From AM Radio to Digital I/Q Modulation COS 463: Wireless Networks Lecture 12 Kyle Jamieson [Parts adapted from H. Balakrishnan, M. Perrott, C. Terman] Roadmap 1. Analog I/Q modulation 2. Discrete-time processing of continuous signals – The Digital Abstraction – Quantization: Discretizing values – Sequences: Discretizing time 3. Digital I/Q Modulation 2 Review: AM Modulation & Demodulation AM Modulation and Demodulation Transmitter: Transmitter OutputReceiver: Receiver Output Lowpass Filter x(t) y(t) y(t) w(t) r(t) H(j2πf) π π 2cos(2 fot) 2cos(2 fot) • Multiplication (i.e., mixing) operation shifts in • Multiplicationfrequency(i.e. mixing) operation shifts in frequency – Also creates undesired high frequency components at • Low-passreceiver filtering passes only the desired baseband signal at the receiver • Works• Lowpass with cos filteringor sin carrier passes signal only the desired baseband signal at receiver What can go wrong here? 3 6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 2 Impact of a 90º phase shift Impact of 90 Degree Phase Shift Transmitter: Transmitter OutputReceiver: Receiver Output Lowpass Filter x(t) y(t) y(t) w(t) r(t) H(j2πf) π π 2cos(2 fot) 2sin(2 fot) ∿ ∿ Local oscillator • If receiver cosine wave turns into a sine wave, we • Supposesuddenly receiver receive uses sine, noinstead baseband of cosine: signal! no output – We apparentlysin( 2πneedf0t) tocos synchronize(2πf0t) = ½ thesin( 4πphasef0t) of the transmitter and receiver local oscillators • Need to synchronize• This is called phase coherent of transmitter demodulation and receiver local oscillators (coherent demodulation) • Some key questions: – How do we analyze this issue? – What would be the impact of a small frequency offset? 4 6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 3 Coherent Demodulation: Frequency-domain analysis Frequency Domain Analysis W(j2πf) H(j2πf) X(j2πf) Y(j2πf) f f f 0 -fo 0 fo -2fo -fo 0 fo 2fo x(t) y(t) y(t) w(t) r(t) Transmitter Output H(j2πf) Receiver Output Lowpass π π ∿ 2cos(2 fot) ∿ 2cos(2 fot) Filter 1111 f f -fo 0 fo -fo 0 fo • Transmitter• When and transmitter receiver oscillators and phasereceiver-synchronized local oscillators are • Demodulatedmatched copies in phase:add constructively at baseband (close to f = 0) – Demodulated signal constructively adds at baseband 5 6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 4 90º Phase Shift: Frequency-domain analysis Impact of 90 Degree Phase Shift W(j2πf) H(j2πf) X(j2πf) Y(j2πf) 2fo f f f -f 0 f -2fo -fo 0 0 o o fo x(t) y(t) y(t) w(t) r(t) Transmitter Output H(j2πf) Receiver Output = 0 Lowpass π π 2cos(2 fot) 2sin(2 fot) Filter 11j f1 f f -fo 0 fo -f1 0 -j • Transmitter, receiver oscillators are offset in phase by 90º • Demodulated• When transmittercopies add destructively and receiverat baseband local oscillators(close to f = 0) are – Zero90 outputdegreefrom offset receiver in phase: – Demodulated signal destructively adds at baseband • But: Opportunity to create two separate channels! What would happen with a small frequency offset? 6 6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 5 In-Phase/Quadrature Modulation • Modulate each with a I/Qcosine &Demodulation sine, then sum the result 11 11 Transmitter Output Receiver Output π It(j2 f) f f π Ir(j2 f) -fo 0 fo π -fo 0 fo 1 11Yi(j2 f) 2 it(t) ir(t) In-Phaseπ (I) f f H(j2 f) 0 -f 0 f component (real) f π o o π Lowpass 2cos(2 fot) y(t) y(t) 2cos(2 fot) 0 π π Qt(j2 f) Qr(j2 f) π π 2sin(2 fot) π 2sin(2 fot) 1 j Yq(j2 f) 2 qt(t) qr(t) H(j2πf) fo Quadrature (Q) f f f componentLowpass (imaginary) 0 j -fo 0 j 0 -j f fo o f f -fo 0 -fo 0 -j -j • I,• Q Demodulatesignals occupy thewith same both frequencya cosine band and sine wave – One– Both is real I (andfor cos Q ),channelsone is imaginary are recovered! (for sin) • I/Q modulation allows twice the amount of 7 information to be sent compared to basic AM modulation with same bandwidth What can go wrong here? 6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 7 In-Phase/Quadrature Demodulation Transmitter: I/Q DemodulationReceiver: 11 11 Transmitter Output Receiver Output π It(j2 f) f f π Ir(j2 f) -fo 0 fo π -fo 0 fo 1 11Yi(j2 f) 2 it(t) ir(t) π f f H(j2 f) 0 -f 0 f f π o o π Lowpass 2cos(2 fot) y(t) y(t) 2cos(2 fot) 0 π π Qt(j2 f) Qr(j2 f) π π 2sin(2 fot) π 2sin(2 fot) 1 j Yq(j2 f) 2 qt(t) qr(t) H(j2πf) fo f f f Lowpass 0 j -fo 0 j 0 -j f fo o f f -fo 0 -fo 0 -j -j • Demodulate• Demodulate with both