1

1

Modulation with continuouswave

•representations in time and frequency for two types of continuous wave modulation:

–Amplitude modulation, AM -amplitude

–Angle modulation,•frequency modulation (FM) - instantaneous frequency•phase modulation (PM) - instantaneous phase

2

• Purpose of a communication system: – transport a signal (a message) over a channel– deliver a reliable estimate to a user

• Example: – radio system: efficient in a freq. range > 30 kHz, – baseband signals = audio signals (0-20kHz)– Frequency shifting => modulation

• The message signal that contains information, generated by sources of information, is a baseband signal

• Modulation – information transfer from the modulating wave to carrier.

2

3

• Modulation / demodulation– (1) shifting frequency range of message signal into

another one- suitable for transmission over the channel

– (2) corresponding shift back to the original frequency range after reception of the signal

• Two most common used forms of carriers– Sinusoidal wave– Periodic pulse wave

• two main classes of modulation– Continuous wave (CW)– Pulse modulation

4

Amplitude modulation

• The amplitude of a carrier sine wave is modified according to a message signal= information

Angle modulation• instantaneous frequency / phase of the

carrier sine wave varies with the message– Frequency modulation– Phase modulation

3

5

Essential components of a communication system, using

continuous-wave (CW) modulation

The noise from the channel decreases performance of the overall scheme

6

Amplitude modulation vs Angle modulation (exponential)

4

7

Amplitude modulation( )( )( ) ( ) ( )

-1

Sinusoidal carrier wave: cos

Modulating signal:

AM signal: 1 cos .

V - amplitude sensitivity of the modulator

c c

c a c

a

c t A t

x t

s t A k x t t

k

= ω

= + ω⎡ ⎤⎣ ⎦⎡ ⎤⎣ ⎦

Modulation degree or p

( ) [ ]max

(index):

100 % ;0 1; (message=sine wave)

= maximum frequency of the modulating signala a m

M

m k x t m m k A

f

= ⋅ < ≤ =

ercentage

8

varying percentage of modulation

1) |kax(t)|≤12) |kax(t)|>1: overmodulation; envelope distortions, phase

inversion in the carrier

( ) ( )( )

The amplitude of a harmonic signal is positive:

1 0 1

If 1 c a a

a

A k x t k x t t

k x t

+ ≥ ⇒ ≤ ∀⎡ ⎤⎣ ⎦> ⇒ overmodulation

m >1

5

9

Measuring modulation index

0 1 2 3 4 5 6

x 10-4

-1.5

-1

-0.5

0

0.5

1

1.5

timp

ampl

itudi

ne

Amin

Amax

max min

max min

100 [%]A AmA A

−= ⋅

+*works for modulating wave = sine !!!

10

AM Spectrum

( ) { } ( ){ }

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

cos cos

1 ,2

.2

.2 2

The notation S(f) is used in communicati

c c c a c

c c c c a c c

a cc c c c c

c a cc c c c

S A t A k x t t

A A k X

k AS A X X

A k AS f f f f f X f f X f f

ω = ω + ω =

= π δ ω−ω + δ ω+ω + ω ∗π δ ω−ω + δ ω+ω⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦π

ω = π δ ω−ω + δ ω+ω + ω−ω + ω+ω⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

= δ − + δ + + − + +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

F F

ons.

6

11

( ) ( ) ( ) ( ) ( )2

a cc c c c c

k AS A X Xω = π δ ω−ω + δ ω+ω + ω−ω + ω+ω⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

Magnitude spectrum for thebaseband signal and AM signal

12

Condition to recover correctly the message signal & Bandwidth

• Upper and lower sidebands do notoverlap if

• Bandwidth of the modulated signal, BTis double of the bandwidth of themessage (modulating) signal, B

