AM Modulation -- Radio 1

http://www.technologyuk.net/telecommunications/telecom_principles/amplitude_modulation.shtml

LO audio baseband mf f f =

Amplitude Modulation – Early RadioEE 442 – Spring Semester

Lecture 6

Modulation 2

Why Use Modulation?

Most input signals, commonly created by transducers, can’t be sentdirectly over the communication channel. We refer to these signalsas baseband signals (i.e., messages or information).

Instead, a carrier wave, whose properties are better suited to thetransmission requirements, is modified (modulated) to representthe signal.

Modulation is the systematic alteration of the carrier wave so that It carriers the message or information intended to be communicated.

A. Bruce Carlson, Communication Systems: An Introduction to Signals and Noise in Electrical Communication, 2nd ed., McGraw-Hill Book Company, New York, 1975; pp. 5-7.

Modulation 3

Reasons for Using Modulation

Modulation for ease of radiation – Antennas must be greater thanone-tenth of the wavelength (/10); thus, low frequency basebandsignals would require overly large antennas.

Modulation for frequency assignment – The FCC assigns frequency bands to each radio application. Different carrier frequencies areused to meet this need.

Modulation for multiplexing – This allows for multiple signals to becarried on a single transmission medium (multiplexing is one formof modulation).

Modulation to overcome equipment limitations – Modulation is usedto place signals in a portion of the spectrum where equipment limitations are minimal or most easily met.

Modulation to reduce noise and interference – Some types ofmodulation are useful for reducing noise and interference.

A. Bruce Carlson, Communication Systems: An Introduction to Signals and Noise in Electrical Communication, 2nd ed., McGraw-Hill Book Company, New York, 1975; pp. 5-7.

Modulation 4

Modulation Options

PM

FM

AM

AM Modulation -- Radio 5

Baseband versus Carrier Communication

Baseband communication is the transmission of a message as generated is Transmitted without frequency translation.

Carrier communication requires the modulation of the message onto a carrier signal to transmit it over a different frequency band. We usemodulators to do this frequency translation.

(Note: “Pulse modulated” signals, such as PAM, PWM, PPM, PCM and DMare actually baseband digital signal coding (and not the result of frequencyconversion).

Use of Sinusoidal Carrier Signal: Using a sine waveform there are three parameters which we can use to “modulate” a message onto the carrier –they are the amplitude, frequency and phase of the sinusoidal carrier.

Signal

AM

FM

AM Modulation -- Radio 6

Amplitude modulation (AM) is a modulation technique where the amplitude of a high-frequency sine wave (called a radio frequency) is varied in direct proportion to the modulating signal m(t). The modulating signal contains the intended message or information –sometimes consisting of audio data, as in AM radio broadcasting, or two-way radio communications.

The high-frequency sinusoidal waveform (i.e., carrier) is modulated by combining it with the message signal using a multiplier or mixer (Note: mixing is a nonlinear operation because it generates new frequencies).

Amplitude Modulation Description

Agbo & Sadiku; Section 3.2, pp. 84 to 99

AM Modulation -- Radio 7

Amplitude Modulation in Pictures

100 kHz carrier modulated by a 5kHz audio tone

100 kHz carrier modulated by an audio signal

(frequencies up to 6 kHz)

5 kHz Audio tone

Tone-modulatedAM signal

Voice-modulatedAM signal

Frequency Domain Time Domain

AM Modulation -- Radio 8

As with any technology there are advantages and disadvantages to be considered.

Advantages• It is simple to implement• It can be demodulated using a circuit consisting of very few components• AM receivers are inexpensive because no specialized components are required

Disadvantages• It is not efficient with respect to power usage• It is not efficient in bandwidth; requires a bandwidth equal to twice the highest audio frequency• It is prone to high levels of noise because most noise is amplitude based and AM detectors are sensitive to it

Amplitude Modulation Advantages & Disadvantages

AM Modulation -- Radio 9

Example: Voice Signal – 300 Hz to 3400 Hz Baseband

time

amp

litu

de

m(t)

Symbol m(t) represents the source’s message signal.

Time Domain Display

AM Modulation -- Radio 10

Frequency Domain

Voice Band for Telephone Communication

Voice Channel0 Hz – 4 kHz

Voice Bandwidth300 Hz – 3.4 kHz

f0 Hz 300 Hz 3.4 kHz 4 kHz 7 kHz

PSTN or POTS

Pow

er

PSTN → Public SwitchedTelephone Network

AM Modulation -- Radio 11

3,400 Hz

Speechsignal

Speechspectra

Representative Voice Spectrum for Human Speech

For the telephone AT&T determined many years ago that speech couldbe easily recognized when the lowest frequencies and frequencies above 3.4 kilohertz were cutoff.

Waveform as received froma microphone convertingacoustic energy into electricalenergy.

Fast Fourier transform of the above speech waveformshowing energy over range of 0 Hz to 12 kHz.

Time t

Frequency f (in Hz)

AM Modulation -- Radio 12

A crystal radio receiver, also called a crystal set or cat's whisker receiver, is a very simple radio receiver, popular in the early days of radio. It needs no other power source but that received solely from the power of radio waves received by a wire antenna. It gets its name from its most important component, known as a crystal detector, originally made from a piece of crystalline mineral such as galena. This component is now called a diode.

Early AM Crystal Radio Receiver (Minimalist Radio)

Note: 1N34A is agermanium diode

that half-wave rectifies signal

Earphones

LC TunedCircuit

Demodulator

13AM Modulation -- Radio

Foxhole Radio (as used in World war I)

http://bizarrelabs.com/foxhole.htm

Ground

Coil – 120 turns of wire

Cold water pipe

Safety pinRazor blade

Earphones

AM Modulation -- Radio 14

http://www.peeblesoriginals.com/projects/images/foxhole1.jpg

Foxhole Radio (as built in a modern shop today)

AM Modulation -- Radio 15

Crystal Radio Receiver from 1922

Diagram from 1922 showing the circuit of a crystal radio. This common circuit did not use a tuning capacitor, but used the capacitance of the antenna to form the tuned circuit with the coil.

Galena (lead sulfide) was probably the most

common crystal used in “cat's whisker” detectors.

https://en.wikipedia.org/wiki/Crystal_radio

AM Modulation -- Radio 16

Amplitude Modulation (DSB with Carrier)

Amplitude Modulation: The amplitude of a carrier signal is varied linearly with a time-varying message signal.

( ) ( ) ( )

= +

+

= + = +

CNote: Keep & fixed.Carrier signal: ( ) cos( )

Only amplitude is allowed to vary in AM: ( )

( ) ( ) cos cos ( ) cos

C C

C C C

AM C C C C C

c t A t

A A A m t

t A m t t A t m t t

( )m t

cos( )C CA t

( ) cos( )C CA m t t+

AM Modulation -- Radio 17

Expressions for AM-DSB with Carrier

( ) ( ) ( )

= +

+

= + = +

CNote: Keep & fixed.Carrier signal: ( ) cos( )

Only amplitude is allowed to vary in AM: ( )

( ) ( ) cos cos ( ) cos

C C

C C C

AM C C C C C

c t A t

A A A m t

t A m t t A t m t t

( ) ( ) ( )

= + = +

=

Some books write (such as in Haykin & Moher, 2009)

( ) 1 ( ) cos cos ( ) cos

where is the "amplitude sensitivity."

And other books write (Leon W. Couch, II, 2013)

( ) 1

AM C am C C C C am C

am

AM C

t A k m t t A t A k m t t

k

t A ( ) ( ) ( ) + = + ( ) cos cos ( ) cosC C C C Cm t t A t A m t t

But other forms exist for this equation:

We will generally use this format.

AM Modulation -- Radio 18

Amplitude Modulation (DSB with Carrier) Illustrated

http://hyperphysics.phy-astr.gsu.edu/hbase/Audio/bcast.html

AM Modulation -- Radio 19

Phasor View of Amplitude Modulation

https://inspirehep.net/record/1093258/plots

Modulated oscillation is a sum of

these three vectors an is given by

the red vector. In the case of

amplitude modulation (AM), the

modulated oscillation vector is

always in phase with the carrier field

while its length oscillates with the

modulation frequency. The time

dependence of its projection onto

the real axis gives the signal

strength as drawn to the right of the

corresponding phasor diagram.

