Chapter 5
Angle Modulation
Updated: 4/6/15
Outline
• 
Angle Modulation
Review: Modulation Concept
•  Modulation is the process by which a message or
information-bearing signal is transformed into another signal
to facilitate transmission over a communication channel
–  Requires an auxiliary signal called carrier
–  The modulation process is performed to accomplish several
objectives
•  Modulation Objectives
–  Frequency translation
•  Designating various frequency spectrum for difference applications
–  Channelization
•  E.g., assigning difference channels for uploading and downloading
–  Practical Equipment Design
•  Antenna size (λf=c)
–  Noise Performance
•  Assigning higher BW to ensure higher noise performance
•  E.g., FM has 200-KHz channel BW compared to 10KHz for AM
Review: Bandpass Signal & AM Modulation
•  Remember for bandpass waveform we have
e
•  The voltage (or current) spectrum of the bandpass signal is
•  The PSD will be
•  In case of Ordinary AM (DSB – FC) modulation:
•  In this case Ac is the power level of the carrier signal with no
modulation;
•  Therefore:
Review: Voltage/Current Spectrum in AM
•  We know for AM:
•  The voltage or Current Spectrum will be
Amax – Amin
m=
Amax + Amin
Angle Modulation – Basic Concepts
Φi(t)
Definitions:
Phase
deviation sensitivity
(rad/V) (excess phase) - radian
•  θ(t) is the
instantaneous
phase deviation
•  θ’(t) is the instantaneous frequency deviation – radian/sec
•  Φi(t)=ωct + θ(t) is the instantaneous phase (exact) - radian
•  fi(t)=(1/2p)dΦ
Freq. deviation
sensitivity
in
rad/sec
ι(t)/dt = d(ω
ct + θ(t))/dt
•  This is the instantaneous frequency (exact) – radian/sec
à Note that θ’(t) Referred as the instantaneous frequency deviation
Angle Modulation Representation
Constant called Phase deviation sensitivity (rad/V)
Constant called Freq. deviation sensitivity in ((rad/sec)/V)
In PM: θ(t) is proportional to m(t)
à θ(t) = Dp . m(t)
àθ’(t) = Dp . d [m(t)] / dt
à Max. Instant. Frequency Deviation at Zero Crossing!
In FM: θ’(t) is proportional to m(t) à θ’(t) = Df . m(t)
à Max. Instant. Frequency Deviation at max[m(t)]
or Df/2π = Hz/V
Frequency VS Phase Modulation
Frequency
Modulation
θ’(t) = Df . m(t)
Phase
Modulation
θ’(t) = Dp . d [m(t)] / dt
Frequency VS Phase Modulation
Max. Instant. Frequency
Deviation at max[m(t)]
Max. Instant. Frequency
Deviation at Zero Crossing
Frequency
Modulation
θ’(t) = Df . m(t)
Phase
Modulation
θ’(t) = Dp . d [m(t)] / dt
Generation of FM from PM & Vice Versa
Frequency Deviation
•  In general
Frequency deviation from
the carrier frequency
–  For FM
–  Thus, in case of FM
The instantaneous freq.
varies about carrier
freq. proportional to m(t)
–  For PM
–  Thus, in case of PM
,
p
Derivative
of m(t)
Maximum Frequency Deviation
Angle Modulation Using MATLAB
Assuming the Modulating Signal is Sinusoid
In general (Vp=Vm):
s(t) = Vc cos(ω c t + θ (t))
sPM (t) = Vc cos(ω c t + Dp m(t))
sFM (t) = Vc cos(ω c t +
∫ D m(τ )dτ )
f
If the modulating signal is sinusoid:
m(t) = Vm cos(ω m t)
sPM (t) = Vc cos(ω c t + DpVm cos(ω m t))
D f Vm
sFM (t) = Vc cos(ω c t +
sin(ω m t))
ωm
The modulation index can be defined as (pay attention to units):
sPM (t) = Vc cos(ω c t + m p cos(ω m t));→ m p = β p = Dp max[m(t)] = DpVm
D f Vm 1 ΔF
sFM (t) = Vc cos(ω c t + m f sin(ω m t));→ m f = β f =
. =
2 π fm
B
Note that the Peak Phase Deviation is
the same as modulation index in PM
Peak Freq. Deviation=ΔF
BW of m(t)
Assuming the Modulating Signal is Sinusoid
m p = β p = Dp max[m(t)] = DpVm
mf = β f =
D f Vm 1 ΔF
. =
2 π fm
B
Notes:
•  Vm is proportional to ΔF (peak frequency
deviation)
•  Vm is proportional to Β (bandwidth of the
modulating signal)
•  Vm directly impacts the BW but no impact on the
total signal spectral power –
•  This is difference from AM!
