Costas Receiver and Quadrature-Carrier Multiplexing Costas Receiver and Quadrature-Carrier
Multiplexing Reference:
Sections 3.4 and 3.5 of Professor Deepa Kundur
University of Toronto S. Haykin and M. Moher, Introduction to Analog & Digital Communications, 2nd ed., John Wiley & Sons, Inc., 2007. ISBN-13 978-0-471-43222-7.
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3.4 Costas Receiver 3.4 Costas Receiver Coherent Demodulation Output
1 v (t) = A A0 cos(φ)m(t) 0 2 c c
1 Costas Receiver I To maximize v0(t), would like φ ≈ 0.
0 -1
I For v0(t) to be proportional to m(t), φ should be constant.
Thus, we wish to synchronize the local oscillator. 0 Let Ac = 1 for simplicity.
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Coherent Demodulation
Product Low-pass Demodulated Product Low-pass Demodulated Modulator lter Signal Modulator lter Signal v (t) local oscillator output 0
Voltage-controlled Phase Voltage-controlled Phase Oscillator Discriminator Oscillator Discriminator DSB-SC wave DSB-SC wave
-90 degree -90 degree Phase Shifter Phase Shifter
Product Low-pass Product Low-pass Modulator Filter Modulator Filter
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)CostasReceiver and Quadrature-Carrier Multiplexing 5 / 40 Professor Deepa Kundur (University of Toronto)CostasReceiver and Quadrature-Carrier Multiplexing 6 / 40
3.4 Costas Receiver 3.4 Costas Receiver Costas Receiver: Phase Lock Circuit Costas Receiver: Phase Lock Circuit φ > 0: Freq of local oscillator needs to temporarily decrease Coherent Demodulation
Product Low-pass Demodulated carrier local oscillator Modulator lter Signal v (t) local oscillator output 0
Voltage-controlled Phase t Oscillator Discriminator DSB-SC wave
-90 degree 0 Phase Shifter local oscillator phase is ahead of carrier and must decrease Product Low-pass its frequency to Modulator Filter wait for carrier t Circuit for Phase Locking
Goals: (1) Coherent demodulation of DSB-SC input signal. 0 (2) Tweak the local oscillator phase such that φ = 0. Professor Deepa Kundur (University of Toronto)CostasReceiver and Quadrature-Carrier Multiplexing 7 / 40 Professor Deepa Kundur (University of Toronto)CostasReceiver and Quadrature-Carrier Multiplexing 8 / 40 3.4 Costas Receiver 3.4 Costas Receiver Costas Receiver: Phase Lock Circuit Costas Receiver: In-Phase and Quadrature-Phase φ < 0: Freq of local oscillator needs to temporarily increase I-Channel (in-phase coherent detector)
carrier local oscillator Product Low-pass Demodulated Modulator lter Signal vI (t)
Voltage-controlled Phase t Oscillator Discriminator DSB-SC wave
0 -90 degree Phase Shifter local oscillator phase lags behind carrier and must increase its v (t) frequency to Product Q Low-pass catch up Modulator Filter t Q-Channel (quadrature-phase coherent detector)
0
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3.4 Costas Receiver 3.4 Costas Receiver Costas Receiver: In-Phase Coherent Detector Costas Receiver: In-Phase Coherent Detector
s(t) = Ac cos(2πfc t)m(t) Product Low-pass Demodulated Modulator lter Signal v (t) = s(t) · cos(2πf t + φ) vI (t) I c
