2/6/01

Types of Communication Analog: continuous variables with noise ⇒

P{error = 0} = 0 (imperfect)

Digital: decisions, discrete choices, quantized, noise ⇒ P{error} → 0 (usually perfect)

The channel can be radio, optical, acoustic, a memory device (recorder), or other objects of interest as in radar, sonar, lidar, or other scientific observations.

A1

Message S1, S2,…or SM

Modulator ⇒ v(t)

channel; adds noise and distortion

M-ary messages, where M can be infinite

Demodulator; hypothesize H1…HN (chooses one)

message S1, S2,…or SM

Lec14.10-1

Received

2/6/01

Optimum Demodulator for Binary Messages

P2OKERRORS2

P1ERROROKS1

H2H1

Probability a prioriHypothesis:

Message:

A2 How to define V1,V2?

Demodulator design

V1 V2

vb

va

2-D case

E.G.

"Vv "Vv

22

11 ⇒∈

⇒∈

vb va

vc

v

t

Lec14.10-2

measured v

H " H "

2/6/01

Optimum Demodulator for Binary Messages

va

How to define V1,V2?

V1 V2

vb

va

2-D case

E.G.

"Vv "Vv

22

11 ⇒∈

⇒∈

vb

vc

v

t

Minimize { } { }2 1e 1 1 2 2V VP P P S P S∆ = = +∫ ∫

{ } { }11 2 2 1 1VP S S= + −⎡ ⎤⎣ ⎦∫

{ } { } 1 21 V 1V 1 =+ ∫∫Note:

replace with ∫ 1V

A3 Lec14.10-3

measured v

H " H "

error p v dv p v dv

P p v P p v dv

vd S v p vd S v p

2/6/01

Optimum Demodulator for Binary Messages

{ } { }[ ]∫ −+= 1V 11221e dvPP

A4

{ } { }1 1 1 2 2i i , S S∋ >

Very general solution ↑ [i.e., choose maximum a posteriori

Lec14.10-4

S v p P S v p P

error To m nim ze P choose V P p v P p v

P (“MAP” estimate)]

2/6/01

Example: Binary Scalar Signal Case

A5

S 2 n21 ∆∆∆ =σ==

{ } 2)Av(

1 e 1 −− π

=∴ { }2 e 1 − π

=

:P21 = { }2p { }1p

0 AA/2 Decision threshold if 21 PP =

1P2P

v

1P 2P

0 AA/2 v

2H 21 PP >Threshold if

{ }p v S(bias choise toward H2 and a priori information)

Lec14.10-5

noise Gaussian , N , volts O S , volts A

N2 N 2

S v p N2 v N 2

S v p

P If S v S v

2/6/01

Rule For Defining V1 : (Binary Scalar Case)

A6

{ } { } "

P P

1 1 2

2

1 ⇒>= ∆A

( ) "Pn 112 ⇒> AAA

(binary case) “Likelihood ratio”

or (equivalently)

For additive Gaussian noise,

( )[ ] ( ) ( )12 ?222 PAv-n AAA >−=+=

( ) ( ) 2

2 1 1 2 1

A P A NV i , P2A 2 A

+∴ > > +

A A ��

{ } { }1 1 1 2 2S S∋ > { } 2

2 1 p v S e

2 N −=

π

Lec14.10-6

" V S v p S v p

" V P n

P n N2 vA 2 N2 N2 A -v

bias

2N n P choose f v or v n P ���

Choose V P p v P p v v 2N

2/6/01

Binary Vector Signal Case For better performance, use multiple independent samples:

{ } { } 1

2?

2

1 P P

p p

>= ∆A

2S { } { }

m 1 2 m 1 i 1

i 1 S S

= = π (independent noise samples)

{ } ( )Sv ii

2 i1ie 1 −−

π =Where

{ } ( )

( )m 2i 1ii 1

v S

i m 1p v S e

2 N =

− −∑ =

π B1

v(t)

0 t

1S

1 2 m

Lec14.10-7

S v S v

Here P v ,v ,...,v p v

N2

N 2 S v p

2N

2/6/01

Binary Vector Signal Case

B2

{ } ( )

( )Sv

mi

m

1i

2 i1i

e 1 p ∑ −− −

π =

Thus the test becomes:

( ) ( )i i

m m ?2 2 2i 2 i 1

1i 1 i 1

P1n v S v S n2N P = =

⎡ ⎤ = − − − >⎢ ⎥

⎣ ⎦ ∑ ∑A A A

2 2Sv −

2 1Sv −

2 2 2 1 2 1 2 2 1 1But v S v S 2v S S S S S S− − − • + • + • − •

( ) 1 1 2 2 21 21 1

PS S S Sv V i v S S N n2 P

∗ ∗ • − • ⎛ ⎞∈ • − > + ⎜ ⎟⎝ ⎠

ATherefore

if energy E1 = E2 if P2 = P1

{ } { }

?1 2 12

p v S P = Pp v S ∆

>A

Lec14.10-8

N2

N 2 S v

2v = −

ff

Bias = 0 Bias = 0

2/6/01

Binary Vector Signal Case

B3

Multiple hypothesis generalization:

This “matched filter” receiver minimizes Perror

?i iii i i j i

S SH if f v S N n P all f2 ∆

≠ • = • − + >A

∑τ

m

∑τ

m

operator H1

H2

+ -

×

×

1S

2S

v i=1

i=1

( ) 1 1 2 2 21 1 21 1

PS S S SS S N n2 P • − • ⎛ ⎞• − > + ⎜ ⎟

⎝ ⎠ A

Lec14.10-9

Choose

V iff v

2/6/01

Graphical Representation of Received Signals

B4

2 2 i i

i 1 A S P

= = ∑

)t(1S

11S

12S

13S

t

3-D Case:

n

n

1S

2S

0

1V

2V

Lec14.10-10

verage energy

2/6/01

Design of Signals iS

1ESenergyAverage 2 ==

B5

E.G. consider +1

-1

vs.

21 SS −=

2S

2S1 +=

0S2 =

( ) 21 PE ==

0 1S

2S

3S

4S

boundaries

b

n

E.G. 2-D space

( ) 12111

4321 SS

: =

( ) 1312111 SS =

3-D space 1S

2S

3S4S

a

b ratio

a bBetter

Lec14.10-11

P 2

decision

S , S , S , S , S for S , S ,

2/6/01

Design of Signals iS

B6

2-D space:

16-ary signals

161

or magnitude/phase

vs

2iS

1iS

n-Dimensional sphere packing optimization unsolved

equilateral triangle slightly lower average

p{error}

19

Lec14.10-12

S ,..., S

signal energy for same

2/6/01

:eP∆ =

Binary case: For additive Gaussian noise, optimum is

( ) 1 2

2 2

2 1

211 P P nN

2 SS

SSvif A+ −

>−•

nSvWhere +=

)t(nN so 2 === ∆ [ ]( )No ×

( ) ⎪⎭

⎪⎬ ⎫

⎪⎩

⎪⎨ ⎧

+ −

<−•= 1 2

2 2

2 1

21 P P nN

2 SS

SSP 1 A

( ) { } 2

1 2 21 2 1

S S P p n S S N n b2 P

⎧ ⎫− −⎪ ⎪ = • − < + =⎨ ⎬ ⎪ ⎪⎩ ⎭

A

D1 Lec14.10-13

p{error} of n Calculatio

" H "

B kT B N sideband double ,B2 Hz W 2 1­

S e v p

p y < −

y • 2B[GRVZM] -b • 2B

Lec14.10-14 2/6/01

Duality of Continuous and Sampled Signals

( ) [ ]

{ } bypP P nN

2 SS

SSnpP

B2b 1 2

2 21

GRVZMB2y 21Se 1 − <=

⎪ ⎪ ⎭

⎪ ⎪⎬

⎫

⎪ ⎪ ⎩

⎪ ⎪⎨

⎧

+ −−

<−•=

•− • ��� ��� �

A�����

[ ]∫ −•= ∆ T

o 21 dt)t(S)t(S)t(ny

[ ] ( )∫ −−= ∆ T

o 12

o2 21 PPn

2 Ndt)t(S)t(S

2 1b A

Conversion to continuous signals assuming nyquist sampling is helpful here, 11 S]Tt0)[t(S ↔<< (2BT samples, sampling theorem)

[ ] ( ) ⎪ ⎭

⎪ ⎬

⎫

⎪ ⎩

⎪ ⎨

⎧

⎥ ⎥ ⎦

⎤

⎢ ⎢ ⎣

⎡ −==σ ∑

=

∆ 2BT2

1j j2j1j

22 y SSn

B2 1EyE

-B B0

noise [ ] 1o WHz

2 N −

f

D2

2/6/01

Calculation of Pe, continued

[ ] ( ) ⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥ ⎥ ⎦

⎤

⎢ ⎢ ⎣

⎡ −= ∑

=

∆ 2

1j j2j1j

22 y SSn 1E

( ) ( )⎪⎭

⎪⎬ ⎫

⎪⎩

⎪⎨ ⎧

−−⎟⎠ ⎞

⎜⎝ ⎛= ∑ ∑

= =1i 1j 2j12i1ji

2

ji SSSSE1

i j ijN⎡ ⎤ = δ⎣ ⎦

[ ]∫ −=−⎟⎠ ⎞

⎜⎝ ⎛ T

o

2 21

o2 21

2 2 y dt)t(S)t(S

2 NS1

NoB [ ]∫ −

T

o

2 21 dt)t(S)t(S

D3 Lec14.10-15

= σ BT2

B2 y E

BT2 BT2 n n

B2

where E n n

= σ S N B2

B2

2/6/01

Calculation of Pe, continued

D4

)GRVZM(e 2

1)y(p 2 y

2 2y 2 y

σ−

πσ =

[ ]∫ −=−⎟⎠ ⎞

⎜⎝ ⎛=σ

T

o

2 21

o2 21

2 2 y dt)t(S)t(S

2 NS1

NoB [ ]∫ −

T

o

2 21 dt)t(S)t(S

2 2 y

1

b y 2e S 2

y

1P e 2

− − σ

−∞

= πσ

∫

-b 0

P(y)

y

Therefore:

