Costas Receiver and Quadrature-Carrier Multiplexing · Costas Receiver and Quadrature-Carrier Multiplexing Professor Deepa Kundur University of Toronto Professor Deepa Kundur (University

Costas Receiver and Quadrature-Carrier

Multiplexing

Professor Deepa Kundur

University of Toronto

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 1 / 40

Costas Receiver and Quadrature-Carrier

Multiplexing

Reference:

Sections 3.4 and 3.5 of

S. Haykin and M. Moher, Introduction to Analog & DigitalCommunications, 2nd ed., John Wiley & Sons, Inc., 2007. ISBN-13978-0-471-43222-7.

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 2 / 40

3.4 Costas Receiver

Costas Receiver

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 3 / 40

3.4 Costas Receiver

Coherent Demodulation Output

v0(t) =1

2AcA

′c cos(φ)m(t)

I To maximize v0(t), would like φ ≈ 0.

0

1

-1

I For v0(t) to be proportional to m(t), φ should be constant.

Thus, we wish to synchronize the local oscillator.Let A′c = 1 for simplicity.

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 4 / 40

3.4 Costas Receiver

Costas Receiver

-90 degreePhase Shifter

Voltage-controlledOscillator

ProductModulator

Low-passFilter

PhaseDiscriminator

ProductModulator

Low-pass�lter

DSB-SC wave

DemodulatedSignal

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 5 / 40

3.4 Costas Receiver

Costas Receiver: Coherent Demodulation

-90 degreePhase Shifter

Voltage-controlledOscillator

ProductModulator

Low-passFilter

PhaseDiscriminator

ProductModulator

Low-pass�lter

DSB-SC wave

DemodulatedSignal

Coherent Demodulation

v (t)0local oscillator output

Goals: (1) Coherent demodulation of DSB-SC input signal.

(2) Tweak the local oscillator phase such that φ = 0.Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 6 / 40

3.4 Costas Receiver

Costas Receiver: Phase Lock Circuit

-90 degreePhase Shifter

Voltage-controlledOscillator

ProductModulator

Low-passFilter

PhaseDiscriminator

ProductModulator

Low-pass�lter

DSB-SC wave

DemodulatedSignal

Coherent Demodulation

Circuit for Phase Locking

v (t)0local oscillator output

Goals: (1) Coherent demodulation of DSB-SC input signal.

(2) Tweak the local oscillator phase such that φ = 0.Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 7 / 40

3.4 Costas Receiver

Costas Receiver: Phase Lock Circuitφ > 0: Freq of local oscillator needs to temporarily decrease

0

t

0

t

local oscillatorcarrier

local oscillator phase is ahead of carrier and must decrease its frequency to wait for carrier

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 8 / 40

3.4 Costas Receiver

Costas Receiver: Phase Lock Circuitφ < 0: Freq of local oscillator needs to temporarily increase

0

t

0

t

local oscillatorcarrier

local oscillator phase lags behind carrier and must increase its frequency to catch up

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 9 / 40

3.4 Costas Receiver

Costas Receiver: In-Phase and Quadrature-Phase

-90 degreePhase Shifter

Voltage-controlledOscillator

ProductModulator

Low-passFilter

PhaseDiscriminator

ProductModulator

Low-pass�lter

DSB-SC wave

DemodulatedSignal

Q-Channel (quadrature-phase coherent detector)

I-Channel (in-phase coherent detector)

v (t)I

v (t)Q

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 10 / 40

3.4 Costas Receiver

Costas Receiver: In-Phase Coherent Detector

-90 degreePhase Shifter

Voltage-controlledOscillator

ProductModulator

Low-passFilter

PhaseDiscriminator

ProductModulator

Low-pass�lter

DSB-SC wave

DemodulatedSignalv (t)I

v (t)Q

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 11 / 40

3.4 Costas Receiver

Costas Receiver: In-Phase Coherent Detector

s(t) = Ac cos(2πfct)m(t)

vI (t) = s(t) · cos(2πfct + φ)

Recall

cos(2πfct + φ) =e jφ

2e j2πfc t +

e−jφ

2e−j2πfc t

e jφ

2δ(f − fc) +

e−jφ

2δ(f + fc)

VI (f ) = S(f ) ?