Voltage-controlled Phase Recall Oscillator Discriminator DSB-SC wave ejφ e−jφ ejφ e−jφ cos(2πf t + φ)= ej2πfc t + e−j2πfc t δ(f − f ) + δ(f + f ) c 2 2 2 c 2 c -90 degree Phase Shifter ejφ e−jφ VI (f ) = S(f ) ? δ(f − fc ) + δ(f + fc ) Product vQ (t) Low-pass 2 2 Modulator Filter ejφ e−jφ = S(f ) ? δ(f − f ) + S(f ) ? δ(f + f ) 2 c 2 c ejφ e−jφ = S(f − f ) + S(f + f ) 2 c 2 c
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ejφ e−jφ repetitions repetitions V (f ) = S(f − f ) + S(f + f ) I 2 c 2 c cos(φ) + j sin(φ) cos(−φ) + j sin(−φ) = S(f − f ) + S(f + f ) 2 c 2 c repetitions repetitions cos(φ) + j sin(φ) cos(φ) − j sin(φ) = S(f − f ) + S(f + f ) 2 c 2 c cos(φ) sin(φ) = [S(f − fc ) + S(f + fc )] + j [S(f − fc ) − S(f + fc )] 2 2 repetitions repetitions | {z } | {z } ≈1/2 small for φ 1. baseband components baseband components add coherently cancel out
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3.4 Costas Receiver 3.4 Costas Receiver Costas Receiver: In-Phase Coherent Detector Costas Receiver: In-Phase Coherent Detector
ejφ e−jφ VI (f ) = S(f − fc ) + S(f + fc ) baseband components 2 2 add coherently cos(φ) + j sin(φ) cos(−φ) + j sin(−φ) = S(f − f ) + S(f + f ) 2 c 2 c cos(φ) + j sin(φ) cos(φ) − j sin(φ) = S(f − f ) + S(f + f ) 2 c 2 c cos(φ) sin(φ) = S(f − fc ) + S(f + fc ) + j S(f − fc ) − S(f + fc ) 2 | {z } 2 | {z } baseband components | {z } significant baseband | {z } negligible baseband cancel out ≈1/2 small for φ 1. The closer φ is to zero, the more significant the baseband term of vI (t) and vice versa.
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Product Low-pass Demodulated Modulator lter Signal vI (t)
Voltage-controlled Phase Oscillator Discriminator DSB-SC wave
-90 degree Phase Shifter
Product vQ (t) Low-pass Modulator Filter
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3.4 Costas Receiver 3.4 Costas Receiver Costas Receiver: Quadrature-Phase Detector Costas Receiver: Quadrature-Phase Detector
jφ −jφ s(t) = Ac cos(2πfc t)m(t) e e VQ (f ) = S(f − fc )− S(f + fc ) vQ (t) = s(t) · sin(2πfc t + φ) 2j 2j cos(φ) + j sin(φ) cos(−φ) + j sin(−φ) = S(f − f )− S(f + f ) 2j c 2j c Recall cos(φ) + j sin(φ) cos(φ) − j sin(φ) jφ −jφ jφ −jφ e e e e = S(f − fc )− S(f + fc ) sin(2πf t + φ)= ej2πfc t − e−j2πfc t δ(f − f ) − δ(f + f ) 2j 2j c 2j 2j 2j c 2j c cos(φ) sin(φ) = [S(f − f )−S(f + f )] + j [S(f − f )+S(f + f )] 2j c c 2j c c ejφ e−jφ V (f ) = S(f ) ? δ(f − f ) − δ(f + f ) | {z } | {z } Q 2j c 2j c ≈1/2j small ejφ e−jφ = S(f ) ? δ(f − f ) − S(f ) ? δ(f + f ) for φ 1. 2j c 2j c ejφ e−jφ = S(f − f ) − S(f + f ) 2j c 2j c
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ejφ e−jφ V (f ) = S(f − f )− S(f + f ) baseband components Q 2j c 2j c cancel out cos(φ) + j sin(φ) cos(−φ) + j sin(−φ) = S(f − f )− S(f + f ) 2j c 2j c cos(φ) + j sin(φ) cos(φ) − j sin(φ) = S(f − f )− S(f + f ) 2j c 2j c cos(φ) sin(φ) = S(f − fc )−S(f + fc ) + j S(f − fc )+S(f + fc ) baseband components 2j | {z } 2j | {z } | {z } negligible baseband | {z } significant baseband add coherently ≈1/2j small for φ 1. The closer φ is to zero, the more negligible the baseband term of vQ (t) and vice versa.
