3 Properties of an Analog Signal: amplitude frequency phase

Angle Modulation – FM & PM - often referred to as simply

FM, although there are actual distinctions between the two

Angle Modulation

Angle Modulation vs. Amplitude Modulation Advantages:

- Noise Immunity- Noise Performance and Signal-to-

Noise Improvement- Capture Effect- Power Utilization and Efficiency

Disadvantages:- Bandwidth- Circuit Complexity and cost

Angle Modulation- first introduced in 1931 as an alternative

to amplitude modulation- Major E.H. Armstrong, 1936- July 1939, Alpine, New Jersey- commercial radio broadcasting,

television sound transmission, two-way mobile radio, cellular radio, microwave and satellite communications systems

Angle Modulation- results whenever the phase angle () of a

sinusoidal wave is varied with respect to time.

m(t) = Vc cos [ ct + (t) ] eq. 1

where: m(t) = angle-modulated wave Vc = peak-carrier amplitude (volts)

c = carrier radian frequency (rad/sec, 2fc) (t) = instantaneous phase deviation

(radians)

- with angle modulation, it is necessary that (t) be a prescribed function of the modulating signal.

(t) = F [m(t)] eq.2

where: m(t) = Vm sin(mt) – modulating signalm = angular velocity of the modulating

signal (rad/sec, 2fm)fm = modulating signal frequency (Hz)Vm = peak amplitude of the modulating

signal (volts)

Difference between FM & PM- lies in which property of the carrier is

directly varied by the modulating signal and which property is indirectly varied.

Direct Frequency Modulation (FM) : Varying the frequency of a constant-amplitude carrier directly proportional to the amplitude of the modulating signal at a rate equal to the frequency of the modulating signal.

Direct Phase Modulation (PM): Varying the phase of a constant-amplitude carrier directly proportional to the amplitude of the modulating signal at a rate equal to the frequency of the modulating signal.

Angle-Modulated Wave in the Frequency Domain

f – frequency shiftMagnitude and Direction – proportional to the amplitude and

polarity of the modulating signalRate at which frequency changes – equal to the frequency of

the modulating signal

Angle Modulation in Time Domain(Phase Changing with Time) - phase deviation , reference

angular displacement of the carrier in radian in respect to the reference phase

f – frequency deviation, relative displacement of the carrier frequency in hertz in respect to its unmodulated value

& f – magnitude is proportional to the amplitude of the modulating signal and rate at which the changes are occurring is equal to the modulating frequency

Angle Modulation in the Time Domain (Frequency Changing with Time)

f- is changed or deviated over a period of time

Tmin – maximum frequency

Tmax – minimum frequency

Angle Modulation in the Time Domain

Resultant Angle-Modulated Waveform – carrier rests frequency and an infinite no. of pairs of side frequencies displaced on either side of the carrier by an integral multiple of the modulating signal frequency.

Mathematical Analysis1. Instantaneous Phase Deviation

- the instantaneous change in the phase of the carrier at a given instant of time and indicates how much the phase of the carrier is changing with respect to its reference phase.instantaneous phase deviation = (t) rad eq 3

2. Instantaneous Phase- the precise phase of the carrier at a given instant of timeinstantaneous phase = ct + (t) rad eq

4

where: ct = carrier reference phase (radians)

= [ 2 (rad/cycle)] [ fc (cycles/sec)] [t (sec)]

fc = carrier frequency (Hz)

(t) = instantaneous phase deviation (radians)

3. Instantaneous Frequency Deviation- the instantaneous change in

the frequency of the carrier and is defined as the first time derivative of the instantaneous phase deviationinstantaneous frequency deviation = ’(t) rad/sec eq 4 ’(t) rad/sec cycles

2 rad/cycle sec“ ‘ “ = first derivative with respect to time

= = = Hz

4. Instantaneous Frequency- the precise frequency of the

carrier at a given instant of time and is defined as the first time derivative of the instantaneous phase.instantaneous frequency = i(t) = d/dt [ct + (t)] eq 6a

= c(t)+ ’(t) rad/sec eq 6b

instantaneous frequency = fit

i(t) = ( 2 rad/cycle) (fc cycles/sec) + ’(t)

2 fc + ’(t) rad/sec fc + ’(t) cycles

2 rad/cycle 2 secfc + ’(t) 2 eq 6c

= =

Hz=

Deviation Sensitivity Phase Modulation – can be defined as angle

modulation in which instantaneous phase deviation [(t)] is proportional to the amplitude of the modulating signal voltage and the instantaneous frequency deviation is proportional to the slope or first derivative of the modulating signal.

