FM_lecture1.pdf

Dr. M. Venu Gopala Rao, Professor, Dept. of ECE, KL University

Angle / Exponential Modulation Lecture -1 1.0 Introduction. 1.1 Instantaneous Frequency. 1.2 Phase Modulation (PM). 1.3 Frequency Modulation (FM). 1.4 Relation between Phase Modulation and Frequency Modulation. 1.5 Single-tone Angle Modulation. 1.5.1 Single-tone Phase Modulation. 1.5.2 Single-tone Frequency Modulation. 1.6 Phase and Frequency deviation. 1.6.1 Units for Phase / Frequency deviation and Modulation indices. 1.6.2 Summary of PM and FM equations. 1.7 Narrowband Frequency Modulation. 1.8 Bandwidth of Single-tone Narrowband Frequency Modulation. 1.9 Phasor diagram for NBFM. 1.10 Spectrum of Narrowband Angle Modulation for Baseband signal 1.11 References

Objectives:  Define and mathematically describe angle modulation  Explain the difference between frequency and phase modulation  Describe direct and indirect frequency modulation  Describe direct and indirect phase modulation  Define deviation sensitivity  Describe FM and PM waveforms  Define phase deviation and modulation index  Explain frequency deviation and percent modulation  Analyze the frequency content of an angle modulated waveform  Describe the Narrowband PM and FM  Determine the bandwidth requirement for Narrowband PM and FM 1

Dr. M. Venu Gopala Rao, Professor, Dept. of ECE, KL University

Angle / Exponential Modulation 1.0. Introduction: In the previous chapters, we studied the different AM techniques in which the amplitude of some carrier signal is modified according to the message signal. The frequency and phase of the carrier of the carrier signal in all AM modulation techniques were constant. In this chapter, we will study a different method for transmitting information by changing the angle (changing the phase or frequency) of the carrier signal and keeping its amplitude constant usually referred to as Angle Modulation. The phase of carrier is varied in accordance with amplitude of the message signal referred to as phase modulation (PM). In other case the frequency of the carrier is varied in accordance with amplitude of modulating signal is called frequency modulation (FM). The PM and FM are non-linear function of modulating signal which makes them are non-linear modulation process. 1.1 Instantaneous Frequency: Let a generalized sinusoidal carrier c(t )  Ac cos (t ) , where  (t ) is a generalized angle and is a function of time. That is  (t )  ct  0 . Then the carrier signal is represented by

c(t )  Ac cos( (t ))  Ac cos(ct  0 )

Fig 1 Concept of instantaneous frequency

where c(t ) is the instantaneous value (voltage or current), Ac is maximum amplitude, c is the angular velocity in rad / sec and  0 is phase angle in radians. It should be noted that ct represents an angle in radians. A hypothetical case general angle of  (t ) happens to be tangential to the angle ct  0 at some instant time ‘t’ as shown in Fig 1. The crucial part is that around ‘t’ over a small interval t  0 , the signal c(t )  Ac cos (t ) and Ac cos(ct  0 ) are identical.

That is c(t )  Ac cos(ct  0 )

t1  t  t2

Over this small interval t , the angular frequency of c(t ) is c . Because (ct  0 ) is tangential to  (t ) , the angular frequency of c(t ) is the slope of its angle  (t ) over this small interval. Therefore the generalized angle  (t ) instantaneous frequency i are related by

i (t ) 

t d  (t ) or  (t )   i ( ) d  dt

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Dr. M. Venu Gopala Rao, Professor, Dept. of ECE, KL University

With this background we will discuss the phase modulation and frequency modulations are as follows. 1.2 Phase Modulation (PM): In PM, the phase of a constant amplitude carrier wave varies according to the amplitude of modulating signal m(t ) at a rate equal to the frequency of the modulating signal. By assuming  0  0 ,  (t )  ct  k p m(t ) , where k p is phase modulation sensitivity constant. Then the resulting Phase Modulated wave is expressed as sPM (t )  Ac cos ct  k p m(t ) 

