1. demodulation of am - Çankaya Üniversitesiece376.cankaya.edu.tr/uploads/files/ece...

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1. Demodulation of AM HTE - 21.01.2013

In general demodulation is established by frequency shifting of the modulated signal. Inmathematical formulation, we use the following notation

a) Modulating (message) signal

cos 2 , in practice cos 2 (1.1)m i i ii

m t a f t m t a f t

We will usually adopt the first expression for m t , since whatever happens to m t will also be

applicable to individual cosine terms of the summation on the right hand side.

b) Carrier signal

cos 2 (1.2)c cc t A f t

The carrier is mostly a single sinusoidal signal except in multi carrier cases like orthogonalfrequency division multiplexing. In most cases, c mf f

c) Modulated signal

in DSB(SC) - AM (1.3)u t m t c t

d) Received signal

assuming AWGN channel, : Noise signal

if noise is to be ignored (1.4)

r t u t n t n t

r t u t

The simplest form of modulation is multiplying modulating signal m t directly by the carrier c t as

shown in (1.3). By setting the modulating signal to a single sinusoid, with this operation, we get

cos 2 cos 2

cos 2 cos 2 (1.5)2

c m c

cc m c m

u t m t c t aA f t f taA f f t f f t

Corresponding frequency domain expression is

4

(1.6)

cc m c m

c m c m

aAU f F u t f f f f f f

f f f f f f

The above is valid for double side band suppressed carrier, DSB(SC). Now if we add a normalized DClevel of unity to the message signal, we get full amplitude modulation (AM), thus


1 1 cos 2 cos 2

cos 2 cos 2 cos 2 (1.7)2

carrier lower sideband upp

c m c

cc c c m c m

u t m t c t A a f t f t

aAA f t f f t f f t

er sideband

So the corresponding time waveforms and spectrums for (1.6) and (1.7) will be

Fig. 1.1a Time waveforms of full AM and DSB(SC).

Fig. 1.1b Frequency spectrums of full AM and DSB(SC).

Fig. 1.1 Time waveforms and frequency spectrums for DSB(SC) and full AM.

Demodulation of AM : The simplest form of demodulation in AM is to multiply the received signal bythe same carrier generated at the receiver. Since the transmitter and receiver are far apart (as shownbelow), the carrier generated at the receiver will have a phase difference, thus

t

t

Ac (1 + a )

- Ac (1 + a )

u ( t ) = Ac [ 1 + acos (2fmt ) ] cos ( 2fct ) full AM

u ( t ) = Ac acos (2fmt ) cos ( 2fct ) DSB (SC)

- Ac a

Ac a

f = fcf = - fc

f

f

U ( f ) - full AM

U ( f ) - DSB(SC)

0.5Ac ( f + fc ) 0.5Ac

( f - fc )

0.25Ac a ( f + fc + fm )

0.25Ac a ( f + fc + fm )

0.25Ac a ( f + fc - fm )

0.25Ac a ( f + fc - fm )

0.25Ac a ( f - fc + fm )

0.25Ac a ( f - fc + fm )

0.25Ac a ( f - fc - fm )

0.25Ac a ( f - fc - fm )


Fig. 1.2 Block diagram of transmitter and receiver locations.

cos 2 (1.8)r c cc t A f t

After multiplying the received signal by (1.8), for DSB(SC) we get (excluding noise)

cos 2 cos 2

cos cos 4 (1.10)2 2

r c c c

c cc

r t Am t f t f tA Am t m t f t

In terms of frequency spectrum, (1.10) will look like

Fig. 1.3 Frequency spectrum of the time signal in (1.10)

From Fig. 1.3, we see that the recovery of the message signal m t can be achieved by low pass

filtering the first term on the second line of (1.10), after this operation, we will have

ReceiverTransmitter

Located far apart

DemodulationModulation

c ( t ) cr ( t )

f- 2 fc

Low pass filtering

2 fc fm0- fm

M ( f + 2fc ) M ( f - 2fc )M ( f )

F [rr( t )]


cos (1.11)2c

r

Am t m t

As seen (1.11) contains the message signal m t multiplied by cos . Hence the message signal

m t can be recovered from (1.11) provided that 0 . At other values of , the amplitude of

