FM Demodulation Techniques & PLL Updated: 4/26/15 Sections: 4-11 to 4-15
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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.
– A noncoherent detector has only one input, namely, the modulated signal port.
– Example: The envelope detector is an example of a noncoherent detector. • Demodulator Classification
– Frequency Discrimination • Noncoherent demodulator • FMàAMàEDàm(t)
– Phase Shift Discrimination • Noncoherent demodulator • FMàPMàm(t)
– Phase-Locked Loop (PLL) Detector • Coherent demodulator • Superior performance; complex and expensive
Let’s look at each!
Frequency Discrimination • Components
– Bandpass Limiter: Consists of Hard Limiter & BP Filter – Discriminator (frequency discriminator gain: KFD V/rad - assume unity) – Envelope Detector
THE OUTPUT WILL BE:
DC Component can be blocked by an AC coupled circuit
Note: Df=Kf Freq. deviation sensitivity
Frequency Discrimination - Discriminator • How the discriminator operates:
– Generally, has a gain of KFD V/rad – In freq. domain: H(w) = jw KFD – In time domain: v2(t) d[v1(t)]/dt
Frequency Discrimination
FM Wave
Output of Tuned Circuit (discriminator)
Frequency Discrimination – Slope Detector • In practice the differentiator can be approximated by a
slope detector that has a linear frequency-to-amplitude transfer characteristic over the bandwidth BW –One drawback is that it is narrow band
Tuned Circuit
BT is Carson’s BW
Transfer Curve
Output
Slope Detector Transfer Characteristics
Frequency Discrimination – Slope Detector
Frequency Discrimination – Slope Detector • Major Limitations:
– It is inefficient, as it is linear in very limited frequency range. – It reacts to all amplitude changes (needs a limiter). – It is relatively difficult to tune, as tuned circuit must be tuned to
different frequency than carrier frequency.
Transfer Curve
Frequency Discrim. – Balanced Slope Detector Envelope Detector
• Also called balanced discriminator • Uses two tuned circuits each set to a
fixed frequency – f1 = 3ΔF + fc & f2 = 3ΔF – fc
• The center-tapped transformer feeds the tuned circuits
– Tuned circuits are 180 degrees out of phase
• When fi>fcà Then output of T’(+Ve) > output of T’’ (-Ve) à max voltage across D1 (net voltage positive)
• When fi<fcà Then output of T’(+Ve) < output of T’’ (-Ve) à max voltage across D2 (net voltage negative)
• When f=fcà voltage across D1=D2 (the net voltage will be zero)
D1
D2
T’: fc+ΔF
T’’: Fc-ΔF
Frequency Discrim. – Balanced Slope Detector • Uses two tuned circuits each set to a fixed frequency
– f1 = 3ΔF + fc & f2 = 3ΔF - fc
After the Limiter
K1 and K2 are constant depending on values of the series capacitors and parallel resonant circuits
90 Degree out of phase
Balanced Slope Detector - Transfer Curve
Useful Range
Major Advantage: Larger Range We still like to pull it to +/-δf !
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. Shift – The tuned circuità additional phase shift proportional to instantaneous frequency
deviation from fc
Another approach
Balanced zero-crossing FM detector • This is a hybrid circuit
– Analog and digital combination
IF fi > fc à Tc>Ti Qdc > Qdc à Vout > 0
IF fi < fc à Tc<Ti Qdc < Qdc à Vout < 0
Linear frequency-to-voltage Characteristic: C[fi(t) – fc]
For the case of FM: fi(t) = (1/2p)Df m(t)
Free-running fc PW changes
Phase-Locked Loops • Applications: Frequency synthesizer, TV, Demodulators, clock recovery
circuits, multipliers, etc. • 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
V1(t) = Km Vin(t).Vo(t) Km is the gain of the multiplier
Km
Km
Kv
- Coherent demodulator - Out of phase 90 deg.