In this case the instantaneous angular frequency is given by

i (t ) 

d d  (t )  ct  k p m(t )   c  k p m(t ) dt dt

Hence in PM the instantaneous angular frequency i varies linearly with the derivative of the modulating signal. 1.3 Frequency Modulation (FM): In frequency modulation, the frequency of constantamplitude carrier varies according to the amplitude of modulating signal m(t ) at a rate equal to the frequency of the modulating signal. Thus the instantaneous angular frequency of FM wave is

i (t )  c  k f m(t ) or fi (t )  f c  k f m(t ) where k f is frequency modulation sensitivity constant. Then the angle  (t )  

t



i ( ) d  

t

  2 k f m( ) d  c

 ct  2 k f

t

 m( ) d

With this notation the mathematical expression of FM wave is represented by t sFM (t )  Ac cos ct  2 k f  m( ) d    

1.4 Relationship between PM and FM: PM and FM are closely related to each other. Comparing sPM (t ) and sFM (t ) reveals that an FM signal may be regarded as a PM signal in which the modulating wave is

t

 m( ) d

in place of m(t ) . This means that an FM signal

can be generated b y first integrating m(t ) and then using the result as the input to a phase modulator, as in Fig 2(a). Conversely, a PM signal can be generated by first differentiating 3

Dr. M. Venu Gopala Rao, Professor, Dept. of ECE, KL University

m(t) and then using the result as the input to a frequency modulator, as in Fig 2(b). We may thus deduce all the properties of PM signals from those of FM signals and vice versa. A consequence of allowing the angle  (t ) to become dependent on the message signal m(t) as in sPM (t ) or on its integral as in sFM (t ) is that the zero crossings of a PM signal or FM signal no longer have a perfect regularity in their spacing; (zero crossings refer to the instants of time at which a wave-form changes from a negative to a value or vice versa) positive

Fig 2 Phase and frequency modulation

(Refer Fig 3). This is one important feature that distinguishes both FM and PM signals from an AM signal. Another important difference is that the envelope of a PM or FM signal is constant (equal to the carrier amplitude), whereas the envelope of an AM signal is dependent on the message signal.

Fig 3. The AM, PM and FM waveforms

1.5 Single-Tone Angle Modulation: Let the message signal be m(t )  Am cos 2 f mt , where

Am is amplitude and f m is the frequency of the message signal. 1.5.1 Single-Tone Phase Modulation (PM): The phase modulated signal is represented by sPM (t )  Ac cos ct  k p m(t )  For a single-tone modulating signal, the PM wave is represented by 4

Dr. M. Venu Gopala Rao, Professor, Dept. of ECE, KL University

sPM (t )  Ac cos ct  k p Am cos 2 f mt   Ac cos ct   p cos 2 f mt 

where  p  k p Am is called phase modulation index. Phase deviation   k p max m(t )  k p Am . In Phase modulation, both modulation index and phase deviation are same.

1.5.2 Single-Tone Frequency Modulation (PM): The frequency modulated signal is represented by t sFM (t )  Ac cos ct  2 k f  m( ) d    

For a single-tone modulating signal, the FM wave is represented by t s FM (t )  Ac cos ct  2 k f  Am cos 2 f mt dt     2 k f Am    Ac cos ct  sin 2 f mt  2 f m  

 Ac cos ct   f sin 2 f mt  where  f 

k f Am fm



f is called frequency modulation index, and fm

where again f  k f Am is known as frequency deviation. In frequency modulation,  f is the modulation index and represents the maximum phase shift of the carrier, and f is the maximum frequency deviation of the carrier. The maximum frequency deviation in FM broadcasting is 75 KHz, and frequency spacing is 200 KHz. Brief Summary: Phase Modulation:

P.M :

Ac cos ct   p cos 2 f mt 

Phase Modulation index  p  k p Am radians; Frequency Modulation: F.M :

Phase deviation   k p Am radians.