0C Mω −ω >

2TB B=

(message's bandwidth)c Mf f B>> =

7

13

AM advantages and disadvantages

+simple implementation– used from the

beginning in radio transmission

– cheaper

– bandwidth is 2x bandwidth of modulating wave

– low energyefficiency– AM spectrum: the carrier

~ no information ⇒waste of power

– Solution: suppress one of the sidebands and carrier ⇒ linear AM

14

Modulator

• nonlinear device, i.e. diode

8

15

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

1 2

1

1

cos cos

11 2 cos 2 12 2 1

c c c c

n

cn

u t A t x t u t A t x t g t

g t n tn

−∞

=

= ω + ⇒ = ω +⎡ ⎤⎣ ⎦

−= + − ω⎡ ⎤⎣ ⎦π −∑

( ) ( ) ( ){ }( ) ( ) ( ) ( )

( )

1

21

1

1

1cos cos 2 cos 2 2

2 2 1

12 cos 2 1 2 2 1For , in the neighborhood of

2cos cos 2

AM signal - separated by band-pass

nc c

c c cn

n

cn

c M c

cc c

A Au t t n t n tn

x tx t n t

n

A t x t t

−∞

=

−∞

=

−≅ ω + ω + − ω +⎡ ⎤⎣ ⎦π −

−+ + − ω⎡ ⎤⎣ ⎦π −

ω >> ω ω

ω + ωπ

∑

∑

filtering centered on cω

16

Demodulator : envelopedetector

•low-pass filtering of u2(t) -> capacitor•removal of the DC component

9

17http://www.discip.crdp.ac-caen.fr/phch/lycee/terminale/modem/modem.htm

http://www.discip.crdp.ac-caen.fr/phch/lycee/terminale/modem/modem.htm

18

http://www.discip.crdp.ac-caen.fr/phch/lycee/terminale/modem/modem.htm

10

19

http://www.discip.crdp.ac-caen.fr/phch/lycee/terminale/modem/modem.htm

20

http://www.discip.crdp.ac-caen.fr/phch/lycee/terminale/modem/modem.htm

11

21

Power of AM signal( )

( ) [ ] ( )

( ) ( ) ( )

( )

2

2 2 2 2 2 22

cos

cos cos

cos cos cos 2 2

Power of the modulating signal 2

Power of the AM signal : 22 8 8 4

m m

c a m m c

c cc c c m c m

mm

c c c cs

x t A t

s t A K A t tmA mAA t t t

AP

A m A m A AP m

= ω

= + ω ω

= ω + ω −ω + ω +ω⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

=

= + + = +

22

( )

2 2

22

2

max

For detection, use only one sideband, amplitude 2.

Useful power 8

Efficiency at receiver ;0 12 2

1Maximum efficiency 100 16.67% (m=1)6

P

c

cu

u

S

mA

m AP

P m mP m

=

η = = < ≤+

η = ⋅ ≅

2 2

2

max2

ower from both sidebands = useful , / 4,

double efficiency : , 33.33%2

uP m A

mm

=

η = η =+

12

23

observation

• This amplitude modulation is not linear

( ) ( ) ( ) ( )( )

( ) ( )( ) ( ) ( ){ } ( ) ( )

1 1 2 2

1 2

1 2

1 2 1 2 1 2

1 cos ; 1 cos ;

If results from the modulation with the sum

+ we have

1 cos

c a c c a c

c a c

s t A k x t t s t A k x t t

s t

x t x t

s t A k x t x t t s t s t

+

+

= + = +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

= + + ≠ +⎡ ⎤⎣ ⎦

ω ω

ω

24

Linear Amplitude Modulation

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

( )( )

cos

cos cos sin sin

cos sin ,

canonical form of a bandpass signal in phase component (I-channel),

in quadrature component (Q-chann

c

c c

I c Q c

I

Q

s t a t t t

a t t t a t t t

s t t s t t

s t

s t

= ω + φ⎡ ⎤⎣ ⎦= φ ω − φ ω⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦= ω − ω

−

−

( ) ( ) ( )el).