[e jφ

2δ(f − fc) +

e−jφ

2δ(f + fc)

]=

e jφ

2S(f ) ? δ(f − fc) +

e−jφ

2S(f ) ? δ(f + fc)

=e jφ

2S(f − fc) +

e−jφ

2S(f + fc)

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 12 / 40

3.4 Costas Receiver

Costas Receiver: In-Phase Coherent Detector

VI (f ) =e jφ

2S(f − fc) +

e−jφ

2S(f + fc)

=cos(φ) + j sin(φ)

2S(f − fc) +

cos(−φ) + j sin(−φ)2

S(f + fc)

=cos(φ) + j sin(φ)

2S(f − fc) +

cos(φ) − j sin(φ)2

S(f + fc)

=cos(φ)

2︸ ︷︷ ︸≈1/2

[S(f − fc) + S(f + fc)] + jsin(φ)

2︸ ︷︷ ︸small

[S(f − fc) − S(f + fc)]

for φ� 1.

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 13 / 40

3.4 Costas Receiver

Costas Receiver: In-Phase Coherent Detector

repetitions

repetitions

repetitions

baseband componentsadd coherently

repetitions

repetitions

repetitions

baseband componentscancel out

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 14 / 40

3.4 Costas Receiver

Costas Receiver: In-Phase Coherent Detector

baseband componentsadd coherently

baseband componentscancel out

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 15 / 40

3.4 Costas Receiver

Costas Receiver: In-Phase Coherent Detector

VI (f ) =e jφ

2S(f − fc) +

e−jφ

2S(f + fc)

=cos(φ) + j sin(φ)

2S(f − fc) +

cos(−φ) + j sin(−φ)2

S(f + fc)

=cos(φ) + j sin(φ)

2S(f − fc) +

cos(φ) − j sin(φ)2

S(f + fc)

=cos(φ)

2︸ ︷︷ ︸≈1/2

S(f − fc) + S(f + fc)︸ ︷︷ ︸significant baseband

+ j sin(φ)2︸ ︷︷ ︸small

S(f − fc) − S(f + fc)︸ ︷︷ ︸negligible baseband

for φ� 1.

The closer φ is to zero, the more significant the baseband term ofvI (t) and vice versa.

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 16 / 40

3.4 Costas Receiver

Costas Receiver: In-Phase Coherent Detector

For φ small.Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 17 / 40

3.4 Costas Receiver

Costas Receiver: Quadrature-Phase Detector

-90 degreePhase Shifter

Voltage-controlledOscillator

ProductModulator

Low-passFilter

PhaseDiscriminator

ProductModulator

Low-pass�lter

DSB-SC wave

DemodulatedSignalv (t)I

v (t)Q

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 18 / 40

3.4 Costas Receiver

Costas Receiver: Quadrature-Phase Detector

s(t) = Ac cos(2πfct)m(t)

vQ(t) = s(t) · sin(2πfct + φ)

Recall

sin(2πfct + φ) =e jφ

2je j2πfc t − e

−jφ

2je−j2πfc t

e jφ

2jδ(f − fc) −

e−jφ

2jδ(f + fc)

VQ(f ) = S(f ) ?

