Signal and Systems Prof. H. Sameti Chapter 8: Complex Exponential Amplitude Modulation Sinusoidal AM...

Signal and SystemsProf. H. Sameti

Chapter 8:• Complex Exponential Amplitude Modulation• Sinusoidal AM • Demodulation of Sinusoidal AM• Single-Sideband (SSB) AM• Frequency-Division Multiplexing• Superheterodyne Receivers• AM with an Arbitrary Periodic Carrier• Pulse Train Carrier and Time-Division Multiplexing• Sinusoidal Frequency Modulation• DT Sinusoidal AM• DT Sampling, Decimation, and Interpolation

Book Chapter8: Section1

2

The Concept of Modulation

Why? More efficient to transmit E&M signals at higher frequencies Transmitting multiple signals through the same medium using different

carriers Transmitting through “channels” with limited pass-bands Others…

How? Many methods Focus here for the most part on Amplitude Modulation (AM)

Computer Engineering Department, Signals and Systems


3

Amplitude Modulation (AM) of a Complex Exponential Carrier


carrier frequency( ) ,

( ) ( )

1( ) ( ) ( )

21

( ) 2 ( )2

( ( ))

c

c

j tc

j t

c

c

c t e

y t x t e

Y j X j C j

X j

X j


4

Demodulation of Complex Exponential AM


cos sincj tc ce t j t

Corresponds to two separate modulation channels (quadratures) with carriers 90˚ out of phase.


5

Sinusoidal AM


1( ) ( ) { ( ) ( )}

21 1

( ( )) ( ( ))2 2

c c

c c

Y j X j

X j X j

Drawn assuming

c M


6

Synchronous Demodulation of Sinusoidal AM

Suppose = 0 for now, θ ⇒Local oscillator is in phase with

the carrier.



7

Synchronous Demodulation in the Time Domain


2

High-Frequency Signals filterd out by the LRF

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

2

Then ( ) ( )

Now suppose there is phase difference, i.e. 0, then

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

c c c

c c c

w t y t t x t t x t t

r t x t

w t y t t x t t t

HF signal

0

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

2 2

Now ( ) ( ) cos

Two special cases:

1) 2, the local oscillator is 90 out of phase with the carrier, ( ) 0,signal

unrecoverable.

2) ( ) slowly var

cx t x t t

r t x t

r t

t

ying with time, ( ) cos[ ( )] ( ), time-varying "gain".r t t x t


8

Synchronous Demodulation (with phase error) in the Frequency Domain


Demodulating signal has phase difference w.r.t. the modulating si

cos(

gnal

)1 1

2 2

( ) ( )

c cj t j tj jc

j jc c

t e e e e

F

e e

Again, the low-frequency signal ( ) when 2.M


9

Alternative: Asynchronous Demodulation


Assume ,so signal envelope looks like ( )c M x t

0 DSB/SC (Double Side Band, Suppressed Carrier)

0 DSB/WC (Double Side Band, With Carrier)

A

A


10

Asynchronous Demodulation (continued)Envelope Detector


In order for it to function properly, the envelope function must be positive for all time, i.e. A+ x(t) > 0 for all t.

Demo: Envelope detection for asynchronous demodulation.

Advantages of asynchronous demodulation: — Simpler in design and implementation.

Disadvantages of asynchronous demodulation: — Requires extra transmitting power [Acosωct]2to make sure

A+ x(t) > 0 Maximum power efficiency = 1/3 (P8.27)⇒


11

Double-Sideband (DSB) and Single-Sideband (SSB) AM


Since x(t) and y(t) are real, from Conjugate symmetry both LSB and USB signals carry exactly the same information.

DSB, occupies 2ωMbandwidth in > 0ω

Each sideband approach only occupies ωM bandwidth in > 0ω


12

Single Sideband Modulation


Can also get SSB/SC or SSB/WC


13

Frequency-Division Multiplexing (FDM)

(Examples: Radio-station signals and analog cell phones)


All the channels can share the same medium.


14

FDM in the Frequency-Domain



15

Demultiplexing and Demodulation


ωa needs to be tunable

Channels must not overlap Bandwidth Allocation⇒ It is difficult (and expensive) to design a highly selective

band-pass filter with a tunable center frequency Solution –Superheterodyne Receivers


16

The Superheterodyne Receiver

Operation principle: Down convert from ωc to ωIF, and use a coarse tunable BPF for the front

end. Use a sharp-cutoff fixed BPF at ωIF to get rid of other signals.