linear modulation and - linear dependent on I Qs t s t x t⇔

13

25

Linear Amplitude Modulation

( )x t ( )x t

( )tx21 ( ) ( ){ }x̂ t x t=H

( )tx21 ( ) ( ){ }x̂ t x t=H

( )tx21 ( )t'x

21

( )tx21 ( )t'x

21

−the vestige of the inferior sideband is transmitted

the vestige of the superior sideband is transmitted

with vestigial sideband VSB

inferiorsidebandis transmitted

Superior sidebandis transmitted

Single sideband SSB

-message0Double sideband suppressedcarrier DSB-SC

ObservationsIn quadrature component

In phase component

Modulation type

( )1 ˆ2

x t

( )1 ˆ2

x t−

DSB-SC –supressed carrier, bandwith is the same as full AM

VSB – large bandwidth signals

26

Observations

• in phase component sI(t) depends only on themessage signal

• in quadrature component sQ(t) = filteredversion of the message signal. The spectral modification of s(t) compared to x(t) is givenonly by sQ(t)

• The purpose of sQ(t) if it exists, is to interferewith the in phase component to eliminate /reduce the power from a sideband of themodulated signal

14

27

Double sideband-suppressed carriermodulation

( ) ( ) ( ) ( ) ( )[ ].XXAStcostxAts ccc

cc ω+ω+ω−ω=ω⇒ω=2

28

• carrier multiplied withthe message signal

• carrier absent in thespectrum

• But S(ω) has spectral components in ω=ωc !

• transmitted band= 2x band of themodulating wave

15

29

Coherent (Synchronous) Detection• reconstruct modulating signal x(t)

30

( ) ( ) ( ) ( ) ( )

( )

( )

( ) ( )

( )

cos cos cos

cos cos 22 2

centered in 2 , 2 , 2 base band ,

c c c c

c cc

C M C MM M

v t s t t A x t t tA Ax t x t t

c

= ω + θ = ω ω + θ

= θ + ω + θ

ω ω −ω ω +ω−ω ω

( ) ( )0 cos (LPF output)2

Desynchronisation between local oscillators - receiver & emission unit phase error decreasing of detector response.

maximum for 0; zero for . 2

Local osci

cAv t x t= θ

⇒ θ⇒π

θ = θ = ±

llator of the receiver with the local oscillatorthat generates the carrier signal and

synchronizedin frequency in phase

16

31

Quadrature-Carrier multiplexing

• Also known as Quadrature-amplitude modulation (QAM)

• Transmit two DSB-SC modulated waves on thesame banwidth ⇒ bandwidth-conservation scheme

• Modulators with quadrature phase:– carriers in quadrature, same frequency, differ in phase

by ±π/2 (±900)• Demodulator: two coherent detectors with 90

degree phase shift• θ=±π/2: output of synchronous detector = null

(quadrature effect)

32

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

1 2

, - independently modulating signals.

cos sinc c c c

x t x t

s t A x t t A x t tω ω= +

17

33

Single side band modulation – SSB• one sideband transmitted• frequency-discrimination scheme with 2 steps• Product modulator ⇒ double sideband-suppressed

carrier• Bandpass filter : passes the sideband selected for

transmission and suppresses the remaining sidebands• separation lower and upper sideband ⇒ energy gap

in the spectrum of the message signal x(t)• Speech signals (telephony): energy gap= -300, 300

Hz• modulated signal: energy gap 2ωm

34

Restrictions for BPF of sideband selection:1. selected sideband passing band of the filter, 2. unwanted sideband stop band of the filter,filter's transition bandwidth 2 . Demodulation- synchron

m

⊂⊂

< ωous detection.

18

35

Vestigial sideband modulation VSB

• Transmitted: modified version of one sidebandand appropriately chosen vestige of the othersideband.