Black vector Carrier signalRed vector AM modulated signal

Example showing tone modulation

AM Modulation -- Radio 20

Phasor Expression of Amplitude Modulation

( ) Re 12 2

m m

C

j t j tj t

AM

e et e

− = + +

Ct+

mt+

mt−

Tone signal cos(mt)

Carrier cos(Ct)

C+ C m +C m −

Tone modulation

ComplexExponentialFormat

Spectrum

AM Modulation -- Radio 21

Phasor Interpretation of AM DSB with Carrier (continued)

https://www.slideshare.net/azizulhoque539/eeng-3810-chapter-4

Tone modulation

Time t

AM Modulation -- Radio 22Agbo & Sadiku; Section 3.2.1; pp. 89 to 91

Double-Sideband Amplitude Modulation Spectrum

Baseband

CarrierCarrier

Message

SidebandMessage

https://en.wikipedia.org/wiki/Amplitude_modulation

( ) ( ) ( )

( )

= + = +

= − + + + − + +1 12 2

( ) ( ) cos cos ( ) cos

The spectrum is found from the Fourier transform of ( )

( ) ( ) ( ) ( ) ( )

AM C C C C C

AM AM

AM C C C C

t m t t A t A m t t

t

FT t M M A

CarrierMessage

Frequency Shifting

Property

AM Modulation -- Radio 23

AM Modulation Index Basics – Definition

The amplitude modulation (AM) modulation index can be defined as the ratio of the peak

value of the message signal to the amplitude AC of the carrier signal.

When expressed as a percentage it is the same as the depth of modulation. In other words

it can be expressed as:

where AC is the carrier signal amplitude, and

mp is the peak modulation amplitude (maximum change in the RF amplitude

relative to its un-modulated value.

Example: An AM modulation index of 0.5 means the signal increases by a factor of 0.5,

and decreases to 0.5, centered around its unmodulated level. See drawings below.

= + ( ) ( ) cos( )AM C Ct A m t t

=Modulation Indexp

C

m

A

Agbo & SadikuPage 85-86

50% modulation illustrated

AM Modulation -- Radio 24

AM Modulation Index Basics – Examples

50% Tone Modulation

100% Tone Modulation

150% Tone Modulation

Overmodulationor

Envelope Distortion

50%

100%

150%

= + ( ) 1 cos( ) cos( )AM C m Ct A t t

=p

C

m

A

https://en.wikipedia.org/wiki/Amplitude_modulation

AM Modulation -- Radio 25

AM Overmodulation → Envelope Distortion

= = 1.50p

C

m

A

https://upload.wikimedia.org/wikipedia/commons/d/da/AM_150%25_modulation_depth.png

AM Modulation -- Radio 26

Power Efficiency of Amplitude Modulation

Agbo & Sadiku; Section 3.2.2; pp. 91 to 92

2

/2

2 2

/2

G

iven the

{Let power in one

M

sideban

A signal: ( ) cos( ) ( ) cos( )

The power in the carrier is 2

The power in the sidebands (modulated message) is

1lim ( )cos (

}

)

d

AM c c c

cc

T

S c

T

S

T

t A t m t t

AP

P m t t d

P

tT

→

−

= +

=

= =

/2

2

/2

/2

2

/2

/2

2

/2

1 1lim ( ) 1 cos(2 )

2

But ( ) cos(2 ) 0, and so

1 1 1we are left with lim ( )

2 2

That is, is one-half the total message power .

In AM the power in the mess

T

cT

T

T

c

T

T

S mT

T

S m

m t t dtT

m t t dt

P m t dt PT

P P

→−

−

→−

+

=

=

age (useful power) is the power in the

two sidebands. Next, we define .power efficiency

AM Modulation -- Radio 27

Power Efficiency in Amplitude Modulation (continued)

The power efficiency of a modulated signal is the ratio of the power inthe message part of the signal relative to the total power of themodulated signal.

212

12

2

messsage power sideband powerPower efficiency = =

total power total power

In symbols, = , and2

=

mS CC

C S C m

m

C m

PP AP

P P P P

P

A P

= =+ +

+

AM Modulation -- Radio 28

Power Efficiency in Amplitude Modulation (continued)

In general, the form of Pm is complicated and not known precisely. However,we can study AM power efficiency using a tone modulation message.

( )22 2 2

2 2 2

For tone modulation, ( ) cos( ) cos( ) ,

and2 2 2 2

p C C C

p pC

m

C p

m t m t A t

m mAP

A m

= =

= = = =+ +

0.25 0.0303 or 3.03 %

0.5 0.111 or 11.1 %

1.0 0.333 or 33.3 %

Example: (For double-sideband with carrier is AM)

2

22

=

+Modulation index =

Conclusion: AM power efficiency is very low (highly undesirable).

AM Modulation -- Radio 29

Power Efficiency in Amplitude Modulation (continued)

In general, the form of Pm is complicated and not known precisely. In practice,We find Pm by the integral,

/2 /2

2 2 2

/2 /2

/2

2

/2

/2

2 2

/2

1 1lim ( ) cos ( ) lim ( ) 1 cos (2 )

2

1lim ( ) .

2

If ( ) is a , then we have

( ) cos ( ) , then

1 1

single to

i

ne

lim cos ( ) l m

T T

m C CT T

T T

T

mT

T

p C

T

m p CT T

T

P m t t dt m t t dtT T

P m t dtT

m t

m t m t

P m t dtT

→ →− −

→−

→ →−

= = +

=

=

= =

( ) ( )

/2

2

/2

/2

2 2 2

/2

2

1 cos (2 )2

1 1 ( / 2) ( / 2) 1lim

2 2 2

1and ( )

2 4

T

p C

T

T

m p p p

TT

mS p

m t dtT

T TP m dt m m

T T

PP m

−

−→

+

− − = = =

= =

AM Modulation -- Radio 30

Can We Reduce Transmitted Power in AM?

1. Do we need to transmit the carrier, which is the majority of thetransmitted power, in AM?

No, but we will need to have a more sophisticated demodulationscheme in the receiver. We will do this with “double-sidebandsuppressed carrier” modulation.

2. Do we have to transmit both sidebands in AM?

No, because both sidebands contain identical information. Wewill do this by introducing “single-sideband modulation” (withoutcarrier).

AM Modulation -- Radio 31

Preview: Categories of Amplitude Modulation

Baseband spectrum (message spectrum M(f))

Conventional AM (Double-SideBand With Carrier)

Double-Sideband-Suppressed Carrier (DSB-SC)

Single-Sideband /Upper Sideband SSB/USB

Single-Sideband /Lower Sideband SSB/LSB

f

f

f

f

fAlso Vestigial Sideband and Amplitude Companded SSB

Special cases of AM:

DSB-w/C

DSB-SC

AM Modulation -- Radio 32

Generation of Amplitude Modulated Signals

Agbo & Sadiku; Section 3.2.3; pp. 93 to 95

Agbo & Sadiku present two methods for AM generation:

1. Nonlinear AM modulatorAlmost any nonlinearity will work, but a very inexpensive but strongly nonlinear device is the diode. Transistors are also nonlinear and work well as modulators (but more complicated).

2. Switching AM modulatorSwitching is an easily attained function with diodes and transistors in electronic circuits.

There is also a third method:

3. Electronic multipliers (such as Gilbert cells)can be used as modulators.

AM Modulation -- Radio 33

Diode Operation Applied to AM Modulators & Demodulators

1. As nonlinear circuit components (primarily the “square law” part )

2. As “on-off” switches (they have to be driven hard to do this)

Cu

rre

nt

(mA

)

Voltage (V)

UseTaylor’sseries

approx.

( )/ 1DqV kT

D satI I e= −

AM Modulation -- Radio 34

Using Nonlinearity For Modulation (i.e., AM Generation)

R

+

_y(t)

m(t)+

_

Accos(Ct)+

_

Diode

BPFFilter(c)

+

_x(t)

The diode is the nonlinear component (it has an exponentialcharacteristic). Using a Taylor’s series we can express the diodecurrent iD as (with only first two terms of the Taylor’s series),

2

1 2

2 2

1 2 1 2

( ) ( ) ( ); ( ) is diode voltage.

The voltage across resistor R is given by

( ) ( ) ( ) ( ) ( ) ( )

D D D D

D D D D D

i t b v t b v t v t

x t i t R b Rv t b Rv t a v t a v t

= +

= = + = +

“Square Law”behavior

Di

AM Modulation -- Radio 35

Using Nonlinearity For Modulation (continued)

2

1 2

22 2

1 2

21

1

We now can evaluate voltage ( )

( ) ( ) cos ( ) ( ) cos ( )

( ) ( ) ( ) 1 cos (2 )2

2 ( )1 cos ( )

Applying the bandpass filter about , the output voltage (

C C C C

CC

C C

C

x t

x t a m t A t a m t A t

a Ax t a m t a m t t

a m ta A t

a

y

= + + +

= + + +

+ +

21

1

) is

2 ( )( ) ( ) 1 cos ( )AM C C

t

a m ty t t a A t

a

= = +

Note: For to be less than unity, we require 2

1

2 ( )1.

a m t

a

(Eq. 3.23)

AM Modulation -- Radio 36

Using Nonlinearity For Modulation (continued)

Comments:

1. Can use a general nonlinear element (not just a “square law” device)2. The filter can be as simple as a LC resonator3. This is a about the simplest of all modulators (it is unbalanced)

R

m(t)+

_

Accos(Ct)+

_

GeneralNonlinearElement

+

_y(t)C

Tuned to radianfrequency C

1C

LC =

AM Modulation -- Radio 37

Using General Nonlinearity For Modulation

fRF = 0.8fLO = 1.0

LO

RF

LO - RF LO + RF

Vout()

2LO

-R

F

3RF 3LO

2LO

+ R

F

2RF 2LO

1 2 30

2RF

–LO

2RF

+ L

O

vout RF

and LO

(linear)

v2out (

RF+

LO ), (

LO -

RF ), 2

RFand 2

LO(square law)

v3out (2

RF +

LO), (2

RF -

LO), (2

LO+

RF), (2

LO -

RF), 3

RF & 3

LO

Conclusion: Nonlinearity generates new frequencies.

