Wireless PHY: Digital Demodulation and Wireless Channels Y. Richard Yang 09/13/2012

Outline Admin and recap Digital demodulation Wireless channels

Admin Assignment 1 posted

Recap Demodulation Low pass filter and FIR Convolution Theorem Digital modulation/demodulation ASK, FSK, PSK General representation

Recap: gi() for BPSK 1: 0: Note: g1(t) = -g0(t) g1(t) = cos(2πfct) t in [0, T] 0: g0(t) = -cos(2πfct) t in [0, T] Note: g1(t) = -g0(t) cos(2πfct)[0, T] 1 -1 g0(t) g1(t)

Recap: Signaling Functions gi() for QPSK 11: cos(2πfct + π/4) t in [0, T] 10: cos(2πfct + 3π/4) t in [0, T] 00: cos(2πfct - 3π/4) t in [0, T] 01: cos(2πfct - π/4) t in [0, T] Q I 11 01 10 00

Recap: QPSK Signaling Functions as Sum of cos(2πfct), sin(2πfct) 11 00 10 11: cos(π/4 + 2πfct) t in [0, T] -> cos(π/4) cos(2πfct) + -sin(π/4) sin(2πfct) 10: cos(3π/4 + 2πfct) t in [0, T] -> cos(3π/4) cos(2πfct) + -sin(3π/4) sin(2πfct) 00: cos(- 3π/4 + 2πfct) t in [0, T] sin(3π/4) sin(2πfct) 01: cos(- π/4 + 2πfct) t in [0, T] sin(π/4) sin(2πfct) 01 [cos(3π/4), sin(3π/4)] [cos(π/4), sin(π/4)] cos(2πfct) [cos(3π/4), -sin(3π/4)] [-sin(π/4), cos(π/4)] We call sin(2πfct) and cos(2πfct) the bases.

Recap: Demodulation/Decoding Considered a simple setting: sender uses a single signaling function g(), and can have two actions send g() or nothing (send 0) How does receiver use the received sequence x(t) in [0, T] to detect if sends g() or nothing?

Recap: Design xT x2 x1 x0 h0 h1 h2 hT * Streaming algorithm: use all data points in [0, T] As each sample xi comes in, multiply it by a factor hT-i-1 and accumulate to a sum y At time T, makes a decision based on the accumulated sum at time T: y[T] xT x2 x1 x0 h0 h1 h2 hT *

Determining the Best h where w is noise, Design objective: maximize peak pulse signal-to-noise ratio

Determining the Best h Assume Gaussian noise, one can derive Using Fourier Analysis and Convolution Theorem:

Determining the Best h Apply Schwartz inequality By considering

Determining the Best h

Determining Best h to Use xT x2 x1 x0 h0 h1 h2 hT * xT x2 x1 x0 gT g2 g1 g0 *

Summary of Progress After this “complex” math, the implementation/interpretation is actually the following quite simple alg: precompute auto correlation: <g, g> compute the correlation between received x and signaling function g, denoted as <x, g> if <x, g> is closer to <g, g> output sends g else output sends nothing

Applying Scheme to BPSK Consider g1 alone, compute <x, g1>, check if close to <g1, g1>: |<x, g1> - <g1, g1>| Consider g0 alone, compute <x, g0>, check if close to <g0, g0>: |<x, g0> - <g0, g0>| Pick closer if |<x, g1> - <g1, g1>| < |<x, g0> - <g0, g0>| pick 1 else pick 0 cos(2πfct)[0, T] 1 -1 g1(t) g0(t)

Applying Scheme to BPSK since g0 = -g1 <x, g0> = - <x, g1> <g0, g0> = - <g0, g1> rewrite as if |<x, g1> - <g1, g1>| < |<x, g1> - <g0, g1>| pick 1 else pick 0 cos(2πfct)[0, T] 1 -1 g1(t) g0(t)

Interpretation For any signal s, <s, g1> computes the coordinate of s when using g1 as a base cleaner if g1 is normalized, but we do not worry about it yet <x, g1(t)> g1=cos(2πfct)[0, T] <g0(t), g1(t)> <g1(t), g1(t)> =-<g1(t), g1(t)>

Applying Scheme to QPSK: Attempt 1 Consider g00 alone, compute <x, g00> … Consider g01 alone, compute <x, g01> … Consider g10 alone, compute <x, g10> … Consider g11 alone, compute <x, g11> … Issues Complexity: Need to compute M correlation, where M is number of signaling functions Think of 64-QAM Objective the previous scheme is defined for a single signaling function, does it work for M?