V1(t)
Vo(t)
Vin(t)
PLL Characteristics
http://www2.ensc.sfu.ca/people/faculty/ho/ENSC327/Pre_13_PLL.pdf
Analog PLL
Vp = KmAiAo/2=Kd
When locked, that is when no phase error à exactly 90 deg. Diff (90 deg. out of phase)
Phase detector constant gain V/rad
Analog PLL
Locked in frequency
Analog PLL – Linear Model (Transfer Function) Open loop transfer function: G(f) = Kv Kd F(f)/jw
G(f)
Vo(t)
VCO Gain
Phase Detector Gain Phase Detector
Analog PLL – Linear Model (Transfer Function) Open loop transfer function: G(f) = Kv Kd F(f)/jw Thus:
G(f)
Θin ( f )−Θo( f ) =Θe( f )
Θo( f ) =Θe( f ) ⋅G( f )→Θin ( f ) =Θo( f )G( f )+1G( f )
H ( f ) = Θo( f )Θi ( f )
=G( f )G( f )+1
=Kd ⋅Kv ⋅F( f ) / jω1+Kd ⋅Kv ⋅F( f ) / jω
=Kd ⋅Kv ⋅F( f )
jω +Kd ⋅Kv ⋅F( f )
Loop Gain: Kd Kv
Phase Detector
Remember: G(f) is Open loop transfer function
Loop Gain: Kd Kv
Analog PLL – Linear Model (Phase Error Function)
He( f ) =Θe( f )Θi ( f )
=Θin ( f )−Θo( f )
Θi ( f )=1− Θo( f )
Θi ( f )=1−H ( f )
He( f ) =jω
jω +Kd ⋅Kv ⋅F( f )→Θe( f ) = He( f ) ⋅Θi ( f )
What is the steady-state error? We use Final Value Theorem of the Laplace Transform
Θe(∞) = lims→0 sΘe(s);s = jω
Θe(∞) = lims→0Θi (s) ⋅s2
s+Kd ⋅Kv ⋅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!
Phase Error Transfer Function
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) = 1àFirst Order PLL He( f ) =1−H ( f )
He( f ) =jω
jω +Kd ⋅Kv
H ( f ) = Kd ⋅Kv
jω +Kd ⋅Kv
PLL Basic Operation
Analog Loop Filter – First Order • Example 1: Assume the loop is locked and we have a phase
step change. Calculate the steady-state phase error:
• Example 2: Assume the loop is locked and we have a frequency step change. Calculate the SS phase error:
θin (t) = Δθ ⋅u(t)→Θin (s) = Δθ / s
Θe(∞) = lims→os ⋅ Δθ
s+Kd ⋅Kv
= 0
ωin (t) =ωc +Δω ⋅u(t)→θin (t) = Δω ⋅ tΘin ( f ) = Δω / ( jω)
2;s = jωΘin (s) = Δω / (s)
2
Θe(∞) = lims→os2
s+Kd ⋅Kv
Θin (s) =Δω
Kd ⋅Kv
Note that the larger K The smaller the error will be!
Θe(∞) = lims→0 sΘe(s);s = jω
Θe(∞) = lims→0Θi (s) ⋅s2
s+Kd ⋅Kv ⋅F(s)
Remember:
Indicating no phase error!