2 3

out DC in in inv v Gv Av Bv= + + + +

Input signals: cos( ) cos( )in RF RF LO LOv A t B t = +

IF

Taylor’s series

tone carrier

AM Modulation -- Radio 38

Switching Amplitude Modulator – The Switch

1 2 1 1( ) cos( ) cos(3 ) cos(5 )

2 3 5C C Cp t t t t

= + − + −

Ron = 0

Roff is infinite

No Capacitance

(t)

By driving a diode with sufficient AC voltage it acts like a switch:

1

periodCf

T=

Forward bias Reverse bias

No current flowSwitch open

Current flowSwitch closed

AM Modulation -- Radio 39

1 2 1 1( ) cos( ) cos(3 ) cos(5 )

2 3 5p t t t t

= + − + −

FourierSeriesrepresentation

Switching Amplitude Modulator – Pulse Spectrum Generated

1

p(t)

02

T

2

T−

2

T

2

− time

Infinitely long pulse train

P()

1

T

2

T

1

T

− 3

T0

Fourier series ofthe pulse trainof period T

1

1

−

Duty CycleT

=

Shown for aduty cycle of 1/4

2

Uses the “Frequency Shifting” Property of the Fourier Transform.

AM Modulation -- Radio 40

Switching Modulator – Generating m(t)cos(Ct)

Pulse trainp(t)

m(t)p(t)

Reference: Lathi & Ding,4th ed., 2009; Fig. 4.4.

AM Modulation -- Radio 41

A “hopelessly unsophisticated” mixer.

− Tom Lee (Stanford University)

The unbalanced single-diode mixer has no isolation and no conversion gain.

Single-diode mixers have been used in many applications --

(1) Detectors for radar in WW II(2) Early UHF Television tuners(3) Crystal radio detectors(4) mm-wave & sub-mm-wave

receivers

Diode Mixer For Modulation and Demodulation

IFRF +

LO

Filter

Diode

This is the same circuitused in the Foxhole radio

for demodulation.

AM Modulation -- Radio 42

AM Demodulation

Section 3.2.4 (pp. 95 to 99)

Coherent (i.e., synchronous) demodulation (or detection) is a methodto recover the message signal from the received modulated signal thatrequires a carrier at the receiver. This carrier signal must match infrequency and phase the received signal.

But . . . Amplitude Modulation has the advantage of not requiring coherent detection methods. Non-coherent methods can be used which are much simpler to implement.

1. AM Envelope Detector

2. AM Rectifier Detector

AM Modulation -- Radio 43

AM Envelope Detector Circuit

Two conditions must be met for an envelope detector to work:

(1) Narrowband [meaning fc >> bandwidth of m(t)](2) AC + m(t) 0

Incoming AMmodulated signal

Rectified AMmodulated signal

Capacitor stores energyfrom the peaks of the

rectified signal

Envelope Detection requires the an RC network with time constant = RC

Key idea: Capacitor captures the voltage peaks of rectified waveform

RC

AM Modulation -- Radio 44

Choosing the RC Time Constant in Envelope Detector

Time constant = RC too short.t

t Time constant = RC too long.

1

Design criteria is 2 2 CB fRC

How the envelope is captured

Reference: Lathi & Ding,4th ed., 2009; Fig. 4.11.

B Hz is bandwidth of message signal.

AM Modulation -- Radio 45

The user has a choice of changing the filter to meet their needs.

Practical Demodulation of an AM Signal

V V

ModulatedRF signal

inputIF output

• • •

•

•

•

IF

output

RF

AM input

Diode

DC blockingEnvelopedetection

FrequencySelectivity

AM Modulation -- Radio 46

AM Rectifier Detection

( )

= +

= + + − + −

= + +

1 13 5

( ) ( ( ))cos( ) ( )

1 2( ( ))cos( ) (cos( ) (cos(3 ) (cos(5 ) ....

2

1( ) other terms.

Note: Multiplication with ( ) allows

= dc term + baseband term

rect C C

C C C C C

C

V t A m t t p t

A m t t t t t

A m t

p t rectifier detection to act essentially as a

synchronous detection without a carrier being generated at the receiver.

DC componentis removed by

capacitor C

LPF

Reference: Lathi & Ding,4th ed., 2009; Fig. 4.10.

AM Modulation -- Radio 47

Double-Sideband Suppressed Carrier AM

Conventional AM transmits both the message and carrier signals.

Hence, the its power efficiency is low,

If (AC)2 approaches zero, then approaches 100%.

2100%m

C m

P

A P =

+

( ) ( ) cos( )

For DSB-SC (double sideband -- suppressed carrier) we have

( ) ( ) cos( ) with a FT pair ( ) ( )

1( )cos( ) ( ) ( ) ( )

2

AM C C

DSB SC C

C DSB SC C C

t A m t t

t m t t m t M

FT m t t M M

−

−

= +

=

= = − + +

AM Modulation -- Radio 48

The penalty for using DSB-SC is a complex detection scheme is needed.We can’t use simple envelope detection.

m(t) ( )DSB SC t − Phase reversals

( ) ( ) cos( )DSB SC Ct m t t − =

Double-Sideband Suppressed Carrier (continued)

AM Modulation -- Radio 49

Double-Sideband Suppressed Carrier Generation

( )m t

( )m t

( )m t−

( ) cos( )Cm t t

t →

t →

Note phasereversal

( )m t( ) cos( )Cm t t

cos( )Ct

(Modulator)

Figure 3.11 in Agbo & Sadiku

Modulation:

DSB-SC Output

AM Modulation -- Radio 50

( )DSB SC −

C m− + 0C−C m− − C m+ + C+C m+ −

TransmittedDSB-SC Signal

USBLSB

DSB-SC has USB and LSB spectra but no carrier impulses at C.

( )M

0 m+ m−

( ) ( )m t M

Double-Sideband Suppressed Carrier Spectrum

( )m t( ) cos( )Cm t t

cos( )Ct

(Modulator)

AM Modulation -- Radio 51

Double-Sideband Suppressed Carrier (continued)

Demodulation:

( )m t

( ) cos( )Cm t t

2cos( )Ct

(Modulator)

( )x t

Low-passfilter

2

2

( ) 2 ( ) cos ( ) ( ) ( ) cos(2 )

1Used the identity: cos 1 cos(2 )

2

The Fourier transform of ( ) is

1( ) ( ) ( 2 ) ( 2 )

2

C C

C C

x t m t t m t m t t

x t

X M M M

= = +

= +

= + − + +

Frequency and phasemust match.

AM Modulation -- Radio 52

DSB-SC Synchronous Demodulation

http://www.amalgamate2000.com/radio-hobbies/radio/dsbsc____demodulation_by_the_squ.htm

Function is to providecarrier recovery

AM Modulation -- Radio 53

DSB-SC Synchronous Demodulation (continued)

x2 2DSB-SC signal

BPF

Carrier recovered

m(t)

Squaring Component

Mixer

Divide by 2

2 1cos 1 cos(2 )

2t t = +

AM Modulation -- Radio 54

Double-Sideband Suppressed Carrier (continued)

Let us examine the spectrum of the demodulated DSB-SC signal.

The message sidebands are shifted from being centered at C

back to the about the origin ( = 0) and 2C. This is illustrated

below.

Note: We now need coherent (i.e., synchronous) detection!

( )X

02 C− 2 C+

Low-pass filterSelects baseband

m+ m−

We filter out the signals centered at 2C.

Message

recovered

AM Modulation -- Radio 55

Double-Sideband Suppressed Carrier (continued)

Example 3.5 (from page 101 of Agbo & Sadiku):

We are given a carrier signal of Ac cos(Ct) and a tone message signal of

m(t) = Am cos(m

t)

Therefore, the AM signal is

( ) cos( ) cos( ) cos( ) cos( )2

1This comes from the identity: cos( ) cos( ) = cos( ) cos( )

2

Next, we take the Fourier transform to get

( ) ( ) (2

C mDSB SC C m C m C m C m

C mDSB SC C m

A At A A t t t t

A A

−

−

= = − + +

− + +

= − + + ) ( ) ( )C m C m C m + − + − − + + +

AM Modulation -- Radio 56

( )DSB SC −

C m − + 0C−C m − −

( )M

0 m+m−

C m + +C+

C m + −

2

C mA A

Double-Sideband Suppressed Carrier (continued)

Tone Modulation

No carrier present

AM Modulation -- Radio 57

Analog Product Modulator

https://en.wikibooks.org/wiki/Electronics/Analog_multipliers

Log

Log

Anti-Log

BufferOutput

X

Y

outV kX Y=

DSB-SC is used primarily today for point-to-point communicationswhere a small number of receivers are involved.