17

AM with an Arbitrary Periodic Carrier

Computer Engineering Department, Signal and Systems

C(t) – periodic with period T, carrier frequency ωc = 2 /TπRemember: periodic in t discrete in ω

( ) 2 ( )k ck

C j a k

(𝑎𝑘= 1𝑇

𝑓𝑜𝑟 𝑖𝑚𝑝𝑢𝑙𝑠𝑒 𝑡𝑟𝑎𝑖𝑛)1

( ) ( )* ( ) ( )* ( )2 k c

k

Y j X j C j X j a k

( ( ))k ck

a X j k

⇓


18

Modulating a (Periodic) Rectangular Pulse Train


)().()( tctxty


19

Modulating a Rectangular Pulse Train Carrier, cont’d


( ) 2 ( )k ck

C j a k

𝑎𝑛𝑑

𝑎0=𝛥𝑇, 𝑎𝑘=

sin(𝑘𝜔𝑐𝛥2 )

𝜋 𝑘

For rectangular pulse1

( ) ( )* ( )2

Y j X j C j

Drawn assuming:

Nyquist rate is met


20

Observations


1) We get a similar picture with any c(t) that is periodic with period T 2) As long as ωc= 2 /π T > 2ωM, there is no overlap in the shifted and

scaled replicas of X(j ). Consequently, assuming ω a0≠0:

x(t) can be recovered by passing y(t) through a LPF 3) Pulse Train Modulation is the basis for Time-Division Multiplexing Assign time slots instead of frequency slots to different channels, e.g. AT&T wireless phones4) Really only need samples{x(nT)} when ωc> 2ωM Pulse Amplitude ⇒

Modulation


21

Sinusoidal Frequency Modulation (FM)


))(cos()( tAty

Amplitude fixed

Phase modulation: Frequency modulation:

Instantaneous ω

X(t) is signal To betransmitted


22

Sinusoidal FM (continued)


Transmitted power does not depend on x(t): average power = A2/2 Bandwidth of y(t) can depend on amplitude of x(t) Demodulation Demodulation a) Direct tracking of the phase (θ t) (by using phase-locked loop) b) Use of an LTI system that acts like a differentiator

H(j ) —Tunable band-limited differentiator, over the bandwidth of ω y(t)

/

( )( ) ( ( )) sin ( )c f

d dt

dy tu t w k x t A tdt

𝐻 ( 𝑗𝜔 )≅ 𝑗𝜔⇓

…looks like AM envelope detection


23

DT Sinusoidal AM


Multiplication ↔Periodic convolution

Example#1:

[ ] cj nc n e

( ) 2 ( 2 )jc

k

c e k

1( ) ( ) ( )

2j j jY e X e C e


24

Example#2: Sinusoidal AM


[ ] cos cc n n

( ) ( 2 ) ( 2 )jc c

k

C e k k

0 and

2c M

c M c M

/ 2M c M

M

π

1( ) ( ) ( )

2j j jY e X e C e

i.e.,

No overlap ofshifted spectra

Drawn assuming:


25

Example #2 (continued):Demodulation


Possible as long as there is no overlap of shifted replicas of X(ejω):

i.e., 0

and 2c M

c M c M

M c M

1( ) ( ) ( )

2j j jW e Y e C e


26

Example #3: An arbitrary periodic DT carrier


2 / 2[ ] [ ], jk N

k ck N

c n a e c n NN

2( ) 2j

kk

kC e a

N

1( ) ( ) ( )

2j j jY e X e C e

1

0

1 2( )* 2

2

Nj

kk

kX e a

N

1

( 2 / )

0

( )N

j k Nk

k

a X e

- Periodic convolution

- Regular convolution


27

Example #3 (continued):



28

DT Sampling


Motivation: Reducing the number of data points to be stored or transmitted, e.g. in CD music recording.


29

DT Sampling (continued)


2 2[ ] [ ] ( ) ( ), j

s sk k

p n n kN P e kN N

k

p kNnkNxnpnxnw ][][][].[][ Note:

𝑥𝑝 [𝑛 ]={𝑥 [𝑛 ] , 𝑖𝑓 𝑛𝑖𝑠𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒𝑜𝑓 𝑁0 ,𝑜𝑡 h𝑒𝑟𝑤𝑖𝑠𝑒

⇒Pick one out of N

1( )

0

1 1( ) ( ) ( ) ( )

2s

Nj kj j j

pX e X e P e X eN

- periodic with period


30

DT Sampling Theorem


We can reconstruct x[n] if ωs= 2 /π N > 2ωM

Drawn assuming ωs > 2ωM

Nyquist rate is met ⇒ωM<

/Nπ

Drawn assuming ωs < 2ωM

Aliasing!


31

Decimation — Downsampling


xp[n] has (n-1) zero values between nonzero values:Why keep them around?

Useful to think of this as sampling followed by discarding the zero values


32

Illustration of Decimation in the Time-Domain (for N= 3)



33

Decimation in the Frequency Domain


( ) [ ] ( [ ] [ ])j jkb b b p

k

X e x k e x k x kN

[ ] Let n kN or k n/Njkp

k

x kN e

( / )

an integermultiple of N

[ ] j n Np

n

x n e

( / )p[ ] (Since x [n kN] 0)j N n

pn

x n e

¿ 𝑋𝑝(𝑒 𝑗 ( 𝜔𝑁 ))

Squeeze in timeExpand in frequency

- Still periodic with period 2π since Xp(ejω) is periodic with 2 /Nπ


34

Illustration of Decimation in the Frequency Domain



35

The Reverse Operation: Upsampling(e.g.CD playback)


Signal and Systems Prof. H. Sameti Chapter 8: Complex Exponential Amplitude Modulation Sinusoidal AM...

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