(1.11) will be reduced. In particular, if / 2 , then it will be impossible to recover the message

signal m t from (1.11). So we conclude that for demodulation to be successful, some

synchronisation mechanism is required between the carrier used at transmitter (to performmodulation) and the carrier used at receiver (to perform demodulation). One way of achieving this isto add a small amount of carrier to the DSB signal before transmission. Then by some phase lockedloop setup, it will be to phase synchronize the carrier generated locally at the receiver with the onesent from the transmitter. The effects of variations in can also be illustrated by the following

MATLAB code, AMModDemod_Exp1.m. As can be seen from this MATLAB file, a setting of / 2 will make the demodulation impossible.

Exercise 1.1 : By using the above given formulation, predict if demodulation can be successful at . In general for a range of 0 2 , find which values of make the recovery of the

message signal difficult or impossible and which values of are harmless for the demodulation.

Fig. 1.4 Adding a small amount of carrier to DSB signal before transmission.

The above is called phase coherent or synchronous demodulation which is essential for thedemodulation of DSB signal. In the case of full AM however, there is an alternative method calledenvelope detection (incoherent detection). This is based on the idea that in full AM, the envelope of

the modulated signal follows (traces) the message signal m t as shown below.

c ( t )

DSB

Osc

Ac

cos ( 2fct )

+X

Small amount ofcarrier added

Oscillator for carrier

m ( t )

Transmittedmodulated signal

Message signal


Fig. 1.5 Illustration of the envelope in full AM.

Mathematically, envelope detection functions as follows

Fig. 1.6 Input and output characteristics of a diode (Germanium).

t

u ( t ) = Ac [ 1 + a cos (2fmt ) ] cos ( 2fct ) full AM

Envelope a cos (2fm

t )

xDiode input voltage

y

y = k x2

Diode output current


Fig. 1.7 Circuit diagram of envelope detector.

If we identify, xwith r t , then the output (from the diode) will be

21 cos 2 (1.12)c cy k A m t f t

The squaring action in (1.12) will produce sinusoidal components around 2 cf which will be filtered by

the RC filter in Fig. 1.5. The remaining terms will then be

2 21 (1.13)2m cy kA m t m t

We see here m t as well as 2m t are generated. It is easy to get the message signal m t from

(1.13) so long as 212

m t m t or simply 1m t . In this case (1.13) will approximate to

2 (1.14)m cy kA m t

Referring to Fig. 1.7, we can get another operational view of envelope detector, which is particularlyapplicable to silicon diode. Here we envisage that, the diode 1D conducts only the positive cycles of

the modulated signal, thus act as a half wave rectifier, then the obtained envelope is low pass filteredto get the message signal as shown below in Fig. 1.8.

C Rr ( t ) ym m ( t )

Low pass filterCd - DC blocking

D1

1 / fc << RC << 1 / fm


Fig. 1.8 Another operational view of envelope detector.

2. Generation and Demodulation of PM and FMPM and FM are angle modulations. In this case, the modulated signal can be represented as

cos (2.1)cu t A t

Here the angle, t is the quantity to be modulated by the message signal. The (instantaneous)

frequency of t can be retrieved from

1 (2.2)2i

df t tdt

Assume that t consists of two parts such that

2 (2.3)ct f t t

With this arrangement, (2.1) and (2.2) will become

1cos 2 , (2.4)2c c i c

du t A f t t f t f tdt

where if t is known as instantaneous frequency. Now it is possible to arrive at PM or FM,

depending on how t is related to the message signal

PM , Phase deviation related parameter (2.5)

2 FM , Frequency deviation related parameter

p p

t

f f

k m t kt

k m d k

Half wave rectified signal

Capacitor discharge

t


(2.5) can alternatively be written as

PM (2.6)

2 FM

p

f

dk m td dttdt k m t

Now we define modulation indices (an indication of the depth of modulation) for PM and FM as

, max PM

max , FM (2.7)

p p p p

fff f

m m

k a k m t

k m tk af f

where mf corresponds to the highest frequency in m t . The first definition on the two line refer to

the message signal being a single sinusoid, while the second expressions are valid for the case of the

modulating signal, m t being in the form of a summation as given on the right hand side of (1.1).

