Module-3 : Transmission Lecture-3 (27/4/00)

Module-3 : Transmission Lecture-3 (27/4/00)

Marc Moonen

Dept. E.E./ESAT, K.U.Leuven

marc.moonen@esat.kuleuven.ac.be

www.esat.kuleuven.ac.be/sista/~moonen/

Lecture-3: Transmitter Design

Overview

• Transmitter : Constellation + Transmit filter

• Preliminaries : Passband vs. baseband transmission

• Constellations for linear modulation

->M-PAM / M-PSK / M-QAM

->BER performance in AWGN channel for transmission of 1 symbol (Gray coding, Matched filter reception)

• Transmission pulses :

->Zero-ISI-forcing design procedure for transmit pulse

(and receiver front-end filter), Nyquist pulses, RRC pulses

Lecture-3: Transmitter Design

Lecture partly adopted from

Module T2

`Digital Communication Principles’

M.Engels, M. Moeneclaey, G. Van Der Plas

1998 Postgraduate Course on Telecommunications

Special thanks to Prof. Marc Moeneclaey

Transmitter: Constellation + Transmit Filter

PS: channel coding (!) not considered here

s

k

E

a .

r(t)

aˆ

transmit pulse

s(t)

n(t)

p(t) +

AWGN

transmitter receiver (to be defined)

h(t)

channel

...

constellation

transmit filter (linear modulation)

 ^



k

s k

s

a p t kT

E t

s ( ) . . ( )

Transmitter: Constellation + Transmit Filter

-> s(t) with infinite bandwidth, not the greatest choice for p(t)..

-> implementation: upsampling/digital filtering/D-to-A/S&H/...

s

k

E

a .

transmit pulse

s(t)

p(t)

transmitter discrete-time

symbol sequence

continuous-time transmit signal t

Example: p(t)

t

Preliminaries: Passband vs. baseband transmission (I)

Baseband transmission

• transmitted signal is (linear modulation)

• transmitted signals have to be real, hence = real, p(t)=real

• baseband means for

B f -B

0 )

( f 

S

| f |  B

) ( f S

 ^



k

s k

s

LP

t E a p t kT

s ( ) . . ( )

a

Preliminaries : Passband vs. baseband transmission (II)

Baseband transmission model/definitions

g(t)=p(t)*h(t)*f(t)

everything is real here!

s

k

E

a .

r(t)

aˆ

transmit pulse

s(t)

n(t)

p(t) + f(t)

front-end filter

AWGN

1/Ts

transmitter receiver

(first version, see also Lecture4)

h(t)

channel

Preliminaries : Passband vs. baseband transmission (III)

Bandpass transmission

transmitted signal is modulated baseband signal

) (t s _LP

) .

2

cos(  f

t

B f -B

) ( f S

) (t s _BP

-fo f

) ( f S

fo fo+B

`e nv elo pe ’

Preliminaries : Passband vs. baseband transmission (IV)

Bandpass transmission:

• note that modulated signal has 2x larger bandwidth, hence inefficient

scheme !

• solution = accommodate 2 baseband signals in 1

bandpass signal :

I =`in-phase signal’

Q=`quadrature signal’

such that energy in BP is

2

Preliminaries : Passband vs. baseband transmission (V)

• Convenient notation for `two-signals-in-one’ is complex notation :

• re-construct `complex envelope’ from BP-signal

(mathematics omitted)

`c om pl ex e nv el op e’

) ( .

) ( )

( t s t j s t

s





low-pass filter

Assignment 2.1

• Prove for yourself that this is indeed a correct complex-envelope reconstruction procedure!

low-pass filter

Preliminaries : Passband vs. baseband transmission (VI)

Passband transmission model/definitions

a convenient and consistent (baseband) model can be

obtained, based on complex envelope signals, that does not have the modulation/demodulation steps:

aˆk

f(t)

front-end filter

1/Ts

receiver (first version) r(t)

n’(t) +

AWGN

s

k E

a .

transmit pulse

s(t) p(t)

transmitter

h’(t)

channel

Preliminaries : Passband vs. baseband transmission (V)

aˆ

f(t)

front-end filter

1/Ts

receiver (first version) r(t)

n’(t) +

AWGN

s

k

E

a .

transmit pulse

s(t)

p(t)

transmitter

h’(t)

channel

=complex symbols

=usually a complex filter )

( )

(

' t e

h t h 

=complex AWGN

=complex

=real-valued transmit pulse

Q k I

k

k

a j a

a 

 .

Preliminaries : Passband vs. baseband transmission (VI)

• In the sequel, we will always use this baseband-

equivalent model, with minor notational changes (h(t) and n(t), i.o. h’(t) and n’(t)).

Hence no major difference between baseband and passband transmission/models (except that many things (e.g. symbols) can become complex-valued).

• PS: modulation/demodulation steps are transparent (hence may be omitted in baseband model) only if receiver achieves perfect carrier synchronization

(frequency fo & phase).

Synchronization not addressed here

(see e.g. Lee & Messerschmitt, Chapter 16).

Constellations for linear modulation (I)

Transmitted signal (envelope) is:

Constellations:

PAM PSK QAM

pulse amplitude modulation phase-shift keying quadrature amplitude modulation

4-PAM (2bits) 8-PSK (3bits) 16-QAM (4bits) ps: complex constellations for passband transmission

I

R I

R

I

R

 ^



k

s k

s

a p t kT

E t

s ( ) . . ( )

Constellations for linear modulation (II)

M-PAM pulse amplitude modulation

• energy-normalized iff

• then distance between nearest neighbors is

larger d -> noise immunity (see below) I

R





k

A A M A

a   ,  3 ,...,  (  1 )

a

1 ) 3

(

  M M A

1 ) 12

(

  M M d

d

Constellations for linear modulation (III)

M-PSK phase-shift keying

• energy-normalized iff ….

