Communication Systems, 5e

(1)

Communication Systems, 5e

Chapter 4: Linear CW Modulation A. Bruce Carlson

Paul B. Crilly

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Chapter 4: Linear CW Modulation

• Bandpass signals and systems

• Double-sideband amplitude modulation

• Modulation and transmitters

• Suppressed-sideband amplitude modulation

• Frequency conversion and demodulation

(3)

A family of AM Waveforms

• Double-Sideband, Carrier (AM in text)

• Double-Sideband, Suppressed Carrier (DSB in text)

  ^t ^A   ^t ^cos  ² ^f ^t 

v    

_c



  ^t ^A ^m   ^t

A 

_c



  ^t ^A  ¹ ^m   ^t 

A 

_c

   

(4)

Conventional AM

• Baseband

• Bandpass

• Fourier Domain

𝐴 𝑡 𝐴 ⋅ 1 𝜇 ⋅ 𝑚 𝑡

𝑣 𝑡 𝐴 ⋅ 1 𝜇 ⋅ 𝑚 𝑡 ⋅ cos 2𝜋 ⋅ 𝑓 ⋅ 𝑡

𝐴 ⋅ cos 2𝜋 ⋅ 𝑓 ⋅ 𝑡 𝐴 ⋅ 𝜇 ⋅ 𝑚 𝑡 ⋅ cos 2𝜋

𝑉 𝑓 𝐴

2 ⋅ 𝛿 𝑓 𝑓 𝛿 𝑓 𝑓 𝐴

2 ⋅ 𝜇 ⋅ 𝑀 𝑓 𝑓 𝑀 𝑓 𝑓

carrier message

(5)

AM Sidebands

• Assume that the message is a cosine wave

– messages are typically bounded by +/- 1.0

• The spectral response becomes

𝑚 𝑡 cos 2𝜋 ⋅ 𝑓 ⋅ 𝑡

𝑉 𝑓 𝐴

2 ⋅ 𝛿 𝑓 𝑓 𝛿 𝑓 𝑓 𝐴

4 ⋅ 𝜇 ⋅ 𝛿 𝑓 𝑓 𝑓 𝛿 𝑓 𝑓 𝑓

𝐴

4 ⋅ 𝜇 ⋅ 𝛿 𝑓 𝑓 𝑓 𝛿 𝑓 𝑓 𝑓

𝑀 𝑓 1

2 ⋅ 𝛿 𝑓 𝑓 𝛿 𝑓 𝑓

carrier

sidebands

(6)

AM Positive Frequencies

• For the positive frequency segment of the spectrum

𝑉 𝑓 𝐴

4 ⋅ 𝜇 ⋅ 𝛿 𝑓 𝑓 𝑓 𝐴

2 ⋅ 𝛿 𝑓 𝑓 𝐴

4 ⋅ 𝜇 ⋅ 𝛿 𝑓 𝑓 𝑓

Lower sideband Carrier Upper sideband

Mag

fc m

c f

f  f_c f_m A_c2

A_c4

(7)

DSB (with Suppressed Carrier)

• Baseband

• Bandpass

• Fourier Domain

  ^t ^A ^m   ^t

A 

_c



  ^t ^A ^m   ^t ^cos  ² ^f ^t 

v 

_c

   

_c



 

^c

 ^M  ^f ^f

_c

 ^M  ^f ^f

_c

 

2 f A

V     

Message only, no carrier

(8)

DSB Sidebands

• Assume that the message is a cosine wave

– messages are typically bounded by +/- 1.0

• The spectral response becomes

  ^t ^cos  ² ^f ^t 

m   

_m



       

   



_c _m _c _m



c

m c

c

f f

f 4 f

A

f f

f 4 f

f A V

































    ^f ^f

_m

  ^f ^f

_m

 

2 f 1

M       

sidebands

(9)

DSB Positive Frequencies

• For the positive frequency segment of the spectrum

 

