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

•  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:

e

Review: Voltage/Current Spectrum in AM •  We know for AM: •  The voltage or Current Spectrum will be

Amax – Amin Amax + Amin

m =

Angle Modulation – Basic Concepts

à Note that θ’(t) Referred as the instantaneous frequency deviation

Phase deviation sensitivity (rad/V)

Freq. deviation sensitivity in rad/sec

Definitions: •  θ(t) is the instantaneous phase deviation (excess phase) - radian •  θ’(t) is the instantaneous frequency deviation – radian/sec •  Φi(t)=ωct + θ(t) is the instantaneous phase (exact) - radian •  fi(t)=(1/2p)dΦι(t)/dt = d(ωct + θ(t))/dt

•  This is the instantaneous frequency (exact) – radian/sec

Φi(t)

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) àθ’(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

Phase Modulation

θ’(t) = Dp . d [m(t)] / dt

θ’(t) = Df . m(t)

Frequency VS Phase Modulation

Frequency Modulation

Phase Modulation

θ’(t) = Dp . d [m(t)] / dt

θ’(t) = Df . m(t)

Max. Instant. Frequency Deviation at max[m(t)]

Max. Instant. Frequency Deviation at Zero Crossing

Generation of FM from PM & Vice Versa

Frequency Deviation •  In general

–  For FM

–  Thus, in case of FM

–  For PM

–  Thus, in case of PM p

,

The instantaneous freq. varies about carrier freq. proportional to m(t)

Frequency deviation from the carrier frequency

Derivative of m(t)

Maximum Frequency Deviation

Angle Modulation Using MATLAB

Assuming the Modulating Signal is Sinusoid s(t) =Vc cos(ωct +θ(t))sPM (t) =Vc cos(ωct +Dpm(t))

sFM (t) =Vc cos(ωct + Dfm(τ )d∫ τ )

m(t) =Vm cos(ωmt)sPM (t) =Vc cos(ωct +DpVm cos(ωmt))

sFM (t) =Vc cos(ωct +DfVmωm

sin(ωmt))

In general (Vp=Vm):

If the modulating signal is sinusoid:

sPM (t) =Vc cos(ωct +mp cos(ωmt));→mp = βp = Dpmax[m(t)]= DpVm

sFM (t) =Vc cos(ωct +mf sin(ωmt));→mf = β f =DfVm2π

. 1fm=ΔFB

The modulation index can be defined as (pay attention to units):

Peak Freq. Deviation=ΔF BW of m(t) Note that the Peak Phase Deviation is the same as modulation index in PM

Assuming the Modulating Signal is Sinusoid

Note K = Dp & K1=Df ;Vm = max [m(t)]=max [Vm(t)] = Modulating Signal

mp = βp = Dpmax[m(t)]= DpVm

mf = β f =DfVm2π

. 1fm=ΔFB

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!

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(ωct +mp cos(ωmt));→mp = βp = Dpmax[m(t)]= DpVm

sFM (t) =Vc cos(ωct +mf sin(ωmt));→mf = β f =DfVm2π

. 1fm=ΔFB

Summary

Note K = D = Sensitivity; Vm = max [m(t)]=max [Vm(t)] = Modulating Signal m modulation index; ΔF=Δf;

=Df =Dp

Spectra of Angle-Modulated Signals

s(t) =Vc cos(ωct +θ(t))sPM (t) =Vc cos(ωct +Dpm(t))

sFM (t) =Vc cos(ωct + Dfm(τ )d∫ τ )

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 FM: For PM:

Complex envelope: Using Fourier Series

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

Note: wmt= θ dθ = wmdt dt=dθ/wm Change Limits: Tm/2àπ/-π

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

•  The Bessel Function:

S(t)

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

β=0.5

Normalized

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