• Well suited for large bandwidth signals

• Product modulator: double sideband-suppressedcarrier

• Bandpass filter for a VSB modulation (differencebetween SSB and VSB modulations)

• Commercial television

36

Vestigial sideband modulation

( ) ( ) 1=ω+ω+ω−ω cc HH

BPF Freq. response

19

37

38

20

39

Frequency translation

2 1

2 1

2 1

1 2

Up conversion

Down conversion l

l

ω > ωω = ω −ω

ω < ωω = ω −ω

• Change the carrierfrequency of themodulated signal fromω1 to ω2

• Mixer: product modulator+bandpassfilter

40

Up conversion, ω2>ω1

Spectrum of the modulated signal with up conversion

Image signal spectrum

21

41

Down conversion, ω2<ω1

Image signal spectrum

Spectrum of the modulatedsignal with down conversion

42

Frequency Division Multiplexing

• Telephony systems: 300Hz-3400Hz• Goal: transmit simultaneously several

vocal signals on the same channel:– FDM-frequency division multiplexing– TDM-time division multiplexing

• FDM, using AM-SSB• Distance between carriers 4kHz• BPFs – bandwidth limitation at 4kHz

22

43

Frequency Division Multiplexing

LPF -remove high frequencycomponents

AM Modulators modulate thesignals on different carrierfrequencies

44

23

45

Angular Modulation

• modulate a carrier : alter its angle –phase, according to the message; amplitude ~constant

• Advantage: signal more robust against noise and interference.

• Disadvantage: increase in bandwidth

46

Angular Modulation

( )( ) ( )

( )

Modulated signal- rotating vector with amplitude and angle :

cos Its angular velocity:

.

c

i

c i

i

At

s t A t

dt

θ

= θ

ω =

instantaneous frequency of the modulated signal

( )i tdtθ

24

47

Angular Modulation

( ) ( )[ ]( ) ( )

rad/V - phase sensitivity.

cos

i c p

p

c c p

t t k x t

k

s t A t k x t

θ = ω +

⎡ ⎤= ω +⎣ ⎦

Phase modulation (PM)( ) ( )

[ ]

( ) ( )

( ) ( )

0

0

2

Hz/V - frequency sensitivity.

2

cos 2

i c f

f

t

i c f

t

c c f

t k x t

k

θ t ω t k x τ dτ

s t A t k x d

ω = ω + π

= + π

⎡ ⎤⇒ = ω + π τ τ⎢ ⎥

⎣ ⎦

∫

∫

Frequency modulation (FM).

( ) ( )0

FM signal generated using =PM signal generated using .t

x t x dτ τ∫

48

Frequency Modulation( )( )

modulating signal: cos

instantaneous frequency 2 cos

- 2

- maximum instantaneous difference between FM modulated carrier fr

m m

i c f m m

f m

x t A t

t k A t

k A

= ω

ω = ω + π ω

Δω = πfrequency deviation

( ) ( ) [ ]

equency and nominal carrier frequency;

2- .

cos cos sin : freq. deviation proportio

f m

m m

c i c c m

k A

s t A θ t A t tf

πΔωβ = =

ω ω

= = ω +β ω

Δ

modulation index

Essential characteristic nal withmodulating signal amplitude; does not depend on its frequency. 1 radian - . 1 radian - .

β <<β >>

narrow band modulationwide band modulation

25

49

Narrow Band Frequency Modulation( ) ( ) ( )

( ) ( )

( )

cos cos sin sin sin sin .

If rad cos sin 1 and sin sin sin 36

cos sin sin .

c c m c c m

m m m

c c c c m

s t A t t A t t

t t t

s t A t A t t

ω β ω ω β ωπβ β ω β ω β ω

ω β ω ω

= −

< ⇒ ≅ ≅

⇒ = −

50

Phasorial representation of the FM and AM signals

• Narrow band FM and AM – same bandwidth

( )

( ) ( )

cos1 cos cos2

FM c c

c c m c m

s t A t

A t t

≅ ω

+ β ω +ω − ω −ω⎡ ⎤⎣ ⎦

( ) [ ]

( ) ( )0

1 cos coscos

1 cos cos2

AM c m c

c c

c m c m

s t A m t tA t

mA t t

= + ω ω

= ω +

ω +ω + ω −ω⎡ ⎤⎣ ⎦

FM AM

26

51

Narrow Band FM Spectrum – general case

( ) ( )( )

( ) ( )

( )( ) ( )( )

( ) ( )

0

0

cos 2

cos cos 2 sin sin 2 .

