EE 179, Lecture 11, Handout #17
Angle Modulation
◮
Time-varying frequency
◮
Introduction to angle modulation
◮
Relationship between FM and PM
◮
FM bandwidth
Many signals have time-varying frequency distributions (music, speech,
video). Visualizations of time-varying frequency:
◮
spectrogram
◮
spectrum analyzer
◮
graphics equalizer
EE 179, April 23, 2014
Lecture 11, Page 1
Spectrogram
This spectrogram shows a whistler in the VLF band.
EE 179, April 23, 2014
Lecture 11, Page 2
Spectrum Analyzer
EE 179, April 23, 2014
Lecture 11, Page 3
Spectrum Analyzer (cont.)
EE 179, April 23, 2014
Lecture 11, Page 4
Graphic Equalizer
EE 179, April 23, 2014
Lecture 11, Page 5
Instantaneous Frequency
◮
In general, the frequency of a signal at an instant in time depends on the
entire signal (Hilbert transform)
◮
For generalized sinusoids, we can use a simpler approach. Suppose
ϕ(t) = A cos θ(t) .
Then θ(t) is the generalized angle. For a true sinusoid,
θ(t) = ωc t + θ0 ,
linear with slope ωc and offset θ0
◮
The generalized angle is not limited to [0, 2π]. Wrapping introduces
discontinuities.
◮
Matlab has unwrap function to remove discontinuities.
EE 179, April 23, 2014
Lecture 11, Page 6
Instantaneous Frequency (cont.)
◮
Instananeous frequency is derivative of generalized angle:
ωi (t) =
◮
dθ
= θ′ (t)
dt
By Fundamental Theorem of Calculus,
Z t
Z t
ωi (u) du
ωi (u) du = ωi (0) +
θ(t) =
−∞
0
◮
We can modulate a generalized sinusoid by using a signal m(t) to vary
either θ(t) or ωi (t).
◮
In either case, the frequency of the modulated signal changes as a
function of m(t).
EE 179, April 23, 2014
Lecture 11, Page 7
Instantaneous Frequency (cont.)
EE 179, April 23, 2014
Lecture 11, Page 8
Phase Modulation (PM) & Frequency Modulation (FM)
◮
In PM, phase is varied linearly with m(t):
θ(t) = ωc t + kp m(t) ⇒ ϕPM (t) = cos ωc t + kp m(t)
The instantaneous frequency is
ωi (t) =
dθ
= ωc + kp ṁ(t)
dt
If m(t) varies rapidly, then the frequency deviations are larger.
◮
In FM, frequency deviation is linear in m(t):
ωi (t) = ωc + kf m(t)
The angle is
θ(t) =
EE 179, April 23, 2014
Z
t
ωc + kf m(t) du = ωc t + kf
−∞
Z
t
m(u) du
−∞
Lecture 11, Page 9
Relationship Between FM and PM
◮
◮
Phase modulation of m(t) = frequency modulation of ṁ(t).
R
Frequency modulation of m(t) = phase modulation of m(u) du.
EE 179, April 23, 2014
Lecture 11, Page 10
Generalized Angle Modulation
◮
We can vary phase using a linear transform of message signal:
ϕEM (t) = A cos ωc t + h(t) ∗ m(t)
Z t
m(u)h(t − u) du
= A cos ωc t +
−∞
Note that the impulse response h(t) is causal.
◮
PM (h(t) = kp δ(t)) and FM (h(t) = kf u(t)) are special cases.
◮
We recover m(t) from phase by using inverse filter H −1 (s).
E.g., for FM the inverse of integration is differention.
EE 179, April 23, 2014
Lecture 11, Page 11
FM Example: kf = 2π × 105 , fc = 100 MHz
Instantaneous frequency:
kf
m(t) = 108 + 105 m(t)
2π
= 99.9 MHz
fi = fc +
fi,min
fi,max = 100.1 MHz
Is the modulated signal confined to the frequency band 99.9− − 10.1+ ?
No. The bandwidth is approximately 300 KHz.
EE 179, April 23, 2014
Lecture 11, Page 12
PM Example: kp = 10π, fc = 100 MHz
Instantaneous frequency:
kp
ṁ(t) = 108 + 5ṁ(t)
2π
= 99.9 MHz
fi = fc +
fi,min
fi,max = 100.1 MHz
FM modulation of m(t) is same as PM modulation of ṁ(t).
EE 179, April 23, 2014
Lecture 11, Page 13
Frequency Shift Keying (FSK)
The binary message can be detected using two filters tuned for the two
frequencies.
FSK is a historically important digital modulation scheme. The Bell 103
modem used frequencies 1070 and 1270 for originating station.
EE 179, April 23, 2014
Lecture 11, Page 14
Phase Shift Keying (PSK)
The binary message can be detected by correlating with sinusoids of a fixed
frequency but different phases.
The sinusoids are 180◦ out of phase but synchronized with the carrier.
EE 179, April 23, 2014
Lecture 11, Page 15
PSK: Direct Modulation
If kp = π/2 and m(t) takes on only values +1 and −1, then
ϕFM (t) = A cos ωc t + kp m(t)
π
= A cos ωc t + m(t)
2
(
A sin ωc t
when m(t) = −1
=
−A sin ωc t when m(t) = 1
The figure on the previous slide corresponds to bit duration 7/fc , where
fc is the carrier frequency.
Phase changes of ±π are possible at the end of each bit period.
The transmitter must low pass filter the PSK signal.
EE 179, April 23, 2014
Lecture 11, Page 16