Decoding for QPSK using bases 4 signaling functions g00(), g01(), g10(), g11() For each signaling function, computes correlation with the bases (cos(), sin()), e.g., g00: [a00, b00] What is the meaning of a00, b00? For received signal x, computes ax=<x, cos> and bx=<x, sin> (how many correlation do we do now?) Question: what is the meaning of a00, b00

QPSK Demodulation/Decoding sin(2πfct) [a00,b00] [a01,b01] [ax,bx] cos(2πfct) [a10,b10] [a11,b11] Q: how to decode?

Look into Noise Assume sender sends gm(t) [0, T] Receiver receives x(t) [0, T] Consider one sample where w[i] is noise Assume white noise, i.e., prob w[i] = z is

Likelihood What is the likelihood (prob.) of observing x[i]? it is the prob. of noise being w[i] = x[i] – g[i] What is the likelihood (prob.) of observing the whole sequence x? the product of the probabilities

Likelihood Detection Suppose we know Maxim likelihood detection picks the m with the highest P{x|gm}. From the expression We pick m with the lowest ||x-gm||2

Back to QPSK Yry: Ignored noise effect

QPSK Demodulation/Decoding sin(2πfct) [a00,b00] [a01,b01] [ax,bx] cos(2πfct) [a10,b10] [a11,b11] Q: what does maximum likelihood det pick?

General Matched Filter Detection: Implementation for Multiple Sig Func. Basic idea consider each gm[0,T] as a point (with coordinates) in a space compute the coordinate of the received signal x[0,T] check the distance between gm[0,T] and the received signal x[0,T] pick m* that gives the lowest distance value

Computing Coordinates Pick orthogonal bases {f1(t), f2(t), …, fN(t)} for {g1(t), g2(t), …, gM(t)} Compute the coordinate of gm[0,T] as cm = [cm1, cm2, …, cmN], where Compute the coordinate of the received signal x[0,T] as x = [x1, x2, …, xN] Compute the distance between r and cm every cm and pick m* that gives the lowest distance value

Example: Matched Filter => Correlation Detector received signal x

BPSK vs QPSK fc: carrier freq. Rb: freq. of data 10dB = 10; 20dB =100 11 10 00 01

Spectral Density = bit rate ------------------- width of spectrum used BPSK vs QPSK A major metric of modulation performance is spectral density (SD) Q: what is the SD of BPSK vs that of QPSK? Q: Why would any one use BPSK, given higher QAM? Spectral Density = bit rate ------------------- width of spectrum used

Context Previous demodulation considers only additive noise, and does not consider wireless channel’s effects We next study its effects

Outline Admin and recap Digital demodulation Wireless channels

Signal Propagation

Antennas: Isotropic Radiator Isotropic radiator: a single point equal radiation in all directions (three dimensional) only a theoretical reference antenna Radiation pattern: measurement of radiation around an antenna z y z ideal isotropic radiator y x x Q: how does power level decrease as a function of d, the distance from the transmitter to the receiver?

Free-Space Isotropic Signal Propagation In free space, receiving power proportional to 1/d² (d = distance between transmitter and receiver) Suppose transmitted signal is cos(2ft), the received signal is Pr: received power Pt: transmitted power Gr, Gt: receiver and transmitter antenna gain  (=c/f): wave length Sometime we write path loss in log scale: Lp = 10 log(Pt) – 10log(Pr)

Log Scale for Large Span dB = 10 log(times) Log Scale for Large Span ~100B 10,000 times 10,000 x 1,000 40 dB 40 + 30 = 70 dB ~10M 1000 times 30 dB ~10K Slim/Gates Obama

