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Lecture 17:Modulation, Broadcast Radio

Module by: Thanh Dinh Vu, Truc Pham-Dinh, Anh Tuan Hoang, Tam Huynh-Ngoc

Summary: Modulation is widely used to encode a signal so as to more effectively utilize it. Modulation is fundamental to electronic communication systems—radio, TV, satellite communications, cell phones, etc.

Lecture #17:
MODULATION, BROADCAST RADIO
Motivation:
  • Modulation is widely used to encode a signal so as to more effectively utilize it.
  • Modulation is fundamental to electronic communication systems—radio, TV, satellite communications, cell phones, etc.
Outline:
  • General description of modulation
  • Amplitude modulation
  • Broadcast AM radio
  • Conclusions
I. GENERAL DESCRIPTION OF MODULATION
1/ Overview
The word modulation in an electronic context means to recode a signal for the purpose of more effectively manipulating that signal. For example, suppose we have some signal xm(t) that we wish to process in some way.
For example, we wish to
  • transmit it through a channel,
  • filter it,
  • amplify it,
  • display it,
  • record it.
However, it is not efficient, convenient, economical, or possible to do so directly. Then we encode the signal and process the encoded signal to improve some aspect of the processing.
2/ Wave-parameter modulation
Modulation can involve varying some feature of a CT signal to encode the signal. Varying the amplitude of a sinusoid (amplitude modulation or AM) or its frequency (frequency modulation or FM) in proportion to a signal is called wave-parameter modulation.
Figure 1
3/ Pulse-parameter modulation
Modulation can also encode the CT signal with the parameters of pulses called pulse-parameter modulation. A number of different pulse-parameter modulation schemes are shown below.
Figure 2
In PWM, the width of pulses encodes the amplitude of the CT signal. In PAM the amplitude of pulses encodes the CT signal. In PCM the amplitude of the quantized CT signal is encoded as a binary number that is represented by a pulse code.
4/ Example of the use of modulation — pigeon telemetry
An ornithologist wishes to record the sounds made by a Lahore pigeon (shown below) while in flight.
Figure 3
Typical pigeon sounds have a spectrum in the frequency range 0.1-3 kHz. Since the pigeon is in flight, we need to make a small (light weight) system consisting of a microphone and a telemetering system that will transmit the sound information.
One might simply transduce the audio signal from the microphone and transmit the electrical signal to the ground. A question arises — what size antenna is needed to transmit the signal in an energetically efficiently manner?
For energetic efficiency, the dimensions of the antenna cannot be orders of magnitude smaller than the wavelength of the transmitted signal. The wavelength λ of the transmitted signal is
λ = c f ≈ 3 × 10 8 m / s 3 × 10 3 Hz ≈ 100 km λ = c f ≈ 3 × 10 8 m / s 3 × 10 3 Hz ≈ 100 km size 12{λ= { {c} over {f} } approx { {3 times "10" rSup { size 8{8} } m/s} over {3 times "10" rSup { size 8{3} } ital "Hz"} } approx "100" ital "km"} {}
If we make the antenna λ/10, then the antenna dimensions are at least 10 km. Thus, the antenna dimensions exceed that of the pigeon by a factor of more than 104!
Figure 4
On the left is a scale drawing of the pigeon (in red) and the antenna (in dark blue).
The pigeon will not get off the ground!
One solution is to move the spectrum of the transduced pigeon sounds to a high frequency, to transmit this modulated signal to the ground, and then to demodulate to audio frequencies.
Figure 5
{}{}If the signal is transmitted at a carrier frequency fc=600Mhzfc=600Mhz size 12{f rSub { size 8{c} } ="600" ital "Mhz"} {}, then the λ≈3×108 m/s6×108Hz≈0.5mλ≈3×108 m/s6×108Hz≈0.5m size 12{λ approx { {3 times "10" rSup { size 8{8} } " m/s"} over {6 times "10" rSup { size 8{8} } ` ital "Hz"} } approx 0 "." 5`m} {} so that an antenna whose length is λ/10≈5λ/10≈5 size 12{λ/"10" approx 5} {} cm which is much more manageable for the pigeon.
5/ Narrow-band signals
The modulated transduced pigeon sound has a spectrum that is centered about the carrier frequency fcfc size 12{f rSub { size 8{c} } } {} and has a bandwidth of 2fm2fm size 12{2f rSub { size 8{m} } } {} where fmfm size 12{f rSub { size 8{m} } } {} is the maximum frequency of the pigeon sound.
