Angle Modulation

Angle modulation is defined as a process in which the angle of the carrier wave c(t) is varied according to the baseband signal m(t).

In this method of modulation the amplitude of the carrier wave is maintained constant.

An important feature of angle modulation is that it can provide better discrimination against noise and interference than amplitude modulation. This improvement in the performance is achieved at the expense of increased transmission bandwidth; that is; angle modulation provides us with a practical means of exchanging channel bandwidth for improved noise performance. Such a trade-off is not possible with amplitude modulation.

Basic Definition of Angle ModulationLet θi(t) denote the angle of a modulated sinusoidal carrier, then resulting angle-modulated wave as: where Ac is the carrier amplitude.

A complete oscillation occurs whenever θi(t) changes by 2π radians. If θi(t) increases with time, the average frequency from t to t+∆t, is given by

Thus, the instantaneous frequency of the angle-modulated signal s(t) as follows:

There are an infinite number of ways in which the angle θi(t) may be varied in some manner with the message (baseband) signal. However, we shall consider only two commonly used methods, phase modulation and frequency modulation.

In the simple case of an unmodulated carrier, the angle θi(t) isThe constant φc is the value of θi(t) at t=0.

ctcfti φπθ +=2)(

Phase modulation (PM) is that form of angle modulation in which the angle θi(t) is varied linearly with the message signal m(t), as shown by

Where, 2πfct: the angle of the unmodulatedcarrier; kp: phase sensitivity of the modulator

The phase-modulated signal s(t) is thus described in the time domain by

Frequency Modulation (FM) is that form of angle modulation in which the instantaneous frequency fi(t) is varied linearly with the message signal m(t), as shown by Where, fc: the frequency of

unmodulated carrier, kf: frequency sensitivity of the modulator

Integrating Eq. (3.50) with respect to time and multiplying the result by 2π, we get

The frequency-modulated signal is therefore describe in the time domain by

Comparing Eq. (3.49) with (3.52) reveals that an FM signal may be regarded as a PM signal in which the modulating wave is

in place of m(t). This means that an FM signal can be generated by first integrating m(t) and then using the result as the input to a phase modulator, as in Fig. 3.30a.

Conversely, a PM signal can be generated first differentiating m(t) and then using the result as the input to a frequency modulator, as in Fig. 3.30b. We may thus deduce all the properties of PM signals from those of FM signals and vice versa.

∫t dttm0 )(

AM

PM

FM

Frequency ModulationThe FM signal s(t) defined by Eq. (3.52) is a nonlinear function of the

modulating signal m(t), which makes frequency modulation a nonlinear modulation process.

To analyze simple way, consider a sinusoidal modulating signal defined by: m(t)=Amcos(2πfmt) (3.53)

The instantaneous frequency of the resulting FM signal equals

fi(t)= fc + KfAmcos(2πfmt)=fc + ∆fcos(2πfmt)Frequency deviation, ∆f=KfAm, representing the maximum departure of the instantaneous frequency of the FM signal from the carrier frequency fc.

Using Eq. (3.54), the angle θi(t) of the FM signal is obtained as

)2sin()/(20

2)( tmfmfftcft

dtifti πππθ ∆+=∫= )2sin(2)( tmftcfti πβπθ +=

Modulation index, mff /∆=β

β represents the phase deviation of the FM signal, that is, the maximum departure of the angle θi(t) from the angle 2πfct of the unmodulated carrier.

The FM signal itself is given by(3.59)

Depending on the value of the modulation index β, we may distinguish two cases of frequency modulation:

Narrow-band FM, for which β is small compared to one radian.Wide-band FM, for which β is large compared to one radian.

)]2sin(2cos[)( tmftcfcAts πβπ +=

Narrow Band FMEq. (3.59) can be written as follows:

(3.60))]2sin(sin[)2sin()]2sin(cos[)2cos()( tmftcfcAtmftcfcAts πβππβπ −=

1)]2sin(cos[ ≈tmfπβ )2sin()]2sin(sin[ tmftmf πβπβ ≈

Assuming that the modulation index β is small compared to one radian, we may use the following approximations:

and

Hence, Eq (3.60) simplifies to

)2sin()2sin()2cos()( tmftcfcAtcfcAts ππβπ −= (3.61)

Using the Eq. (3.61), the block diagram of a method for generating a narrow-band FM signal can be drown as Fig. 3.31.

Comparison Between FM and AM System:

FM has several advantages over AM, a few of them are being listed here:

1. The amplitude of the frequency modulated wave in FM remains independent of the depth of modulation, while in case of AM thisamplitude is dependent on the modulation factor (or index).

2. The envelope of FM signal is constant (equal to the carrier amplitude), whereas the envelope of an AM signal is dependent onthe message signal.

3. All the transmitted power in FM is useful, whereas in case of AM, most of the transmitted power is carrier which does not indicateany modulation changes.

4. There is large decrease in noise in case of FM which in other words indicates a rise in signal to noise ratio.

5. Noise in FM can be further reduced by increasing the deviation. This facility is not available in case of AM systems.

FM has several disadvantages over AM, a few of them are being listed here:

1. A large bandwidth, probably 7 to 15 times of that of AM is needed for FM system.

2. FM transmitting and receiving equipment is more complex and hence is more expensive.

3. As the reception is limited to line of sight, the area covered by FM is much smaller than AM.

End of the Angle Modulation