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FM Demodulation Techniques
& PLL
Updated: 4/26/15

Sections: 4-11 to 4-15

Outline
• 

FM Demodulation Techniques

FM Demodulator Classification
•  Coherent & Non-coherent
–  A coherent detector has two inputs—one for a reference signal, such as the
synchronized oscillator signal, and one for the modulated signal that is to be
demodulated
...

–  Example: The envelope detector is an example of a noncoherent detector
...
deviation sensitivity

THE OUTPUT WILL BE:

DC Component can be blocked
by an AC coupled circuit

Frequency Discrimination - Discriminator
•  How the discriminator operates:
–  Generally, has a gain of KFD V/rad
–  In freq
...

–  It reacts to all amplitude changes (needs a limiter)
...


Transfer
Curve

Frequency Discrim
...
– Balanced Slope Detector
•  Uses two tuned circuits each set to a fixed frequency
–  f1 = 3ΔF + fc & f2 = 3ΔF - fc
90 Degree out of phase

After the
Limiter

K1 and K2 are constant
depending on values of the
series capacitors and
parallel resonant circuits

Balanced Slope Detector - Transfer Curve
Major Advantage: Larger Range
We still like to pull it to +/-δf !

Useful
Range

Phase Shift Discriminator – Quadrature Detector
• 
• 
• 

Very common in TV receivers
It uses a phase shift circuit
It converts the instantaneous frequency deviation in an FM signal to phase shift
and then detects the changes of phase
–  Cs results in -90 deg
...

•  Basic Idea: A negative feedback control system
•  Basic Components: PD, Loop Filter (LPF), VCO
•  Types: Analog / Digital
•  Operation: when it is locked it will track the input frequency: wout=win
Mixer

Basic Operation
•  as

Km

Kv

- Coherent demodulator
- Out of phase 90 deg
...
Vo(t)
Km is the gain of the multiplier

PLL Characteristics

http://www2
...
sfu
...
pdf

Analog PLL
When locked, that is when no phase error à exactly 90 deg
...
out of phase)

Phase detector constant gain V/rad

Vp = KmAiAo/2=Kd

Analog PLL

Locked in frequency

Analog PLL – Linear Model (Transfer Function)
Open loop transfer function:

Phase Detector

Phase Detector Gain

G(f) = Kv Kd F(f)/jw
G(f)
Vo(t)

VCO Gain

Analog PLL – Linear Model (Transfer Function)
Open loop transfer function:

Phase Detector

G(f) = Kv Kd F(f)/jw
Loop Gain: Kd Kv

G(f)

Thus:

Θin ( f ) − Θo ( f ) = Θe ( f )
G( f ) +1
Θo ( f ) = Θe ( f )⋅ G( f ) → Θin ( f ) = Θo ( f )
Loop Gain: Kd Kv
G( f )
Θo ( f )
G( f )
K d ⋅ K v ⋅ F( f ) / jω
K d ⋅ K v ⋅ F( f )
H( f ) =
=
=
=
Θi ( f ) G( f ) +1 1+ K d ⋅ K v ⋅ F( f ) / jω jω + K d ⋅ K v ⋅ F( f )
Remember: G(f) is Open loop transfer function

Analog PLL – Linear Model (Phase Error Function)
Θe ( f ) Θin ( f ) − Θo ( f )
Θo ( f )
He ( f ) =
=
= 1−
= 1− H ( f )
Θi ( f )
Θi ( f )
Θi ( f )

Phase Error
Transfer Function

jω
He ( f ) =
jω + K d ⋅ K v ⋅ F( f )
→ Θe ( f ) = H e ( f )⋅ Θi ( f )
What is the steady-state error?
We use Final Value Theorem of the Laplace Transform

Θe (∞) = lim s→0 sΘe (s);s = jω
s2
Θe (∞) = lim s→0 Θi (s)⋅
s + K d ⋅ K v ⋅ F(s)

Note that ideally we want this
to be zero – this has to do with
K and F(s) – loop filter
characteristics!
à Lets look at special cases!

Analog Loop Filter
•  There are e number of options for the loop filter
•  In the case of first-order PLL we assume F(s) = 1 (All-pass
filter)

Analog Loop Filter – First Order
•  We assume All-pass filter:
–  F(f) = 1àFirst Order PLL

H e ( f ) = 1− H ( f )
jω
He ( f ) =
jω + K d ⋅ K v
Kd ⋅ Kv
H( f ) =
jω + K d ⋅ K v

PLL Basic Operation

Analog Loop Filter – First Order
•  Example 1: Assume the loop is locked and we have a phase
step change
...
Calculate the SS phase error:
ωin (t) = ω c + Δω ⋅ u(t)→ θ in (t) = Δω ⋅ t
2

Θin ( f ) = Δω / ( jω ) ;s = jω

Note that the larger K
The smaller the error will be!

