Oct 04, 2005 CS477: Analog and Digital Communications

1

Exponential Modulation

Analog and Digital Communications

Autumn 2005-2006

Oct 04, 2005 CS477: Analog and Digital Communications 2

Exponential Modulation Modulation in which information signal is

encoded in carrier frequency or phase Amplitude of the transmitted signal is

constant This modulation is robust to channel gain

variations The modulation is non-linear

Bandwidth of the transmitted signals depends upon the amplitude of the information signal

Oct 04, 2005 CS477: Analog and Digital Communications 3

Exponential Modulation

xc(t) = Accos(2ùf ct + þ(t))

Transmitted/Modulated signal:

Total instantaneous angle:

òc(t) = 2ùf ct + þ(t)

xc(t) = Accosòc(t) = Ac<fej òc(t)g

To obtain angle or exponential modulation:

Make instantaneous phase dependent on the information signal

Oct 04, 2005 CS477: Analog and Digital Communications 4

Instantaneous Frequency

f (t) ñ 2ù1 á dt

dòc(t) = 2ù1òçc(t)

Instantaneous frequency:

For constant phase signals:

òc(t) = 2ùf ct + þ

f (t) = 2ù1òçc(t) = f c

For signals with time varying phase:

f (t) = f c + 2ù1þç(t)

Oct 04, 2005 CS477: Analog and Digital Communications 5

Phase Modulation (PM)

þ(t) = þÉ x(t); þÉ ô 180î

Make phase proportional to the information signal:

f (t) = f c + 2ù1þç(t) = f c + 2ù

þÉxç(t)

xc(t) = Accos(2ùf ct + þÉ x(t))

Instantaneous frequency is dependent onthe derivative of the information signal

is called phase deviationþÉ

Oct 04, 2005 CS477: Analog and Digital Communications 6

Frequency Modulation (FM)

f (t) = f c + f É x(t); f É ô f c

Make instantaneous frequency dependent on the information signal

òc(t) = 2ùR

tf (t)dt = 2ùf ct + 2ùf É

R

tx(ü)dü

is called frequency deviationf É

What is the transmitted signal?

xc(t) = Accos(2ùf ct + 2ùf ÉR

tx(ü)dü)

Oct 04, 2005 CS477: Analog and Digital Communications 7

PM to FM and Vice Versa

xc(t) = Accos(2ùf ct + 2ùf ÉR

tx(ü)dü)

xc(t) = Accos(2ùf ct + þÉ x(t))

(For FM)

(For PM)

RPM Modulator

FM OutputInputþÉ = 2ùf É

Oct 04, 2005 CS477: Analog and Digital Communications 8

Peak and Average Powers Peak power is:

Average power is:

Peak to average power ratio is 2 Compare with DSB and AM Low peak to average power ratio is

good for circuit designers

Pmax = A2max = A2

c

Pavg = 21A2

c

Oct 04, 2005 CS477: Analog and Digital Communications 9

AM, FM, and PM Comparison

Oct 04, 2005 CS477: Analog and Digital Communications 10

Spectral Analysis of FM Exponential modulation is non-linear

Bandwidth of the transmitted signals depends upon the amplitude of the information signal

Hard to analyze with Fourier transforms Narrowband approximation makes

analysis a little easier NBFM and NBPM For narrowband approximation to hold:

jþ(t)j ü 1 rad

Oct 04, 2005 CS477: Analog and Digital Communications 11

NBFM and NBPM

xc(t) = xci(t) cos2ùf ct à xcq(t) sin2ùf ct

Consider the bandpass transmitted signal:

xc(t) = Accos(2ùf ct + þ(t))

xci(t) = Accosþ(t) = Ac[1à 2!þ2(t) + ááá]

xcq(t) = Acsinþ(t) = Ac[þ(t) à 3!þ3(t) + ááá]

For narrowband approximation:

xci(t) ù Ac; xcq(t) ù Acþ(t)

Oct 04, 2005 CS477: Analog and Digital Communications 12

NBFM and NBPM

xc(t) ù Accos2ùf ct à Acþ(t) sin2ùf ct

Thus, for NBFM and NBPM:

Ð(f ) =þÉ X(f ) PMà j f É X(f )=f FM

ú

X c(f ) ù 21Acî (f à f c) à 2

jAcÐ(f à f c); f > 0

þ(t) =þÉ x(t) PM2ùf É

Rx(t) FM

ú

Oct 04, 2005 CS477: Analog and Digital Communications 13

Tone Modulation

x(t) =Amsin2ùf mt PMAmcos2ùf mt FM

ú

þ(t) =þÉ x(t) PM2ùf É

Rx(t) FM

ú

) þ(t) = ì sin2ùf mt

ì =þÉ Am PM( f m

Am)f É FM

ú

ì ü 1Thus, for NBFM and NBPM:

Oct 04, 2005 CS477: Analog and Digital Communications 14

Tone Modulation with NB

xc(t) ù Accos2ùf ct à Acþ(t) sin2ùf ct

X c(f ) ù Acî (f à f c) à 2A cì î (f à (f c à f m))

xc(t) ù Accos2ùf ct à Acì sin2ùf mt sin2ùf ct

= Accos2ùf ct à 2A cì cos2ù(f c à f m)t

+ 2A cì cos2ù(f c + f m)t

+ 2A cì î (f à (f c + f m)); f > 0

Oct 04, 2005 CS477: Analog and Digital Communications 15

Tone Modulation with NB

X c+(f ) ù Acî (f à f c) à 2A cì î (f à (f c à f m))

xc(t) ù Accos2ùf ct à 2A cì cos2ù(f c à f m)t

+ 2A cì cos2ù(f c + f m)t

+ 2A cì î (f à (f c + f m))

f c f c+ f m

2Acì

fà 2

Acì

f cà f m

A c

Oct 04, 2005 CS477: Analog and Digital Communications 16

Tone Modulation with WB

xc(t) = Accos(2ùf ct + þ(t))

xc(t) = Accos(2ùf ct + ì sin2ùf mt)

= Ac[cos(ì sin2ùf mt) cos2ùf ct

cos(ì sin2ùf mt) sin(ì sin2ùf mt)and are periodicNOTE:

à sin(ì sin2ùf mt) sin2ùf ct]