witha sine both and aa cosine cosine and sine wave – Both– Both I and I Qand channels Q channels are recovered! are recovered! • I/Q modulation allows twice the amount of 8 information to be sent compared to basic AM modulation with same bandwidth What can go wrong here? 6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 7 I/Q Demodulation: 90º Phase Shift Transmitter:Impact of 90 DegreeReceiver: Phase Shift 11 j Transmitter Output f Receiver Output π o It(j2 f) f f π Ir(j2 f) -fo 0 fo π -fo 0 1 Yi(j2 f) 11 -j 2 it(t) ir(t) π f f H(j2 f) 0 -f 0 f f π o o π Lowpass 2cos(2 fot) y(t) y(t) 2sin(2 fot) 0 π π Qt(j2 f) Qr(j2 f) π π 2sin(2 fot) π -2cos(2 fot) 1 j Yq(j2 f) qt(t) qr(t) H(j2πf) fo f f f Lowpass 0 j -fo 0 0 -j f -fo fo -2 o f f -fo 0 0 -j -1 -1 • I and Q channels get swapped at receiver •– KeyI and observation: Q channels No information get swapped is lost! at receiver – Key observation: no information is lost! • Questions 9 – What would happen with a small frequency offset? – What would happen with a large frequency offset? 6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 8 Summary of Analog I/Q modulation Summary of Analog I/Q Modulation • Frequency domain view Baseband Input Receiver Output π π It(j2 f) Ir(j2 f) 1 2 it(t) ir(t) H(j2πf) f f 0 Lowpass 2cos(2πf t) π 0 π o 2cos(2 fot) Qt(j2 f) π π Q (j2πf) 2sin(2 fot) 2sin(2 fot) r 1 2 qt(t) qr(t) H(j2πf) f f 0 Lowpass 0 • Time domain view Baseband Input Receiver Output t t it(t) ir(t) H(j2πf) Lowpass π π 2cos(2 fot) 2cos(2 fot) 2sin(2πf t) 2sin(2πf t) t o o t qt(t) qr(t) H(j2πf) Lowpass 10 6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 9 I/Q modulation: Wrap-up • I/Q modulation allows twice the amount of information to be sent compared to basic AM • Impact of phase offset is to swap I/Q • Impact of frequency offset is I/Q swapping (small offset) – Or catastrophic corruption (large offset) of received signal 11 Roadmap 1. Analog I/Q modulation 2. Discrete-time processing of continuous signals – The Digital Abstraction – Quantization: Discretizing values – Sequences: Discretizing time 3. Digital I/Q Modulation 12 Information Processing in the Analog Domain First let’s introduce some processing blocks: Information ProcesRepresentingsing in t hinformatione Analo withg D voltageomain • Encoding of pixel at location (x, y) in a Black & White picture: – 0 Volts = Black First let’s introduce some– 1 Volt =processing White blocks: v – 0.37Copy V = 37% Gray v – etc... • Encoding of a picture: – Scan points in prescribed order – Generate a continuous voltage v waveformCopy (baseband signalv ) v INV 1-v 13 v INV 1-v 6.082 Spring 2007 The Digital Abstraction, Slide 3 6.082 Spring 2007 The Digital Abstraction, Slide 3 Summary of Analog I/Q DemodulationSummary of Analog I/Q Demodulation • Frequency domain view • Frequency domain view Information Processing in the Analog DomaIinnformBasebandat iInputon Processing in BasebandReceiverthe Outpu Input At nalog Domain Receiver Output π I (j2π πf) I (j2πf) It(j2 f) Ir(j2t f) r 1 1 2 i (t) i (t) 2 it(t) ir(t) t r H(j2πf) H(j2πf) Let’s build a P2P television networkf f f First let’s introduce some processing blocks: f Lowpass First let’s 0introduce some processingLowpass 0 0 blocks:π 0 2cos(2πf t) π 2cos(2 fot) 2cos(2πf t) π o 2cos(2 fot) Q (j2πf) o π Qt(j2 f) π π Q (j2t πf) 2sin(2πf t) 2sin(2πf t) Qr(j2 f) 2sin(2 fot) 2sin(2 fot) r o o 1 1 2 q(t) q(t) 2 qt(t) qr(t) t r Let’s build aH(j2 πf)system! H(j2πf) f f f One hop: f Lowpass 0 Lowpass 0 0 0 • Time domain view • Time domain view Baseband Input Receiver Output Encode Basebandx(t) Input Receiver OutputDecode v Copy v t × × t t t v Copy v i (t) i (t) it(t) ir(t) t r H(j2πf) H(j2πf) Copy LowpassINV Lowpass π 2cos(2πf t) π 2cos(2 fot) 2cos(2πf t) ∿ o 2cos(2 fot) ∿ o π π 2sin(2πf t) 2sin(2πf t) 2sin(2 fot) 2sin(2 fot) Information Processing in the Analog Domain t o o t t t q(t) q(t) qt(t) qr(t) t r H(j2πf) H(j2πf) Lowpass Lowpass First let’s introduce some processing blocks: input6.082 Spring 2007 Copy 6.082 SpringINV 2007 Digital Modulation (Part I), Slide 4 Digital Modulation (Part I), Slide 4 Many hops: (In (Reality) Theory) v INV 1-v Copyv INV INV 1-v Encode v Copy v Copy INV Decode ? 6.082 Spring 2007 The Digi6ta.0l 8A2b Sstprraicntgi o2n0,0 S7lide 3 output The Digital Abstraction, Slide 3 14 v INV 1-v 6.082 Spring 2007 The Digital Abstraction, Slide 5 6.082 Spring 2007 The Digital Abstraction, Slide 3 Why did our network fail? 1.