[e jφ

2jδ(f − fc) −

e−jφ

2jδ(f + fc)

]=

e jφ

2jS(f ) ? δ(f − fc) −

e−jφ

2jS(f ) ? δ(f + fc)

=e jφ

2jS(f − fc) −

e−jφ

2jS(f + fc)

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 19 / 40

3.4 Costas Receiver

Costas Receiver: Quadrature-Phase Detector

VQ(f ) =e jφ

2jS(f − fc)−

e−jφ

2jS(f + fc)

=cos(φ) + j sin(φ)

2jS(f − fc)−

cos(−φ) + j sin(−φ)2j

S(f + fc)

=cos(φ) + j sin(φ)

2jS(f − fc)−

cos(φ) − j sin(φ)2j

S(f + fc)

=cos(φ)

2j︸ ︷︷ ︸≈1/2j

[S(f − fc)−S(f + fc)] + jsin(φ)

2j︸ ︷︷ ︸small

[S(f − fc)+S(f + fc)]

for φ� 1.

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 20 / 40

3.4 Costas Receiver

Costas Receiver: Quadrature-Phase Detector

baseband componentscancel out

baseband componentsadd coherently

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 21 / 40

3.4 Costas Receiver

Costas Receiver: Quadrature-Phase Detector

VQ(f ) =e jφ

2jS(f − fc)−

e−jφ

2jS(f + fc)

=cos(φ) + j sin(φ)

2jS(f − fc)−

cos(−φ) + j sin(−φ)2j

S(f + fc)

=cos(φ) + j sin(φ)

2jS(f − fc)−

cos(φ) − j sin(φ)2j

S(f + fc)

=cos(φ)

2j︸ ︷︷ ︸≈1/2j

S(f − fc)−S(f + fc)︸ ︷︷ ︸negligible baseband

+ j sin(φ)2j︸ ︷︷ ︸small

S(f − fc)+S(f + fc)︸ ︷︷ ︸significant baseband

for φ� 1.

The closer φ is to zero, the more negligible the baseband term ofvQ(t) and vice versa.

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 22 / 40

3.4 Costas Receiver

Costas Receiver: Quadrature-Phase Detector

For φ small.Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 23 / 40

3.4 Costas Receiver

Costas Receiver: vI (t) and vQ(t)

-90 degreePhase Shifter

Voltage-controlledOscillator

ProductModulator

Low-passFilter

PhaseDiscriminator

ProductModulator

Low-pass�lter

DSB-SC wave

DemodulatedSignal

spectrum

spectrum

spectrum

v (t)0v (t)I

v (t)Q

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 24 / 40

3.4 Costas Receiver

Costas Receiver: vI (t) and vQ(t)

-90 degreePhase Shifter

Voltage-controlledOscillator

ProductModulator

Low-passFilter

PhaseDiscriminator

ProductModulator

Low-pass�lter

DSB-SC wave

DemodulatedSignal

spectrum

spectrum

spectrum

v (t)I

v (t)Q

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 25 / 40

3.4 Costas Receiver

Costas Receiver: vI (t) and vQ(t)

-90 degreePhase Shifter

Voltage-controlledOscillator

ProductModulator

Low-passFilter

PhaseDiscriminator

ProductModulator

Low-pass�lter

DSB-SC wave

DemodulatedSignal

spectrum

spectrum

spectrum

v (t)I

v (t)Q

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 26 / 40

3.4 Costas Receiver

Costas Receiver: Low-Pass Filter

ProductModulator

-90 degreePhase Shifter

Voltage-controlledOscillator

ProductModulator

Low-passFilter

PhaseDiscriminator

Low-pass�lter

DSB-SC wave

DemodulatedSignal

spectrum

spectrum

spectrum

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 27 / 40

3.4 Costas Receiver

Costas Receiver: Local Oscillator Control

-90 degreePhase Shifter

Voltage-controlledOscillator

ProductModulator

Low-passFilter

PhaseDiscriminator

ProductModulator

Low-pass�lter

DSB-SC wave

DemodulatedSignal

g(t)

v (t)0

v’ (t)0

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 28 / 40

3.4 Costas Receiver

Costas Receiver: Phase Discriminator

Two components in sequence:

(1) multiplier

v0(t) · v ′0(t) =Ac2

cos(φ)m(t) · Ac2

sin(φ)m(t)

=A2c4

cos(φ)︸ ︷︷ ︸≈1

sin(φ)︸ ︷︷ ︸≈φ

m2(t)

=A2c4φm2(t)

for φ� 1.

