Ton Duc Thang University
Faculty of Electrical and Electronics Engineering
Dept. of Electronics and Telecommunications Engineering
ANALOG AND DIGITAL
COMMUNICATIONS
(402072)
Duc N. M. Dang, PhD.
CHAPTER 3
ANGLE MODULATION
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Chapter 3
ANGLE MODULATION
• Basic Definitions & Properties
• Relationship between PM and FM waves
• Narrow-Band Frequency Modulation
• Wide-Band Frequency Modulation
• Transmission Bandwidth of FM waves
• Generation of FM waves
• Demodulation of FM signals
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OBJECTIVES
• Understand angle modulation is a nonlinear
process.
• Understand the transmission bandwidth of
an angle-modulated wave may assume an
infinite extent.
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Chapter 3
ANGLE MODULATION
• Basic Definitions & Properties
• Relationship between PM and FM waves
• Narrow-Band Frequency Modulation
• Wide-Band Frequency Modulation
• Transmission Bandwidth of FM waves
• Generation of FM waves
• Demodulation of FM signals
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BASIC DEFINITIONS
• Angel modulation: the angle of the carrier
wave is varied according to the information-
bearing signal.
• Angle-modulated wave:
𝑠(𝑡) = 𝐴𝑐 cos[𝜃𝑖 (𝑡) 𝜃𝑖 (𝑡) = 2𝜋𝑓𝑐 𝑡 + 𝜑𝑐 , for 𝑚(𝑡) = 0
• Instantaneous frequency:
𝜃𝑡 (𝑡 + 𝛥𝑡) − 𝜃𝑖 (𝑡) 1 𝑑𝜃𝑖 (𝑡)
𝑓𝑖 (𝑡) = lim 𝑓𝛥𝑡 (𝑡) = lim =
𝛥𝑡→0 𝛥𝑡→0 2𝜋𝛥𝑡 2𝜋 𝑑𝑡
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BASIC DEFINITIONS & PROPERTIES
• Phase modulation (PM): instantaneous angle
is varied linearly with the message signal
𝜃𝑖 (𝑡) = 2𝜋𝑓𝑐 𝑡 + 𝑘𝑝 𝑚(𝑡
𝑠(𝑡) = 𝐴𝑐 cos 2𝜋𝑓𝑐 𝑡 + 𝑘𝑝 𝑚(𝑡
• Frequency modulation (FM): instantaneous
frequency is varied linearly with the message
𝑡
signal𝑓 (𝑡) = 𝑓 + 𝑘 𝑚(𝑡) 𝜃 (𝑡) = 2𝜋 𝑓 (𝜏) 𝑑𝜏
𝑖 𝑐 𝑓 𝑖 𝑖
0
𝑡
𝑠(𝑡) = 𝐴𝑐 cos 2𝜋𝑓𝑐 𝑡 + 2𝜋𝑘𝑓 𝑚(𝜏)𝑑𝜏
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0 7
BASIC DEFINITIONS & PROPERTIES
Summary of Basic Definitions in Angle Modulation
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BASIC DEFINITIONS & PROPERTIES
Amplitude-modulated signal
Carrier wave
Phase-modulated signal
Sinusoidal modulating signal
Frequency modulated signal
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BASIC DEFINITIONS & PROPERTIES
• Properties:
1. Constancy of transmitted power
2. Nonlinearity of the modulation process
3. Irregularity of zero-crossings
4. Visualization difficulty of message waveform
5. Tradeoff of increased transmission bandwidth
for improved noise performance
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Chapter 3
ANGLE MODULATION
• Basic Definitions & Properties
• Relationship between PM and FM waves
• Narrow-Band Frequency Modulation
• Wide-Band Frequency Modulation
• Transmission Bandwidth of FM waves
• Generation of FM waves
• Demodulation of FM signals
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RELATIONSHIP BETWEEN
PM AND FM WAVES
Scheme for generating an FM wave
by using a phase modulator.