Narrow band modulation, 236

cos 2 sin .

t

y t x dt

c c f y t A

c c f c c f

f

c c c f c

s t A t k x d

A t k y t A t k y t

k A

s t A t A k y t t

= τ τ

≤

∫⎛ ⎞= ω + π τ τ =⎜ ⎟

⎝ ⎠

= ω π − ω π

ππ ≤ ⇒

≅ ω − π ω

∫

Ex: π/10=0.314, sin(π /10)=0.309

( ) ( ) ( ) ( ) ( )c cc c c c

c c

X XS A A

ω−ω ω+ω⎡ ⎤ω = π δ ω−ω + δ ω+ω + π −⎡ ⎤ ⎢ ⎥⎣ ⎦ ω−ω ω+ω⎣ ⎦

52

( ) ( ) ( ) ( )

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

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

0

1

,

2

t

c c c c f c c

c cc c c c

c c

a cMA c c c c c

y t x d Y Xj

XS A A k

j jX X

S A A

k AS A X X

= τ τ ↔ ω = ω ⇒∫ω

ω πω = π δ ω−ω + δ ω+ω − ∗ δ ω−ω −δ ω+ω⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ω

ω−ω ω+ω⎡ ⎤ω = π δ ω−ω + δ ω+ω + π −⎡ ⎤ ⎢ ⎥⎣ ⎦ ω−ω ω+ω⎣ ⎦

ω = π δ ω−ω + δ ω+ω + ω−ω + ω+ω⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

27

53

Wide Band Frequency Modulation( ) [ ]

( ) ( )

( ) ( ){ }( )

( ){ }

( ) ( ) ( )

( )

sin

sin

cos sin

cos cos sin sin sin sin

Re Re ,

- complex envelope of the FM signal

-Bessel function of firs

j tmc

c m c

m

c c m

c c m c c m

s t A ej t t j t

c

jn tc n

n

n

s t A t t

A t t A t t

s t A e s t e

s t A J e s t

J x

β ω=ω +β ω ω

∞ω

=−∞

= ω +β ω

= ω β ω − ω β ω

⇒ = =

= β∑

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

t kind, order and variable .

cos cos 2

.2

c n c m c n c mn n

cn c m c m

n

n x

s t A J t n t A J f nf t

AS J n n

∞ ∞

=−∞ =−∞

∞

=−∞

= β ω + ω = β π +

ω = β δ ω−ω − ω + δ ω+ω + ω⎡ ⎤⎣ ⎦

∑ ∑

∑

54

( ) ( ) ( )

( ) ( ) ( )

( )

0 1

2

properties of Bessel's functions

1. 1 for any ,2. For small , we have:

1 ; ; 0 ; 2 ; 1 ;2

3. 1.

nn n

n

nn

J J n Z

J J J n

J

−

∞

=−∞

= − ∈

≅ ≅ ≅ > <<

=∑

β ββ

ββ β β β

β

First five Besselfunctions, J0(β)-J4(β)

28

55

( ) ( ) ( ) ( ) . 2

Remarks 1. FM Spectrum: component on the carrier, and an infinite setof components on the sidebands at a distance of , 2 , ... 2. 1 (narrow bandw

cn c m c m

n

c

m m c

AS J n n∞

=−∞

= − − + + +⎡ ⎤⎣ ⎦

±

<<

∑ω β δ ω ω ω δ ω ω ω

ωω ω ω

β ( ) ( )

( )

0 1

c

0

idth FM), only and have significative values carrier ( ) and two lateral bands .3. The amplitude of the component on depends on the factor

not constant. The :

c m

c

J J

J⇒ ±

⇒power is constant

β βω ω ω

ω β

( )2 2 21 1 2 2c n c

nP A J A

∞

=−∞

= =∑ β

56

Example 1

const; variable variable

variableSpectral components separated by (const).

m

m f m

m

fA f k A

f

=

⇒ Δ = ⋅

⇒ β

2 f m

m m

k Aπωβω ωΔ

= =

The amplitude of the modulating signal affects the FM spectrum

29

57

Example 2

[ ]

const const

variable variable+ number of sprectral components in the interval , increasesFM bandwidth 2

m f m

m

c c

A f k A

f

f f f f

f→∞

= ⇒ Δ = ⋅

⇒ =

−Δ + Δ

⎯⎯⎯→ Δβ

β

2 f m

m m

k Aπωβω ωΔ

= =

The frequency of the modulating signal affects the FM spectrum

58

The transmission bandwidth of FM signals( ) ( ) ( ) ( ) .

