From AM to Digital I/Q

COS 463: 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 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) Information Processing in the Analog Domain 2sin(2 fot) 2sin(2 fot) o o t 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. Transmitter doesn’t work right 2. Receiver doesn’t work right 3. Theory is imperfect 4. System architecture isn’t right

• Answer: All of the above!

• Noise, transmitter/receiver imperfections inevitable, so must design system to tolerate some amount of error

15 Analog communications issues

• Problem: It’s hard to distinguish legitimate analog waveforms from corrupted ones – Every waveform is potentially legitimate!

• Small errors accumulate at each hop

• Endpoints can’t help, so need to eliminate errors at each hop

16 Plan:Plan: MixedMixed SignalSignal A Architecturerchitecture

Volts Digital Volts Freq Signal Freq Phase bits Processing bits Phase

Analog store Digital to xmit/rcv to Digital modify Analog

ANALOG DIGITAL ANALOG

Continuous time, Discrete time, Continuous values Discrete values

6.082 Spring 2007 The Digital Abstraction, Slide 16 17 Discrete time, Discrete values

• Continuous-time, continuous-valued waveform is being converted into a discrete-valued sequence

– Only certain values are ”allowed” to be used

• Questions yet to be answered:

1. How many discrete values should we use?

2. How rapidly in time should we sample the waveform?

18 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

19 Digital Signaling II Encoding InformationDigital Si gDigitallynaling II Encoding Attempt #E2n:coding Attempt #2: 0 0 1 1 • First encoding attempt: Forbidden Zone Forbidden Zone volts volts 0 VL VH VDD

0 VL VH VDD This avoids “close calls”, but now we have to consider noise (i.e. • But, what happensu inna vtheoida bpresencele perturbati oofns tnoise?o our signaling voltage) This avoids “close calls”, but now we have to consider noise (i.e. unavoidable perturbations to our NsOiIgSnEaling voltage) DEVICE DEVICE NOISE #1 #2 So an output voltage just below VL might become an DEVICE DEVICE illegal input voltage in the forbidden zone! #1 #2V V±NSo an output voltage just below VL might become an illegal input voltage in the 6.082 Spring 2007 The Digital Abstraction, Slide 10 V V±N forbidden zone!

20 6.082 Spring 2007 The Digital Abstraction, Slide 10 Big Idea: Noise Margins Big Idea: Noise Margins

Let’s leave room for bad things to happen! So we’ll design devices restore marginally valid input signals. They must accept marginal inputs and provide unquestionable outputs (i.e., to leave room for noise).

0OUT 1OUT

OUTPUTS: Forbidden Zone volts

0 VOL VOH VDD

0IN 1IN

INPUTS: volts

0 VOL VIL VIH VOH VDD

Noise Margins 6.082 Spring 2007 The Digital Abstraction, Slide 11 21 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

22 Discrete Values If we use N bits to encode the magnitude of one of the discrete-time samples, we can capture 2N possible values.

So we’ll divide up the range of possible sample values into 2Quantization:N intervals and c hHowoose tmanyhe inde xdiscrete of the e nvalues?closing interval as the encoding for the sample value.

VMAX 7 15 3 14 sample voltage 6 13 1 12 5 11 2 10 4 9 8 3 7 1 6 2 5 4 0 1 3 2 0 1

0 Analog value converted to digital index 0 VMIN quantized value 1 11 110 1101 1-bit 2-bit 3-bit 4-bit

6.082 Spring 2007 The Digital Abstraction, Slide 18

23 Quantization Error

Note that when we quantize the scaled sample values we may be off by up Quantizationto ±½ step from th Errore true sampled values.

