# Connexions

### Lenses

What is a lens?

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

#### Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
• VOCW

This module is included inLens: Vietnam OpenCourseWare's Lens
By: Vietnam OpenCourseWare

Click the "VOCW" link to see all content affiliated with them.

### Recently Viewed

This feature requires Javascript to be enabled.

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:

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
• 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.

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.

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.

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!

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.

{}{}If the signal is transmitted at a carrier frequency fc=600Mhzfc=600Mhz size 12{f rSub { size 8{c} } ="600" ital "Mhz"} {}, then the λ3×108 m/s6×108Hz0.5mλ3×108 m/s6×108Hz0.5m 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 "." 5m} {} so that an antenna whose length is λ/105λ/105 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.

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 105105 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

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.

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

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.

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 ( t ) cos 2 ( 2πf c t ) = x m ( t ) × cos ( 2πf c t ) × cos ( 2πf c t ) size 12{y $$t$$ =x rSub { size 8{m} } $$t$$ "cos" rSup { size 8{2} } $$2πf rSub { size 8{c} } t$$ =x rSub { size 8{m} } $$t$$ times "cos" $$2πf rSub { size 8{c} } t$$ times "cos" $$2πf rSub { size 8{c} } t$$ } {}

can be written as

Y ( f ) = X m ( f ) F { cos ( 2πf c t ) } F { cos ( 2πf c t ) } Y ( f ) = X m ( f ) F { cos ( 2πf c t ) } F { cos ( 2πf c t ) } size 12{Y $$f$$ =X rSub { size 8{m} } $$f$$ *F lbrace "cos" $$2πf rSub { size 8{c} } t$$ rbrace *F lbrace "cos" $$2πf rSub { size 8{c} } t$$ rbrace } {}

The Fourier transform of F{cos(2πfct)}F{cos(2πfct)}F{cos(2πfct)}F{cos(2πfct)} size 12{F lbrace "cos" $$2πf rSub { size 8{c} } t$$ rbrace *F lbrace "cos" $$2πf rSub { size 8{c} } t$$ rbrace } {} is shown below.

Thus, using Fourier transform properties it is easy to derive trigonometric identities.

Two-minute miniquiz problem

Problem 23-1 — AM, suppressed carrier radio

A slight alternative to the AM suppressed carrier radio system is shown below.

Using an appropriate low pass filter, H(f), to detect Xm(f)Xm(f) size 12{X rSub { size 8{m} } $$f$$ } {}, determine the spectrum Ym(f)Ym(f) size 12{Y rSub { size 8{m} } $$f$$ } {}.

Solution

We need to convolve the spectrum of the modulated function X(f) with the Fourier transform of Sin(2πfct)Sin(2πfct) size 12{ ital "Sin" $$2πf rSub { size 8{c} } t$$ } {}.

Thus, the output is zero.

The following reviews the results for suppressed carrier radio.

Suppressed carrier AM requires that the transmitter and receiver be perfectly synchronized. A small difference in frequency of transmitter and receiver oscillators results in a drift in the phase difference between the oscillators which causes variations in the amplitude of the detected signal, called signal strength fading.

4/ Synchronous detection

Synchronous detection is effectively used when the oscillators in the modulator and detector are the same. For example in the chopper-stabilized DC amplifier shown below.

In DC amplifiers, the signal component at DC interacts with the biasing of the transistors in the amplifier complicating the design. In a chopper stabilized amplifier, the DC signal is modulated, amplified by an AC amplifier, and then detected. Thus, amplification at DC is achieved with an AC amplifier.

1/ Brief history

1864 James Clerk Maxwell published his equations of electromagnetism.

1887 Heinrich Hertz proved that waves travel through the “ether” by creating a spark in a gap between two wires and picking up a voltage in a loop of wire — the first transmitter and receiver of electromagnetic waves.

Brief history, cont’d

1896 Guglielmo Marconi took out patents on a system of wireless telegraphy.

1906 Lee de Forrest invented the triode vacuum tube for sensitive detection of telegraphy signals. Despite the fact that De Forrest did not understand how the “audion” worked this invention began the modern electronic era.

1906 Reginald Aubrey Fessenden transmitted voice and music over radio waves using a 100,000 Hz alternator designed by Charles Steinmetz at General Electric Company. This was the beginning of broadcasting of audio over the airways.

1912 Edwin Howard Armstrong analyzed the operation of the audion tube and made the first vacuum tube amplifier as part of a sensitive receiver of wireless telegraphy. He used regeneration now called feedback. Noting that the vacuum tube circuit could be made to oscillate, he used this to make the first electronic transmitter. Armstrong’s work ushered in the modern era of radio transmission and reception.

1916 David Sarnoff, working for the Marconi Company, envisaged “music boxes” (radios) as consumer products and the system of radio broadcast as we have it today.

1918 Armstrong enlisted in the army during World War I and worked for the Signal Corps in Paris. He developed the superheterodyne receiver which became, and is to this day, the basis of AM radio receivers.