Figure 6
For the pigeon sound we have fmfm size 12{f rSub { size 8{m} } } {} = 3 kHz and fcfc size 12{f rSub { size 8{c} } } {} = 600 MHz. Thus, the bandwidth is only 10−510−5 size 12{"10" rSup { size 8{ - 5} } } {}of the carrier frequency — an example of a narrowband signal.
An arbitrary narrowband signal can be expressed as
x ( t ) = x c ( t ) cos ( 2πf c t ) + x s ( t ) sin ( 2πf c t ) x ( t ) = x c ( t ) cos ( 2πf c t ) + x s ( t ) sin ( 2πf c t ) size 12{x \( t \) =x rSub { size 8{c} } \( t \) "cos" \( 2πf rSub { size 8{c} } t \) +x rSub { size 8{s} } \( t \) "sin" \( 2πf rSub { size 8{c} } t \) } {}
where xc(t)xc(t) size 12{x rSub { size 8{c} } \( t \) } {}and xs(t)xs(t) size 12{x rSub { size 8{s} } \( t \) } {} are lowpass time functions. We can expand x(t) as follows
x ( t ) = 1 2 ( x c ( t ) + 1 j x s ( t ) ) e j2πf c t + 1 2 ( x c ( t ) − 1 j x s ( t ) ) e − j2πf c t , x ( t ) = R { ( x c ( t ) + 1 j x s ( t ) ) e j2πf c t } , x ( t ) = a ( t ) cos ( 2πf c t + ϕ ( t ) ) , x ( t ) = 1 2 ( x c ( t ) + 1 j x s ( t ) ) e j2πf c t + 1 2 ( x c ( t ) − 1 j x s ( t ) ) e − j2πf c t , x ( t ) = R { ( x c ( t ) + 1 j x s ( t ) ) e j2πf c t } , x ( t ) = a ( t ) cos ( 2πf c t + ϕ ( t ) ) , alignl { stack { size 12{x \( t \) = { {1} over {2} } \( x rSub { size 8{c} } \( t \) + { {1} over {j} } x rSub { size 8{s} } \( t \) \) e rSup { size 8{j2πf rSub { size 6{c} } t} } + { {1} over {2} } \( x rSub {c} size 12{ \( t \) - { {1} over {j} } x rSub {s} } size 12{ \( t \) \) e rSup { - j2πf rSub { size 6{c} } t} } size 12{,}} {} # size 12{x \( t \) =R lbrace \( x rSub { size 8{c} } \( t \) + { {1} over {j} } x rSub { size 8{s} } \( t \) \) e rSup { size 8{j2πf rSub { size 6{c} } t} } rbrace ,} {} # size 12{x \( t \) =a \( t \) "cos" \( 2πf rSub { size 8{c} } t+ϕ \( t \) \) ,} {} } } {}
Where
a ( t ) = x c 2 ( t ) + x s 2 ( t ) and ϕ ( t ) = − tan − 1 x s ( t ) x c ( t ) . a ( t ) = x c 2 ( t ) + x s 2 ( t ) and ϕ ( t ) = − tan − 1 x s ( t ) x c ( t ) . size 12{a \( t \) = sqrt {x rSub { size 8{c} } rSup { size 8{2} } \( t \) +x rSub { size 8{s} } rSup { size 8{2} } \( t \) } ~ matrix { {} # {} } ital "and"~ matrix { {} # {} } ϕ \( t \) = - "tan" rSup { size 8{ - 1} } { {x rSub { size 8{s} } \( t \) } over {x rSub { size 8{c} } \( t \) } } "." } {}
An arbitrary narrowband signal can be written as
x ( t ) = a ( t ) Cos ( 2πf c t + ϕ ( t ) ) x ( t ) = a ( t ) Cos ( 2πf c t + ϕ ( t ) ) size 12{x \( t \) =a \( t \) ` ital "Cos"` \( 2πf rSub { size 8{c} } t+ϕ \( t \) \) } {}
Thus, a general narrowband signal contains both amplitude and phase/frequency modulation. In amplitude modulation (AM) ϕ(t)ϕ(t) size 12{ϕ \( t \) } {} is constant; in phase/frequency modulation (PM or FM) a(t) is constant.