^c



_c _m



^c

 ^f ^f

_c

^f

_m



4 f A

f 4 f

f A

V          

No signal carrier, two sidebands

Lower sideband Upper sideband

Mag

fc m

c f

f  f_c f_m A_c4 A_c4

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Phasor Analysis AM

• Given a tone message …

𝑠 𝑡 𝐴 ⋅ 1 𝜇 ⋅ cos 2𝜋 ⋅ 𝑓 ⋅ 𝑡 ⋅ cos 2𝜋 ⋅ 𝑓 ⋅ 𝑡

• A positive frequency phasor can be defined and drawn

𝐴 𝑡 1 𝜇 ⋅ cos 2𝜋 ⋅ 𝑓 ⋅ 𝑡

𝑠 𝑡 𝐴

2 ⋅ exp 𝑗 ⋅ 2𝜋 ⋅ 𝑓 ⋅ 𝑡 𝐴

2 ⋅ 𝜇 ⋅ exp 𝑗 ⋅ 2𝜋 ⋅ 𝑓 𝑓 ⋅ 𝑡 𝐴

2 ⋅ 𝜇 ⋅ exp 𝑗 ⋅ 2𝜋 ⋅ 𝑓 𝑓 ⋅ 𝑡 𝑠 𝑡 𝐴 ⋅ cos 2𝜋 ⋅ 𝑓 ⋅ 𝑡 𝐴

2 ⋅ 𝜇 ⋅ cos 2𝜋 ⋅ 𝑓 𝑓 ⋅ 𝑡 𝐴

2 ⋅ 𝜇 ⋅ cos 2𝜋 ⋅ 𝑓 𝑓 ⋅ 𝑡

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Phasor Analysis AM (2)

• A positive frequency phasor can be defined and drawn

𝑠 𝑡 𝐴

2 ⋅ exp 𝑗 ⋅ 2𝜋 ⋅ 𝑓 ⋅ 𝑡 𝐴

2 ⋅ 𝜇 ⋅ exp 𝑗 ⋅ 2𝜋 ⋅ 𝑓 𝑓 ⋅ 𝑡 𝐴

2 ⋅ 𝜇 ⋅ exp 𝑗 ⋅ 2𝜋 ⋅ 𝑓 𝑓 ⋅ 𝑡

The message phasors add so only the magnitude of the carrier phasor appears to change.

im a g

real A

^c

2 A

^c

4  

A

^c

4   f

c

f

m



f

m



Sideband phaser sum

𝑠 𝑡 𝐴

2 ⋅ exp 𝑗 ⋅ 2𝜋 ⋅ 𝑓 ⋅ 𝑡 · 1 𝜇 ⋅ exp 𝑗 ⋅ 2𝜋 ⋅ 𝑓 ⋅ 𝑡 exp 𝑗 ⋅ 2𝜋 ⋅ 𝑓 ⋅ 𝑡

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Phasor Analysis DSB

• Given a tone message …

𝑚 𝑡 𝐴 ⋅ cos 2𝜋 ⋅ 𝑓 ⋅ 𝑡

𝑠 𝑡 𝐴 ⋅ 𝐴 ⋅ cos 2𝜋 ⋅ 𝑓 ⋅ 𝑡 ⋅ cos 2𝜋 ⋅ 𝑓 ⋅ 𝑡

𝑠 𝑡 𝐴 ⋅ 𝐴

2 ⋅ cos 2𝜋 ⋅ 𝑓 𝑓 ⋅ 𝑡 cos 2𝜋 ⋅ 𝑓 𝑓 ⋅ 𝑡

• A positive frequency phasor can be defined and drawn

𝑠 𝑡 𝐴 ⋅ 𝐴

4 ⋅ exp 𝑗 ⋅ 2𝜋 ⋅ 𝑓 𝑓 ⋅ 𝑡 exp 𝑗 ⋅ 2𝜋 ⋅ 𝑓 𝑓 ⋅ 𝑡

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Phasor Analysis DSB (2)

