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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
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)
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)
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
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