Lecture 3 Vladimir Stojanović - MIT OpenCourseWare · PDF file6.973 Communication System Design – Spring 2006 Massachusetts Institute of Technology Cite as: Vladimir Stojanovic,

6.973 Communication System Design – Spring 2006Massachusetts Institute of Technology

Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology.

Downloaded on [DD Month YYYY].

Bandlimited communication systems

Lecture 3Vladimir Stojanović

Passband channel exampleTwo-ray wireless channel (multi-path – 1+0.9D)

Multi-path creates notching in frequency domainJust slide the frequency window to bb

Add single-sided noiseCite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.

MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

6.973 Communication System Design 2

Bandlimited channel example

0 2 4 6 8 10

-60

-50

-40

-30

-20

-10

0

frequency [GHz]

Atte

nuat

ion

[dB

]

0 1 2 3

0

0.2

0.4

0.6

0.8

1

nspu

lse

resp

onse

Tsymbol=160ps

Low-pass channel causes pulse attenuation and dispersionNotches cause ripples in time domainMakes it hard to transmit successive messages




Inter-Symbol Interference (ISI)

0 2 4 6 8 10 12 14 16 180

0.2

0.4

0.6

0.8

1

Symbol time

Am

plitu

de

Error!

Middle sample is corrupted by 0.2 trailing ISI (from the previous symbol), and 0.1 leading ISI (from the next symbol) resulting in 0.3 total ISIAs a result middle symbol is detected in error




Bandlimited communication systemsBlock detector vs. symbol-by-symbol

Block of K symbols – MK messagesMAP/ML detector complexity grows exponentially

MK basis functions (branches in the matched filter)Sequence detection can bound that growth

Simpler detector is “Symbol-By-Symbol”Optimal for AWGN channel




BlockDetector

R SBSdetector

yp(t)

yp(t)

φk(KT - t)

φ2(KT - t)

φ1(KT - t)

X

ϕp(T - t)

t = KT

t = kT, k = 0,...,K - 1

Xk k = 0,...,K -1

yp(t)

.

.

.

Figure by MIT OpenCourseWare.

Symbol-by-symbol detection

Suffers significantly from Intersymbol-interference (channel memory), so need to remove ISI to get almost AWGN channelNeed to adapt basis functions to the particular channel, to avoid ISIAlternatively, use equalization to remove ISI




Bandlimitedchannel

Receiver

x (t)n (t)

SamplerEstimate of inputsymbol at time k

Input symbolat time k

xk

yk zk

xk+ Matchedfilter

SBSdetectorRmod h(t)


Vector channel - revisited




x(t) h(t)

h(t)xkn

np (t)

np (t)

xp (t)

yp (t)

yp (t)

yp (t)

xp (t)xn (t)

np (t)

pn(t)

ϕn(t)

xp (t)

xkn

+

+

+


Mean-distortionTreat ISI as noise

Peak-distortionTreat worst-case ISI as constellation offset



ISI impact


pulse response p(t)

sample attimes kT

discretetime

receiverxk

xp,k xp(t)

np(t)

yp(t) ykxky(t)

Σ

q(t) = ϕp(t)*ϕp(-t)

||p|| ϕp(t) ϕp (-t)





Matched Filter BoundYou can’t do better with successive transmissions than with one-shotMatched filter collects the pulse energy ||p||2Then calculate performance as on AWGN

Example – binary transmission

Will use MFB to compare different ISI compensation techniques

10

Nyquist criterion – 6.011 revisitedA channel specified by pulse response p(t) is ISI free if

Nyquist frequency: w=pi/T or f=1/2TCite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.

MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

6.973 Communication System Design

Raised-cosine pulsesCan have “excess” bandwidth as long as there is symmetry that “fills” the aliased spectrum flat




2

Basic equalization concepts

Zero-forcing equalizationFlattens equalized channel transfer function

H(D)=Q(D)*W(D) Wzfe(D)=1/(Q(D)||p||)

X =

Channel Q(w) Equalizer W(w) Equalized

=>

Q(D) W(D)kx ˆkxky




Linear equalizationZero-forcing not good on channels with nulls

Equalizer enhances noiseRemember, Pe depends on both noise and ISIBalance noise and ISI in the mean-square sense

Minimizing MMSE wrt. WkSame as using the orthogonality principle




ZFE vs. MMSE - LE




0ω

Q(e-jωT )WZFE (e

-jωT )

WMMSE-LE (e-jωT )

Tπ 0

ωTπ

SNR||p||


Example: ZFE vs. MMSE LE1+0.9D channel

Equalizer response

zfemmse




Fractional equalizers

Oversampling in the receiverCan merge matched filter and equalizer

Can reconstruct original signal from oversampled signal (as long as original is band-limited)




xk +p ϕp(t)xp,k xp(t)

np(t)

yp(t) y(t)Fractionally-spaced equalizer

Wk

gain T

anti-aliasfilter

sampleat timesk(T/l)

zk

yk

Figure by MIT OCW.


