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.