One can buy commercial ICs that perform this function.

Vout

AM Modulation -- Radio 58

http://www.iitk.ac.in/eclub/ee381/AnalogMultipliers.pdf

Gilbert Cell Multiplier

VCC

-VEE

IEE

RCRC

Vin1

Vin2

C Cm

th

qI Ig

kT V= =

Vout

1 2out EE in inV I K V V=

AM Modulation -- Radio 59

Non-Linear DSB-SC Modulator

22 2 2

1 1 2 1

1

22 2 2

2 1 2 1

1

1 2 1 2

2

2 ( )( ) ( ) ( ) 1 cos(2 ) 1 cos( )

2

2 ( )( ) ( ) ( ) 1 cos(2 ) 1 cos( )

2

( ) ( ) ( ) 2 ( ) 4 ( ) cos( )

( ) 4

CC C C

CC C C

C C

DSB SC C

a A a m ts t a m t a m t t a A t

a

a A a m ts t a m t a m t t a A t

a

s t s t s t a m t a A m t t

t a A m

−

= + + + + +

= − + + + + −

= − = +

= ( ) cos( ) BPF selected thisCt t

BPF

( )DSB SC t −

Refer to slide 35 for equations.

AM Modulation -- Radio 60

Non-Linear DSB-SC Modulator (continued)

2( ) 4 ( ) cos( )DSB SC C Ct a A m t t − =

Note that this expression contains no carrier signal. Why?

Answer: The modulator is a balanced configuration and this results

in the carrier signal being cancelled (that assumes perfect balance,

of course).

Definition: A balanced modulator does not output either the carrier

component or the message component. When both are missing, we

say it is double-balanced.

AM Modulation -- Radio 61

Switching DSB-SC Modulators

Agbo & Sadiku present three switching modulators:

1. Series-bridge modulator

2. Shunt-bridge modulator, and

3. Ring modulator

We are only going to discuss the ring modulator because it is the

most widely used and contains the fundamental principle of

operation for all of them. It is important you understand how it

works.

Note: Examples of series-bridge and shunt-bridge modulators areshown in Figure 3.16 (a) and (b), page 106 of Agbo and Sadiku.

AM Modulation -- Radio 62

Ring Modulator for DSB-SC Generation

“Ring”

AM Modulation -- Radio 63

Double-Balanced Diode Ring Modulator

( ) ( )

− + −

1 1

3 5

4cos( ) cos(3 cos(5C C Ct t t

cos( )C CA t

( )cos( )Ck m t t( )m t ( )x t

( )p t

( )m t = ( ) cos( )i Cv p t t

The LO is driven hard enoughto operate the diodes

as on/off switches.

T1 T2

BPF

D1

D2

D4

D3

a b

Bipolar square wave:

1:1 1:1

AM Modulation -- Radio 64

Assume the diodes act as perfect switches (either “on” or “off”) and are controlled by the RF carrier signal (requires large amplitude).

T1T2

D1

D2

D3

D4

RF carrier signal

AC cos(ct)

m(t)

DSB-SC:

( ) cos( )Cm t t

Double-Balanced Diode Ring Modulator (continued)

AM Modulation -- Radio 65

Double-Balanced Diode Ring Modulator (continued)

Operation in the positive half-cycle of the carrier signal

T1T2

D1

D2

m(t) = 0

+

currents

These currents cancel in the primary, thus,

no output.

Diodes D3 &D4 are Off

Positive Half-Cycle:

AC cos(ct)

AM Modulation -- Radio 66

Double-Balanced Diode Ring Modulator (continued)

AC cos(ct)

Operation in the negative half-cycle of the carrier signal

T1T2

D3

D4

m(t) = 0

+

currents

These currents cancel in the

primary sono output.

Diodes D1 &D2 are Off

Negative Half-Cycle:

AM Modulation -- Radio 67

Double-Balanced Diode Ring Modulator (continued)

AC cos(ct)

Operation in the positive half-cycle of the carrier signal passes message signal m(t) to output.

Diodes D1 and D2 are “on” and the secondary of T1 is applied directly to T2.

T1T2

D1

D2

+_

+_

+

_

+_

+_

+

m(t)

+

_

+ m(t)output

AM Modulation -- Radio 68

Double-Balanced Diode Ring Modulator (continued)

AC cos(ct)

Diodes D3 and D4 are “on” and the secondary of T1 is applied directly to T2.

T1T2

D3

D4

+_

+_

+_

+_

+

_

+

m(t)

+

_

- m(t)output

Operation in the negative half-cycle of the carrier signal inverts message signal m(t) at the output.

AM Modulation -- Radio 69

Double-Balanced Diode Ring Modulator (continued)

1 2&D D on3 4&D D on

DSB-SC signal at primary of T2

m(t)

OUTPUT

LO rectangular waveform

t

https://electronicspost.com/ring-modulator-for-the-double-sideband-suppressed-carrier-generation/

AM Modulation -- Radio 70

Double-Balanced Diode Ring Modulator Waveforms

( )m t

cos(2 )C CA f t

( )cos( )Ck m t t

( ) ( )bipolarp t m t

D1 & D2 are “on”

D3 & D4 are “on”

After filtering

AM Modulation -- Radio 71

Requirements:

1. The carrier signal is higher in amplitude than the modulating signal m(t).

2. The carrier signal must be of sufficient amplitude to fully switch the diodes between “on” and “off” states.

3. The carrier signal switches the diodes on and off at a rate higher than the highest frequency contained in m(t).

4. The message signal m(t) is chopped into segments, alternating between two amplitudes; +m(t) and –m(t).

Double-Balanced Diode Ring Modulator

( )m t

AM Modulation -- Radio 72

Double-Balanced Diode Ring Modulator

( )

( )

1( ) 2 ( ) 2 ( ) 1

2

1 2 1 1( ) 2 cos cos(3 ) cos(5 )

2 3 5

4 1 1( ) cos cos(3 ) cos(5 )

3 5

Therefore, the ou

This is a bipolar square wave train.bipolar

bipolar

bipolar

C C C

C C C

p t p t p t

p t t t t

p t t t t

= − = −

= + − + −

= − + −

( )

( )

tput is found by the product,

4 ( ) ( )( ) ( ) ( ) ( ) cos cos(3 ) cos(5 )

3 5

Upon passing through the BPF,

4( ) ( ) cos

bipolar C C C

DSB SC C

m t m tx t m t p t m t t t t

t m t t

−

= = − + −

=

The mathematics behind the Diode Ring Modulator:

t

+1

-1

pbipolar(t)

. . .

. . .

. . .

AM Modulation -- Radio 73

Double-Balanced Diode Ring Modulator/Mixer

LO RF

IF

It is inexpensive and easy to build a ring mixer.

AB

C

D

GG

Trifilar-Wound Toroid

A

B

C

G

D

G

http://kambing.ui.ac.id/onnopurbo/orari-diklat/teknik/qrp/Broadband%20Transformers.htm

AM Modulation -- Radio 74

Commercial Diode Ring Mixer (Mini-Circuits)

Mixer in surface-mount package

Ring of FET devicesoperated as nonlinear

resistances

It is even easierto buy a ring mixer

component.

AM Modulation -- Radio 75

Mixers

Frequency mixing → frequency conversion → heterodyning

( )IF t( )RF t

cos( )LOt

(Mixer)

( )x t

Band-passfilter

RF

LO

IF

A mixer translates the modulation around one carrier frequency to another frequency. In a receiver, this is usually from a higher RF frequency to a lower IF frequency. In a transmitter, it’s the inverse.

We know that a LTI circuit can’t perform frequency translation. Mixers can be realized with either time-varying circuits or non-linear circuits.

AM Modulation -- Radio 76

Digression: What is Heterodyning?

Heterodyning is a signal processing technique invented in 1901 by Canadian inventor-engineer Reginald Fessenden that creates new frequencies by combining or mixing two frequencies using a nonlinear device. Edwin Armstrong invented the superheterodynereceiver in 1918.

Using an electronic circuit to combine an input radio frequency signal(RF) with another signal that is locally generated (LO) to produce new frequencies (IF): one being the sum of the two frequencies and the other being the difference of the two frequencies.

http://www.yourdictionary.com/heterodyning#SgeI5zTvo7VV96o7.99

Applications of heterodyning:1. Used in communications to generate new frequencies. 2. Move modulated signals from one frequency channel to another.3. Used in the superheterodyne radio receivers able to select from

multiple communication channels.