Ac cos ct   f cos 2 f mt 

Frequency Modulation index  f 

k f Am fm



f ; Frequency deviation f  k f Am Hz fm

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Dr. M. Venu Gopala Rao, Professor, Dept. of ECE, KL University

1.6 Phase and Frequency Deviation: Let a single-tone angle modulated signal is represented by

s(t )  Ac cos[ωct   sin ωmt ] PM: Suppose this equation represents the Phase Modulation, then  is the peak amplitude of the phase information. In this case 

is the maximum phase deviation 

( max{ sin ωmt} ), usually referred to as modulation index  p . FM: Suppose this equation represents the Frequency Modulation, then  is the modulation index (  f ). In this case the maximum frequency deviation f   f m (  f f m ).

f  max{ f m sin ωmt} 

d  dt

In this case of the angle modulated signal the generalized angle is

 (t )  ωct   sin ωmt The instantaneous frequency ωi (t ) 

d  (t )  ωc   ωm cos ωmt or fi  fc   fm cos ωmt dt

Then the maximum frequency deviation f   f m . 1.6.1 Units for Phase / Frequency deviation and Modulation indices: In Phase Modulation, the phase deviation is given by   k p m(t ) radians. Similarly in Frequency Modulation the frequency deviation is given by ω 

d   k f m(t ) radians/sec. or f  k f m(t ) Hz. dt

where k p and k f are constants and are deviation sensitivities of the Phase and Frequency Modulations respectively. The deviation sensitivities are the output versus input transfer function for the modulation, which gives the relationship between the parameter changes in respect to specified changes in the input signal.  For a phase modulation, changes would occur in the phase of the output frequency in respect to changes in the amplitude of the input modulating signal voltage. Therefore the deviation sensitivity for a phase modulator is    Radians kp     V  V

 For a frequency modulation, changes would occur in the output frequency in respect 6

Dr. M. Venu Gopala Rao, Professor, Dept. of ECE, KL University

to changes in the input modulating signal voltage. Therefore the deviation sensitivity for a frequency modulator is  ω  Radians / sec  f  Cycles / sec or  Hz / V kf    V V  V   V 

 Modulation index for PM is defined as  p  k p Am for a single-tone modulating signal m(t )  Am cos 2 f mt . Then the units for modulation index  p is defined as Radians V  Radians . V

k f Am  Modulation index for FM is defined as  f  for a single-tone modulating ωm signal m(t )  Am cos 2 f mt . Then the units for modulation index  f

is defined as

Radians / sec V Cycles / sec V or (unit less). V Radians / sec V Cycles / sec

1.6.1: Summary of Phase Modulation and Frequency Modulation Phase Modulation A Generalized PM equation

sPM (t )  Ac cos ct  k p m(t )  where k p rad/sec

is phase modulation

Frequency Modulation A Generalized FM equation t sFM (t )  Ac cos ct  2 k f  m( ) d    

where k f Hz/Volt is frequency modulation

sensitivity constant.

sensitivity constant

Single-Tone case m(t )  Am cos 2 f mt

Single-Tone case m(t )  Am cos 2 f mt

sPM (t )  Ac cos ct  k p Am cos 2 f mt 

k f Am   s FM (t )  Ac cos  ct  sin 2 f mt  fm  

 Ac cos ct   p cos 2 f mt  Phase modulation index

 p  k p Am radians Phase deviation   k p Am radians In Phase modulation,  p  

 Ac cos  ct   f sin 2 f mt  Frequency Modulation index

f 

k f Am fm



f (unit less) fm

Frequency deviation f  k f Am Hz.