The complete PM and FM expressions and the instantaneous frequencies for a general m t then

become

cos 2 , PM2

cos 2 2 , FM (2.8)

fPM c c p iPM c

t

FM c c f iFM c f

k du t A f t k m t f t f m tdt

u t A f t k m d f t f k m t

In order to highlight the implications of (2.6) and (2.7), we show the following waveforms (copieddirectly from Proakis 2002)


Fig. 2.1 FM and PM waveforms for square and triangular modulating signals.

By using (2.4), (2.6) and (2.8), we obtain the following instantaneous frequency expressions for theFM and PM signals of Fig. 2.1.

1

1

1

For FM signal modulated with Square waveform

1 = = constant 0 221 = = constant 2 4

2

c c f c f high

i

c c f c f low

m t

df t f k m t f k f tdtf tdf t f k m t f k f tdt

2

2

2

For FM signal modulated with Triangular waveform

1 = = increasing with time 0 221 = = decreasing with time 2 4

2

c c f c f

i

c c f c f

m t

df t f k m t f k t tdtf tdf t f k m t f k t tdt

(2.9)


1

1

1

For PM signal modulated with Square waveform

1 = constant 0 22 21 = constant 2 4

2 2Discontiniuty at 0, w

pc c c

pi c c c

m t

kd df t f m t f tdt dt

kd df t f t f m t f tdt dt

t

2

2

hich makes 2 = 2

For PM signal modulated with Triangular waveform

1 = = constant 0 22 2 21

2

p pc c c high

i

c

t t

m t

k kd df t f m t f f tdt dtf tdf tdt

1

(2.10)= = constant 2 4

2 2p p

c c low

k kdf m t f f tdt

As understood from (2.9) and (2.10), provided that2p

f

kk

, then we will obtain equivalent

waveforms, for FM signal modulated with 1m t and the PM signal modulated with 2m t . From this

point onwards, we concentrate on FM and set f .

2.1 Spectral Components of FM

For a single sinusoidal message signal

cos 2 (2.11)mm t a f t

The FM expression will become

cos 2 sin 2 Re exp 2 exp sin 2 (2.12)c c m c c mu t A f t f t A j f t j f t

We know that

exp sin 2 exp 2 (2.13)m n mn

j f t J j nf t

By using (2.13) in (2.12) we get the following

cos 2 (2.14)c n c mn

u t A J f nf t

We conclude from (2.14) that FM frequency spectrum extends from minus infinity to plus infinity intheory and the spectral components are placed around cf (carrier) at frequency intervals of mnf


where the respective amplitudes are determined by the values of Bessel function nJ . A typical

(one sided) FM spectrum is shown in Fig. 2.2.

Fig. 2.2 Typical FM spectrum for a single sinusoidal modulating signal.

Of course we cannot tolerate an single message signal to occupy an infinite bandwidth, besides at

higher orders of the Bessel function, i.e. higher n , the magnitudes of nJ start to become

smaller. A reasonable estimate of bandwidth of FM that will accommodate 98 % of the power isgiven by the following formulation

2 1 (2.15)FM mB f

Example 2.1.1 : We are given a carrier of 10cos 2 cc t f t and a message signal of

cos 20m t t . To generate FM from these two signals, we set 50fk . Find the related FM

expression, modulation index and the required bandwidth to transmit this signal. Plot the resultingFM waveform and the frequency spectrum.

Solution : From (2.4) and (2.5) or (2.6) and (2.8), we have

10cos 2 10cos 2 2 cos 20

50 10cos 2 sin 20 (2.16)10

t

c c f

c

u t f t t f t k d

f t t

By taking 50, 1, 10 Hzf mk a f , we get from (2.7)

50 1 5 (2.17)10

f

m

k af

Finally using (2.15), we find the required bandwidth as

0.5 A

c J0(

)( f

- f c

)

0.5 A

c J1(

)( f

- f c -

f m)

0.5 A

c J1(

)( f

- f c +

f m)

0.5 A

c J2(

)( f

- f c +

2 f m

)

0.5 A

c J2(

)( f

- f c -

2 f m

)

0.5 A

c J3(

)( f

- f c -

3 f m

)

0.5 A

c J3(

)( f

- f c +

3 f m

)

0.5 A

c J4(

)( f

- f c +

4 f m

)

0.5 A

c J4(

)( f

- f c -

4 f m

)

0.5 A

c J5(

)( f

- f c -

5 f m

)

0.5 A

c J5(

)( f

- f c +

5 f m

)

U ( f ) - FM

f = fc

Carrier Upper side bandsLower side bands

f


2 1 =120 Hz (2.17)FM mB f

Note that this is six times the bandwidth of an AM signal, since in AM, 2 =20 HzAM mB f . Time

waveform and the frequency spectrum of the FM signal are left as class exercise.