AM Modulation -- Radio 77

Frequency Translation By Mixing

Let ( ) ( ) cos( ) and ( ) ( ) cos( )

The local oscillator (LO) is proportional to cos( )

The mixer (or multiplier) output ( ) is given by

( ) 2 ( ) cos( ) cos( )

Choos. ing

RF C IF IF

LO

C LO

LO C IF

t m t t t m t t

t

x t

x t m t t t

A

= =

=

= −

( ) ( )

( ) ( )

, we have

( ) ( ) cos ( ) cos ( )

( ) ( ) cos (2 )

Note: Used even property of cosines [ . ., cos( ) cos( )]

Choosing , then we have

( ) ( ) co

( ) cos

.

s (

C IF C C IF C

C IF

LO C IF

C

F

IF

I

x t m t t t

x t m t t

i e

t

B

x t m t

m t

= − − + − +

= + −

− =

= +

= + −

( ) ( )

( ) ( )

) cos ( )

( ) ( ) co( ) (2 )cos sI

C C IF C

F C IF

t t

x t m t tm t t

+ + +

= + +

AM Modulation -- Radio 78

Frequency Conversion From C to IF

With a Mixer

Multiplying a modulated signal by a sinusoidal moves the frequency band to sum and difference frequencies.

Note: Super-heterodyning: c + IF ; Sub-heterodyning: c - IF

Example: We want to convert from frequency C to frequency IF .

( )x t = ( ) ( ) cos( )RF ct m t t = ( ) ( ) cos( )IF IFt m t t

( ) 2 cos ( )c IF t

= 2 f

Input frequency c Output frequency fIF

X()Filtered out

BPF response

IF −2 C IF +2 C IF2 C

→

Negative not shown

Bandpass filter (tuned

to IF)

AM Modulation -- Radio 79

Mixer Example (Example 3.7 on Page 110)

Example: Derive the relationship between LO and C so that centering the bandpass filter of the mixer is at LO - C and also ensure that IF is less than (i.e., below) C.

Answer:

We know that LO = C IF in general. We must meet two conditions:

(1) IF = LO - C and (2) IF < C

Start by assuming LO = C + IF that meets the first condition; then thesecond condition, IF < C , implies that

LO - C < C → LO < 2C

AM Modulation -- Radio 80

Superheterodyne Receiver

The superheterodyne, the circuit used in virtually all modern

radios, was invented by Edwin Armstrong in 1918 while he

worked in a US Army Signal Corps laboratory in Paris during

World War I. This is one of the receivers constructed at that

laboratory, shown in a 1920 article in an amateur radio

magazine. It is constructed in two parts. The righthand

section consists of the mixer and local oscillator. Now on

display at the Smithsonian.

https://commons.wikimedia.org/wiki/File:Prototype_Armstrong_superheterodyne_receiver_1920.jpg

Digression

AM Modulation -- Radio 81

An advertisement for the RCA Radiola AR-812 radio – it was the first

commercially produced superheterodyne radio receiver. The

superheterodyne receiver circuit was invented by US engineer Edwin

Armstrong in 1918 during World War I. The rights were purchased by

RCA, and the AR-812 was released March 4, 1924. It used 6 UV-199

triodes: a mixer, a local oscillator, two IF and two audio amplifier stages,

with an IF of 45 kHz, and was priced at $289. It was semi-portable, with

compartments for the batteries in back and a handle on top, and it

weighed 30 pounds.

https://en.wikipedia.org/wiki/Superheterodyne_receiver

First Commercial Superheterodyne Receiver

AM Modulation -- Radio 82

Advantages of the Superheterodyne Receiver

Superior selectivity in signal receptionUses fixed frequency filters allowing for excellent adjacentchannel signal rejection and also helps with image rejection.

Able to receive multiple modulation schemesAbility to incorporate a wide variety of different types ofdemodulators.

Able to receive very high frequency signalsMost of the receiver’s signal processing is done a lowerfrequencies using mixers to down convert higherfrequency incoming signals (easier to filter at lowerfrequencies, lower cost components, etc.).

The word heterodyne is derived from

the Greek roots hetero -"different", and - dyne "power".

AM Modulation -- Radio 83

Image Signals in Mixers (1)

RF IF

LO

frequency

Signal band

LORF

frequency

IF band

IF

RF

spec

trum

Desireddown-conversion

IF = LO - RF

This converts thespectrum at the RFcarrier frequency

down to the spectrumcentered at the IF

frequency.

There is no signal shown inthis part of the spectrum.

AM Modulation -- Radio 84

Image Signals in Mixers (2) – Now an Image Signal Appears

Now both the spectrum at the RF

carrier frequency andthe undesired imagespectrum are down

converted to the spectrum centered at

the IF frequency.

RF IF

LO

frequency

Signal band

Image band

LORF image

frequency

IF band

IF

RF

spec

trum

IF = LO - RF

IF = image - LO

and

But, now both signalsappear in the IF band.

The image spectrum is not wanted.

AM Modulation -- Radio 85

Image Signals in Mixers (3)

RF IF

LO

frequency

Signal band

LORF

frequency

IF band

IF

RF

spec

trum

Again, the desireddown-conversion

IF = RF - LO

This converts thespectrum at the RFcarrier frequency

down to the spectrumcentered at the IF

frequency.

Suppose the LO frequency isBelow the RF frequency.

AM Modulation -- Radio 86

Image Signals in Mixers (4) – With Image Signal

As before both the spectrum at the RF

carrier frequency andthe undesired imagespectrum are down

converted to the spectrum centered at

the IF frequency.

RF IF

LO

frequency

Signal band

LORF

frequency

IF band

IF

RF

spec

trum

IF = RF - LO

The LO frequency isbelow the RF frequency.

image

Again, both signalsappear in the IF band.

IF = LO - image

and

AM Modulation -- Radio 87

Image Rejection in a Single Mixer Heterodyne Receiver

https://en.wikipedia.org/wiki/Superheterodyne_receiver

A

A

B

B

C

C

D

D

RF channels

AM Modulation -- Radio 88

Simultaneous Selectively and Image Rejection

High Frequency IF

Making the IF frequency higher enables image signals to be far fromthe wanted signal; thus, RF filtering is relatively easier and increasesthe rejection level of image signals.

Low Frequency IF

The advantage of a low frequency IF allows for filters providing muchbetter adjacent channel rejection by filtering – this improves the selectivity of the receiver. And low frequency filters are less expensivealso.

Solution:

These two conflicting requirements can not be simultaneously satisfied by a single IF frequency. A solution is to use two IF frequenciesvia a double-conversion superheterodyne architecture.

AM Modulation -- Radio 89

A Double-Conversion Superheterodyne Receiver

https://commons.wikimedia.org/wiki/File:Double-conversion_superheterodyne_receiver_block_diagram.svg

Many high performance spectrum analyzer receivers use triple-conversion or quadruple-conversion superheterodyne architectures.

A good reference is https://www.keysight.com/upload/cmc_upload/All/5952-0292EN.pdf

AM Modulation -- Radio 90

Simplified Swept-Tuned Spectrum Analyzer Block Diagram

http://www.microwavejournal.com/articles/print/27120-part-3-overcoming-rfmicrowave-interference-challenges-in-the-field-using-rtsa

Keysight Application Note

September 15, 2016

https://www.keysight.com/upload/cmc_upload/All/5952-0292EN.pdf

AM Modulation -- Radio 91

Elenco AM/FM Dual-Radio Receiver Kit

AM

FM

Antenna

Model AM/FM-108CK Superhet Radio

Antenna

https://www.elenco.com/product/amfm-radio-kitcombo-ic-transistor/

Battery

Speaker

AM Modulation -- Radio 92

AM and FM Broadcast Bands

http://hyperphysics.phy-astr.gsu.edu/hbase/Audio/radio.html

10 kHz bandwidth from540 kHz to 1720 kHz(118 possible bands)

200 kHz bandwidth from88.1 MHz to 108.1 MHz

(100 possible bands)

UH

F TV

AM Modulation -- Radio 93

Example: AM Broadcast Stations

http://ypropubmedia.com/online-radio-advertising-services-delhi/

AM Modulation -- Radio 94

Example: Aircraft Communication (Airband)

https://aviatorsattic.com/product-category/pilot-amt-library/pilots-

library/communication/

Airband – 108 MHz to 137 MHz

108 to 117.95 MHz has 200 channels of 50 kHz each for navigational aids such as VOR beacons and precision

approach systems (ILS localizer).

118 to 136.975 MHz is divided into 2,280 channels (each 8.33 kHz of

bandwidth) for AM voice transmissions.

Instrument Landing System localizer

VHF Omni-Directional Range

AM Modulation -- Radio 95

Another Mixer Example (Practice Problem 3.8, Page 110)

For a frequency converter the carrier frequency of the output signal is 425 kHz and the carrier frequency of the AM input signal ranges from 500 kHz to 1500 kHz. Find the tuning ratio of the local oscillator

If the frequency of the local oscillator is given by (a) IF = LO - C and(b) IF = C + LO .