ISI channel modelOversampled channel representation (3x e.g)




Finite length equalizer formulation

Write convolution as multiply with Toeplitz matrix

yk zkxk p w

k kz wPx=




ZFE and MMSE solutionZero forcing equalizer (ZFE)

Minimum-mean square error (MMSE) equalizer

=>

( ) 11 1 , 1 [00...1...00]T T T T

k k k zfe zfez x wPx w P w P PP−

−∆ ∆ ∆ ∆= = ⇒ = ⇒ = =




Decision feedback equalizer

Feed-forward equalizerMatched + whitening filter + remove pre-cursor ISI

Feed-back equalizerRemoves trailing ISITo get w, first puncture the channel matrix to emulate the effect of feedback on the equalized pulse response wPThen, get b from the causal taps of equalized pulse response wP

yk zkxk p w

b

0 2 4 6 8 10 12 14 16

18

0

0.2

0.4

0.6

0.8

1

Symbol time

Am

plitu

de

Feedbackequalization

xk-∆




MMSE DFESelects the feedback taps




Basic multitone modulation

Best performance if basis functions are tailored to the channel

Use each tone as a basis functionEach tone transmits narrow QAM signal and satisfies Nyquistcriterion – i.e. no ISI per tonePut less energy where channel is bad or where there is more noise




Frequency Frequency Frequency

Frequency Frequency Frequency

Bits/chan

Bits/chan

Bits/chan

Bits/chanRF

xtalk

Gain

Gain

=


A bit of history1948 Shannon constructs capacity bounds

AWGN channel with linear ISI – effectively uses multi-tone modulation

Analog multi-tone1958 Collins Kineplex modem (first voiceband modem) – analog multitone1964 Holsinger’s MIT thesis – modem that approximates Shannon’s “water-filling”1967 Saltzberg, 1973 Bell Labs, 1980 IBM …

Digital multi-tone ~ 1990sDMT for DSL - Major push by prof. Cioffi’s group at StanfordUse DSP power to improve the robustness and algorithms for discrete multi-tone modulationWe will mostly focus on this type of modulation




Basic multitone transmission

Each tone sees AWGN channel (no ISI)N QAM-like symbols (complex)1 PAM symbol




X0

X1

XN-1

XN

ϕ0(t)

ϕ1(t)

ϕ0(-t)

ϕ1(-t)

ϕN-1(t) ϕN-1(-t)

ϕN (-t)ϕN (t) ϕp,n(t) = ϕn(t) * h(t)

X X

X

X

X

X

e j2πf1t

e j2πfN-1t

e j2πfN t

e-j2πf1t

e-j2πfN-1t

e-j2πfN t

ℜ

ℜ

ℜ

h(t) ++

n(t)p sh pa ls ie t

Yn = Hn . Xn+ Nn

Y0

Y1

YN-1

YN

bitsT=kNT' =kT

N = 2N

+bits...

bn = 12log2 1+

SNRnΓ ((ϕn(t) =

. sinc1T T (( t

ENCODER

DECODER


X( f )N = 2N

X0

Y0 Y

1 Y2

X1

f2

f1

f2

f1

X2

XN-1

fN-1

fN

XN

YN-1

fN-1

YN

fN

H( f )

Y( f )

Yn Hn Xn≈

input

output

.

The effect of the channel

Each channel can be treated as AWGNWith only one basis function – hence simple symbol-by-symbol detector is optimal





Gap review




3

2.5

2

1.5

1

0.5

02 4 6 8 10 12 14 16

bits

/dim

ensi

onSNR (dB)

0 dB

3 dB

6 dB

9 dB

Achievable bit rate for various gaps

Illustration of bit rates versus SNR for various gaps.

22b 1 (dB)

b .5 1 2 3 4 5

SNR for Pe = 10-6 (dB) 8.8

8.8 8.8 8.8 8.8 8.8 8.8

13.5 20.5 26.8 32.9 38.9

0 4.7 11.7 18.0 24.1 30.1

(dB)

Table of SNR Gaps for Pe = 10-6.



Example – simplified multitone1+0.9D channel

With gap 4.4 dB

Put unit energy per dimension (simply guessed)Same as baseband DFE

Data rate 1bit/dimensionRe-calculate the necessary SNR – margin

SNRmfb=10dBSNRmultitone=8.8dB (with more tones to better approx no-ISI case)SNRdfe=7.1dBCan do even better with multitone, if allocated energy properly




Find optimum energy allocation that maximizes b for given total energy constraint

b is a convex function in energy/dimension

Use Lagrange multipliers to solve for εn

=>

ddεn




Water-filling derivation

Water-filling spectrumFlip the channel and pour in energy like water

Channel




E0 E1 E2 E3

L

g0

L

g1

L

g3

L

g2L

g4

L

g5

Energy

constant

0 1 2 3 4 5

Subchannelindex

+ =σ2

H 2 constant.n+ = constantnn

nng


Water-fill loading algorithms

Rate maximization

Margin maximization




Rate-adaptive loadingSolve through matrix inversion

Solve iteratively





Water-filling example (rate-adaptive)1+0.9D again (Gap=1, so calculating capacity)

Try 8 dim first

Try 7 dim nextCapacity=1.55bits/dim

=>



Ener

gy

1.29

0 1 2 3 4 5 6 7n

1.29 1.29 1.29 1.29 1.29

2.968

1.24

E0 E1 E1 E2 E2 E3 E3

1.23 1.23 1.19 1.19 .96 .96

2.968

1.29

119.94

117.03

117.03

110

110

1.0552


33

SummaryBandlimited communication

Block vs. symbol-by-symbol detectorTry to make bandlimited channel look AWGN

Use complex block detectors to orthogonalize basis functions (MAP)Simplify with equalization+sbs detectorGenerate basis functions that don’t loose orthogonality when passing through frequency selective channle (multitone modulation)

EqualizationZFE removes ISI but enhances noiseTrade-off by designing MMSE equalizerDFE removes trailing ISI without noise enhancement

MultitoneOptimal transmission with proper allocation of energy/dimension (waterfilling)

Next – practical loading algorithms and DMT/OFDM, Vector coding



6.973 Communication System Design

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Lecture 3 Vladimir Stojanović - MIT OpenCourseWare · PDF file6.973 Communication System Design – Spring 2006 Massachusetts Institute of Technology Cite as: Vladimir Stojanovic,