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Chapter 5. Angle
Husheng Li
The University of Tennessee
Phase and Frequency
 Consider the standard CW signal
 We define the total instantaneous angle
Phase and Frequency
 Phase modulation (PM)
 Frequency modulation (FM)
Characteristics of Angle
 The amplitude of an angle
modulated wave is constant.
 The message resides in the
zero crossings alone, providing
the carrier frequency is large.
 The modulated wave does
not resemble the message
Narrowband PM and FM
 We can expand the signal (using Taylor’s expansion)
 The spectrum is given by
 Hence, the signal has a bandwidth of 2W.
Example of Narrow Band
Angle Modulation
 Both PM and FM have carrier component.
Tone Modulation
 We can allow a 90 degree difference in the
modulating tones:
 Βis called the modulation index for PM or FM with
tone modulation.
Spectrum of Narrowband
Tone Modulation
 When the modulation index is very small, we
 The spectrum is given by
Spectrum of Arbitrary
Modulation Index
 For a single tone signal with arbitrary modulation
index, the modulated signal can be written as
where j_n(β) is the Bessel function.
Bessel Functions
Characteristic of FM
Homework 5
 Deadline Oct. 14, 2013
Spectrum with Different
Modulation Indices
We can either fix
fmor fix Am fD
 Consider the case of multiple tones, e.g.,
x(t) = A1 cosw1t + A2 cosw2t
 The modulated signal can be written as
Periodic Modulation
 When the signal is periodic, the Fourier series are
given by
 The modulated signal can be written as
Transmission Bandwidth
 The generation and transmission of pure FM
requires infinite bandwidth. Hence, our questions
is: how much of the modulated signal spectrum is
 The Bessel function falls off rapidly for
n / b >1
 There are M significant sideband pairs and 2M+1
significant lines all told. The bandwidth can be
given by
Arbitrary Modulated Signal
 For arbitrary modulating signal, the required
bandwidth is given by
(deviation ratio)
 An approximation:
Carson’s rule
Case of Phase Modulation
 We can also define the phase deviation.
 We have
Linear Distortion
 We consider an angle-modulated bandpass
signal applied to a linear system:
 The lowpass equivalent output spectrum is
Nonlinear Distortion
 The output of signal through a nonlinear system is
given by
Example: Clipper
 A clipper has only two outputs
 The output signal is given by
Procedure of Clipper
Direct FM
 In direct FM, we use VCO to generate the
frequency modulated by the signal.
Phase Modulator
 Although we seldom transmit a PM wave, we are
still interested in phase modulators because (1)
the implementation is relatively easy; (2) the
carrier can be supplied by a stable frequency
source; (3) integrating the input signal to a phase
modulator produces an FM output.
Switching-circuit Modulator
 Larger phase shifts can be achieved by the
switching-circuit modulator:
Indirect FM Transmitter
 The integrator and phase modulator constitute a
narrowband frequency modulator that
generates an initial NBFM signal with
instantaneous frequency:
Triangular-Wave FM
 Triangular-wave FM is a modern and rather novel
method for frequency modulation that
overcomes the inherent problems of
conventional CVOs and indirect FM systems.
Frequency Detection
 A frequency detector, often called a
discriminator, produces an output voltage that
should vary linearly with the instantaneous
frequency of the input.
 Almost every circuit falls into one of the following
four categories:
 FM-to-AM conversion
 Phase-shift discrimination
 Zero-crossing detection
 Frequency feedback
FM-to-AM Conversion
 Any device of circuit whose output equals the
time derivative of the input produces FM-to-AM
PHASE-SHIFT Discriminators
 Phase-shift discriminators
involve circuits with linear
phase response, in contrast
to the linear amplitude
response for slope
Quadrature Detector
 A phase-shift discriminator built with a network
having group delay and carrier delay:
Zero Crossing Detector
 Interference refers to the contamination of an
information-bearing signal by another similar
signal, usually from a human source.
 Interfering sinusoids: consider a receiver tuned to
some carrier frequency. The total received signal
Demodulated Output
 Consider a weak interference. The demodulated
output is
 The fact that detected FM interference is most
severe at large values of |f_i| suggests a method
for improving system performance with selective
postdetection filtering, called deemphasis