•  Then what is the spectral impact of Vm? à
If impact the individual spectral lines!
Note K = Dp & K1=Df ;Vm = max [m(t)]=max [Vm(t)] = Modulating Signal
Example (C0)
•  Assume Df = 10π (rad/sec/V); Dp=π/2 rad/V, fc=10Hz,
fm=1Hz, Vc=1Volt.
–  Determine XFM(t) and plot it
–  Determine XPM(t) and plot it
Example (C0) - Answer
•  Assume Df = 10π (rad/sec/V); Dp=π/2 rad/V, fc=10Hz,
fm=1Hz, Vc=1Volt.
–  Determine XFM(t) and plot it
–  Determine XPM(t) and plot it
Transitions
See
Notes
Example (C)
•  Assume Df = 5KHz/V and m(t) = 2cos(2π.2000t)
–  Determine the peak frequency for FM
–  Determine the modulation index for FM
–  If Dp=2.5 rad/V, determine the peak phase deviation
sPM (t) = Vc cos(ω c t + m p cos(ω m t));→ m p = β p = Dp max[m(t)] = DpVm
D f Vm 1 ΔF
sFM (t) = Vc cos(ω c t + m f sin(ω m t));→ m f = β f =
. =
2 π fm
B
Summary
=Df
=Dp
Note K = D = Sensitivity; Vm = max [m(t)]=max [Vm(t)] = Modulating Signal
m modulation index; ΔF=Δf;
Spectra of Angle-Modulated Signals
s(t) = Vc cos(ω c t + θ (t))
sPM (t) = Vc cos(ω c t + Dp m(t))
sFM (t) = Vc cos(ω c t +
∫ D m(τ )dτ )
f
Example: Spectrum of a PM or FM Signal with Sinusoidal
Modulation
So, what is the expression for angle modulation in frequency domain (assume
m(t) is sinusoidal:
For PM:
For FM:
Complex envelope:
Using Fourier Series
Note:
wmt= θ
dθ = wmdt
dt=dθ/wm
Change Limits:
Tm/2àπ/-π
Jn (β) is Bessel function of the first kind of the nth order;
Cannot be evaluated in closed form, but it has been evaluated numerically
Bessel Function
Carson’s Rule: shown that 98% of the total
power is contained in the bandwidth
(sometimes we use 99% rule)
Zero crossing points;
Used to determine the modulation index
Bessel Function for Angle Modulation
•  In general the modulated signal (s(t)) is
S(t)
•  The Bessel Function:
Bessel Function for Angle Modulation
S(t)
S(t)
Example (A)
•  Assume FM modulation with modulation index of 1
•  m(t) =Vmsin(2.pi.1000t) and Vc(t)= =10sin(2.pi.500.103t)
•  Find the following:
–  Number of sets of significant side frequencies (G(f))
–  Amplitude of freq. components
–  Draw the frequency component
Example (B)
•  Plot the spectrum from the modulated FM
signal for β=0.5, 1, 2
Normalized
β=0.5
Bessel Function Using MATLAB
Narrowband Angle Modulation
Note: m=|θ(t)|
NBPM / NBFM & WB Angle Modulation
Wideband Angle Modulation
Frequency-division multiplexing (FDM)
Stereo FM Modulator
Stereo FM De-Modulator
References
•  Leon W. Couch II, Digital and Analog Communication
Systems, 8th edition, Pearson / Prentice, Chapter 5
•  Electronic Communications System: Fundamentals Through
Advanced, Fifth Edition by Wayne Tomasi – Chapter 7
(https://www.goodreads.com/book/show/209442.Electronic_Communications_System)
See
Notes