The red shaded region shows 57 the error we’ve introduced

56

55

54

53

54 55 56 55 55 • Introduce at most ½ interval length of quantization error

6.082 Spring 2007 The Digital Abstraction, Slide 19 24 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

25 Need for ContinuousThe Need for-to- DiscreteSampling Conversion Real World Matlab Real(USRP world Board)(Hack RF) Computer (Python) xc(t) A-to-D x[n] Converter xc(t) x[n]

t n 1 • TheContinuous boundary-time signal between analog andDiscrete digital-time sequence – Real world is filled with continuous-time signals • The– boundaryComputers between (i.e. analog Matlab) and operate digital on sequences

• – CrossingReal world thefilled withanalog-to-digital continuous-time signals boundary requires – samplingComputers ofoperate the oncontinuous-time discrete-time sequences signals • Key questions • Crossing– How analog do we to digitalanalyze boundary the sampling requires process? conversion between the two

– What can go wrong? 26 6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 2 TheHow Need Fast for to Sample?Sampling Real World Matlab Real(USRP world Board)(Hack RF) Computer (Python) xc(t) A-to-D x[n] Converter xc(t) x[n]

t n 1 • ContinuousThe boundary-time (CT) signalbetween analog andDiscrete digital-time (DT) sequence – Real world is filled with continuous-time signals • Too– slowComputers (undersampling (i.e. Matlab)): operate on sequences – DT sequence loses information about CT signal • Crossing the analog-to-digital boundary requires • Toosampling fast (oversampling of the ):continuous-time signals • – KeyDT sequence questions contains redundant, useless information – How do we analyze the sampling process?

– What can go wrong? 27 6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 2 Modelling Continuous-to-Discrete Conversion An Analytical Model for Sampling p(t) 1

t p(t) T

xc(t) xp(t) Impulse Train x[n] to Sequence xc(t) xp(t) x[n]

t t n T 1 • Two•-stepTwo process step :process – Sample continuous-time signal every T seconds 1. Sample continuous-time signal every T seconds • Model as multiplication of signal with impulse train – Model as multiplication of signal with impulse train – Create sequence from amplitude of scaled impulses → 2. Create sequence• Model asfrom rescaling amplitudeof timeof scaled axis (impulsesT 1) – Model •asNotation: rescaling replaceof time axisimpulses (T à with1) stem symbols Can we model this in the frequency domain? 28

6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 3 Fourier Transform of Impulse Train Fourier Transform of Impulse Train

p(t) P(j2 πf) 1 1 T t t T -2 -1 0 1 2 T T T T

• •ImpulseImpulse train intrain time correspondsin time correspondsto impulse train in to frequency impulse train in frequency – –SpacingSpacing in time in oftime T seconds of T seconds corresponds corresponds to 1/T Hz spacing to spacing in frequency in frequency of 1/T Hz – Scale factor of 1/T for impulses in frequency domain – Note: this is painful to derive, so we won’t …

• The above transform pair allows us to see the 29 following with pictures – Sampling operation in frequency domain – Intuitive comparison of FT, DTFT, and Fourier Series

6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 4 Frequency Domain view of Sampling Frequency Domain View of Sampling

xc(t) p(t) xp(t) 1

t t t T T

π π Xc(j2 f) P(j2 f) Xp(j2 πf) 1 A A T T

f t f -2 -1 0 1 2 -2 -1 0 1 2 T T T T T T T T • Recall that multiplication in time corresponds to • Samplingconvolution in time leadsin frequencyto a periodic Fourier Transform, with period 1/T

• We see that sampling in time leads to a periodic 30 Fourier Transform with period 1/T

6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 5 Frequency Domain View of a Sequence Frequency Domain View of Output Sequence

xp (t) x[n]

t n T 1

j2 πλ Xp (j2 πf) X(e ) A A T T

f λ -2 -1 0 1 2 -2 -1 012 T T T T

• •ConversionScaling to in a sequencetime leads amounts to to scaling setting T in= 1 frequency • Resulting– Compression/expansion Fourier Transform is now inperiodic time leadswith a periodto expansion/ of one compression in frequency → • Conversion to sequence amounts to T 1 31 – Resulting Fourier Transform is now periodic with period 1 – Note that we are now essentially dealing with the DTFT

6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 6 Summary: Continuous-to-Discrete Conversion Summary of Sampling Process p(t) 1

t p(t) T

xc(t) xp(t) Impulse Train x[n] to Sequence xc(t) xp(t) x[n]

t t n T 1

π j2 πλ Xc(j2 f) Xp(j2 πf) X(e ) A A A T T

f f λ 0 -2 -1 0 1 2 -2 -1 012 T T T T

• Sampling• Sampling leads toleads periodicity to periodicityin frequency indomain frequency domain