2For , the transmission bandwidth 2 ; centered on .

: nearly all (~98%) of the power of a FM signal lies within a bandwidth of:

cn c m c m

n

T c

T

AS J n n

B f f

B

∞

=−∞

ω = β δ ω−ω − ω + δ ω+ω + ω⎡ ⎤⎣ ⎦

β→∞ → Δ

∑

Carson's rule

1 2 2 2 1T mB f f f ⎛ ⎞≅ Δ + = Δ +⎜ ⎟β⎝ ⎠

: of transmission band.The : of transmission bandThe transmission bandwidth is found between the two estimates

−−

Carson's rule under estimationuniversal curve over estimation

30

59

Equivalent definition of thetransmission bandwidth

The frequency interval where the spectral components of the FM signal . have a value superior to 1% of the carrier amplitude

( )

max

max

max

2 , where for each is satisfied the condition

0,01.

The value depends on .

T m

n

B n fn n

J

n

=<

>β

β

60

Non harmonic modulating wave

( )( )max max

- modulating signal, maximum frequency (same as )

max , frequency deviation

/ (same as ).Carson's rule: replace with and with and the unive

m

f

m

x t W f

A x t f k A

D f WD f W

= ⇒ Δ =

⇒ = Δ deviation ratio ββ

rsal curve for any modulating signal

31

61

Example 3

( )

North America, radio transmissions:7575 kHz ; 15 kHz ; 5.15

Carson's rule : 2 180 kHz.Universal curve : D 5 3, 2 240 kHz.In practice a transmission bandwidth of 200 kHzis used.

T

T

f W D

B f WB f

Δ = = = =

= Δ + =

= ⇒ = Δ =

62

Frequency Modulated Signals’Generation

There are 2 methods, direct - based on a voltage controlled oscillator - 555 timer indirect - 1. narrow band FM 2. frequency multiplication to set the frequency deviation. The second method high frequency stability

FM radio broadcasting⇒

⇒

32

63

FM Signal Generation

narrow band FM signal ⇒ wide band FM signal by frequency multiplication

64

• Integrate the modulating signal x(t) (fromthe baseband)

• The phase of an oscillator is modulated -> frequency stability required for radio transmission

• maximum frequency deviation= small ⇒reduces distortions ⇒ narrow bandmodulation

33

65

( ) ( ) ( ) ( )

( ) ( )

( ) ( )( )

( ) ( )

21 2

0

0

... ;

cos 2

. The pass-band of the band-pass filter

is time larger than the bandwidth of the signal .

' 'cos 2 with the

nn

t

c c f

i c f

t

c c f

v t a s t a s t a s t

s t A t k x d

f t f k x t

n s t

s t A n t nk x d

ω π τ τ

ω π τ τ

= + + +

⎡ ⎤= +⎢ ⎥

⎣ ⎦= +

⎡ ⎤= +⎢ ⎥

⎣ ⎦

∫

∫( ) ( )

instantaneous

frequency: ' .i c ff t nf nk x t= +

66

• The frequency multiplier is a nonlineardevice followed by a bandpass filter

• The nonlinear device is memoryless in the sense that it doesn’t have in itsstructure reactive elements

34

67

Demodulation

• reconstruction of modulating wave• characteristic of transfer inverse of the

characteristic of transfer of the FM modulator• 1. directly: frequency discriminator: output

proportional with the instantaneousfrequency of the FM signal.