Scheme for generating a PM wave by
using a frequency modulator
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Chapter 3
ANGLE MODULATION
• Basic Definitions & Properties
• Relationship between PM and FM waves
• Narrow-Band Frequency Modulation
• Wide-Band Frequency Modulation
• Transmission Bandwidth of FM waves
• Generation of FM waves
• Demodulation of FM signals
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NARROW-BAND FREQUENCY
MODULATION
• Sinusoidal modulating wave:
𝑚(𝑡) = 𝐴𝑚 cos(2𝜋𝑓𝑚 𝑡)
• Instantaneous frequency of FM wave:
𝑓𝑖 (𝑡) = 𝑓𝑐 + 𝑘𝑓 𝐴𝑚 cos(2𝜋𝑓𝑚 𝑡) = 𝑓𝑐 + 𝛥𝑓cos(2𝜋𝑓𝑚 𝑡)
frequency deviation
• Phase deviation of FM wave:
𝛥𝑓
𝜃𝑖 (𝑡) = 2𝜋𝑓𝑐 𝑡 + sin(2𝜋𝑓𝑚 𝑡)= 2𝜋𝑓𝑐 𝑡 + 𝛽sin(2𝜋𝑓𝑚 𝑡)
𝑓𝑚 modulation index
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NARROW-BAND FREQUENCY
MODULATION
• FM wave:
𝑠(𝑡) = 𝐴𝑐 cos(2𝜋𝑓𝑐 𝑡)cos 𝛽sin(2𝜋𝑓𝑚 𝑡) − 𝐴𝑐 sin(2𝜋𝑓𝑐 𝑡)sin 𝛽sin(2𝜋𝑓𝑚 𝑡)
• If the modulation index is small compared
to one radian:
𝑠(𝑡) ≈ 𝐴𝑐 cos(2𝜋𝑓𝑐 𝑡) − 𝛽𝐴𝑐 sin(2𝜋𝑓𝑐 𝑡)sin(2𝜋𝑓𝑚 𝑡)
1
𝑠(𝑡) ≈ 𝐴𝑐 cos(2𝜋𝑓𝑐 𝑡) + 𝛽𝐴𝑐 cos 2𝜋(𝑓𝑐 + 𝑓𝑚 )𝑡 − cos 2𝜋(𝑓𝑐 − 𝑓𝑚 )𝑡
2
1
𝑠𝐴𝑀 (𝑡) = 𝐴𝑐 cos(2𝜋𝑓𝑐 𝑡) + 𝜇𝐴𝑐 cos 2𝜋(𝑓𝑐 + 𝑓𝑚 )𝑡 + cos 2𝜋(𝑓𝑐 − 𝑓𝑚 )𝑡
2
A narrow-band FM wave requires essentially the same
transmission bandwidth as the AM wave.