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Dr. M. Venu Gopala Rao, Professor, Dept. of ECE, KL University

1.7 Narrowband F.M.: To simplify the analysis of F.M., we first assume that  f 

 2

(  f  0.2 ), and modulating signal m(t )  Am cos 2 f mt . A standard FM signal is represented by sFM (t )  Ac cos ct   f sin 2 f mt  , where

f 

k f Am fm



f is called frequency modulation index, where again f  k f Am is known fm

as frequency deviation. Then by expanding, we get sFM (t )  Ac cos ct   f sin 2 f mt   Ac cos ct  cos(  f sin 2 f mt )  sin ct  sin(  f sin 2 f mt ) 

For  f   / 2 , cos(  f sin 2 f mt )  1 and sin( f sin 2 f mt )   f sin 2 f mt . Therefore, the narrowband FM is described by

s NBFM (t )  Ac cos ct  sin ct   f sin 2 f mt   Ac cos ct  Ac  f sin ct  sin 2 f mt Ac  f  Ac cos ct  cos(c  m )  cos(c  m ) 2 Ac  f Ac  f  Ac cos ct  cos(c  m )  cos(c  m ) 2 2 The single-tone Amplitude Modulation equation is given by

s AM (t )  Ac cos ct 

 Ac 2

cos(c  m )t 

 Ac 2

cos(c  m )t

The equation sNBFM (t ) resembles the AM ( s AM (t ) ) except that in narrowband FM, the phase of LSB signal reversed and the resultant sideband vector sum is always in-phase quadrature with the carrier. Thus the FM gives rise to phase variations with very small amplitude change (  f   / 2 ), while AM gives amplitude variations with no phase deviation. The frequency spectrum of Narrow Band Frequency Modulation is represented by Ac  f A S FM ( f )  c  ( f  f c )   ( f  f c )    ( f  f c  f m )   ( f  f c  f m ) 2 4 Ac  f   ( f  f c  f m )   ( f  f c  f m ) 4

The spectrum of this narrowband FM wave is shown in Fig 4. For comparison AM spectrum is also shown in figure. The AM, PM and FM signals are shown in Fig 4. 8

Dr. M. Venu Gopala Rao, Professor, Dept. of ECE, KL University

Fig 4. Spectrum Top: Amplitude Modulation, Bottom: Narrowband Modulation.

1.8 Bandwidth of Single-tone Narrowband Frequency Modulation From the spectrum of Single-tone Narrowband Frequency Modulation, it can be observed that the bandwidth of the Narrowband Frequency Modulation is equal to AM bandwidth, which is two times of message signal bandwidth. For a single-tone modulation frequency f m Hz then the bandwidth of the Narrowband Frequency Modulation is 2 f m Hz. Similar statement holds for Single-tone Narrowband Phase Modulation. 1.9 Phasor Diagram for Narrow Band FM signals: The phasor diagram describes or understanding an assortment of the sidebands in FM signal of constant amplitude. The diagram will also make clear the difference between AM and NBFM. Let us consider a NBFM (i. e.,  f   / 2 ) signal described by the equation s NBFM (t )  Ac cos ct 

Ac  f Ac  f cos(c  m )  cos(c  m ) 2 2

The NBFM signal is represented with Phasor Diagram is shown in Fig 5(a), in which the carrier phasor has been assumed to be the reference. It should be noted that resultant of the phasors corresponding to the two side frequencies is always perpendicular to the carrier phasor. As a result it produces a resultant phasor representing a NBFM which is approximately of the same amplitude as the carrier phasor but out of phase with respect to it. It is interesting to compare the phasor diagram of this NBFM with that of conventional AM shown in Fig 5(b). It can be easily verified that in the case of AM, the resultant of the two side frequency phasors is always in-phase with the carrier phasor. The effect is that the resultant 9

Dr. M. Venu Gopala Rao, Professor, Dept. of ECE, KL University

phasor representing the AM wave has amplitude significantly different from the carrier phasor amplitude and is always in-phase with it.