• 2. indirectly: PLL circuit (Phase-locked loop)

68

FM quadrature demodulator

The block diagram of the demodulator

35

69

• The quadrature demodulator converts the FM signal:

into a PM signal, and a PM detector is used to recover the message signal, x(t)

( ) ( )0

cos 2 2

10.7 10700

t

c

c

s t A f t x d

f MHz kHz

⎡ ⎤= π + π τ τ⎢ ⎥

⎣ ⎦= =

∫

70

• 1. The phase shifter converts FM modulation into PM modulation but preserves the FM modulation

• 2. The analog multiplier serves as a phase detector, PD, and produces an output being linearly proportional to PM. PD is not sensitive to FM

• 3. The low-pass filter suppresses thespectral components with highfrequency (2fc)

36

71

The phase shift is linearly proportional to the instantaneous frequency deviation around the carrier frequency, 10700kHz.

72

FM

( ) ( )

( ) ( )3

34.4290 10700150

10700 [rad], f [kHz]2

4 10

o of f

f f−

= − −

= − −

+

+ ⋅

φ

πφ

37

73

The phase shifted signal is :

( )

( )

0

0

3

3

( ) cos 2 10700 2 ( ) 107002

sin 2 10,700 2 ( ) 10700

4 10

4 10

t

t

s t A t k x d f

A t k x d f

ππ π τ τ

π π τ τ

−

−

⎡ ⎤= + − −⎢ ⎥

⎣ ⎦⎡ ⎤

= + −⎢ ⎥⎣ ⎦

+ ⋅

+ ⋅

∫

∫

74

But the amplitude response of the phase shifter is

38

75

FM

/20log 20log 0.771 [dB]/

M M

m m

A A AA A A

= =

-2

1.093 .

The mean gain can be taken as:

20log 26.57 [dB] 4.7 10

M

m

A A cstA

A A AA

≅ ⇒ ≅

≅ − ⇒ ≅ ⋅

Maximum gain

Minimum gainThe amplitude varies only a little, and therefore we canconsider the output amplitude of the output from the phaseshifter is constant:

76

The phase detector is implemented by an analog multiplier:

( ) ( ) ( )

( )

( ) ( )

0

0

0

3

-2 2 3

-2 2 3

( ) ( )

cos 2 10700 2 sin 2 10700 2 10700

2.35 10 10700

2.35 10 sin 2 21.400 4 10700

4 10

sin 4 10

4 10

t

t

ts t s t

AA t x d t k x d f

A f

A t k x d f

π π τ τ π π τ τ

π π τ τ

−

−

−

=

⎡ ⎤ ⎡ ⎤= + + −⎢ ⎥ ⎢ ⎥

⎣ ⎦⎣ ⎦⎡ ⎤= ⋅ −⎣ ⎦⎡ ⎤

+ ⋅ + −⎢ ⎥⎣ ⎦

+ ⋅

⋅

+ ⋅

∫ ∫

∫

The low-pass filter suppresses the second component, centered at 21.4 MHz. The first component, a base-band component is retained:

( )-2 2 3ˆ( ) 2.35 10 10,700sin 4 10x t A f−⎡ ⎤= ⋅ −⎣ ⎦⋅

39

77

3 3( 10,700) 75 0.3 [rad]4 10 4 10f− −− ≤ ⋅ =⋅ ⋅

0.3 sinfor α α α≤ ≅

( )( )

-2 2 3

6 2

ˆ( ) 2.35 10 10700

94.12 10 10700

4 10x t A f

A f

−

−

= ⋅ −

= ⋅ −

⋅

78

• The instantaneous frequency is

• And therefore

• We have obtained a FM demodulator. The circuit configuration presented is almost exclusively used to implement a modern FM demodulator (discriminator).

10700 ( ) [kHz]f kx t= +

-6 2x̂(t) 94.2 10 ( )A kx t≅ ⋅

40

79

• The transfer function obtained is called an S curve

Linear portion of thecharacteristic

80

Stereo FM Signals Multiplexing

Stereo - 2 different signals are transmitted using the same carrier. The stereo radio broadcasting satisfies the conditions:1. It is realized inside the broadcasting FM channel alocated,2. It is compatible with the mono receivers.