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NARROW-BAND FREQUENCY
MODULATION
Block diagram of an indirect method
for generating a narrow-band FM wave
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Chapter 3
ANGLE MODULATION
• Basic Definitions & Properties
• Relationship between PM and FM waves
• Narrow-Band Frequency Modulation
• Wide-Band Frequency Modulation
• Transmission Bandwidth of FM waves
• Generation of FM waves
• Demodulation of FM signals
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WIDE-BAND FREQUENCY
MODULATION
• Assume that the carrier frequency fc is large
enough
𝑠(𝑡) = Re 𝐴𝑐 exp(𝑗2𝜋𝑓𝑐 𝑡 + 𝑗𝛽sin(2𝜋𝑓𝑚 𝑡))
~
= Re 𝑠 (𝑡)exp(𝑗2𝜋𝑓𝑐 𝑡
• The complex envelope
∞
~
𝑠 (𝑡) = 𝐴𝑐 exp[𝑗𝛽sin 2𝜋𝑓𝑚 𝑡 = 𝑐𝑛 exp(𝑗2𝜋𝑛𝑓𝑚 𝑡)
𝑛=−∞
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WIDE-BAND FREQUENCY
MODULATION
• nth order Bessel function:
𝜋
1
𝐽𝑛 (𝛽) = exp[𝑗(𝛽sin𝑥 − 𝑛𝑥) 𝑑𝑥
2𝜋 −𝜋
• The complex envelope of the FM wave:
∞
~
𝑠 (𝑡) = 𝐴𝑐 𝐽𝑛 (𝛽)exp(𝑗2𝜋𝑛𝑓𝑚 𝑡)
𝑛=−∞
∞
𝑠(𝑡) = Re 𝐴𝑐 𝐽𝑛 (𝛽)exp[𝑗2𝜋 𝑓𝑐 + 𝑛𝑓𝑚 𝑡
𝑛=−∞
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WIDE-BAND FREQUENCY
MODULATION
• Properties of single-tone FM for arbitrary
modulation index β:
𝐽𝑛 𝛽 = 𝐽−𝑛 𝛽 , for 𝑛 even
𝐽𝑛 (𝛽) = −𝐽−𝑛 (𝛽), for 𝑛 odd
𝛽
𝐽0 (𝛽) ≈ 1, 𝐽1 (𝛽) ≈ , 𝐽𝑛 (𝛽) ≈ 0, 𝑛>2
2
∞
𝐽𝑛2 (𝛽) = 1
𝑛=−∞
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WIDE-BAND FREQUENCY
MODULATION
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Chapter 3
ANGLE MODULATION
• Basic Definitions & Properties
• Relationship between PM and FM waves
• Narrow-Band Frequency Modulation
• Wide-Band Frequency Modulation
• Transmission Bandwidth of FM waves
• Generation of FM waves
• Demodulation of FM signals
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TRANSMISSION BANDWIDTH OF
FM WAVES
• Carson’s Rule:
• The FM wave is effectively limited to a finite
number of significant side-frequencies
compatible with a specified amount of
distortion.
• An approximate rule for the transmission
bandwidth of an FM wave
1
𝐵𝑇 ≈ 2𝛥𝑓 + 2𝑓𝑚 = 2𝛥𝑓 1 +
𝛽
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Chapter 3
ANGLE MODULATION
• Basic Definitions & Properties
• Relationship between PM and FM waves
• Narrow-Band Frequency Modulation
• Wide-Band Frequency Modulation
• Transmission Bandwidth of FM waves
• Generation of FM waves
• Demodulation of FM signals
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GENERATION OF FM WAVES
• Direct method:
• A sinusoidal oscillator, with one of the reactive
elements in the tank circuit of the oscillator
being directly controllable by the message
signal
• The tendency for the carrier frequency to drift,
which is usually unacceptable for commercial
radio applications.
• To overcome this limitation, frequency
stabilization of the FM generator is required,
which is realized through the use of feed-back
around the oscillator
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GENERATION OF FM WAVES
• Indirect Method: Armstrong Modulator
• Frequency multiplier
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Chapter 3
ANGLE MODULATION
• Basic Definitions & Properties
• Relationship between PM and FM waves
• Narrow-Band Frequency Modulation
• Wide-Band Frequency Modulation
• Transmission Bandwidth of FM waves
• Generation of FM waves
• Demodulation of FM signals
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DEMODULATION OF FM SIGNALS
• Frequency Discriminator
Block diagram of balanced frequency discriminator
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DEMODULATION OF FM SIGNALS
Phase-locked loop
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SUMMARY
• In this chapter, we have learnt:
• PM and FM waves
• Narrow-Band/ Wide-Band Frequency Modulation
• Transmission Bandwidth of FM waves
• Generation of FM waves
• Demodulation of FM signals
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ASSIGNMENTS
• Reading assignment:
• [1]: p. 152-189
• Problems:
• [1]: 4.12, 4.13, 4.14
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