Path Loss in dB dB = 10 log(times) 40 dB 40 + 30 = 70 dB 30 dB 10 W 10,000 x 1,000 40 dB 40 + 30 = 70 dB power 1 mW 1000 times 30 dB 1 uW source d1 d2

dBm (Absolute Measure of Power) dBm = 10 log (P/1mW) dBm (Absolute Measure of Power) 10 W 40 dBm 10,000 times 10,000 x 1,000 40 dB 40 + 30 = 70 dB power 1 mW 1000 times 30 dB 1 uW -30 dBm source d1 d2

Number in Perspective (Typical #) Data rate (Mbps) Receive threshold (dBm) Signal/Noise (dB) 6 -82 6.02 9 -81 7.78 12 -79 9.03 18 -77 10.79 24 -74 17.04 36 -70 18.8 48 -66 24.05 54 -65 24.56

Exercise: 915MHz WLAN (free space) Transmit power (Pt) = 24.5 dBm Receive sensitivity = -64.5 dBm Receiving distance (Pr) = Gt=Gr=1

Two-ray Ground Reflection Model Single line-of-sight is not typical. Two paths (direct and reflect) cancel each other and reduce signal strength Pr: received power Pt: transmitted power Gr, Gt: receiver and transmitter antenna gain hr, ht: receiver and transmitter height

Exercise: 915MHz WLAN (Two-ray ground reflect) Transmit power (Pt) = 24.5 dBm Receive sensitivity = -64.5 dBm Receiving distance (Pr) = Gt=Gr=hr=ht=1

Real Antennas Q: Assume frequency 1 Ghz,  = ? Real antennas are not isotropic radiators Some simple antennas: quarter wave /4 on car roofs or half wave dipole /2  size of antenna proportional to wavelength for better transmission/receiving /4 /2 Q: Assume frequency 1 Ghz,  = ?

Figure for Thought: Real Measurements

Signal Propagation: Complexity Receiving power additionally influenced by shadowing (e.g., through a wall or a door) refraction depending on the density of a medium reflection at large obstacles scattering at small obstacles diffraction at edges diffraction reflection refraction scattering shadow fading

Signal Propagation: Complexity Details of signal propagation are very complicated We want to understand the key characteristics that are important to our understanding

Outline Admin and recap Digital demodulation Wireless channels Intro shadowing

Shadowing Signal strength loss after passing through obstacles Same distance, but different levels of shadowing: It is a random, large-scale effect depending on the environment

Example Shadowing Effects i.e. reduces to ¼ of signal 10 log(1/4) = -6.02

JTC Indoor Model for PCS: Path Loss Shadowing path loss follows a log-normal distribution (i.e. L is normal distribution) with mean: A: an environment dependent fixed loss factor (dB) B: the distance dependent loss coefficient, d : separation distance between the base station and mobile terminal, in meters Lf : a floor penetration loss factor (dB) n: the number of floors between base station and mobile terminal

JTC Model at 1.8 GHz

Outline Admin and recap Digital demodulation Wireless channels Intro Shadowing Multipath

Multipath Signal can take many different paths between sender and receiver due to reflection, scattering, diffraction

Multipath Example: Outdoor Example: reflection from the ground or building ground

Multipath Effect (A Simple Example) Assume transmitter sends out signal cos(2 fc t) d1 d2 phase difference:

Multipath Effect (A Simple Example) Where do the two waves totally destruct? Q: where do the two waves construct?

Option 1: Change Location If receiver moves to the right by /4: d1’ = d1 + /4; d2’ = d2 - /4; -> By moving a quarter of wavelength, destructive turns into constructive. Assume f = 1G, how far do we move?

Option 2: Change Frequency

Multipath Delay Spread RMS: root-mean-square

Multipath Effect (moving receiver) example d d1 d2 Suppose d1=r0+vt d2=2d-r0-vt d1d2

Derivation See http://www.sosmath.com/trig/Trig5/trig5/trig5.html for cos(u)-cos(v)

Derivation See http://www.sosmath.com/trig/Trig5/trig5/trig5.html for cos(u)-cos(v)

Derivation See http://www.sosmath.com/trig/Trig5/trig5/trig5.html for cos(u)-cos(v)

Derivation See http://www.sosmath.com/trig/Trig5/trig5/trig5.html for cos(u)-cos(v)

Derivation See http://www.sosmath.com/trig/Trig5/trig5/trig5.html for cos(u)-cos(v)

Derivation See http://www.sosmath.com/trig/Trig5/trig5/trig5.html for cos(u)-cos(v)

Waveform v = 65 miles/h, fc = 1 GHz: fc v/c = 109 * 30 / 3x108 = 100 Hz 10 ms deep fade Q: How far does a car drive in ½ of a cycle?