We need to avoid overlap of replicated 32 signals in frequency domain (i.e., aliasing) 6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 7 TheThe SamplingSampling TheoremTheorem

xc(t) p(t) xp(t) 1

t t t T T

π π Xc(j2 f) P(j2 f) Xp(j2 πf) 1 A A T T

f t f -fbw fbw -2 -1 0 1 2 -2 -1 0 1 2 T T T T T T T T -fbw fbw - 1 + f 1 - f T bw T bw • •NoOverlap overlap in infrequency frequency domain domain (aliasing) (i.e., when aliasing) is "#⁄ − & ≥ & or #⁄ ≥ 2& avoided$ '( if:'( $ '( • We refer to the minimum 1/T that avoids aliasing as the Nyquist sampling frequency of a signal

• We refer to the minimum 1/T that avoids aliasing 33 as the Nyquist sampling frequency 6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 8 Sine Wave Example: Sampling Above Nyquist Rate Example: Sample a Sine Wave

t p(t) T

xc(t) xp(t) Impulse Train x[n] to Sequence

t t n T 1

Xc(j2πf) Xp(j2πf) X(j2πλ)

f f λ -1/T 0 1/T -1/T 0 1/T -1 0 1

• Time domain:Sample Resulting rate sequence is well maintains above sameNyquist period rate as input signal • Time domain: resulting sequence maintains the

same period as the input continuous-time signal 34 • Frequency domain: no aliasing 6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 9 Sine Wave Example: Sampling at Nyquist Rate Increase Input Frequency Further …

t p(t) T

xc(t) xp(t) Impulse Train x[n] to Sequence

t t n T 1

Xc(j2πf) Xp(j2πf) X(j2πλ)

f f λ -1/T 0 1/T -1/T 0 1/T -1 0 1

• Time Domain: ResultingSample sequence rate is maintains at Nyquist same rateperiod as input signal • Time domain: resulting sequence still maintains the same period as the input continuous-time signal 35 • Frequency domain: no aliasing 6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 10 Sine Wave Example: Sampling at Half the Nyquist Rate Increase Input Frequency Further …

t p(t) T

xc(t) xp(t) Impulse Train x[n] to Sequence

t t n T 1

Xc(j2πf) Xp(j2πf) X(j2πλ)

f f λ -1/T 0 1/T -1/T 0 1/T -1 0 1

• Sequence nowSample appears rate as constant is at (zero half frequency)the Nyquist signal rate

• Frequency• Time domain: domain: Aliasing resulting to 0 sequence now appears as a DC signal! 36 • Frequency domain: aliasing to DC 6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 11 Sine Wave Example: Sampling Below the Nyquist Rate Increase Input Frequency Further …

t p(t) T

xc(t) xp(t) Impulse Train x[n] to Sequence

t t n T 1

Xc(j2πf) Xp(j2πf) X(j2πλ)

f f λ -1/T 0 1/T -1/T 0 1/T -1 0 1

• ResultingSample sequence rate now ais sine well wave below with athe different Nyquist period ratethan the input • •FrequencyTime domain: domain: Aliasing resultingto a lower sequence frequency is now a sine wave with a different period than the input 37 • Frequency domain: aliasing to lower frequency 6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 12 TheThe IssueIssue ofof HighHigh FrequencyFrequency Noise p(t) 1

t p(t) T

Undesired xc(t) xp(t) Impulse Train x[n] Signal or to Sequence Noise π j2 πλ Xc(j2 f) Xp(j2 πf) X(e ) A A A T T

f f λ -1 0 1 -2 -1 0 1 2 -2 -1 012 T T T T T T

• •TypicallyWe typically set sample set rate tothe exceed sample desired rate signal’s to bandwidthbe large enough to accommodate full bandwidth of signal • Real systems can introduce noise/other interference at high frequencies • –RealSampling systems causes oftennoise to introducealias into desired noise frequencyor other band interfering signals at higher frequencies 38 – Sampling causes this noise to alias into the desired signal band 6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 13 Anti-Alias Filtering Anti-Alias Filtering p(t) 1

t Anti-alias p(t) T Lowpass xc(t) xc(t) xp(t) Impulse Train x[n] H(j2 πf) to Sequence H(j2 πf) π j2 πλ Xc(j2 f) Xp(j2 πf) X(e )

f f λ -1 0 1 -2 -1 0 1 2 -2 -1 012 T T T T T T

• •PracticalPractical systems A-to-D use a continuous converters-time filterinclude before a the continuous- sampling operation –timeRemoves filter noise/interference before the >1/2Tsampling in frequency, operation prevents aliasing – Designed to filter out all noise and interfering signals

above 1/(2T) in frequency 39 – Prevents aliasing

6.082 Spring 2007 I/Q Modulation and RC Filtering, Slide 14 Summary: Advantages of Going Digital

• Allows error correction to be achieved – Less sensitivity to radio channel imperfections