Answer: (a) IF = LO - C → LO = C + IF → superheterodyning

(b) IF = C + LO → LO = C - IF → sub-heterodyning

,max

,min

,LO

LO

,max ,max

,min ,min

1500 4252.081

500 425

LO C IF

LO C IF

+ += = =

+ +

,max ,max

,min ,min

1500 42514.33

500 425

LO C IF

LO C IF

− −= = =

− −

AM Modulation -- Radio 96

Quadrature Amplitude Modulation (QAM)

Fact: Both conventional AM and DSB-SC AM are wasteful of bandwidth.

One way to improve of bandwidth efficiency is with quadrature amplitudemodulation (QAM). It involves two data streams: the I-channel and theQ-channel.

Bandwidth efficiency is improved by allowing two signals to share the same bandwidth of a channel. But this can only be done if the twomodulated signals are orthogonal to each other. We now see how this can be accomplished.

A better name for QAM might be quadrature-carrier multiplexing.

https://www.rcrwireless.com/20160901/test-and-measurement/what-is-64-qam-tag6-tag99

AM Modulation -- Radio 97

Quadrature-Carrier Multiplexing (QCM)

Quadrature-carrier multiplexing allows for transmitting two message signals on the same carrier frequency.

(1) Two quadrature carriers are multiplexed together,

(2) Signal mI(t) modulates the carrier cos(Ct), and Signal mQ(t) modulates the carrier sin(Ct).

(3) The two modulated signals are added together & transmitted over the channel as

= + ( ) ( ) cos( ) ( ) sin( )QAM I C Q Ct m t t m t t

AM Modulation -- Radio 98

Quadrature Amplitude Modulation Features

1. QAM transmits two DSB-SC signals in the bandwidth of one DSB-SC signal.

2. Interference between the two modulated signals of the same frequencyis prevented by using two carriers in phase quadrature. This is becausethey are orthogonal to each other.

3. The In-phase (I-phase) channel modulates the cos(Ct) signal and the Quadrature-phase (Q-phase) channel modulates the sin (Ct) signal.

4. The carriers used in the transmitter and receiver are synchronous witheach other. In fact, they must be almost exactly in quadrature with each other, otherwise they experience cochannel interference.

5. Low-pass filters are used to extract the baseband signals mI(t) and mQ(t) in the receiver.

AM Modulation -- Radio 99

https://math.stackexchange.com/questions/474398/waves-of-differing-frequency-are-orthogonal-help-me-understand

Signals are harmonically related.

0 0sin( ) sin(2 ) 0 and cos( ) cos(2 ) 0

T T

C C C Ct t dt t t dt = =

sin(Ct)

sin(2Ct)

t

Illustrating Orthogonality With Sinusoidal Signals - I

AM Modulation -- Radio 100

Illustrating Orthogonality With Sinusoidal Signals - II

Area of product over one period is equal to zero.Therefore, the signals are orthogonal.

http://www.katjaas.nl/sinusoids2/sinusoids2.html

0sin( ) cos( ) 0

T

C Ct t dt =

AM Modulation -- Radio 101

Quadrature Amplitude Modulation (QAM)

cos( ) ( ) cos( ) ( )sin( )

where ( ) cos( ) and ( ) sin( )

C C C C

C C

A t I t t Q t t

I t A Q t A

+ = +

= =

https://www.slideshare.net/ahsanhalini/quadrature-amplitude-modulation-54999195

HSDPA (High-Speed Downlink Packet Access) is a packet-based mobile telephony protocol used in 3G UMTS radio networks.

QAM Can Be Viewed As Combination of ASK and PSK

AM Modulation -- Radio 102

Review: Spectral Lines for Sine and Cosine Signals

AM Modulation -- Radio 103

Quadrature Amplitude Modulation and Demodulation

2 cos( )Ct

2 sin( )Ct

( )Qz t

Transmitter Receiver

Channel

( )Iz t

( )QAM tcos( )Ct

sin( )Ct

− =Note: cos( 90 ) sin( )C Ct t

( )Im t

( )Qm t

( )Im t

( )Qm t

AM Modulation -- Radio 104

Cosine waveform

Cosine waveform delayed 90

Cosine waveform advanced 90

( ) cos ( )x t t=

2( ) cos ( )x t t = −

2( ) cos ( )x t t = +

t

t

t

Illustrating 90 Degree ( /2) Phase Shift of a Cosine Wave

AM Modulation -- Radio 105

Quadrature Amplitude Demodulation (QAM)

= +

= = +

= +

= + +

2

( ) ( )cos( ) ( )sin( )

( ) 2 cos( ) ( ) 2 cos( ) ( )cos( ) ( )sin( )

( ) 2 ( ) cos ( ) 2 ( ) cos( ) sin( )

( ) ( ) ( ) cos(2 ) ( )sin(2 )

We rec

QAM I C Q C

I C QAM C I C Q C

I I C Q C C

I I I C Q C

t m t t m t t

z t t t t m t t m t t

z t m t t m t t t

z t m t m t t m t t

= = +

= + 2

over ( ) by passing ( ) through a LPF.

and

( ) 2 sin( ) ( ) 2 sin( ) ( )cos( ) ( )sin( )

( ) 2 ( ) sin ( ) 2 ( ) sin( ) sin( )

( )

I I

Q C QAM C I C Q C

Q Q C I C C

Q

m t z t

z t t t t m t t m t t

z t m t t m t t t

z t = − +( ) ( ) cos(2 ) ( )sin(2 )

We recover ( ) by passing ( ) through a LPF.

Q Q C I C

Q Q

m t m t t m t t

m t z t Page 111Agbo & Sadiku

AM Modulation -- Radio 106

Quadrature-Amplitude Modulation & Demodulation

( ) ( ) cos( ) ( )sin( )QAM I C Q Ct m t t m t t = +

2 cos( )Ct

2 sin( )Ct

( )Qz t

Transmitter Receiver

Channel

( )Iz t

( )QAM tcos( )Ct

sin( )Ct

( )Im t ( )Im t

( )Qm t ( )Qm t

Analogsignal

Analogsignal

AM Modulation -- Radio 107

Quadrature-Amplitude Demodulation

-/2

cos(LOt)

sin(LOt)

mI(t)

LORF

mQ(t)

Quadrature Downconverter

+C-C

+LO-LO

½ ½

+C-C

+LO

-LO

+½j

-½j

+IF-IF

+IF

-IF -½j

+½j

Re

Im

Im

Re mI(t)

mQ(t)

cos(LOt)

sin(LOt)

j

IF C LO = −

Receiver

AM Modulation -- Radio 108

Re

Im

Re

Im

mI(t)

mQ(t)

Quadrature-Amplitude Demodulation (continued)

This illustrates the orthogonality of the two modulated signals.

In-phase signal occupies thereal axis-frequency plane

Quadrature signal occupies theimaginary axis-frequency plane

AM Modulation -- Radio 109

Quadrature-Amplitude Modulation & Demodulation

( ) ( ) cos( ) ( )sin( )QAM I C Q Ct m t t m t t = +

2 cos( )Ct

2 sin( )Ct

( )Qz tTransmitter Receiver

Channel

( )Iz t

( )QAM tcos( )Ct

sin( )Ct

( )Im t ( )Im t

( )Qm t ( )Qm t

Digitalsignals

Now it becomes a digital communication system

AM Modulation -- Radio 110

Quadrature-Amplitude Modulation (4-QAM)

http://www.ni.com/white-paper/3896/en/

time t

Amplitude

ConstellationDiagram

AM Modulation -- Radio 111

Effect of frequency error and phase errorin synchronous detection

AM Modulation -- Radio 112

QAM: Phase Error in Synchronous Detection

The local carrier in a DSB-SC receiver and a QAM receiver is 2cos(Ct + ),while the signal carrier at the input of each receiver is cos(Ct). Thatmeans the signal carrier and local carrier in the receivers are phase shifted relative to each other. Derive expressions for the demodulatedoutput signals for both receivers. Compare your results for DSB-SC and QAM.

