angle modulation transmission
TRANSCRIPT
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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
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Angle Modulation vs. Amplitude Modulation Advantages:
- Noise Immunity- Noise Performance and Signal-to-
Noise Improvement- Capture Effect- Power Utilization and Efficiency
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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
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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)
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- 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)
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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.
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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
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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
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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
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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.
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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
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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)
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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
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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
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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=
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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.
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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
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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
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Deviation SensitivityPhase Modulator:
K = rad/V ( /V)Frequency Modulator:
K1 = rad/sec /V or rad/V-sec (/ V)
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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
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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
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Equations for Phase and Frequency Modulated Carriers
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FM & PM Waveforms
Unmodulated Carrier
Modulating Signal
Frequency -Modulated Wave
Phase-Modulated Wave
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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.
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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)
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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)
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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
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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)
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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
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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
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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
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Angle-Modulation Summary
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Modulation Index vs. Amplitude
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Frequency Deviation vs. Modulating Frequency
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Phase Deviation vs. Amplitude
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Frequency Deviation vs. Amplitude
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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%