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