Fig 5. Phasor representation of (a) NBFM (b) AM

1.10 Spectrum of Narrow Band Angle Modulation for Base Band Signal: Let a base-band signal represented by m(t ) is band limited to ‘W’ Hz, which is a finite energy signal. AM: Then the conventional A.M. signal is represented by

sAM (t )  Ac [1  ka m(t )]cos ct  Ac cos ct  Ac ka m(t )cos ct

(A1)

Corresponding the spectrum of AM wave is given by A A S AM ( f )  c  ( f  fc )   ( f  fc )  c ka  M ( f  fc )  M ( f  fc )  2 2

(A2)

where M ( f ) is F.T. of baseband signal m(t ) . FM: The frequency modulation signal is represented by

sFM (t )  Ac cos[ct  k f

t

 m(t ) dt ]

Further by expanding this equation,









t t sFM (t )  Ac cos ct  cos k f  m(t ) dt  sin ct  sin k f  m(t ) dt     

t

For | k f  m(t ) dt | 1 , we can approximate the following 



t

cos k f  m(t ) dt 



1 , and

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Dr. M. Venu Gopala Rao, Professor, Dept. of ECE, KL University



t

sin k f  m(t ) dt 



kf

t

 m(t ) dt

With these approximations the Narrow Band FM is expressed as sNBFM (t )  Ac cos ct  k f 



t





m(t ) dt sin ct  

(F1)

Further simplified as sNBFM (t )  Ac cos ct  k f Ac m '(t )sin ct , where m '(t )  

t



m(t ) dt

Then the spectrum of Narrow Band Frequency Modulation is represented by A A S NBFM ( f )  c  ( f  fc )   ( f  fc )  c k f e j /2  M '( f  fc )  M '( f  f c )  2 2

(F2)

PM: The phase modulation signal is represented by sPM (t )  Ac cos[ct  k p m(t )]

Further by expanding this equation,

sPM (t )  Ac cos ct  cos k p m(t )  sin ct  sin k p m(t ) For | k p m(t ) | 1 , we can approximate the following

cos | k p m(t ) | 1 , and sin | k p m(t ) |1 k p m(t ) With these approximations the Narrow Band PM is expressed as sNBPM (t )  Ac cos ct  k p m(t )sin ct   Ac cos ct  Ac k p m(t )sin ct

(P1)

Then the spectrum of Narrow Band Phase Modulation is represented by A A S NBPM ( f )  c  ( f  fc )   ( f  fc )  c k pe j /2  M ( f  fc )  M ( f  fc ) 2 2

(P2)

A comparison of NBPM (equations F1 and F2) or NBFM (equations P1 and P2) with AM (equations A1 and A2) brings out clearly the similarities and differences between two types of modulation. Similarities: (i) Both have the same modulated bandwidth 2W, where W is the highest modulating signal frequency. (ii) The sideband spectrum for FM has a phase shift of  / 2 radians with respect to the carrier, whereas that of AM is in-phase with the carrier. 11

Dr. M. Venu Gopala Rao, Professor, Dept. of ECE, KL University

Differences: In an AM signal, the oscillation frequency is constant and the amplitude varies with time, whereas in an FM signal, the amplitude stays constant and frequency varies with time. 1.11 References: 1. B.P. Lathi and Zhi Ding, “Modern Digital and Analog Communication Systems”, International 4th Edition, Oxford University Press, 2010. 2. Wayne Tomasi, ‘Electronic Communications Systems fundamentals through advanced’, Pearson Education, fifth edition, 2011. 3. H Taub & D. Schilling, Gautam Saha, ”Principles of Communication Systems, TMH, 2007, 3rd Edition. 4. Simon Haykin ,”Principles of Communication Systems “,John Wiley, 2nd Ed. 5. John G. Proakis, Masond, Salehi ,”Fundamentals of Communication Systems “,PEA, 2006. 6. George Kennedy, “ Electronic Communication Systems”, 3rd edition, Tata McGraw-Hill Edition. 7. Masoud Salehi and John G Proakis, ‘Contemporary Communication Systems Using Matlab and Simulink’, 2004.

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