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 29 / 40

3.4 Costas Receiver

Costas Receiver: Phase Discriminator

Two components in sequence:

(2) time-averaging unit

g(t) =1

2T

∫ T−T

A2c4φm2(t)dt

=A2c4φ

1

2T

∫ T−T

m2(t)dt︸ ︷︷ ︸power of m(t); for T large is constant

Therefore, g(t) is proportional to φ, and is the same sign as thephase error φ.

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 30 / 40

3.4 Costas Receiver

Costas Receiver: Voltage-controlled Oscillator

-90 degreePhase Shifter

Voltage-controlledOscillator

ProductModulator

Low-passFilter

PhaseDiscriminator

ProductModulator

Low-pass�lter

DSB-SC wave

DemodulatedSignal

g(t)

v (t)0

v’ (t)0

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 31 / 40

3.4 Costas Receiver

Costas Receiver: Voltage-controlled Oscillator

I If g(t) > 0 (or φ > 0), then the local oscillator will decreasefrom fc proportional to the value of g(t) (or φ).

I If g(t) < 0 (or φ < 0), then the local oscillator will increasefrom fc proportional to the value of g(t) (or φ).

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 32 / 40

3.4 Costas Receiver

Costas Receiver: Voltage-controlled Oscillator

φ > 0: Freq of local oscillator needs to

temporarily decrease

0

t

0

t

local oscillatorcarrier

local oscillator phase is ahead of carrier and must decrease its frequency to wait for carrier

φ < 0: Freq of local oscillator needs to

temporarily increase

0

t

0

t

local oscillatorcarrier

local oscillator phase lags behind carrier and must increase its frequency to catch up

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 33 / 40

3.4 Costas Receiver

Costas Receiver

-90 degreePhase Shifter

Voltage-controlledOscillator

ProductModulator

Low-passFilter

PhaseDiscriminator

ProductModulator

Low-pass�lter

DSB-SC wave

DemodulatedSignal

Coherent Demodulation

Circuit for Phase Locking

v (t)0local oscillator output

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 34 / 40

3.5 Quadrature-Carrier Multiplexing

Quadrature Amplitude Multiplexing

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 35 / 40

3.5 Quadrature-Carrier Multiplexing

Multiplexing and QAM

Multiplexing: to send multiple message simultaneously

Quadrature Amplitude Multiplexing (QAM): (a.k.a quadrature-carriermultiplexing) amplitude modulation scheme that enables twoDSB-SC waves with independent message signals to occupy the samechannel bandwidth (i.e., same frequency channel) yet still beseparated at the receiver.

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 36 / 40

3.5 Quadrature-Carrier Multiplexing

QAM: Transmitter

-90 degreePhase Shifter

ProductModulator

ProductModulator

MultiplexedSignal

Messagesignal

Messagesignal

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 37 / 40

3.5 Quadrature-Carrier Multiplexing

QAM: Transmitter

s(t) = Acm1(t) cos(2πfct) + Acm2(t) sin(2πfct)

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 38 / 40

3.5 Quadrature-Carrier Multiplexing

QAM: Receiver

ProductModulator

Low-passFilter

ProductModulator

Low-pass�lter

MultiplexedSignal

-90 degreePhase Shifter

Professor Deepa Kundur (University of Toronto)Costas Receiver and Quadrature-Carrier Multiplexing 39 / 40

3.5 Quadrature-Carrier Multiplexing

QAM: Receiver

I Costas receiver may be used to synchronize the local oscillatorfor demodulation.

�

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3.4 Costas Receiver3.5 Quadrature-Carrier Multiplexing