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Angle Modulation

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... a signal generator, and wiggling it back and forth ... wiggling parameters: How far you deviate from the center frequency (Df) How fast you wiggle ... – PowerPoint PPT presentation

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Title: Angle Modulation


1
Chapter 4
  • Angle Modulation

2
FM Illustration
The frequency of the carrier is varied around wc
in relation with the message signal. wi(t) wc
kf m(t)
3
Instantaneous Frequency
  • The argument of a cosine function represents an
    angle.
  • The angle could be constant cos(300), or
    varying with time, cos q(t)
  • The instantaneous angular frequency (in rad/sec)
    is the rate of change of the angle. That iswi
    (t) dq (t)/dt .
  • For cos(wc t f), wi (t) wc as expected.

4
Representation of Angle Modulation in Time Domain
  • For an FM signal wi (t) wc kf m(t)
  • For Phase Modulation (PM), the phase of the
    carrier is varied in relation to the message
    signal f(t) kp m(t)

5
Relation Between FM and PM
m(t)
gFM(t)
FM Modulator
m(t)
gPM(t)
PM Modulator
6
Which is Which?
7
FM and PM Modulation
  • kf 2p105 rad/sec/volt 105 Hz/Volt 105
    V-1sec-1
  • kp 10p rad/Volt 5 v-1
  • fc 100 MHz
  • FMfi fc kf m(t)
  • 108 -105 lt fi lt 108 105 99.9 lt fi lt 100.1
    MHZ
  • PM fi fc kp dm(t)/dt
  • 108 -105 lt fi lt 108 -105 99.9 lt fi lt 100.1
    MHZ
  • Power (FM or PM) A2/2

8
Spectrum of FM/PM
  • Unlike Amplitude Modulation, it is not
    straightforward to relate the spectrum of the
    FM/PM modulated signal to that of the modulating
    signal m(t). We can deal with it on a
    case-by-case basis.
  • We are, however, particularly interested in
    finding the bandwidth occupied by an FM/PM
    signal.
  • For that purpose, we will make some assumptions
    and work on simple modulating messages.
  • Because of the close relation between FM and PM,
    we will do the analysis for FM and extend it to
    PM.

9
What is NOT the bandwidth of FM!
  • One may tend to believe that since the modulated
    signal instantaneous frequency is varying between
    by Df around fc, then the bandwidth of the FM
    signal is 2Df. False!
  • In fact, the motivation behind introducing FM was
    to reduce the bandwidth compared to that of
    Amplitude Modulation, which turns out to be
    wrong.
  • What was missing from the picture of bandwidth?

10
FM Visualization
  • Think of holding the frequency knob of a signal
    generator, and wiggling it back and forth to
    modulate the carries in response to some message.
  • There are two wiggling parameters
  • How far you deviate from the center frequency
    (Df)
  • How fast you wiggle (related to Bm)
  • The rate of change of the instantaneous frequency
    was missing!

11
Carsons Rule
  • BFM 2(DfBm)where Df frequency deviation
    kf m(t)max
  • Bm bandwidth of m(t)
  • Define the deviation ratio b Df / Bm.BFM 2(b
    1) Bm
  • The same rule applies to PM bandwidth, BPM
    2(DfBm) 2(b 1) Bm where (Df )PM kp
    dm(t)/dtmax

12
Narrow Band and Wide Band FM
  • When Df ltlt Bm (or b ltlt1), the scheme is called
    Narrow Band (NBFM, NBPM).
  • BNBFM 2Bm (same for NBPM)
  • Therefore, no matter how small we make the
    deviation around fc , the bandwidth of the
    modulated signal does not get smaller than 2Bm.

13
Estimate BFM and BPM
  • kf 2p105 rad/sec/volt 105 Hz/Volt 105
    V-1sec-1
  • kp 5p rad/Volt 2.5 v-1
  • fc 1000 MHz
  • First estimate the Bm.Cn 8/p2n2 for n odd, 0 n
    evenThe 5th harmonic onward can be neglected.Bm
    15 kHz
  • For FMDf 100 kHz BFM 230 KHz
  • For PMDf 50 kHz BFM 130 KHz

14
Repeat if m(t) is Doubled
  • kf 2p105 rad/sec/volt 105 Hz/Volt 105
    V-1sec-1
  • kp 5p rad/Volt 2.5 v-1
  • fc 1000 MHz
  • For FMDf 200 kHz BFM 430 KHz
  • For PMDf 100 kHz BFM 230 KHz
  • Doubling the signal peak has significant effect
    on both FM and PM bandwidth

2
40,000
-40,000
-2
15
Repeat if the period of m(t) is Doubled
  • kf 2p105 rad/sec/volt 105 Hz/Volt 105
    V-1sec-1
  • kp 5p rad/Volt 2.5 v-1
  • fc 1000 MHz
  • Bm 7.5 kHz
  • For FMDf 100 kHz BFM 215 KHz
  • For PMDf 25 kHz BFM 65 KHz
  • Expanding the signal varies its spectrum. This
    has significant effect on PM.

4x10-4
10,000
-10,000
16
Spectrum of NBFM (1/2)
where
17
Spectrum of NBFM (2/2)
  • For NBFM, kf a(t)ltlt 1
  • Bandwidth of a(t) is equal to the bandwidth of
    m(t), Bm.
  • BNBFM 2 Bm (as expected).
  • Similarly for PM (kp m(t)ltlt 1 )
  • BNBPM 2 Bm

18
NBFM Modulator
19
NBPM Modulator
20
Immunity of FM to Non-linearities
21
Frequency Multipliers
22
Generation of WBFM Indirect Method
  • Usually, we are interested in generating an FM
    signal of certain bandwidth (or Df or b) and
    certain fc.
  • In the indirect method, we generate a NBFM with
    small b then use a frequency multiplier to scale
    b to the required value.
  • This way, fc will also be scaled by the same
    factor. We may need a frequency mixer to adjust
    fc.

23
Example From NBFM to WBFM
  • A NBFM modulator is modulating a message signal
    m(t) with bandwidth 5 kHz and is producing an FM
    signal with the following specifications
  • fc1 300 kHz, ?f1 35 Hz.
  • We would like to use this signal to generate a
    WBFM signal with the following specifications
  • fc2 135 MHz, ?f 2 77 kHz.

24
From NBFM to WBFM System 1

25
From NBFM to WBFM System 2

26
Generation of WBFM Direct Method
  • Has poor frequency stability. Requires feedback
    to stabilize it.

27
FM Demodulation Signal Differentiation
28
FM Demodulation Signal Differentiation
29
Frequency Discriminators
  • Any system with a transfer function of the form
    H(w) aw b over the band of the FM signal
    can be used for FM demodulation
  • The differentiator is just one example.

30
Slope Detectors (Demodulators)
31
Phased-Locked Loop (PLL)
  • The multiplier followed by the filter estimates
    the error bewteen the angle of gFM(t) and
    gVCO(t).
  • The error is fed to VCO to adjust the angle.
  • When the angles are locked, the output of the PLL
    would be following m(t) pattern.
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