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. Professor Deepa Kundur (University of Toronto)CostasReceiver and Quadrature-Carrier Multiplexing 1 / 40 Professor Deepa Kundur (University of Toronto)CostasReceiver and Quadrature-Carrier Multiplexing 2 / 40 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. Professor Deepa Kundur (University of Toronto)CostasReceiver and Quadrature-Carrier Multiplexing 3 / 40 Professor Deepa Kundur (University of Toronto)CostasReceiver and Quadrature-Carrier Multiplexing 4 / 40 3.4 Costas Receiver 3.4 Costas Receiver Costas Receiver Costas Receiver: Coherent Demodulation 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 Professor Deepa Kundur (University of Toronto)CostasReceiver and Quadrature-Carrier Multiplexing 9 / 40 Professor Deepa Kundur (University of Toronto)CostasReceiver and Quadrature-Carrier Multiplexing 10 / 40 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 Professor Deepa Kundur (University of Toronto)CostasReceiver and Quadrature-Carrier Multiplexing 11 / 40 Professor Deepa Kundur (University of Toronto)CostasReceiver and Quadrature-Carrier Multiplexing 12 / 40 3.4 Costas Receiver 3.4 Costas Receiver Costas Receiver: In-Phase Coherent Detector Costas Receiver: In-Phase Coherent Detector 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 Professor Deepa Kundur (University of Toronto)CostasReceiver and Quadrature-Carrier Multiplexing 13 / 40 Professor Deepa Kundur (University of Toronto)CostasReceiver and Quadrature-Carrier Multiplexing 14 / 40 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 2 3 2 3 cos(φ) 6 7 sin(φ) 6 7 = 6S(f − fc ) + S(f + fc )7 + j 6S(f − fc ) − S(f + fc )7 2 4| {z }5 2 4| {z }5 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. Professor Deepa Kundur (University of Toronto)CostasReceiver and Quadrature-Carrier Multiplexing 15 / 40 Professor Deepa Kundur (University of Toronto)CostasReceiver and Quadrature-Carrier Multiplexing 16 / 40 3.4 Costas Receiver 3.4 Costas Receiver Costas Receiver: In-Phase Coherent Detector Costas Receiver: Quadrature-Phase Detector 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 For φ small. Professor Deepa Kundur (University of Toronto)CostasReceiver and Quadrature-Carrier Multiplexing 17 / 40 Professor Deepa Kundur (University of Toronto)CostasReceiver and Quadrature-Carrier Multiplexing 18 / 40 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 Professor Deepa Kundur (University of Toronto)CostasReceiver and Quadrature-Carrier Multiplexing 19 / 40 Professor Deepa Kundur (University of Toronto)CostasReceiver and Quadrature-Carrier Multiplexing 20 / 40 3.4 Costas Receiver 3.4 Costas Receiver Costas Receiver: Quadrature-Phase Detector Costas Receiver: Quadrature-Phase Detector 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 2 3 2 3 cos(φ) 6 7 sin(φ) 6 7 = 6S(f − fc )−S(f + fc )7 + j 6S(f − fc )+S(f + fc )7 baseband components 2j 4| {z }5 2j 4| {z }5 | {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. Professor Deepa Kundur (University of Toronto)CostasReceiver and Quadrature-Carrier Multiplexing 21 / 40 Professor Deepa Kundur (University of Toronto)CostasReceiver and Quadrature-Carrier Multiplexing 22 / 40 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 For φ small. Professor Deepa Kundur (University of Toronto)CostasReceiver and Quadrature-Carrier Multiplexing 23 / 40 Professor Deepa Kundur (University of Toronto)CostasReceiver and Quadrature-Carrier Multiplexing 24 / 40 3.4 Costas Receiver 3.4 Costas Receiver 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 Professor Deepa Kundur (University of Toronto)CostasReceiver and Quadrature-Carrier Multiplexing 25 / 40 Professor Deepa Kundur (University of Toronto)CostasReceiver