II. AMPLITUDE MODULATION
1/ AM, suppressed carrier
Figure 7
Perhaps the simplest amplitude modulation scheme is the suppressed carrier scheme in which
x ( t ) = x m ( t ) × cos ( 2πf c t ) x ( t ) = x m ( t ) × cos ( 2πf c t ) size 12{x \( t \) =x rSub { size 8{m} } \( t \) times "cos" \( 2πf rSub { size 8{c} } t \) } {}
Therefore, the Fourier transform is
x ( f ) = x m ( f ) ∗ F { cos ( 2πf c t ) } , x ( f ) = x m ( f ) ∗ 1 2 ( δ ( f − f c ) + δ ( f + f c ) ) , x ( f ) = 1 2 ( x m ( f − f c ) + x m ( f + f c ) ) . x ( f ) = x m ( f ) ∗ F { cos ( 2πf c t ) } , x ( f ) = x m ( f ) ∗ 1 2 ( δ ( f − f c ) + δ ( f + f c ) ) , x ( f ) = 1 2 ( x m ( f − f c ) + x m ( f + f c ) ) . alignl { stack { size 12{x \( f \) =x rSub { size 8{m} } \( f \) *F lbrace "cos" \( 2πf rSub { size 8{c} } t \) rbrace ,} {} # x \( f \) =x rSub { size 8{m} } \( f \) * { {1} over {2} } \( δ \( f - f rSub { size 8{c} } \) +δ \( f+f rSub { size 8{c} } \) \) , {} # x \( f \) = { {1} over {2} } \( x rSub { size 8{m} } \( f - f rSub { size 8{c} } \) +x rSub { size 8{m} } \( f+f rSub { size 8{c} } \) \) "." {} } } {}
The Fourier transform of the modulated signal x(t) can be obtained graphically.
Figure 8
The Fourier transform Xm(f)Xm(f) size 12{X rSub { size 8{m} } \( f \) } {} is repeated at ±fc±fc size 12{ +- f rSub { size 8{c} } } {}.
2/ Demodulation (detection) of AM, suppressed carrier
Figure 9
The original signal xm(t)xm(t) size 12{x rSub { size 8{m} } \( t \) } {} can be recovered by modulating the modulated signal and passing the result through a lowpass filter, a process called demodulation or detection.
Figure 10
3/ AM, suppressed carrier radio
A radio communication system that consists of a transmitter and receiver and which uses suppressed carrier AM is shown below.
Therefore,
x ( t ) = x m ( t ) cos ( 2πf c t ) , y ( t ) = x ( t ) cos ( 2πf c t ) , and y ( t ) = x m ( t ) cos 2 ( 2πf c t ) = x m ( t ) 2 ( 1 + cos ( 4πf c t ) ) . x ( t ) = x m ( t ) cos ( 2πf c t ) , y ( t ) = x ( t ) cos ( 2πf c t ) , and y ( t ) = x m ( t ) cos 2 ( 2πf c t ) = x m ( t ) 2 ( 1 + cos ( 4πf c t ) ) . alignl { stack { size 12{x \( t \) =x rSub { size 8{m} } \( t \) "cos" \( 2πf rSub { size 8{c} } t \) ~,~y \( t \) =x \( t \) "cos" \( 2πf rSub { size 8{c} } t \) ,~ ital "and"} {} # y \( t \) =x rSub { size 8{m} } \( t \) "cos" rSup { size 8{2} } \( 2πf rSub { size 8{c} } t \) = { {x rSub { size 8{m} } \( t \) } over {2} } \( 1+"cos" \( 4πf rSub { size 8{c} } t \) \) "." {} } } {}
Hence,
Y ( f ) = 1 2 X m ( f ) + 1 4 X m ( f − 2f c ) + 1 4 X m ( f + 2f c ) . Y ( f ) = 1 2 X m ( f ) + 1 4 X m ( f − 2f c ) + 1 4 X m ( f + 2f c ) . size 12{Y \( f \) = { {1} over {2} } X rSub { size 8{m} } \( f \) + { {1} over {4} } X rSub { size 8{m} } \( f - 2f rSub { size 8{c} } \) + { {1} over {4} } X rSub { size 8{m} } \( f+2f rSub { size 8{c} } \) "." } {}
The spectrum of y(t) involves the spectrum of cos2(2πfct)cos2(2πfct) size 12{"cos" rSup { size 8{2} } \( 2πf rSub { size 8{c} } t \) } {} which can be found by the trigonometric identity or as shown below.
y ( t ) = x m ( t ) cos 2 ( 2πf c t ) = x m ( t ) × cos ( 2πf c t ) × cos ( 2πf c t ) y ( t ) = x m