• Enables compression of information – More efficient use of the channel

• Supports a wide variety of information content – Voice, text, can all be represented as digital bit streams

40 Roadmap

1. Analog I/Q modulation

2. Discrete-time processing of continuous signals

3. Digital I/Q Modulation

41 DigitalDigital I/QI/Q modulation Modulation Baseband Input Receiver Output 3 3 1 i (t) it(t) r 1 I -1 H(j2πf) -1 -3 -3 t Lowpass t π π 2cos(2 fot) 2cos(2 fot) 2sin(2πf t) 2sin(2πf t) 3 o o 3 1 qt(t) qr(t) 1 -1 H(j2πf) -1 Q -3 -3 t Lowpass t Sample Times •• LeverageLeverage analog analog communication communication channel to sendchannel discrete-tovalued sendsymbols – Example: Send symbol from set { -3, -1, 1, 3 } on both I and Q channels discrete-valued symbols • At– receiver,Example: sample send I/Q waveformssymbol fromevery set symbol {-3,-1,1,3} period on both – AssociateI and Q each channels sampled each I/Q valuesymbol with periodsymbols from same set on both I and Q channels • At receiver, sample I/Q waveforms every symbol period

– Associate each sampled I/Q value with symbols from 42 set {-3,-1,1,3} on both I and Q channels