𝑠 𝑡 𝐴 ⋅ 𝐴

4 ⋅ exp 𝑗 ⋅ 2𝜋 ⋅ 𝑓 𝑓 ⋅ 𝑡 exp 𝑗 ⋅ 2𝜋 ⋅ 𝑓 𝑓 ⋅ 𝑡

• A positive frequency phasor can be defined and drawn

im real

A 4 A

m c



A 4 A

m c

 f

c

f

m



f

m



Sideband phaser sum

𝑠 𝑡 𝐴 ⋅ 𝐴

4 ⋅ exp 𝑗 ⋅ 2𝜋 ⋅ 𝑓 ⋅ 𝑡 · exp 𝑗 ⋅ 2𝜋 ⋅ 𝑓 ⋅ 𝑡 exp 𝑗 ⋅ 2𝜋 ⋅ 𝑓 ⋅ 𝑡

The message phasors add

so only the magnitude

appears to change.

(14)

Power and Trig Math Note

1

𝑇⋅ cos 2𝜋 ⋅ 𝑓 ⋅ 𝑡 ⋅ 𝑑𝑡 1

𝑇⋅ 1 cos 2𝜋 ⋅ 2𝑓 ⋅ 𝑡

2 ⋅ 𝑑𝑡

𝑃 s 𝑡 lim

→

1

𝑡 s 𝑡 ⋅ 𝑑𝑡

1

𝑇⋅ cos 2𝜋 ⋅ 𝑓 ⋅ 𝑡 ⋅ 𝑑𝑡 1

𝑇⋅ 1

2⋅ 𝑑𝑡 1 2⋅1

𝑇⋅ cos 2𝜋 ⋅ 2𝑓 ⋅ 𝑡 ⋅ 𝑑𝑡

1

𝑇⋅ cos 2𝜋 ⋅ 𝑓 ⋅ 𝑡 ⋅ 𝑑𝑡 1 𝑇⋅1

2⋅ 𝑇 2

𝑇 2

1 2⋅1

𝑇⋅sin 2𝜋 ⋅ 2𝑓 ⋅ 𝑡 2𝜋 ⋅ 2𝑓

1

𝑇⋅ cos 2𝜋 ⋅ 𝑓 ⋅ 𝑡 ⋅ 𝑑𝑡 1 2

1 2𝑇⋅ 1

2𝜋 ⋅ 2𝑓 sin 2𝜋 ⋅ 2𝑓 ⋅𝑇

2 sin 2𝜋 ⋅ 2𝑓 ⋅ 𝑇 2

1

𝑇⋅ cos 2𝜋 ⋅ 𝑓 ⋅ 𝑡 ⋅ 𝑑𝑡 1 2

1 2𝑇⋅ 1

2𝜋 ⋅ 2𝑓 sin 2𝜋 sin 2𝜋

1

𝑇⋅ cos 2𝜋 ⋅ 𝑓 ⋅ 𝑡 ⋅ 𝑑𝑡 1 2

Average power in a periodic waveform s(t) Average power in

cos waveform

(15)

Power in AM Waveform

  ^t ^A  ¹ ^m   ^t  ^cos  ² ^f ^t 

s 

_c

      

_c



 

² _c

    

_c



²

s

T

P s t A 1 m t cos 2 f t

S           

   

  ^

c

^

²

2 2 2

c

s

A 1 2 m t m t cos 2 f t

P             

 

   

 

²



_c



²

2 2 c

2 c

2 c s

t f 2 cos t

m A

t f 2 cos t

m 2

A

t f 2 cos A

P

































 zero

mean

message

(16)

Power in AM

• Assume that the message is an independent, zero mean, random process

 

    



 



 











2

2 2 2

2

2 2

2 1 cos

lim 2

1 cos lim

T

T T c

c

T

T T c

c s

dt t

T f t

m A

dt t

T f A

P







 

²

2 2

c 2

s c

m t

2 A 2

P  A    

sb c

m 2

2 c 2

s

A

c

A P P 2 P

P        

Carrier and two

symmetric

sub-bands

(17)

Expected Value Math Note

• To compute power/energy we often take the expected value of the squared signal … particularly if random variables are involved.