Multipath with Mobility

Effect of Small-Scale Fading no small-scale fading

Multipath Can Spread Delay signal at sender LOS pulse Time dispersion: signal is dispersed over time multipath pulses signal at receiver LOS: Line Of Sight

JTC Model: Delay Spread Residential Buildings

Multipath Can Cause ISI Dispersed signal can cause interference between “neighbor” symbols, Inter Symbol Interference (ISI) Assume 300 meters delay spread, the arrival time difference is 300/3x108 = 1 ns if symbol rate > 1 Ms/sec, we will have serious ISI In practice, fractional ISI can already substantially increase loss rate signal at sender LOS pulse multipath pulses signal at receiver LOS: Line Of Sight

Summary: Wireless Channels Channel characteristics change over location, time, and frequency Received Signal Large-scale fading Power power (dB) path loss log (distance) time small-scale fading signal at receiver LOS pulse multipath pulses frequency

Preview: Challenges and Techniques of Wireless Design Performance affected Mitigation techniques Shadow fading (large-scale fading) Fast fading (small-scale, flat fading) Delay spread (small-scale fading) received signal strength use fade margin—increase power or reduce distance bit/packet error rate at deep fade diversity equalization; spread-spectrum; OFDM; directional antenna ISI

Representation of Wireless Channels Received signal at time m is y[m], hl[m] is the strength of the l-th tap, w[m] is the background noise: When inter-symbol interference is small: (also called flat fading channel)

Backup Slides

Received Signal d2 d1 receiver

Multipath Fading with Mobility: A Simple Two-path Example r(t) = r0 + v t, assume transmitter sends out signal cos(2 fc t) r0 More detail see page 16 Eqn. (2.13): http://www.eecs.berkeley.edu/~dtse/Chapters_PDF/Fundamentals_Wireless_Communication_chapter2.pdf

Received Waveform v = 65 miles/h, fc = 1 GHz: 10 ms deep fade v = 65 miles/h, fc = 1 GHz: fc v/c = 109 * 30 / 3x108 = 100 Hz Why is fast multipath fading bad?

Small-Scale Fading

Multipath Can Spread Delay signal at sender LOS pulse Time dispersion: signal is dispersed over time multipath pulses signal at receiver LOS: Line Of Sight

RMS: root-mean-square Delay Spread RMS: root-mean-square

Multipath Can Cause ISI dispersed signal can cause interference between “neighbor” symbols, Inter Symbol Interference (ISI) Assume 300 meters delay spread, the arrival time difference is 300/3x108 = 1 ms if symbol rate > 1 Ms/sec, we will have serious ISI In practice, fractional ISI can already substantially increase loss rate signal at sender LOS pulse multipath pulses signal at receiver LOS: Line Of Sight

Summary: Wireless Channels Channel characteristics change over location, time, and frequency Received Signal Large-scale fading Power power (dB) path loss log (distance) time small-scale fading frequency

Dipole: Radiation Pattern of a Dipole http://www.tpub.com/content/neets/14182/index.htm http://en.wikipedia.org/wiki/Dipole_antenna

Free Space Signal Propagation 1 t at distance d ?

Why Not Digital Signal (revisited) Not good for spectrum usage/sharing The wavelength can be extremely large to build portal devices e.g., T = 1 us -> f=1/T = 1MHz -> wavelength = 3x108/106 = 300m

Exercise Suppose fc = 1 GHz (fc1 = 1 GHz, fc0 = 900 GHz for FSK) Bit rate is 1 Mbps Encode one bit at a time Bit seq: 1 0 0 1 0 1 Q: How does the wave look like for? 11 10 00 01 Q I A t