2.2 Generation of FM

The easiest way to generate FM is to use a circuit element whose reactance will change with theapplied voltage or current. These circuit elements are capacitor and inductor, since

1 for capacitor , 2 for inductor (2.18)2C LX X j fLj fC

One method is based on varactor diode whose (junction) capacitance changes with the voltageapplied, whose circuit diagram is given in Fig. 2.3 (circuit diagram copied directly from Proakis 2002)

Fig. 2.3 FM generation using varactor diode.

When the message signal is set to zero, that is 0m t , we have the carrier frequency generated

by the tuned circuit of andaC L , so that

1 (2.19)2c

a

fC L

When 0m t , the instantaneous frequency will change as follows

1 (2.20)2

i

a

f tL C km t

where km t is the amount of parallel (time variable) capacitance, vC added by the varactor. After

rearrangement, (2.20) will become

1 1 1 (2.21)2 1 1

i c

a

a a

f t fLC k km t m t

C C


Provided 1a

k m tC

, we make and expansion of the denominator of the right hand side of (2.21)

to arrive at

1 (2.22)2i c

a

kf t f m tC

(2.22) is exactly in the form of the instantaneous frequency definition given on the second line of(2.8).

2.3 Demodulation of FM

It is easy to see from (2.8) that the frequency modulations in the FM signal can be converted intoamplitude modulations by time differentiating the FM signal given on the second line of (2.8), thus

cos 2 2

2 2 cos 2 2 (2.23)

t

FM c c f

t

c c f c f

d du t A f t k m ddt dt

A f k m t f t k m d

Upon multiplying the term on the second line of (2.23) by a phase synchronized carrier, i.e. by

cos 2 cf t , it will be possible to fully demodulate the message signal as it was done in the case of

coherent demodulation of AM. Alternatively envelope detection can be used. This way it is importantto realize that FM demodulation involves two distinct stages.

A device to perform this combined task is the phase locked loop (PLL) which is shown in Fig. 2.4

Fig. 2.4 The block diagram of PLL used in the demodulation of FM signal.

The input to PLL is the FM signal, hence

uFM ( t )

Input signal Output signal

VCO

Phasecomparator Loop filter

v ( t )

yv ( t ) v ( t )

e ( t )

g ( t )

G ( f )

e ( t )


cos 2 cos 2 2 (2.24)t

FM c c c c fu t A f t t A f t k m d

Voltage controlled oscillator (VCO) of PLL also acts a FM generator in the following manner

sin 2 sin 2 2 (2.25)t

v v c v c c vy t A f t t A f t k v d

while the instantaneous frequency of the VCO is

1 (2.26)2v c v c v

df t f t f k v tdt

After feeding vy t to phase comparator, that acts as a multiplier plus a rejection filter for frequency

components around 2 cf , hence

Input to phase comparator :

Output from phase comparator : sin2

For small : sin , set (2.27)

FM v

c vv

v v v e

u t y tA Ae t t t

e t t t t t t t t

If we substitute for v t on the third line of (2.27) from (2.25), we get

2 (2.28)t

e v vt t t t k v d

In (2.28), we differentiate both side with respect to time and rearrange as follows

2 (2.29)e v

d dt k v t tdt dt

Now v t is the output of the loop filter, while e t is the input, hence they will be related by the

convolution integral such that

(2.30)ev t g t d

By substituting for v t in (2.29) from (2.30), we get

2 (2.31)e v e

d dt k g t d tdt dt

Frequency domain equivalent of (2.31) is

2 2 2 (2.32)e v ej f f k f G f j f f


Deriving e f from (2.32)

(2.33)

1e

v

ff k G f

jf

On the other hand

(2.34)

1e

v

f G fV f f G f k G f

jf

In (2.34), if the condition 1vk G fjf

is satisfied, then

2 (2.35)2 v

j fV f fk

By using (2.24), we get the time domain equivalent of (2.35),

1 1 2 (2.36)2 2

tf

fv v v

kd dv t t k m d m tk dt k dt k

In summary we can say that if an FM signal is supplied to the input of PLL, then demodulation is

performed by PLL so that we get the message signal, m t back at the output.