– message is assumed to be a zero mean random variable – cosine could include a random phase offset

    ^ ^ ^{ } ^ ^ ^ ^

   

  ^ ^

 

 

   ^{ } ^ ^   ^{ } ^ ^ 

 

       

   

c c

c

c c

A P A

t m E A A

t m E t

m E A

t f t

m E t

f t

m E t

f E

A

t f t

m t

m E

A

t f t

m A

t v E















 



 

       





































2 1 2

1 2 1 2

2 1 2 1

2 cos 2

cos 2

2 cos

2 cos 2

1 2 cos 1

2 2

2

2 2 2

2

2 2 2

2 2

2

2 2 2

2

2 2



















  t A  m   t   f t 

v 

_c

 1     cos 2  

_c



Note: zero

^th

lag of the

autocorrelation function

(18)

Power Levels – Carrier Ratio

• Carrier to one sideband power

• Maximum magnitude of mod index and message

𝑃 𝑃

𝐴 2

1 2 ⋅ 𝐴

2 ⋅ 𝜇 ⋅ 𝑃

𝑃

𝑃 2

𝜇 ⋅ 𝑃

𝑃

𝑃 2

1 ⋅ 1 2 or ^𝑃 ¹ ₂ ^{⋅ 𝑃}

Power ⁴^A

2 c

A_c²²8

Carrier and symmetric

subbands

(19)

• Upper or Lower sideband to Total Signal power ratio

• At the maximum modulation index and message

Total Power Levels

or ^𝑃

^/

¹ ₄ ^{⋅ 𝑃}

𝑃

_/

𝑃

1 2 ⋅ 𝐴

2 ⋅ 𝜇 ⋅ 𝑃 𝐴 2 𝐴

2 ⋅ 𝜇 ⋅ 𝑃

𝑃

_/

𝑃

1 ⋅ 1 2 2 ⋅ 1 ⋅ 1

𝑃

_/

𝑃

𝜇 ⋅ 𝑃

2 2 ⋅ 𝜇 ⋅ 𝑃

(20)

AM Signal Power Summary

• Most of the AM signal power is in the carrier.

– Less than 50% of the total signal power is in the message subbands !

• This seams like a waste of power, but it makes it easier to design an AM radio receiver !

– Lock on to the carrier

– Then demodulate the AM signal

(21)

DSB Power Levels

  ^t ^A ^m   ^t ^cos  ² ^f ^t 

s 

_c

   

_c



 

² _c

  

_c



²

DSB

T

P s t A m t cos 2 f t

S        

 

²



_c



² _c²

 

²



_c



²

2 c

DSB

A m t cos 2 f t A m t cos 2 f t

P            

2 P 1

A

P

_DSB



_c²



_m



• All the power is in the message sidebands

(22)

Comparing AM and DSB

• DSB Power vs. 2 AM Sideband Power

– Assume 100% mod index, max m(t)=1 and the waveform maximum magnitudes are identical – For DSB, Ac=Amax

– For AM, Amax=Ac + Ac= 2ꞏAc

4 4 2 P

A 2 2 P

A P

P

2

m 2

2 max

m 2

max

sb

DSB



 





 



 





 

4+ times more message power

(23)

Comparing AM and DSB (2)

4 4 P

P

2 sb

DSB



 

Why would we use AM instead of DSB?

Power

c freq

m f

c f

f  f_c f_m fc

m 

c f

f 

 f_cf_m

A_c² 4 A_c² ²8

A_c² ²8

A_c²4 A_c² ²8

A_c² ²8

Power

c freq

m f

c f

f  f_c f_m fc

m

c f

f  f_c f_m A_c²8

A_c²8

(24)

General Linear CW Modulation Concepts

• Baseband Signal

• Modulated Bandpass Signal

(25)

Bandpass signal

(a) Spectrum; (b) Waveform

Definition: 𝑉 𝑓

𝑋 𝑓 𝑓 𝑊 𝑓 𝑓 𝑊

0 0 𝑓 𝑓 𝑊 0 𝑓 𝑊 𝑓

Note: An artificial spectrum to show Conjugate symmetry about the modulation frequency

If the baseband signal crosses zero, the modulated signal envelope must go to zero and have a phase

reversal. (DSB w/SC only!)