6.082 Spring 2007 Digital Modulation (Part I), Slide 5 ConstellationConstellation DiagramsDiagrams Baseband Input Receiver Output 3 Constellation Diagrams3 1 i (t) Basebandit(t) Input r 1 Receiver Output I -1 3 H(j2πf) -1 3 -3 1 it(t) i-3r(t) 1 t Lowpass π tI -1 2cos(2πf t) π H(j2 f) -1 -3 o 2cos(2 fot) -3 2sin(2πf t)t π Lowpass t o π 2sin(2 fot) π 3 2cos(2 fot) 2cos(2 fot) 3 π 2sin(2 fot) 2sin(2πf t) 1 3 qt(t) o qr(t) 1 3 H(j2πf) Q -1 1 qt(t) q-1r(t) 1 -3 -1 H(j2πf) -3 -1 Q -3 t Lowpass -3 t t Lowpass t Sample Sample • Plot I/Q samples on x-y axis TimesTimes • Plot I/Q samples• Plot onI/Q x- ysamples axis on x-y axis –– AsExample: samples– Example:are sampled plotted, sampled constellation I/Q value I/Q diagram valueof of Q Q eventually{1,-3} forms displays{1,-3} a allforms dot possible ata dotx=1, symbol at x=1,y=-3 values y=-3 3 3 – As more– Assamples more samples are plotted, are plotted, eventually 1 constellation diagram eventually 1 displays all possible symbol values I • Provides a sense of how easy it is to distinguish I betweendisplays different all symbols possible symbol values -1 • Constellation diagram provides-1 • Constellationa sense diagram of how easy provides it is to -3 -3 a sense distinguishof how easy between it is different to -3 -1 1 3 43 distinguishsymbols between different -3 -1 1 3 6.082 Spring 2007 Digital Modulation (Part I), Slide 7 symbols 6.082 Spring 2007 Digital Modulation (Part I), Slide 7 SendingSending DigitalDigital BitsBits Baseband Input Receiver Output 3 Sending Digital Bits3 1 Basebandit(t) Input ir(t) 1 Receiver Output π I -1 3 H(j2 f) -1 3 -3 1 it(t) -3ir(t) 1 t Lowpass π t I -1 2cos(2πf t) π H(j2 f) -1 -3 o 2cos(2 fot) -3 π t π Lowpass t 2sin(2 fot) π 2sin(2 fot) π 3 2cos(2 fot) 2cos(2 fot) 3 π π 2sin(2 fot) 2sin(2 fot) 1 3 qt(t) qr(t) 1 3 -1 1 q(t) H(j2πf) -1q(t) 1 Q t r Q -3 -1 H(j2πf)-3 -1 -3 t Lowpass -3 t t Lowpass t SampleSample • Assign bits to each I/Q symbol TimesTimes • Assign each• Assign I/Q each symbol I/Q symbolto a to a – Example: I/Q = {1, 3} translates to bits 1110 Q set of digitalset ofbits digital bits Q – Example: I/Q = {1,3} translates 10 – GrayExample: coding I/Qminimizes = {1,3} bit errors translates when symbol 10 errors are madeto bits of 1110 11 to bits of– 1110Gray coding minimizes bit errors11 I 01 – Gray• Example: codingI/Qwhen minimizes = {3,symbol 3} translates errorsbit errors areto bits made 1010 I 01 when(one symbol bit flip from •errorsExample: {1, 3}) are I/Q made = {1,1} translates 00 to bits of 1010 • Example: I/Q = {1,1} translates 00 00 01 11 10 – Only one bit change from 44 to bits of 1010I/Q = {1,3} 00 01 11 10 6.082– SpringOnly 2007 one bit change from Digital Modulation (Part I), Slide 8 I/Q = {1,3} 6.082 Spring 2007 Digital Modulation (Part I), Slide 8 TheThe ImpactImpact ofof Noise Noise Baseband Input Receiver Output 3 The Impact of Noise3 Receiver 1 it(t) ir(t) 1 -1 Baseband Input Noise H(j2πf) -1 Receiver Output I 3 3 -3 Receiver -3 1 t it(t) Lowpass ir(t) 1 t π Noiseπ π I -1 2cos(2 fot) 2cos(2 fot) H(j2 f) -1 -3 π π -3 2sin(2 fot) t 2sin(2 fot) Lowpass t 3 π π 3 2cos(2 fot) 2cos(2 fot) π π 1 qt(t) 2sin(2 fot) 2sin(2 fot) qr(t) 1 -1 3 H(j2πf) -1 3 Q 1 q(t) qr(t) 1 -3 t π -3 Q -1 t Lowpass H(j2 f) -1 t -3 -3 t Lowpass Sample t TimesSample • AtNoise receiver, perturbs noise• Noise perturbs perturbssampled sampled sampled I/Q I/Q values I/Q Times – Constellation points no longer consist of single Q Q valuespoints for eachvalues symbol – Constellation– Constellation points no pointslonger no longer 10 10 consist of single dots for each • Issue:consist What’s of the single best way dots of matchingfor eachreceived 11 symbol 11 I/Q samplessymbol with their corresponding transmitted I symbols? • Issue: what is the best 01 I 01 • Issue: whatway isto thematch best received I/Q 00 way to matchsamples received with their I/Q 00 00 01 11 10 corresponding symbols? 45 samples with their 00 01 11 10 corresponding6.082 Spring 2007 symbols? Digital Modulation (Part I), Slide 9