Solution: (Example 3.9 on pp. 112-113)

For the DSB-SC receiver (NOT QAM):

( )=2cos( ) ( ) 2 ( ) cos( ) cos( )

( ) ( ) cos( ) ( ) cos ( )

Therefore, low-pass filering gives ( ) ( ) cos( )

C DSB SC C C

C

x t t t m t t t

x t m t m t t

y t m t

−+ = +

= + +

=

Phase error =

AM Modulation -- Radio 113

QAM: Phase Error in Synchronous Detection (continued)

1

For the QAM receiver:

( ) 2cos( ) ( ) 2cos( ) ( ) cos( ) ( ) sin( )

( ) 2 ( ) cos( ) cos( ) 2 ( ) sin( ) cos( )

( ) ( ) cos( ) ( ) cos(2 ) ( ) sin( )

I C QAM C I C Q C

I I C C Q C C

I I C Q

z t t t t m t t m t t

z t m t t t m t t t

z t m t m t t m t

= + = + +

= + + +

= + + − +

1

( ) sin(2 )

and

( ) 2sin( ) ( ) 2sin( ) ( ) cos( ) ( ) sin( )

( ) 2 ( ) cos( ) sin( ) 2 ( ) sin( ) sin( )

( ) ( ) sin( ) ( ) sin(2 ) ( ) cos( )

Q C

Q C QAM C I C Q C

Q I C C Q C C

Q I C Q

m t t

z t t t t m t t m t t

z t m t t t m t t t

z t m t m t t m t m

+

= + = + +

= + + +

= + + + − ( ) cos(2 )

The low-pass filters suppress the terms centered at 2

Therefore,

( ) ( ) cos( ) ( ) sin( )

( ) ( ) cos( ) ( ) sin( )

Q C

C

I I Q

Q Q I

t t

y t m t m t

y t m t m t

+

= −

= + Co-channel interference

AM Modulation -- Radio 114

QAM: Frequency Error in Synchronous Detection

Compare the effect of a small frequency error in the local carrier for a DSB-SC receiver and a QAM receiver. The carrier at the transmitter is cos(Ct) and the carrier at the receiver is 2cos(C+ )t.

Answer: (Example 3.10 on pp. 113-114)

( ) ( ) cos( ) ( )sin( )

( ) ( ) cos( ) ( )sin( )

I I Q

Q Q I

y t m t t m t t

y t m t t m t t

= −

= +

Note the similarity to the answer to Example 3.9.You should now be able to guess the answer to aquestion involving both phase error and frequencyerror . (Practice Problem 3.10 on page 114)

AM Modulation -- Radio 115

https://hamradioschool.com/hey-why-is-there-no-transmit-power-when-i-key-the-mic/

Single Sideband (SSB) AMModulation and Demodulation

Filter method; Phasing Method; Weaver Method

AM Modulation -- Radio 116

Single Sideband (SSB) AM

Why single sideband? DSB-SC is spectrally inefficient because it uses twice the bandwidth of the message. SSB addresses that issue.

The signal can be reconstructed from either the upper sideband (USB) orthe lower sideband (LSB).

SSB transmits a bandpass filtered version of the modulated signal.

Reference: Lathi & Ding,4th ed., 2009; Fig. 4.12.

AM Modulation -- Radio 117

Single Sideband (SSB) AM

Multiplication of a USB signal by cos(Ct) shifts the spectrum to left and right.

Reference: Lathi & Ding,4th ed., 2009; Fig. 4.13.

AM Modulation -- Radio 118

Phase-Shift Method to Generate SSB AM

= +( ) ( )cos( ) ( )sin( )

where minus sign applies to USB

and plus sign applies to the LSB.

( ) is ( ) phase delayed by - /2

SSB C h C

h

t m t t m t t

m t m t

Reference: Lathi & Ding,4th ed., 2009; Fig. 4.17.

HilbertTransformer

AM Modulation -- Radio 119

Reference Note: Quadrature Phase-Shifts & Hilbert Transform

sin( )tcos( )t−

2

For a + 90° (or /2) phase shift:

−

sin( ) cos( )

cos( ) sin( )

t t

t t

For a - 90° (or -/2) phase shift:

−

sin( ) cos( )

cos( ) sin( )

t t

t t

+j

-j

( ) sgn( )H j = −

=

sin( )t

+j/2

-j/2

-1/2 -1/2

cos( )t−

Hilbert transform

AM Modulation -- Radio 120

Polyphase Filter – HA5WH Network (Gingell)

BalancedAudioInput

0

90

270

180

Polyphase filters are symmetric RC structures with inputs and outputssymmetrically arranged in relative phases.

Achieves constant phase shifts over 300 Hz to about 3,000 Hz witha 60 dB rejection of the other sideband.

HA5WH Network

AM Modulation -- Radio 121

Using a Polyphase Filter in a SSB Generator

Polyphase Filter

VTO

0

90

180

270

Balanced Modulator

Balanced Modulator

90

0Driver& LPF

https://www.radioexperimenter.us/re-04-1994/phaseshifi-network-analysis-and-optimization.html

The ARRL Handbook for the Radio Amateur, (American Radio Relay League, Newington, 1993), and many previous editions.

AM Modulation -- Radio 122

Phase-Shift Method to Generate SSB AM

• The phasing method uses two balanced mixers to eliminate the carrier.

• The phasing method for SSB generation uses a phase-shift to cancel one of the sidebands.

• The carrier oscillator is applied to the upper balanced modulator along with the modulating signal.

•

• The carrier and modulating signals are both shifted by 90 degrees and applied to another modulator.

• Phase-shifting causes one sideband to cancel when the two modulator outputs are summed together.

AM Modulation -- Radio 123

Phase-Shift Method for Receiving SSB Signals

http://www.panoradio-sdr.de/ssb-demodulation/

https://www.dsprelated.com/showarticle/176.php

Reference:

I

Q

Receiver

HT

AM Modulation -- Radio 124

Synchronous Demodulation of SSB AM

https://www.dsprelated.com/showarticle/176.php

( ) 2 ( ) cos( ) ( ) cos( ) ( ) sin( ) 2cos( )

( ) ( ) ( ) cos(2 ) ( ) sin(2 )

and upon low-pass filtering we have

( ) ( )

SSB C C h C C

C h C

x t t t m t t m t t t

x t m t m t t m t t

x t m t

= =

= +

=

AM Modulation -- Radio

https://www.chegg.com/homework-help/questions-and-answers/design-ssb-modulator-using-phasing-method-weaver-s-ssb-modulator-shown-figure-matlab-simul-q21017269

Weaver’s SSB Modulator

125

Figure: Weaver’s method for generating SSB Signals

AM Modulation -- Radio 126

Hartley Image-Rejection Architecture

cos(Ct)

sin(Ct)

IF-IF

LO-LO

-IF

-IF

IF-IF

IFIF

90

RF

IF

Antenna

http://www.microwavejournal.com/articles/3226-on-the-direct-conversion-receiver-a-tutorial

AM Modulation -- Radio 127

SSB Mixer and Image Rejection Mixer Comparison

https://commons.wikimedia.org/wiki/File:SSB_and_Image_Rejection_Mixer.svg

AM Modulation -- Radio 128

Pulse Amplitude Modulation (PAM) → Digital Signal

How is PAM in digital communication similar to AM in analog communication?

AM Modulation -- Radio 129

Questions

?

AM Modulation -- Radio 130

Questions

1. What is the point of creating a rectified output when using a diode forAM modulation?

It combines the carrier signal with the message signal m(t). See slides 34, 35 & 36 for illustration of this.

2. More about how mixers work. (Three questions asked about mixers.)

Two principles are used in mixers to create new frequencies: (1) Nonlinearitythe I-V characteristics of a nonlinear device do this, and (2) time-varying switching will create new frequencies.

3. Explain the idea of images in mixers again.

See the next four slides.

From Spring 2018 EE 442 Class

AM Modulation -- Radio 131

Image Signals in Mixers (1)

RF IF

LO

frequency

Signal band

LORF

frequency

IF band

IF

RF

spec

trum

Desireddown-conversion

IF = LO - RF

This converts thespectrum at the RFcarrier frequency

down to the spectrumcentered at the IF

frequency.

There is no signal shown inthis part of the spectrum.

AM Modulation -- Radio 132

Image Signals in Mixers (2) – Now an Image Signal Appears

Now both the spectrum at the RF

carrier frequency andthe undesired imagespectrum are down

converted to the spectrum centered at

the IF frequency.

RF IF

LO

frequency

Signal band

Image band

LORF image

frequency

IF band

IF

RF

spec

trum

IF = LO - RF

IF = image - LO

and

But, now both signalsappear in the IF band.

The image spectrum is not wanted.

AM Modulation -- Radio 133

Image Signals in Mixers (3)

RF IF

LO

frequency

Signal band

LORF

frequency

IF band

IF

RF

spec

trum

Again, the desireddown-conversion

IF = RF - LO

This converts thespectrum at the RFcarrier frequency

down to the spectrumcentered at the IF

frequency.

Suppose the LO frequency isBelow the RF frequency.

AM Modulation -- Radio 134

Image Signals in Mixers (4) – With Image Signal

As before both the spectrum at the RF

carrier frequency andthe undesired imagespectrum are down

converted to the spectrum centered at

the IF frequency.

RF IF

LO

frequency

Signal band

LORF

frequency

IF band

IF

RF

spec

trum

IF = RF - LO

The LO frequency isbelow the RF frequency.

image

Again, both signalsappear in the IF band.

IF = LO - image

and

AM Modulation -- Radio 135

More Questions

4. Why does FM take so much more bandwidth than AM?

It is because we force FM (and PM) signals to have constant amplitude. We show this in detail when we cover Angle Modulation.