(26)

Complex Signal Representation

• Conventional AM

• DSB (w/ suppressed carrier)

  ^t ^A  ^m   ^t   ^j ^f ^t 

v 

_c

 1     exp  2  

_c



  ^t ^A ^m   ^t  ^j ^f ^t 

v 

_c

  exp  2  

_c



  ^t ^A  ^j ^f ^t  ^A ^m   ^t  ^j ^f ^t 

v 

_c

 exp  2  

_c

 

_c

    exp  2  

_c



(27)

(a) Rotating phasor;

(b) Phasor diagram with rotation suppressed

CW Phasor Diagrams

    t  A t   j    f  t  j   

v

_complex

exp 2

_c

    t  A t    j  

v

_complex

exp

(28)

Complex Phasor Representation

  ^t  ^j ^f ^t  ^m   ^t  ^j ^f ^t 

v  exp  2  

_c

     exp  2  

_c



  ^t ^m   ^t  ^j ^f ^t 

v   exp  2  

_c



(29)

AM Transmission - Blocks

• How do you make an AM signal?



2 f_ct



cos

 

t

x  _A_c

 

t A



x

 

t

 

f t



s  _c 1 cos 2 _c

 

t s

  ^t ^A  ¹ ^cos  ² ^f ^t   ^cos  ² ^f ^t 

s 

_c

     

_m

   

_c



(30)

AM Simulation

• SW CAD AM Circuits

– AMGen

– AMGenv2

(31)

Circuits: Bandpass Filter

• Passive RLC

 

C L jw

jw R

C L jw

jw f

H

 



 

 || 1

|| 1

 

 

 





 









 





 



L C w

w R j C

L jw jw R

C L jw

f jw

H 1

1 1 1

1 1

1 C w L

 1 

0

 

 

 



 







 

 



 







f f f

Q f w j

w w

w L

R C j f

H

0 0

1 1 1

1 L R C

Q  

Q

BW  f

⁰

(32)

SW CAD Examples

• RLC BPF

– Fc=10 kHz BPF

• Sallen-Key LPF

– 6 ^th Order Butterworth, Fco=40 kHz

(33)

AM Transmission - Nonlinear

• Another way to make an AM signal?

             

   

           

           

  ^t â ^x   ^t  â â ^x   ^t   ^f ^t  â  ^f ^t 

x a a

t f a

t f t

x a

a t

x a t x a

t f t

f t

x t

x a t

f t

x a

t f t

x a t

f t

x a t NL

c c

c

c c



































































































2 2 2 cos

2 cos 2 2

2 cos 2

cos 2

2 cos 2

cos 2

2 cos

2 cos 2

cos

2 2

1 2

2 1

2

2 2

2 1 2

2 1

2 2

2 1

2 2

1

  t  a a 2 x   t  cos  2 f t 

s        

(34)

Spectral components in Non-linear AM Modulation

  ^t â â ^x   ^t â ^x   ^t  â â ^x   ^t   ^f ^t  â  ^f ^t 

NL            

_c

   cos 2   2

_c

 2 2

cos 2 2

2 2

1 2

2 1

2

(35)

DSB Balanced modulator

(36)

DSB Alternate Balanced Modulator

• Differentially balanced input drive



2  ^f_c ^t



cos

AM Mod

 

t x

1

1

c 2

A s

 

t

 

^t ^A ^x

 

^t



^f ^t



s  _c  cos 2  _c 

  t A  x   t   f t 

AM

₁



₁

 1     cos 2   

  t A  x   t   f t 

AM

₂



₂

 1     cos 2   

(37)

  ^ ^  ^ ^ ^  ^ ^  ^ ^ ^ ^  ^ ^ ^cos  ² ^ ^ ⁵ ^ ^f ^ ^t  ^ ^

Communication Systems, 5e