It is worth pointing out that, in PLL of Fig. 2.4, we have first performed multiplication of FMu t the

by locally generated carrier (coming from VCO) and the elimination of high frequency componentaround 2 cf in the phase comparator, then carried out the differentiation the loop filter. This way in

the PLL of Fig. 2.4, we have reversed the demodulation operation in (2.23) in the following manner

Multiplying by local carrier, cos 2 assuming phase locked case

cos 2 cos 4 2 cos 22 2

After removing the first term at 2 , differentiating the sec

c

t tc c

FM c c f f

c

f t

A Au t f t f t k m d k m d

f

ond term and taking the envelope

Envelope cos 2 (2.37)2

tc

f r c f

Ad k m d m t A k m tdt

3. Noise Analysis3.1 Noise Analysis in AM


Since AM systems are known as narrow band systems, we model the noise as narrow band andrepresent it by

cos 2 sin 2 (3.1)c c s cn t n t f t n t f t

where cn t and sn t are known as in phase and quadrature noise components. Being white

Gaussian noise, n t , cn t and sn t have the flat frequency spectral density functions of nS f ,

ncS f and nsS f and their spectral views are shown in Fig. 3.1, where 0N kT with231.38 10k , Boltzman constant and T being the absolute temperature in 0K .

Fig. 3.1 Frequency spectral density appearance of n t , cn t and sn t

From Fig. 3.1, it is clear that the power of n t will be equal to power of cn t and sn t so that

0 00

0 0

0 0

22 2

2

2

f f f f f f f fc m c m c m c m

n n n mf f f f f f f fc m c m c m c m

f fm m

nc nc mf fm m

f fm m

ns ns mf fm m

N NP S f df S f df df df f N

P S f df N df f N

P S f df N df f N

(3.2)

On the other hand the right hand side of (3.1) will also have noise power of 02 mf N , since the powers

of cosine and sine terms will be equal to 0.5.

f = 0f = - f

c

f = - fc - f

m

f = - fc + f

m

f = - fm

f = fc - f

m

f = fc

f

f = fc + f

mf = fm

N0

N0 / 2 N0 / 2

Snc ( f ) , Sns ( f )

Sn ( f ) Sn ( f )


Now the received signal will be

(3.3)r t u t n t

where n t will be as given in (3.1) and for u t , we assume DSB modulation, thus

cos 2 cos 2 sin 2 (3.4)c c c c s cr t Am t f t n t f t n t f t

In the demodulation process, we will multiply r t by the locally generated carrier cos 2 cf t

at the receiver, thus

cos 2 cos cos 42 2

1 cos sin21 cos 4 sin 4 (3.5)2

c cc c

c s

c c s c

A Ar t f t m t m t f t

n t n t

n t f t n t f t

Low pass filtering will reject the frequency components around 2 cf , then we are left with

1cos cos sin (3.6)2 2c

c s

Ay t m t n t n t

Assuming the phase difference between the transmitted carrier and the one locally generated at thereceiver via PLL, that is 0 . This way (3.6) becomes

noise

1 (3.7)2

message signal

m c cy t Am t n t

Now we estimate the power in the (demodulated) message signal and noise

2

02

Signal power : , : Power in4

Noise power : (3.8)4 2

c ms m

nc mnc

A PP P m t

P f NP

So the signal to noise ratio (SNR) at the end of the demodulation process will

2

2 0

Signal powerSNR after demodulation : (3.9)Noise power 2

s c m

nc m

P A PP f N


To make a comparison, let’s consider the SNR in an equivalent baseband system where there is nomodulation. In this case, the signal and noise power (from 3.2)) would be

2

0 0

Signal power :

Noise power : 2 (3.10)

sb c m

f fm m

nb n n mf fm m

P A P

P P S f df N df f N

Subsequently SNR if no modulation is applied is found as

2

0

Signal powerSNR if no modulation : (3.11)Noise power 2

sb c m

nb m

P A PP f N

Comparing (3.11) to (3.9) we see that the SNR in both case is the same. Thus we conclude that DSBprocess contributes neither positively or negatively to the demodulation process. In other words, theSNR at input to the system is the same as the SNR at output from the system.