5. Is there a point where having too many mixers can impact your frequencynegatively?

Issues with using multiple mixers:a. Adds complexity (such as many mixing products come into play)b. Most mixers are lossy and need power gain to continue to

process signals (and mixers add noise of their own to signals)c. Cost (not only of mixers but for LO oscillators and amplifiers)

AM Modulation -- Radio 136

More Questions

6. In an antenna why does the length have to be /4?

It actually does not have to be precisely /4, but making it longer does not have an advantage in maximizing its efficiency. Making it shorterdoes decrease the signal strength received. Also, we may want theantenna to resonate at the operating frequency to increase theefficiency of the antenna.

7. Also, if an AM signal has a wavelength of hundreds of feet, how can itsantenna be so small?

The coil of wire picks up the time-variation of the electromagnetic wave’smagnetic field and induces a current in the coil which becomes the signal at the input of the AM receiver. The current increases with the numberof turns of the coil.

AM Modulation -- Radio 137

More Questions

8. When you have no carrier signal, are you sending the message signal?If so, is there even any modulation of the amplitude or justthe original signal?

In the absence of a carrier signal, then only the message signal can be sentat the frequency band of the message signal. We say the message signal “modulates” the carrier signal.

9. Does the local oscillator change its frequency depending upon the RFfrequency you want to select?

I assume you are referring to the superheterodyne receiver. Yes, the local oscillator frequency is changed to select the RF frequency you want to select. We require that frequency difference between theRF and LO frequencies is a constant.

AM Modulation -- Radio 138

More Questions

10. How is a mixer similar/different from amplitude modulation?

The mathematics of the AM says we use a multiplier to multiply the carriersignal with the message signal. A mixer performs signal multiplication as required in amplitude modulation.

11. Why do we use sub-heterodyning? What are the common applicationsof sub-heterodyning?

When analyzing a system we focus upon the RF frequencies involved andPossible local oscillator frequencies. The focus is upon frequency conversion.Considerations: Do we want to work with higher or lower frequencies? What frequency bands are to be avoided?

12. Which modulation is the most useful today? AM, FM or PM Why?

FM is the most widely used. It has better noise immunity.

AM Modulation -- Radio 139

More Questions

13. How do you build a mixer?

Example are on slides 62 and 73 illustrating the diode ring mixer.

14. How would you handle having your oscillator being 1% off?

A 1% offset in frequency is very large. You might lock it to a referencefrequency (such as a precision crystal oscillator). You might use a phase-lock loop to slave the local oscillator to the correct frequency. You mightuse a pilot signal broadcast with the modulated carrier. There are manyother possible solutions.

15. From our homework assignments what would you consider the mostimportant problems?

All of them. During the review session before the first midterm I will give amore detailed and specific answer to this question.

AM Modulation -- Radio 140

More Questions

16. What happens to an over-modulated signal? Can the signal still be used?

Over-modulation leads to distortion in the message. It can still be used in voice communication if the voice over-modulation is not too severe. The point is that voice can still be understood even with moderate distortion.

17. Why do RLC resonator circuits have a -3 dB frequency corresponding toa bandwidth of B = 1/RC?

( ) ( )

= − + + = + + +

= − =

2 22 2

1 2

2 1

1 1 1 1&

2 2 2 2

1Bandwidth

LC LCRC RC RC RC

BRC

2( )

( ) ( )Ri j L

Hi t R j L j RCL

= =

+ +

AM Modulation -- Radio 141

More Questions

R

+

_y(t)

m(t)+

_

Accos(Ct)+

_

Diode

BPFFilter(c)

+

_x(t)

The diode is the nonlinear component (it has an exponentialcharacteristic). Using a Taylor’s series we can express the diodecurrent iD as (only first two terms of Taylor’s series),

2

1 2

2 2

1 2 1 2

( ) ( ) ( ); ( ) is diode voltage.

The voltage across resistor R is given by

( ) ( ) ( ) ( ) ( ) ( )

D D D D

D D D D D

i t b v t b v t v t

x t i t R b Rv t b Rv t a v t a v t

= +

= = + = +

“Square Law”behavior

( )Di t

18. How does x(t) =iD(t)R come about? Why is it not KVL?

Remember:

AM Modulation -- Radio 142

More Questions (2018)

19. Can you explain using nonlinearity for modulation much like Problem 3 in Homework #3?

( )

( ) ( )

( ) ( ) ( )

2

2

2 2 2

( ) 4 , but 1

Substituting for ( ( ) cos( )) ,

( ) 4 ( ) cos( ) ( ) cos( )

( ) 4 ( ) cos( ) ( ) 2 ( ) cos( ) cos ( )

But we k

D D D

D C C

C C C C

C C C C C C

x t i R v v R R

v m t A t

x t m t A t m t A t

x t m t A t m t A m t t A t

= = + =

= + +

= + + + + +

= + + + + + + +

( )2

2 2now cos ( ) 1 cos(2 )2

CC C C

AA t t = +

Continued next slide →

Modulator

Given relationship

AM Modulation -- Radio 143

( ) ( ) ( )

( ) ( )( )

2 2 2

2 22

( ) 4 ( ) cos( ) ( ) 2 ( ) cos( ) cos ( )

( ) 4 ( ( )) ( ) 2 ( ) cos( ) cos(2 )2 2

The band-pass filter (BPF) passes only terms of cos( )

C C C C C C

C CC C C

C

x t m t A t m t A m t t A t

A Ax t m t m t A m t t t

t

= + + + + + + +

= + + + + + + +

( )

( )

, thus ( ) is

( ) 4 cos( ) 2 ( ( )) cos( )

( ) 4 2 cos( ) ( ) cos( )

( ) cos( ) ( ) cos( )

C C C C

C C C C

C C

y t

y t A t A m t t

y t A A t m t t

y t K t m t t

= + +

= + +

= +

Problem 3 in Homework #3 continued:

More Questions

AM Modulation -- Radio 144

More Questions

20. What is the primary form of noise production in AM systems?

The greatest noise problem in AM channels is interference, noise pickup,& fading in wireless transmission, all of which distort the amplitude of thetransmitted signal.

21. QAM (Several asked about QAM so we need to cover it again)

I will review QAM again after questions are answered.

AM Modulation -- Radio 145

( )= +2

( ) ( )y t A B g t

( )

= + = + = + +

= + +

22( ) ( ) cos( ) 1 cos(2 )

2

( ) cos(2 )2 2

By t A B g t A B t A t

B By t A t

( ) ( )= + + + +2 3

( ) ( ) ( ) ( ) .y t A Bg t C g t D g t other terms

22. Go over Problem 2 in Homework #2

Problem 2 Square Law Device (20 points)

Answer: The square-law device generates a frequency that is the double of the single tone frequency f. To show this we make use of the trigonometric identity:

Answer: The cubic term in the series generates a frequency that is triple of the frequency of g(t), that is,

frequency 3f. This comes from using the trigonometric identity of cos3(x) = ¼[3cos(x) + cos(3x)].

Thus, the cos3(2ft) term gives us both a cos(2ft) term (not so interesting) and a cos(3·2ft) term

(which is a new frequency being introduced).

More Questions

You are given a square-law component with an input to output relationship of

(a) To explore the behavior of this device we let the input signal g(t) be a sinusoidal

tone, that is, g(t) = cos(t).

(b) What frequencies does the cubic term (that is, D[g(t)]3) generate when driven by g(t) = cos(t)?

AM Modulation -- Radio 146

More Questions

23. Why does milliwatts relate to dBm rather than just use milliwatts?

We express milliwatts (mW) in decibels by

We use this because logarithms add rather than multiply in calculations.

Example:

Suppose a signal of 3 dBm power drives an amplifier of gain = 13 dB. What is theoutput power of the amplifier.

Answer: Output power (in dB) = 3 dBm + 13 dB = 16 dBm,

compared to 2 mW (= 3 dBm) multiplied by gain of 20 (= 13 dB) = 40 mW

( )10Power in dBm 10 log dBm Note: logarithm of a ratio1 mW

mWP =

AM Modulation -- Radio 147

More Questions

24. What is the one question you think we should have asked?

Answer: The one for the topic that is troubling you.

AM Modulation -- Radio 148

Additional Slides & Illustrations

https://www.stonybrook.edu/commcms/electrical/news/2017/armstrong_award.php

AM Modulation -- Radio 149

Generating m(t)cos(Ct) using Convolution

Fold, Shift & Multiply

From Convolution Theorem: (t) g(t) Cn( ) G( )

F

F

G( )

Cn( ) (t)

g(t)

Output Spectrum

LO

RF

( ) cos( )Cm t t

AM Modulation -- Radio 150

Gilbert Cell Multiplier Using FET Devices

https://www.sciencedirect.com/topics/engineering/gilbert-cell

AM Modulation -- Radio 151

https://www.exploregate.com/video.aspx?video_id=5

IIP3 = Third-order Input Intercept Point

AM Modulation -- Radio 152

Modern Mechanix (December 1952)

http://blog.modernmechanix.com/dick-tracy-wrist-radio/

Today it is reality!