6.082 Spring 2007 Digital Modulation (Part I), Slide 9 Symbol Selection based on Slicing Symbol Selection Based on Slicing Baseband Input Receiver Output 3 3 Receiver 1 it(t) ir(t) 1 -1 SymbolNoise SelectionH(j2 Basedπf) -1on Slicing I -3 -3 t Baseband Input Lowpass Receiver Output t 3 π 3 2cos(2 fot) 2cos(2πReceiverf t) 1 it(t) o ir(t) 1 π π Noise π I 2sin(2-1 fot) 2sin(2 fot) H(j2 f) -1 3 -3 3 -3 t Lowpass t 2cos(2πf t) π 1 q(t) o 2cos(2qfro(t)t) 1 t π 2sin(2 fot) 2sin(2π πf t) Q -1 3 H(j2 f) o -1 3 -3 1 qt(t) -3 qr(t) 1 t -1 Lowpass H(j2πf) -1 Qt -3 -3 t Lowpass Sample t TimesSample •• ReceiverMatch matches I/Q samplesI/Q• samplesMatch toI/Qto corresponding samples to symbols Times based on decision boundaries Q Q their corresponding 00 01 11 10 their corresponding 00 01 11 10 symbols based on 00 • Decisionsymbols boundaries based are ondecision also called regions slicing levels 00 Decision 01 decision regions – Choose decision regions Boundaries I to minimize symbolDecision 0111 I – Choose decision regionserrors Boundaries 10 to minimize symbol– Decision boundaries are 11 also called slicing levels errors Decision Boundaries 1046 – Decision boundaries6.082 Spring 2007 are Digital Modulation (Part I), Slide 10 also called slicing levels Decision Boundaries 6.082 Spring 2007 Digital Modulation (Part I), Slide 10 TransitioningTransitioning BetweenBetween SymbolsSymbols Baseband InputTransitioning Between SymbolsReceiver Output 3 3 Baseband Input Receiver Receiver Output 1 it(t) ir(t) 1 -1 3 Noise H(j2πf) -1 3 I 1 Receiver i (t) -3 it(t) r -3 1 -1 t Noise LowpassH(j2πf) -1 It π π -3 2cos(2 fot) 2cos(2 fot) -3 t π π Lowpass t 2sin(2 f2cos(2ot) πf t) 2sin(2 fot)π 3 o 2cos(2 fot) 3 π π 2sin(2 fot) 2sin(2 fot) 1 3 qt(t) qr(t) 1 3 -1 H(j2πf) -1 Q 1 qt(t) qr(t) 1 -3 H(j2πf) -3 Q -1 Lowpass -1 -3 t -3 t Lowpass t t Sample SampleTimes Times • Transition behavior influenced by transmit I/Q input Q • Transition• Transition behavior behavior between between Q waveforms and receive filter 00 01 11 10 symbolssymbols is influenced is influenced by byboth both 00 01 11 10 00 –transmitToday:transmitFocus I/Q on I/Q whatinput theinput waveformstransmitter waveforms does and receiveand receive filter filter 01 – Ignore impact of noise for this analysis II – We– willWe focus will focus on impact on impact of of 11 transition behavior at transition behavior at 10 transmitter today 10 transmitter today 47 – Ignore the impact of noise for – Ignore the impact of noise for this analysis 6.082 Springthis 2007 analysis Digital Modulation (Part I), Slide 11 6.082 Spring 2007 Digital Modulation (Part I), Slide 11 Transitions and the Transmitted Spectrum

• Want transmitted spectrum with minimal bandwidth

• Issue: Sharply changing I/Q leads to very wide bandwidth spectrum

48 Impact of Transmit Filter

• Transmit filter shapes the pulses, enables reduced bandwidth for transmitted spectrum

• Next Issue: Can lead to bit errors

49 Impact of Low Bandwidth Transmit Filter

• Lowering the transmit bandwidth leads to increased bit errors

50 Eye Diagrams

• Eye Diagram: Wrap signal back onto itself in periodic time intervals, retaining all “traces”

51 Looking at Many Symbols

• Increasing the number of symbols eventually reveals all possible symbol transition trajectories – Intuitively displays the impact of filtering on transmitted signal

52 Assessing the Quality of an Eye Diagram

Sample times: • Eye diagram allows visual inspection of the impact of sample time and decision boundary choices – Large “eye opening” implies less vulnerability to symbol errors

53 Relating Eye Diagrams to Constellation

• Open eye diagrams lead to tight symbol groupings in constellation

– Assumes proper sample time placement

54 Impact of Low Transmit Bandwidth

• Eye diagrams show increased inter-symbol interference (ISI)

– Also show symbol decision sensitivity to sample timing placement

55 Digital Modulation: Summary

• Digital modulation: Sends discrete-valued symbols through an analog communication channel – Receiver must sample I/Q signals at the appropriate time

– Receiver matches I/Q sample values to corresponding symbols based on decision regions

– Constellation diagrams are a convenient tool to see likelihood of bit errors being made

• Choice of transmit filter: Tradeoff between achieving a minimal bandwidth transmitted spectrum and minimal ISI – Eye diagrams: A convenient tool to see effects of ISI and sensitivity of bit errors to sample time choice

56 Thursday Topic: Digital Communications II: Receiver Processing, Performance

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