( ) ( )

( ) ( )

( ) ( )

The signal represents the part of the base-band disponible for mono reception.The signal is amplitude modulated with 2 sidebands and suppressed carrier. The multiplexed signal :

l r

l r

l

x t x t

x t x t

x t x t

+

−

= ( ) ( ) ( ) cos 4 cos 2 ,

is frequency modulated.r l r p px t x t x t f t K f tπ π+ + − +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

41

81

( ) ( ) ( ) ( ) ( ) cos 4 cos 2l r l r p px t x t x t x t x t f t K f tπ π= + + − +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

82

42

83

Non-linear Effects in Frequency Modulation

• Nonlinearities in electronic circuits– Strong non-linearity which is intentional, for

given applications• Weak non-linearity• We study the effect of weak non-linearity

on FM systems

84

Non-linear Effects in Frequency Modulation

( ) ( ) ( ) ( )2 30 1 2 3

Consider a non-linear communication channel with the input-output transfer characteristic: , having at its input the frequency modulated si

i i iv t a v t a v t a v t= + +

( ) ( ) ( ) ( )

( ) ( ) ( )( )

0

2 20 1 2

3 33

gnal:

cos 2 ; 2

cos 2 cos 2

cos 2 .

t

i c c f

c c c c

c c

v t A f t t t k x d

v t a A f t t a A f t t

a A f t t

π φ φ π τ τ

π φ π φ

π φ

= + =⎡ ⎤⎣ ⎦

⇒ = + + + +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦+ +⎡ ⎤⎣ ⎦

∫

43

85

( ) [ ] ( )

( )

2 3

2 232 2

0 1 3

33

From the trigonometric relations: 1 cos 2 cos3 3cos cos ; cos

2 4we have:

3 cos 2 cos 4 22 4 2

cos 6 3 . 4

For the det

c cc c c c c

cc c

x x xx x

a A a Av t a A a A f t f t t

a A f t t

π π φ

π φ

+ += =

⎛ ⎞= + + + + +⎡ ⎤⎜ ⎟ ⎣ ⎦⎝ ⎠

+ +⎡ ⎤⎣ ⎦

( )0ection of the FM signal from it is necessary its identificationv t

86

( ) ( )

Let be the frequency deviation of the FM signal and the maximum frequency of the modulator signal. Ap-

plying Carson's rule we have the separation condition: 2 2 3 2 .

If thisc c c

fW

f f W f f W f f W

Δ

− Δ + > + Δ + ⇒ > Δ +

( )

( ) ( )

0

30 1 3

condition is satisfied then we can extract from ,using a band-pass filter with central frequency and band-width 2 2 , the term

3 ' cos 2 .4

c

c c c

v tf

f W

v t a A a A f t tπ φ

Δ +

⎛ ⎞= + +⎡ ⎤⎜ ⎟ ⎣ ⎦⎝ ⎠

44

87

The Super-heterodyne ReceiverA radio broadcasting receiver has not only the goal to demodulate the received signal.Other goals:- Selection of the desired carrier frequency,- Filtering, for the separation of the desired signal from other modulated signals,- Amplification, for the compensation of the losses produced by the propagation.

88

fLO=630+455 kHz=1085 kHz

fRF=630 kHz

Radio Timisoara

fIF=455kHz0.3-4.5 kHz

RF- radio frequency;

IF-intermediate frequency

45

89

; . An IF signal is generated in the receiver if the differenceof the local oscillator frequency and of the input carrier frequency equals :

IF LO RF LO RF

IF

f f f f f

f

= − >

± .

only one of these frequencies corresponds to the carrier, the other one is named 2

RF LO IF

RF IF

f f f

f f

= ±

= +image frequency

RF- radio frequency; IF-intermediate frequency

90

• For FM case, after the IF amplifier there islimiter and a bandpass filter

• Detection is made using a frequencydiscriminator