Frequency Modulation – angle modulation in which the instantaneous frequency deviation [’(t)] is proportional to the amplitude of the modulating signal and the instantaneous phase deviation is proportional to the integral of the modulating signal voltage.

For a modulating signal mt:

PM = (t) = K mt rad eq 7

FM = ’(t) = K1 mt rad/sec eq 8

where: K and K1 = constants

= deviation sensitivities of the phase and frequency

modulators, respectively

Deviation Sensitivities

- the output vs. input transfer functions for the modulators, which give the relationship between what output parameter changes in respect to specified changes in the input signal.Frequency Modulator : changes would occur in the output frequency in respect to changes in the amplitude of the input voltagePhase Modulator : changes would occur in the phase of the output frequency in respect to changes in the amplitude of the input voltage

Deviation SensitivityPhase Modulator:

K = rad/V ( /V)Frequency Modulator:

K1 = rad/sec /V or rad/V-sec (/ V)

Phase Modulation is the first integral of the frequency modulation.

PM = (t) = ’(t) dt = K1 m (t) dt

= K1 m(t) dt eq 9

Substituting a modulating signal m(t) = Vm cos (mt) into eq 1

m(t) = Vc cos [ct + (t)] = Vc cos [ct + KVm cos (mt)

eq 10

For FM:m(t)= Vc cos [ct + ’(t) ]

= Vc cos [ct + K1 m(t) dt]

= Vc cos [ct + K1 Vm cos (mt) dt]

= Vc cos [ct + K1 Vm/m sin (mt) ] eq 11

Equations for Phase and Frequency Modulated Carriers

FM & PM Waveforms

Unmodulated Carrier

Modulating Signal

Frequency -Modulated Wave

Phase-Modulated Wave

FM – the maximum frequency deviation (change in the carrier frequency) occurs during the maximum positive and negative peaks of the modulating signal.

PM – the maximum frequency deviation occurs during the zero crossings of the modulating signal.

FM & PM – the rate at which the frequency changes occur is equal to the modulating signal frequency.

Phase Deviation and Modulation Index general form :

m(t) = Vc cos [ct + m cos (mt) ] eq

12

where: m cos (mt) = instantaneous phase deviation, (t)

m = peak phase deviation in radians (phase- modulated carrier

= modulation index (index of modulation)

PM : m = proportional to the amplitude of the modulating signal, independent of its frequency m = KVm eq 13

where: m = modulation index and peak phase deviation (, rad) K = deviation sensitivity (radians/volt) Vm = peak modulating-signal amplitude (volts)

m = K (rad/volt) Vm (volts) = radians

PM equations:m(t) = Vc cos [ct + KVm cos (mt) ]

eq 14a

= Vc cos [ct + cos (mt) ] eq 14b

= Vc cos [ct + m cos (mt) ] eq 14c

FM: m = directly proportional to the amplitude of the modulating signal and inversely proportional to the frequency of the modulating signal.

m = K1 Vm/ m (unitless) eq 15

where: m = modulation index (unitless) K1 = deviation sensitivity (rad/V-sec)

Vm = peak modulating-signal amplitude (V)

m= radian frequency (radians/sec)

K1 (rad/volt-sec) Vm (volt)

m (radians/sec)

In Hertz: m = K1 Vm/ fm (unitless) eq 16

K1 (hertz/volt) Vm (volt)

fm (hertz)

m = = unitless

m = = unitless

Frequency Deviation- the change in frequency that

occurs in the carrier when it is acted on by a modulating-signal frequency.

- peak frequency shift (f) in hertzCarrier Swing – peak-to-peak frequency

deviation - (2f ) f = K1Vm (Hz) eq 17

thus, m = f (Hz) / fm (Hz) (unitless) eq 18

FM equations: m(t) = Vc cos [ct + K1Vm /fm sin (mt) ] eq 19a

m(t) = Vc cos [ct + f / fm sin (mt) ] eq 19b

m(t) = Vc cos [ct + m sin (mt) ] eq 19c

Angle-Modulation Summary

Modulation Index vs. Amplitude

Frequency Deviation vs. Modulating Frequency

Phase Deviation vs. Amplitude

Frequency Deviation vs. Amplitude

Percent Modulation- determined in a different manner than

it was with an amplitude-modulated wave- ratio of the frequency deviation

actually produced to the maximum frequency deviation allowed by law stated in percent form% modulation = f(actual) / f (max) x 100%