1919 The Radio Corporation of America was formed out of General Electric Company and the American Marconi Company with David Sarnoffas commercial manager. Between 1918 and 1923 radio broadcasting became pervasive as inexpensive radios became widely available. Sarnoffb ecame president and perhaps the most powerful person in the burgeoning communications industry.

The important conception was to develop a radio broadcast system that consisted of relatively small number of transmitters each transmitting at different (carrier) frequencies and a large numbers of inexpensive receivers that could be tuned to different transmission frequencies. In order to be sufficiently inexpensive so that everyone could own one, the receivers had to be simple to manufacture. Thus, the system consisted of a small number of expensive transmitters and a large number of inexpensive receivers.

A typical AM radio transmitter has a block diagram shown below.

The modulator produces a signal that has the form

x ( t ) = A ( 1 + x m ( t ) ) cos ( 2πf c t ) x ( t ) = A ( 1 + x m ( t ) ) cos ( 2πf c t ) size 12{x $$t$$ =A $$1+x rSub { size 8{m} } \( t$$ \) "cos" $$2πf rSub { size 8{c} } t$$ } {}

Thus, the transmitted signal has the same spectrum as the suppressed carrier AM signal except that the carrier frequency is broadcast. Thus, synchronization of transmitter and receiver is achieved.

4/ Spectrum of transmitted signal

The typical spectrum of an AM station is shown below with the nomenclature defined for its different components.

The Federal Communication Commission (FCC) allots 10 kHz of bandwidth for each station. Stations in any one locale differ in their carrier frequencies, which are the numbers indicated on the radio dial.

The FCC allocates frequency bands in the radio spectrum (3kHz- 300GHz) for communications purposes. The frequency band from 535 to 1605 kHz is reserved for AM broadcast radio. This is a bandwidth of 1070 kHz. Since each station is allotted a bandwidth of 10 kHz, 107 non-overlapping stations can operate in each locale.

6/ Signal at input to an AM receiver

An AM receiver input spectrum consists of AM signals of different carrier frequencies and different signal strengths resulting from differences in strengths of transmitters and differences in their distances from the receiver.

The receiver allows the listener to tune into a station and minimizes the interference of other stations.

Radio frequency (RF) amplifier. Tunable, broadly frequency selective amplifier that attenuates image station at fc+2fifc+2fi size 12{f rSub { size 8{c} } +2f rSub { size 8{i} } } {} and has a gain of 5-15 dB.

Local oscillator. Provides frequencies fcfc size 12{f rSub { size 8{c} } } {} to the RF amplifier and f0=fc+fif0=fc+fi size 12{f rSub { size 8{0} } =f rSub { size 8{c} } +f rSub { size 8{i} } } {} to the modulator.

Intermediate frequency (IF) amplifier. Fixed frequency (at fifi size 12{f rSub { size 8{i} } } {}), highly frequency selective amplifier with a gain of 30 dB.

Audio amplifier. Amplifier with a gain 15-30 dB.

This design has two important attributes: (1) it segregates sharp frequency selectivity from tuning in different stages which simplifies the design; (2) it distributes overall gain over three frequency ranges which improves the stability of the receiver.

The RF amplifier provides some frequency selectivity about the selected station carrier frequency fcfc size 12{f rSub { size 8{c} } } {}. The modulator shifts the spectrum of the output of the RF amplifier so that the frequency fcfc size 12{f rSub { size 8{c} } } {} is shifted to fifi size 12{f rSub { size 8{i} } } {} with another copy of the spectrum centered on fc+2fifc+2fi size 12{f rSub { size 8{c} } +2f rSub { size 8{i} } } {}. The IF amplifier is sharply tuned and centered on fi.

The spectrum of the output of the IF amplifier is the spectrum of the selected station shifted from fcfc size 12{f rSub { size 8{c} } } {} to fifi size 12{f rSub { size 8{i} } } {} . Hence, the output is an AM signal whose carrier frequency is fifi size 12{f rSub { size 8{i} } } {}.

This system modulates the selected station from fcfc size 12{f rSub { size 8{c} } } {} to fifi size 12{f rSub { size 8{i} } } {} . However, note that a station whose carrier frequency is fc+2fifc+2fi size 12{f rSub { size 8{c} } +2f rSub { size 8{i} } } {} is also modulated down to fifi size 12{f rSub { size 8{i} } } {}. A purpose of the RF amplifier is to attenuate this image station.

The peak (or envelope) detector demodulates the AM signal. A simple circuit that detects an AM signal is shown below.

The peak detector works well when

For these conditions, the filter attenuates the carrier frequency

f c >> 1 RC >> f m f c >> 1 RC >> f m size 12{f rSub { size 8{c} } ">>" { {1} over {2π ital "RC"} } ">>"f rSub { size 8{m} } } {}

but not the frequency of the signal

SIMULINK can be used to study properties of the peak detector. The following is a block diagram of a peak detector that can be found in the matlab folder as peakdetector.m.

IV. CONCLUSIONS

Technological developments, such as the use of modulation for signal transmission, can have enormous social implications. The development of telegraphy, telephony, radio, TV, and now the internet, cable TV, and cellular phones have revolutionized how people relate to each other world wide.

## Content actions

### Give feedback:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks