Lecture IIIntroduction to DigitalCommunications

Following Lecture III next week:4. …Matched Filtering (…continued from L2) (ch. 2 – part 0 “Notes”)

5. Statistical Decision Theory - Hypothesis Testing(ch. 4 – part 0 “Notes”)

2. Antipodal transmission (a special case of PAM)3. Finite Energy Signal Space representations4. Matched Filtering in AWGN for PAM antipodal links• (ch. 2 – part 0 “Notes”)

V4 –just moved to L2 the slides that we did not have time for

Review (and elaboration) of L1(3 slides)

Target parameters for communication system design optimization

• Max Transmission rate (bit-rate, symbol-rate)• Min Error Probability (bit/symbol/block error rate, outage)• Min Power (PSD, SNR)• Min Bandwidth (max spectral efficiency bps/Hz)• Min Complexity (Cost)• Min Delay• (multi-user) Max # of users• (multi-user) Min Mutual Interference• Other parameters

Pulse Modulation and Pulse Amplitude Modulation (PAM)

tt

PAM

P(t)AK

Figure 1.12:

PULSE MODULATOR

( ) ( )

( ) ( )

( )

0

( )

0

1 1

: ( )

: ( )

: ( )M M

H S t

H S t

H S t

…

…

Index

CODER (MAPPER)

Bitstream ( )( )

k

kS t Tk

( ) ( )kk

S t p tA kT

( )S t

( )( ) ( ) ( )S t A p t

LTI

t

PAM

( )p t

“Single shot” @ t=0 – Isolated Pulse Amp. Mod.

0 ( )A t

0 ( )A p t

t

0 0( )A A

( )p t

0 ( )A p tt

End of Review

General PAM Link Analysis

SlicerAK S(t)CODER

PAM

g(t)

SAMP [T]Bits CH. Filter

b(t)

N(t)

RX Filter

f(t)

r(t) q(t) qK

TX Medium/Channel RX

Figure 1.17:

(“multi-shot” analysis in TA classes…)

( )g t

Channel Imp. Resp.

( )b t

PAMPulseshape

0( ) ~ [0, / 2]N t WGN N

( )f t

RX filter Imp. Resp.

SAMPLER

SLICER /DECISION

( )0A Single-shot:

kAMulti-shot: t kT0t

Single-shot:

Antipodal transmission system analysis

Antipodal digital link

LTI Filter

f(t)

Pulse Modulator

N(t)

Slicer“0” g(t)“1” -g(t)

Bitstream

1010111...

White Gaussian noise one-sided spectral density No

RXTX

Medium

Antipodal modulation: Special case of PAM modulation:modulating symbols equal +/-1

0 0( ) ( ); 1A Ag t g t

A0 , A1 , ... AkBitstreamCoder

"0" -1"1" 1

PAMg(t)

s(t)Ak {-1,1}

g(t)

0

1

T

s(t)

0

1

T 2T 3T 4T

bitstream: ...1 0 1 1 0 1....

symbolstream: 1 -1 1 1 -1 1....

waveform s(t) = g(t) - g(t-T) + g(t-2T)+ g(t-3T) - g(t-4T) + g(t-5T)

Example of antipodalTransmitter – flat pulses

Antipodal digital link analysis

RX Filter

f(t)

Pulse Modulator

N(t)

SlicerBitstream

1010111...

White Gaussian noise one-sidedspectral density No

" ":

" "

0

1 :

( )

( )

h t

h t

RX Filter

f(t)Medium

b(t)SlicerPAM MOD

1 q t 0q

0N t ~WGN 0,N 2

t 0

g(t)g(t)

h(t)

N(t) n t 0nf(t)

F t 0signal analysis:

noise analysis:

RXMediumTX

TX+ Medium RX

δ(t) p t

0 0q (0)p p h(t) f(t)h(t)

00 0pq n

Signal propagation through the receive filter

done on the board

Noise propagation through the receive filter

N(t) n t 0nf(t)

F t 02~ [0, ]nN

RX Filter

f(t)

Pulse Modulator

N(t)

SlicerBitstream

1010111...

White Gaussian noise one-sidedspectral density No

" ":

" "

0

1 :

( )

( )

h t

h t

Effective gaussian channel

00 0pq n

0q -axis

0p0p 0

RX Filter

f(t)

Pulse Modulator

N(t)

SlicerBitstream

1010111...

White Gaussian noise one-sidedspectral density No

" ":

" "

0

1 :

( )

( )

h t

h t

The statistics of decision

00 0pq n

-4 -2 2 4

0.1

0.2

0.3

0.4

0p0p 0

)0(0( | )p Hq )1(

0( | )p Hq

DECIDE ( )0H DECIDE ( )1H

0q0n

Effective gaussian scalar channel

Self-read

The gaussian-Q function

The Q-function is the complementary cdf of a normalized gaussian r.v.

22

( )2

zeP z

0

Area Q t

t

Figure 1.36:

The gaussian Q-function^

Gaussian integral functionor Q-function

=Prob. of “upper tail” of normalized gaussian r.v.

22

( )2

zeP z

0

Area Q t

t-4 -2 2 4

0.2

0.4

0.6

0.8

1

5 70.001 3.1{0.5, , 100. 0.022, , , 2 0 }16 .93 1

{ [0], , , [3],[1 [2 [5] ,] }][4]Q QQQ QQ

[ ]Q t

t

Q(t) and some of its upper bounds

t

2z /2

tQ(t) 1/ 2π e dz

2t /2Q(t) 1/ 2π t e

2t /2Q(t) 1/2 e

Figure 1.38:

The dog wagging the (gaussian) tail vs. the tail wagging the dog

22

( )2

zeP z

0

t-t

Q t Q -t 1

~ [0,1]N

^

Deviation oft from the meanmeasured in units of the standard deviation

^ ^

^

0,1

Calculating cdf-s of gaussian variables with the Q-function

^ ^

done on the board

Self-study

Error Probability calculation for the antipodal link

Probability of error (conditioned on the “0” hypothesis)

0q-4 -2 2 4

0.1

0.2

0.3

0.4

)0(0( | )p Hq

0p

0p00n

0 ~ [0,1]Nnnoise indep.

of signal

Probability of error (conditioned on the “1” hypothesis)

Self-read

The two conditional Error Probabilities

(graphical representation)

0p 0p

(0)0

( )q H

p q(1)

0

( )q H

p q

Decide (0)H Decide (1)H

0

(0) (0)0Area Pr q 0 H Pr e H

(1) (1)0< Area Pr q 0 H Pr e H

0

n

Qp

Total error probability (I)

on the board

Total error probability (II)

on the board

Total error probability (III)

This completes the error prob. eval. for the antipodal linkNext: optimize it, i.e. design the system to reduce BER

SNR=

0

neP Q

p

antipodal system

Signal Spaces

Signal Spaces (vector spaces of functions of time)

denotes a vector

The key vector property:Vector

Examples:(0) Arrows(i) D-tuples(ii) functions(iii) r.v.-s

VECTOR SPACES Self-study

Gram-Schmidt

Self-study

Inner product spaces (I)

Inner product examples

D

Norm, energy, distanceThe norm is the “length” of a vectoror the root of its energy

Norm, energy – examples

Cauchy-Schwartz (C-S) inequality

Example: geometric vectors:

C-S inequality – more examples

Correlation coefficient

Correlation coefficient - examples

…back to the error probability of the antipodal link

RX Filter

f(t)

Pulse Modulator

N(t)

SlicerBitstream

1010111...

White Gaussian noise one-sidedspectral density No

" ":

" "

0

1 :

( )

( )

h t

h t

00 0pq n

Total error probability (III) - revisit

Rewrite in terms of vector notation for functions:

| |

Error probability optimization – antipodal transmissionFor given h(t) maximize the s-factor by selecting the receive filter f(t):

Use C-S ineq.:

C-S ineq. becomes C-S eq. (i.e. s-factor is maximized) when:

The effect of scaling the receive and transmit filters

(root) SNR does not change when both signal and noise are scaledby the same factor

Constant gain (in RX) does not matter

…but transmitting more power (making h(t) larger is beneficial – though we run into limits…

s

Matched filters

h(-t) 0p

0t

h(t)

Filter matched to h(t)

Matched filter f(t)=C h(-t) minimizes the error probability

receive filter f(t)

C

Optimal receiver for antipodal transmission is based on a matched receive filter

h(-t)

h(t)Filter matched to h(t)

h(t)

TX

g(t)

Medium

b(t)

Optimum PAM RX

Figure 1.48:

Matched filter f(t)=C h(-t)

What is the value of the optimum (min.) error probability?It corresponds to the max s-factor:

…by maximizing the SNR (or s-factor):

max0

filterEnergySNR

/ 2 NoiseSpectralDensity(2s)h

N

minimizes the antipodal error probability…

Optimal Error Probability for antipodal linkas a function of SNR

Pe

0 0b/N or P/ N W or SNRε

0 0

2 PbPr(e) SN N W

εQ Q Q

P is the average received power (energy per unit time)

W is the bandwidth of the received signal (and the receive filter)

A bound for the optimal probability of error in terms of bandwidth and received power

Symbol rate cannot exceed twice the bandwidthEquality achieved for the so-called Nyquist pulses (see TA class)

Self-study

Antipodal transmission operational point:For 10^-5 Error Probability, SNR must be 9.6 dB

0b N2EQ

Pe

= P

roba

bili

ty o

f bi

t err

or

SNR (dB) NE 0b

9.6 dB

Figure 1.41:

Performance of antipodal receiver using a mismatched filter

f(-t)h(t)

Figure 1.49:

<1

Performancedegraded

Proof:

Causal Matched Filter

f(t) =h(T-t)

t Th(t)

Filter matched to h(t)at time T0

p(t) p(T0)

When the input signal h(t) is causal, the impulse response h(-t) of the matched filter is non-causal.Sufficiently delaying this non-causal response turns it causal.Given the time-invariance, we must also delay our sampling instant

h(-t) 0p

t 0h(t)

Filter matched to h(t)(at time zero)

Use this receiver front-end for optimal antipodal detection

Correlators vs. matched filters

A correlator <h(t)| > (I) (multiply & integrate implementation)

.

Correlatorh rh(t) r t

Correlator

h(t)

r(t) rh“Multiply & Integrate”implementation

waveform in… …number out

A correlator <h(t)| > (II)matched filter & sample implementation

.

0Tt

0h T -t

Correlator

r(t) 0q(T ) h rq(t)

Correlatorh rh(t) r t

A correlator <h(t)| > (III).

Proof that “matched filter & sample” works as a correlator:

Correlators and their implementations

Correlatorr(t) rhh(t)

Correlator

h(t)

r(t) rh

0Tt t-Th 0

Correlator

r(t) rh

0t t-h

Correlator

r(t) rh

Implementations:

“Multiply & Integrate”

“Matched-filter & Sample

“Matched-filter & Sample”

Figure 1.52:

Causal implementation

Non-causal implementation

The abstract view:

May use any of thesefor optimal antipodal detection

Correlator optimization for maximum SNR the “Matched correlator” (I)

tf

Choose f(t) for maximum SNR atthe output of the correlator

Nfhfrf N(t)h(t)r(t)

2N0,WGN~ 0“Signal”

f(t) = ?

What is SNR?

tf f NN(t)

2N0,WGN~ 0

White Noise propagation through the Correlator

Var =?

Correlator optimization for maximum SNR the “Matched correlator” (II)

tf

Choose f(t) for maximum SNR atthe output of the correlator

Nfhfrf N(t)h(t)r(t)

2N0,WGN~ 0“Signal”

Correlator optimization for maximum SNR the “Matched correlator” (III)

tf

Choose f(t) for maximum SNR atthe output of the correlator

Nfhfrf N(t)h(t)r(t)

2N0,WGN~ 0“Signal”

Alternative “tricky” view:Maximize the output signal while constraining

the kernel energy to be fixed

But when the kernel energy is fixed so is the noise variance at the output!So signal is maximized, while the noise is fixed -> SNR is maximized

It all happens when f(t) is “matched” to h(t)

(any scale of f(t) will do)

Self-study

Optimum Antipodal PAM Receiver:The matched correlator view

b(t)δ(t)

tN

g(t) th th

Can use either theMultiply&integrate orMatched filter&sampleimplementations

The SNR is maximizedby the matched correlation receiver,yielding minimum BER

Example: Integrate&dump receiver for flat pulse-shape g(t). Must multiply by a constant and integrate for T seconds, However multiplication is irrelevant (does not affect SNR)So, just integrate for T seconds (before dumping)

RXMediumTX