lecture 1 · web viewsimon haykin ,”principles of communication systems “,john wiley, 2nd ed....

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.

1.7 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

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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 , where is a generalized angle and is a function of time. That is .Then the carrier signal is represented by

where is the instantaneous value (voltage or current), Fig 1 Concept of instantaneous frequency

is maximum amplitude, is the angular velocity in rad / sec and is phase angle in

radians. It should be noted that represents an angle in radians.

A hypothetical case general angle of happens to be tangential to the angle at

some instant time ‘t’ as shown in Fig 1. The crucial part is that around ‘t’ over a small interval

, the signal and are identical.

That is

Over this small interval , the angular frequency of is . Because is tangential to , the angular frequency of is the slope of its angle over this small interval. Therefore the generalized angle instantaneous frequency are related by

or

With this background we will discuss the phase modulation and frequency modulations are as

follows.

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1.2 Phase Modulation (PM): In PM, the phase of a constant amplitude carrier wave varies

according to the amplitude of modulating signal at a rate equal to the frequency of the

modulating signal. By assuming , , where is phase modulation

sensitivity constant.

Then the resulting Phase Modulated wave is expressed as

In this case the instantaneous angular frequency is given by

Hence in PM the instantaneous angular frequency varies linearly with the derivative of the

modulating signal.

1.3 Frequency Modulation (FM): In frequency modulation, the frequency of constant-ampli-

tude carrier varies according to the amplitude of modulating signal at a rate equal to the

frequency of the modulating signal.

Thus the instantaneous angular frequency of FM wave is

or

where is frequency modulation sensitivity constant.

Then the angle

With this notation the mathematical expression of FM wave is represented by

1.4 Relationship between PM and FM: PM and FM are closely related to each other.

Comparing and reveals that an FM signal may be regarded as a PM signal in

which the modulating wave is in place of . This means that an FM signal

can be generated by first integrating 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

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.

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A consequence of allowing the angle to

become dependent on the message signal m(t)

as in or on its integral as in 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 positive value or vice

versa) (Refer Fig 3). This is one Fig 2 Phase and frequency modulation

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 , where is amplitude and is the frequency of the message signal. 1.5.1 Single-Tone Phase Modulation (PM):

The phase modulated signal is represented by For a single-tone modulating signal, the PM wave is represented by

where is called phase modulation index. Phase deviation

. In Phase modulation, both modulation index and phase deviation are same.

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1.5.2 Single-Tone Frequency Modulation (PM):

The frequency modulated signal is represented by

For a single-tone modulating signal, the FM wave is represented by

where is called frequency modulation index, and

where again is known as frequency deviation.

In frequency modulation, is the modulation index and represents the maximum phase

shift of the carrier, and is the maximum frequency deviation of the carrier. The maxim-

um frequency deviation in FM broadcasting is 75 KHz, and frequency spacing is 200 KHz.

Brief Summary:

Phase Modulation: P.M :

Phase Modulation index radians; Phase deviation radians.

Frequency Modulation: F.M :

Frequency Modulation index ; Frequency deviation Hz

1.6 Phase and Frequency Deviation:

Let a single-tone angle modulated signal is represented by

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 (

), usually referred to as modulation index .

FM: Suppose this equation represents the Frequency Modulation, then is the modulation index ( ). In this case the maximum frequency deviation ( ).

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

The instantaneous frequency or

Then the maximum frequency deviation .

1.6.1 Units for Phase / Frequency deviation and Modulation indices:

In Phase Modulation, the phase deviation is given by radians. Similarly in

Frequency Modulation the frequency deviation is given by

radians/sec. or Hz.

where and 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

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

to changes in the input modulating signal voltage. Therefore the deviation sensitivity

for a frequency modulator is

or Hz / V

Modulation index for PM is defined as for a single-tone modulating signal . Then the units for modulation index is defined as

.

Modulation index for FM is defined as for a single-tone modulating

signal . Then the units for modulation index is defined as

or (unit less).

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

), and modulating signal .

A standard FM signal is represented by , where

is called frequency modulation index, where again is known

as frequency deviation. Then by expanding, we get

For , and .

Therefore, the narrowband FM is described by

The single-tone Amplitude Modulation equation is given by

The equation resembles the AM ( ) 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 (

), while AM gives amplitude variations with no phase deviation.

The frequency spectrum of Narrow Band Frequency Modulation is represented by

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.

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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 Hz then the

bandwidth of the Narrowband Frequency Modulation is 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., ) signal described by the equation

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

phasor representing the AM wave has amplitude significantly different from the carrier phasor

amplitude and is always in-phase with it.8


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 is band limited to ‘W’ Hz, which is a finite energy

signal.

AM: Then the conventional A.M. signal is represented by

(A1)

Corresponding the spectrum of AM wave is given by

(A2)

where is F.T. of baseband signal .

FM: The frequency modulation signal is represented by

Further by expanding this equation,

For , we can approximate the following

, and

With these approximations the Narrow Band FM is expressed as

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(F1)

Further simplified as , where

Then the spectrum of Narrow Band Frequency Modulation is represented by

(F2)

PM: The phase modulation signal is represented by

Further by expanding this equation,

For , we can approximate the following

, and

With these approximations the Narrow Band PM is expressed as

(P1)

Then the spectrum of Narrow Band Phase Modulation is represented by

(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 radians with respect to the

carrier, whereas that of AM is in-phase with the carrier.

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.

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1.11 References:

1. H Taub & D. Schilling, Gautam Sahe, ”Principles of Communication Systems, TMH, 2007,

3rd Edition.

2. Simon Haykin ,”Principles of Communication Systems “,John Wiley, 2nd Ed.

3. B.P. Lathi and Zhi Ding, “Modern Digital and Analog Communication Systems”,

International 4th Edition, Oxford University Press, 2010.

4. George Kennedy, “ Electronic Communication Systems”, 3rd edition, Tata McGraw-Hill

Edition.

5. Wayne Tomasi, ‘Electronic Communication Systems- fundamentals through advanced’, 5th

edition, Pearson Education Inc, 2011.

6. John G. Proakis, Masond, Salehi ,”Fundamentals of Communication Systems “, PEA, 2006.

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lecture 1 · web viewsimon haykin ,”principles of communication systems “,john wiley, 2nd ed....

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