Θin (s) = Δω / (s)2
s2
Δω
Θe (∞) = lim s→o
Θin (s) =
s + Kd ⋅ Kv
Kd ⋅ Kv
Indicating a slight phase error!

Analog Loop Filter – First Order
How does the control voltage v2(t) change if the
frequency of the input signal changes?

ωin (t) = ω c + Δω ⋅ u(t)→ θ in (t) = Δω ⋅ t
2

Θin ( f ) = Δω / ( jω ) ;s = jω
Θin (s) = Δω / (s)2
v1 (t) = K d ⋅ vo (t)⋅ vin (t)
V1 ( f ) = K d ⋅ Θe ( f )
V1 ( f ) = K d ⋅ Θin ( f )⋅

jω
; F( f ) = 1
jω + K d ⋅ K v

V1 ( f ) = K d ⋅ Δω / ( jω )2 ⋅
V1 ( f ) =
v1 (t) =

jω
jω + K d ⋅ K v

K d ⋅ Δω
jω ( jω + K d ⋅ K v )

K d ⋅ Δω
(1− e−kt );k = K d ⋅ K v
k

V1(t)

Analog Loop Filter – First Order
How does the control voltage v2(t) change if the
frequency of the input signal changes?

ωin (t) = ω c + Δω ⋅ u(t)→ θ in (t) = Δω ⋅ t
2

Θin ( f ) = Δω / ( jω ) ;s = jω
Θin (s) = Δω / (s)2
v1 (t) = K d ⋅ vo (t)⋅ vin (t)
V1 ( f ) = K d ⋅ Θe ( f )
V1 ( f ) = K d ⋅ Θin ( f )⋅

jω
; F( f ) = 1
jω + K d ⋅ K v

V1 ( f ) = K d ⋅ Δω / ( jω )2 ⋅
V1 ( f ) =
v1 (t) =

jω
jω + K d ⋅ K v

K d ⋅ Δω
jω ( jω + K d ⋅ K v )

K d ⋅ Δω
(1− e−kt );k = K d ⋅ K v
k

V1(t)

Analog Loop Filter – First Order
Where is the demodulated signal if the input is an FM modulated signal?

s(t)= Ac cos(ω c t + θ in (t))

V1(t)

D
θ in (t) = D f ∫ m(τ )d τ ⇒ Θin (s) = f M (s)
s
K
Θ (s)
Θout (s) = V2 (s)⋅ v ⇒ V2 (s) = s ⋅ out
s
Kv
Θout (s) = Θin (s)H (s)
% Df
( s Df Kd Kv
V2 (s) = '
M (s)⋅ H (s)*
=
⋅
M (s)
& s
) Kv Kv s + Kv Kd
2π K f
ω 3−dB = K v K d >> 2π f ⇒ V2 (s) =
M (s)
Kv
v2 (t) =

2π K f
m(t)
Kv

Kv (Hz/V)
Frequency deviation sensitivity Kf (Hz/V);
Or Df (rad/V)

Analog Loop Filter – First Order- Example
Assume s(t) =cos( 1000pi + 50sin(20pi
...
5 V/rad
VCO gain constant Kv=1000pi rad/sec-volt
Answer the following questions:
V1(t)
1
...
 
3
...
 
5
...
 
7
...
 
9
...

Find the maximum freq
...

Frequency Deviation Sensitivity (Df in rad/V)
Calculate the total loop gain
...

Calculate the steady state phase error
...
t)) passing through a PLL
Phase detector gain Kd=0
...
 
2
...
 
4
...
 
6
...
 
8
...
 

What is the modulating frequency?
What is the carrier frequency?
What is the modulation Index
...
Deviation
...

What will be the expression for the modulating signal, m(t)?
Find v2(t)
...


ωin (t) = ω c + Δω ⋅ u(t)→ θ in (t) = Δω ⋅ t
Θin ( f ) = Δω / ( jω )2 ;s = jω
Θin (s) = Δω / (s)2

s(t)= Ac cos(1000π t + 50sin(20π t))
V2 (s) = D f

Kd
M (s)
s + Kv Kd

s2
Δω
2π ⋅10
Θe (∞) = lim s→o
Θin (s) =
=
= 0
...
3o
M (s) ω =20 π s + K v K d jω + 500π

→ 360(0
...
3deg

v2 (t) = m(t)@− 2
...
3o )

Applications of PLL
•  Used as demodulators (FM or AM demodulator)
–  AM coherent Detectors

•  Frequency synthesizer

Frequency Synthesizer Using PLL
The frequency of Vout is locked (synchronized) to the input frequency:

Classically, M and N are integers
...
Couch II, Digital and Analog Communication
Systems, 8th edition, Pearson / Prentice, Chapter 4
•  Contemporary Communication Systems, First Edition by M
F Mesiya– Chapter 5
• 

See
Notes

(http://highered
...
com/sites/0073380369/information_center_view0/)



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