Lec14.10-16

S N B2

B2

dy

2/6/01

Definition of ERFC(A)

“Error function” dxe 1)A(ERF A

A

x2 ∫ −

−∆

π =

D5

“Complementary error function” ERF(A))A(ERFC ∆= -A A x

2 1

(ERFC2 1

1 = where A must be found

y 2 22 y

y

2 2 2 y

A 2A y 2x 2A A 2y

l x y 2 then

1 1(A) e e dy 2

σ − σ−

− − σ

= σ

= = π πσ ∫ ∫

2 ylimits A 2 1 2 ari lσ

Lec14.10-17

-1

= σ

), A P Then S e

If we et

ERF dx

where the new and factor se as fol ows: πσ

2/6/01

Definition of ERFC(A)

∫∫ σ

σ−

σ−

−

−

πσ =

π =

2A

2A2 y

A

A

x y

y

2 y2 dye

2

1dxe 1ERF(A)

2 ylimits A 2 1 2 ari lσ

2yAxy yy σ==σ=

2Ay yσ=becomes a limit where

2 yy 2dy πσπσ=

D6 Lec14.10-18

2 y

where the new and factor se as fol ows: πσ

limit the , 2 x Since

1 becomes 1 so 2 dx Also,

2/6/01

Solution for Pe for Binary Signals

D7

( ) 11 2y e Se 2 e SS 1 2 and P1P ( ) P2A2 P= σ = +=

( )2bA yy σ=σ=

1 21 2 e S e S1If P P , and si P P , then2 = = =

( )e y 1P b 22 = σ

[ ]

( ) [ ]

T 2

1 2 0

T 2

o 1 2 0

1 ) S (t)212

2 N 2 ) )

⎡ ⎤ ⎢ ⎥− ⎢ ⎥

= ⎢ ⎥ ⎢ ⎥

−⎢ ⎥⎢ ⎥⎣ ⎦

∫

∫ Lec14.10-19

1 e ERFC b ERFC P P

A so , 2 b limit the where

nce

ERFC

S (t dt ERFC

S (t S (t dt

2/6/01

e for Binary Signals

[ ]

( ) [ ]

T 2

1 2 0

e T 2

o 1 2 0

1 S (t) S (t)21P 2 2 N 2 ) )

⎡ ⎤ ⎢ ⎥− ⎢ ⎥

= ⎢ ⎥ ⎢ ⎥

−⎢ ⎥ ⎣ ⎦

∫

∫

[ ] ⎥ ⎥ ⎦

⎤

⎢ ⎢ ⎣

⎡ −= ∫

T

0 o

2 21e )t(S)t(S

2 1ERFC

2 1P

[ ] ⎟ ⎟ ⎠

⎞ ⎜ ⎜ ⎝

⎛ −∫

T

0

2 2112e dt)t(S)t(Smaximizes)t(S)t(

thenP)t(Sdt)t(SIf 21 T

0

2 2

T

0

2 1 =∫ +∫

D8 Lec14.10-20

Solution for P

dt ERFC

S (t S (t dt

N dt

− = S let , P minimize To

P for fixed is dt

Examples of Binary Communications Systems⎤

Pe = 1ERFC

⎡⎢1

T ∫ [S1(t) − S2(t)]2 N dt o ⎥ 2 ⎢2 0 ⎥⎣ ⎦

∆P Assume 1 = P2 = 1 define and

T ∫ s2(t)dt = E

2 01

Modulation type s1(t) s2(t) Pe

“OOK” (on-off keying) tcos A oω 0

oavg

o

N2 2 E ERFC 1 2 N4 E ERFC 1

=

“FSK” (frequency-shift keying)

tcos A ω1 tcos A ω2 avg o 1 ERFC E 2N 2

“BPSK” binary phase-shift keying)

tA cos ω tA cos ω− avg o 1 ERFC E N2

Lec14.10-21 2/6/01 D9

2/6/01

Examples of Binary Communications Systems

Cost of communications ∝ cost of energy, Joules per bit (e.g. very low bit rates imply very low power transmitters, small antennas)

Note:

( )oAVGe NEfP =

[ ]J=[J]

D10 Lec14.10-22

Hz W 1-

2/6/01

Probability of Baud Error

D11

Non-coherent FSK: carrier is unsynchronized so that both sine and Such

e(E/No).

e declines for 12~ o −>

0 4 8

10-1

10-2

10-3

10-4

10-5

10-6

10-7

1

BPSK

3 dB

6 dB

FSK OOK (coherent)

FSK non-coherent

Pe

E/No (dB)

Lec14.10-23

cosine terms admitted, increasing noise. “envelope detectors have a different form of P

Note how rapidly P dB 16 N E

12 16 20