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) =ωc +Δω ⋅u(t)→θin (t) = Δω ⋅ tΘin ( f ) = Δω / ( jω)
2;s = jωΘin (s) = Δω / (s)
2
v1(t) = Kd ⋅ vo(t) ⋅ vin (t)V1( f ) = Kd ⋅Θe( f )
V1( f ) = Kd ⋅Θin ( f ) ⋅jω
jω +Kd ⋅Kv
;F( f ) =1
V1( f ) = Kd ⋅ Δω / ( jω)2 ⋅
jωjω +Kd ⋅Kv
V1( f ) =Kd ⋅ Δω
jω( jω +Kd ⋅Kv )
v1(t) =Kd ⋅ Δω
k(1− e−kt );k = Kd ⋅Kv
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) =ωc +Δω ⋅u(t)→θin (t) = Δω ⋅ tΘin ( f ) = Δω / ( jω)
2;s = jωΘin (s) = Δω / (s)
2
v1(t) = Kd ⋅ vo(t) ⋅ vin (t)V1( f ) = Kd ⋅Θe( f )
V1( f ) = Kd ⋅Θin ( f ) ⋅jω
jω +Kd ⋅Kv
;F( f ) =1
V1( f ) = Kd ⋅ Δω / ( jω)2 ⋅
jωjω +Kd ⋅Kv
V1( f ) =Kd ⋅ Δω
jω( jω +Kd ⋅Kv )
v1(t) =Kd ⋅ Δω
k(1− e−kt );k = Kd ⋅Kv
V1(t)
Analog Loop Filter – First Order Where is the demodulated signal if the input is an FM modulated signal?
s(t)= Ac cos(ωct +θin (t))
θin (t) = Df m(τ )dτ∫ ⇒Θin (s) =Df
sM (s)
Θout (s) =V2 (s) ⋅Kv
s⇒V2 (s) = s ⋅
Θout (s)Kv
Θout (s) =Θin (s)H (s)
V2 (s) =Df
sM (s) ⋅H (s)
%
&'
(
)*sKv
=Df
Kv
⋅KdKv
s+KvKd
M (s)
ω3−dB = KvKd >> 2π f ⇒V2 (s) =2πK f
Kv
M (s)
v2 (t) =2πK f
Kv
m(t)
V1(t)
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.t)) passing through a PLL Phase detector gain Kd=0.5 V/rad VCO gain constant Kv=1000pi rad/sec-volt Answer the following questions: 1. What is the modulating frequency? 2. What is the carrier frequency? 3. What is the modulation Index. 4. Find the maximum freq. Deviation. 5. Frequency Deviation Sensitivity (Df in rad/V) 6. Calculate the total loop gain. 7. What will be the expression for the modulating signal, m(t)? 8. Find v2(t). 9. Calculate the steady state phase error.
V1(t)
Analog Loop Filter – First Order- Example Assume s(t) =cos( 1000pi + 50sin(20pi.t)) passing through a PLL Phase detector gain Kd=0.5 V/rad VCO gain constant Kv=1000pi rad/sec-volt Answer the following questions: 1. What is the modulating frequency? 2. What is the carrier frequency? 3. What is the modulation Index. 4. Find the maximum freq. Deviation. 5. Frequency Deviation Sensitivity (Df in rad/V) 6. Calculate the total loop gain. 7. What will be the expression for the modulating signal, m(t)? 8. Find v2(t). 9. Calculate the steady state phase error.
V1(t)
s(t)= Ac cos(1000π t + 50sin(20π t))
V2 (s) = DfKd
s+KvKd
M (s)
V2 (s)M (s) ω=20π
=2πK fKd
s+KvKd
=500π
jω + 500π=1@− 2.3o
v2 (t) =m(t)@− 2.3o = cos(20π t − 2.3o )
ωin (t) =ωc +Δω ⋅u(t)→θin (t) = Δω ⋅ tΘin ( f ) = Δω / ( jω)
2;s = jωΘin (s) = Δω / (s)
2
Θe(∞) = lims→os2
s+Kd ⋅Kv
Θin (s) =Δω
Kd ⋅Kv
=2π ⋅10500π
= 0.04
→ 360(0.04) / 2π = 2.3deg
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. Fractional-N technique can be applied to make N non-integer
References • Leon W. Couch II, Digital and Analog Communication
Systems, 8th edition, Pearson / Prentice, Chapter 4 • Contemporary Communication Systems, First Edition by M
F Mesiya– Chapter 5 • (http://highered.mcgraw-hill.com/sites/0073380369/information_center_view0/)
See Notes
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