximum at ω = ω0 .
H(f ) is linear over a range ωc − ∆ω < ω < ωc + ∆ω < ω0 .
Input to diode is an AM signal, and output is envelope, am(t).
EE 179, April 28, 2014
Lecture 13, Page 14
Advantages of FM
FM is less susceptible to amplifier nonlinearities. If input is
x(t) = A cos(ωc + ψ(t))
and the output is
y(t) = a0 + a1 x(t) + a2 x2 (t) + · · ·
= c0 + c1 cos(ωc t + ψ(t)) + c2 cos(2ωc t + 2ψ(t)) + · · ·
The extra terms have spectrum outside the carrier signal band. They will
be blocked by bandpass filter.
Nonlinearities in AM cause signal distortion. For y(t) = ax(t) + bx3 (t),
y(t) = aM (t) cos ωc t + bm3 (t) cos3 ωc t
= (am(t) + 34 bm3 (t)) cos ωc t + 41 b cos 3ωc t
FM is preferred for high power applications, such as microwave relay towers.
EE 179, April 28, 2014
Lecture 13, Page 15
Advantages of FM (cont.)
FM can adjust to rapid fading (change of amplitude) using AGC (automatic
gain control)
FM is less vulnerable to signal interference from adjacent channels.
Suppose interference is I cos(ωc + ω)t. Then received signal is
r(t) = A cos ωc t + I cos(ωc t + ω)t
= (A + I cos ωt) cos ωc t − I sin ωt sin ωc t
= Er (t) cos(ωc t + ψ(t))
where
ψ(t) = tan
−1
I sin ωt
A + I cos ωt
≈
I
sin ωt (I ≪ A)
A
The output of an ideal frequency modulator is ψ̇(t) for FM is
Iω
cos ωt ,
A
which is inversely proportional to amplitude A.
yd (t) =
EE 179, April 28, 2014
Lecture 13, Page 16
Noise and FM
Suppose that the power spectrum of noise is flat over an FM channel.
E.g., white noise t has constant power spectrum
N0
2
The power of the noise in a frequency band of width 2B is
Z fc +B
N0
2
df = 2BN0
2
fc −B
HZ (f ) =
The transfer function for FM demodulator satisfies |H(f )| = af + b. This
filter increases noise at higher frequencies.
We can reduce high frequency noise by using preemphasis/deemphasis.
EE 179, April 28, 2014
Lecture 13, Page 17
FM Preemphasis and Deemphasis
Pre-emphasis: RLC high pass filter. De-emphasis: RC low pass filter.
EE 179, April 28, 2014
Lecture 13, Page 18
FM Preemphasis and Deemphasis (cont.)
The linear preemphasis range is f1 = 2.1 kHz to f2 = 30 kHz. The
preemphasis filter has transfer function
Hp (f ) =
f2 f1 + j2πf
f1 f2 + j2πf
If f ≪ f1 then Hp (f ) ≈ 1.
If f1 ≪ f ≪ f2 then
Hp (f ) ≈
j2πf
f1
which is a differentiator!
The corresponding deemphasis filter has transfer function.
Hd (f ) =
EE 179, April 28, 2014
1
f1
≈
j2πf + f1
Hp (f )
Lecture 13, Page 19
FM Preemphasis and Deemphasis Filters
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EE 179, April 28, 2014
Lecture 13, Page 20