3.1 Noise Analysis in FM

Similar to DSB, we again utilize narrow band noise, then the received signal is

cos 2 sin 2 (3.12)c c s cr t u t n t u t n t f t n t f t

For this analysis, we ignore the modulation part, hence set u t to carrier, then

cos 2 sin 2 (3.13)c c c s cr t c t n t A n t f t n t f t

(3.13) can be rearranged as

0.52 2 1

cos 2

, tan (3.14)

c

sc c s

c c

r t R t f t t

n tR t A n t n t t

A n t

The arrangement of (3.14) is illustrated in Fig. 3.2.in the form of a phasor diagram.


Fig. 3.2 The phasor diagram for the expression in (3.14)

If we are operating under high SNR conditions (which is usually the case), then

1 , , tan , (3.15)c c s c cn t A n t A x x R t A

Under these conditions r t of (3.14) will become

cos 2 (3.16)sc c

c

n tr t A f t

A

As explained above the message signal will be extracted from (3.16) by differentiating with respect totime and then taking the envelope of the resulting expression, thus

12 sin 2

1Envelope 2 (3.17)

sc c s c

c c

c c sc

n td dr t A f n t f tdt A dt A

d dr t A f n tdt A dt

On the second line of (3.17), the first term of the right hand side corresponds to DC, after droppingthis DC term, we get

(3.18)m s

dr t n tdt

As stated when describing the operational aspects of PLL, the time derivative operator,ddt

corresponds to a frequency response of 2H f j f . This way after the application of (3.18) to

the noise component sn t , its frequency spectral density will have become

)

nc ( t )

ns ( t )R ( t )

Ac


2 2 2

04 (3.19)ncm ncS f H f S f f N

The implication of (3.19) is illustrated in Fig. 3.3.

Fig. 3.3 Noise spectral density during FM demodulation stages.

We see both from (3.19) and Fig. 3.3. because of the act of (time) differentiation, the noise spectraldensity is converted from flat spectral density into parabolic type. After this differentiation, AMdemodulation is to be applied. This means that the original FM (baseband) bandwidth has been

reduced from 2 1 mf to 2 mf . In this process, message signal sidebands at mnf will fold back

into mf , since they are correlated, but the same thing is not valid for noise, because noise spectral

components are uncorrelated. Since AM demodulation will only cover a bandwidth of 2 mf , we will

leave out an important amount of noise outside. This will be particularly so due to the parabolic

nature of noise spectral density function of ncmS f . In the end we can safely claim that there is a

considerable SNR improvement from the input of FM demodulator to its output. That is the SNR atthe output of FM demodulator is larger than the SNR at its input. It is important to realize that thisSNR improvement of FM is achieved at the expense of expanding the message signal bandwidth asillustrated in Fig. 2.2.

To estimate what SNR improvement we have gained, we assume that the message signal power hasremained the same during this demodulation process, so it is sufficient to take the ratio of noise

power when the bandwidth is 2 1 mf to that of the bandwidth being reduced to 2 mf , thus

1

1331 1

3SNR improvement due to noise reduction 1 (3.20)

fm

fmncmf fm m

f fm m

fmncmfm

S f df f

fS f df

f

Sncm

( f )

f = fmf = - f

m

FM bandwidth prior to envelope detection

f = 0

Noise left after envelope detection

f = - ( + 1 ) fm

f = ( + 1 ) fm


Hence the larger the modulation index (effectively meaning the utilization of larger bandwidth), themore SNR improvement we get.

The above text is based on

1) John G. Proakis, Masoud Salehi, “Communication Systems Engineering” 2nd Ed. 2002, ISBN :0-13-061793-8.

2) Bernard Sklar, “Digital Communications Fundamentals and Applications”, 2nd Ed. Prentice Hall2002, ISBN : 0-13-084788-7.

3) My own lecture notes.

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