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3.4 Costas Receiver 3.4 Costas Receiver
Costas Receiver: Quadrature-Phase Detector Costas Receiver: vI (t) and vQ(t) spectrum
Product Low-pass Demodulated Modulator lter Signal vI (t) v0 (t)
Voltage-controlled Phase Oscillator Discriminator DSB-SC wave
spectrum -90 degree Phase Shifter
Product vQ (t) Low-pass Modulator Filter
spectrum
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Costas Receiver: vI (t) and vQ(t) Costas Receiver: vI (t) and vQ(t) spectrum spectrum
Product Low-pass Demodulated Product Low-pass Demodulated Modulator lter Signal Modulator lter Signal vI (t) vI (t)
Voltage-controlled Phase Voltage-controlled Phase Oscillator Discriminator Oscillator Discriminator DSB-SC wave DSB-SC wave
spectrum -90 degree spectrum -90 degree Phase Shifter Phase Shifter
Product vQ (t) Low-pass Product vQ (t) Low-pass Modulator Filter Modulator Filter
spectrum spectrum
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3.4 Costas Receiver 3.4 Costas Receiver Costas Receiver: Low-Pass Filter Costas Receiver: Local Oscillator Control
spectrum
Product Low-pass Demodulated Product Low-pass Demodulated Modulator lter Signal Modulator lter Signal
v0 (t) Voltage-controlled Phase Voltage-controlled g(t) Phase Oscillator Discriminator Oscillator Discriminator DSB-SC wave DSB-SC wave v’0 (t) spectrum -90 degree -90 degree Phase Shifter Phase Shifter
Product Low-pass Product Low-pass Modulator Filter Modulator Filter
spectrum
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(2) time-averaging unit (1) multiplier
0 Ac Ac 1 Z T A2 v0(t) · v0(t)= cos(φ)m(t) · sin(φ)m(t) g(t)= c φm2(t)dt 2 2 2T 4 A2 −T c 2 2 Z T = cos(φ) sin(φ) m (t) Ac 1 2 4 | {z } | {z } = φ m (t)dt ≈1 ≈φ 4 2T −T A2 | {z } = c φm2(t) power of m(t); for T large is constant 4 for φ 1. Therefore, g(t) is proportional to φ, and is the same sign as the phase error φ.
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3.4 Costas Receiver 3.4 Costas Receiver Costas Receiver: Voltage-controlled Oscillator Costas Receiver: Voltage-controlled Oscillator
Product Low-pass Demodulated Modulator lter Signal I If g(t) > 0(or φ > 0), then the local oscillator will decrease v0 (t) Voltage-controlled g(t) Phase from fc proportional to the value of g(t)(or φ). Oscillator Discriminator DSB-SC wave v’0 (t) -90 degree Phase Shifter I If g(t) < 0(or φ < 0), then the local oscillator will increase from fc proportional to the value of g(t)(or φ). Product Low-pass Modulator Filter
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φ > 0: Freq of local oscillator needs to φ < 0: Freq of local oscillator needs to Coherent Demodulation temporarily decrease temporarily increase Product Low-pass Demodulated Modulator lter Signal v (t) local oscillator output 0
Voltage-controlled Phase carrier local oscillator carrier local oscillator Oscillator Discriminator DSB-SC wave
t t -90 degree Phase Shifter
0 0 local oscillator phase is ahead local oscillator phase lags behind of carrier and carrier and must must decrease increase its Product Low-pass its frequency to frequency to wait for carrier catch up Modulator Filter t t Circuit for Phase Locking
0 0
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3.5 Quadrature-Carrier Multiplexing 3.5 Quadrature-Carrier Multiplexing Multiplexing and QAM
Multiplexing: to send multiple message simultaneously
Quadrature Amplitude Multiplexing Quadrature Amplitude Multiplexing(QAM): (a.k.a quadrature-carrier multiplexing) amplitude modulation scheme that enables two DSB-SC waves with independent message signals to occupy the same channel bandwidth (i.e., same frequency channel) yet still be separated at the receiver.
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Message Multiplexed Product signal Signal Modulator
s(t)= Ac m1(t) cos(2πfc t) + Ac m2(t) sin(2πfc t)
-90 degree Phase Shifter
Message Product signal Modulator
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3.5 Quadrature-Carrier Multiplexing 3.5 Quadrature-Carrier Multiplexing QAM: Receiver QAM: Receiver
Product Low-pass Modulator lter
I Costas receiver may be used to synchronize the local oscillator Multiplexed for demodulation. Signal -90 degree Phase Shifter
Product Low-pass Modulator Filter
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