• Then distance between nearest neighbors is

 



 

  

 exp( . 2 ) | m 0 , 1 ,..., M 1 M

j m

a



a

) sin(

. 2 )

( M M

d

_ 

d

I

R

Constellations for linear modulation (IV)

M-QAM quadrature amplitude modulation

• distance between nearest neighbors is

1 ) 6

(  

M M d

d

I

R





k Q k

I

a A A M A

a

,

  ,  3 ,...,  (  1 )

k Q k

I

k

a j a

a 

 .

) (

) (

)

( M d M d M

d





BER Performance for AWGN Channel

BER=(# bit errors)/(# transmitted bits)

g(t)=p(t)*f(t)

n’(t)=n(t)*f(t)

BER for different constellations?

r(t)

aˆ

transmit pulse

s(t)

n(t)

p(t) +

s

k

E

a .

f(t)

front-end filter

AWGN channel

1/Ts

transmitter receiver

r’(t)

BER Performance for AWGN Channel

definitions:

- transmitted signal

- received signal (at front-end filter)

- received signal (at sampler)

g(t) =p(t)*f(t) = transmitted pulse p(t) filtered by front-end filter

n’(t) =n(t)*f(t) = AWGN filtered by front-end filter

) ( ' )

( . .

) (

' t E a g t kT n t

r

k

s k

s

 

 

 ^



k

s k

s

a p t kT

E t

s ( ) . . ( )

) ( )

( . .

)

( t E a p t kT n t

r

k

s k

s

 

 

BER Performance for AWGN Channel

Received signal sampled @ time t=k.Ts is...

1 = useful term

2= `ISI’, intersymbol interference (from symbols other than ) 3= noise term

Strategy :

a) analyze BER in absence of ISI (=`transmission of 1 symbol’)

b) analyze pulses for which ISI-term = 0 (such that analysis under a.

applies)

c) for non-zero ISI, see Lecture 4-5

 



 

 



 

 





2 1 0

) . ( ' )

. ( . )

0 ( . . )

. (

'

m

s m

k k

s

s

E a g a g m T n k T

T k

r    

 

a

Transmission of 1 symbol over AWGN channel (I)

BER for different constellations?

aˆ

transmit

pulse n(t)

p(t) +

E

a .

f(t)

front-end filter

AWGN channel

1/Ts

...take 1 sample at time 0.Ts

transmit 1 symbol at time 0.Ts ...

Transmission of 1 symbol over AWGN channel (II)

Received signal sampled @ time t=0.Ts is..

• `Minimum distance’ decision rule/device :

    







1

0

. ( 0 ) 0 ' ( 0 . ) .

) . 0 (

' T

E

a g n T

r   

n s

s M

i n s

s

i

E g

T r

g E

T

a    r     







. ( 0 )

) .

0 ( min '

) 0 ( .

) .

0 ( ˆ '

1 0 0

Transmission of 1 symbol over AWGN channel (III)

`Minimum distance’ decision rule :

Example : decision regions for 16-QAM I

R

Transmission of 1 symbol over AWGN channel (IV) Preliminaries :BER versus SER (symbol-error-rate)

• aim: each symbol error (1 symbol = n bits) introduces only 1 bit error

• how? : GRAY CODING

make nearest neighbor symbols correspond to groups of n bits that differ only in 1 bit position…

• …hence `nearest neighbor symbol errors’

(=most symbol errors) correspond to 1 bit error

Transmission of 1 symbol over AWGN channel (V)

Gray Coding for 8-PSK

Transmission of 1 symbol over AWGN channel (VI)

Gray Coding for 16-QAM

Transmission of 1 symbol over AWGN channel (VII)

• Computations : skipped

(compute probability that additive noise pushes received sample in wrong decision region)

• Results:

neighbors

of number

average )

(

2 ) . exp(

2 ) 1

(

) (

) 0 (

) ( log ).

( 2 .

. (

log . ) (

2 2

2

2 2

0 2



























M N

u du x

Q

df f

F g

M M

N d Q E

M M BER N

x

b







Transmission of 1 symbol over AWGN channel (VIII)

Interpretation (I) : Eb/No

• Eb= energy-per-bit=Es/n=(signal power)/(bitrate)

• No=noise power per Hz bandwidth

lower BER for higher Eb/No

Transmission of 1 symbol over AWGN channel (IX)

Interpretation (II) : Constellation

for given Eb/No, it is found that…

BER(M-QAM) =< BER(M-PSK) =< BER(M-PAM)

BER(2-PAM) = BER(2-PSK) = BER(4-PSK) = BER(4-QAM)

higher BER for larger M (in each constellation family)

Transmission of 1 symbol over AWGN channel (X) Interpretation (III): front-end filter f(t)

It is proven that

and that is obtained only when

this is known as the `matched filter receiver’

(see also Lecture-4)











df f

F g

2 2

) (

) 0

 (

1 0   

 1



) ( )

( and

) (

) ( i.e.

, ) ( )

( f P

f f t p

t G f P f

F    

Transmission of 1 symbol over AWGN channel (XI)

Interpretation (IV)

with a matched filter receiver, obtained BER is

independent of pulse p(t)

Transmission of 1 symbol over AWGN channel (XII) BER for M-PAM (matched filter reception)

Transmission of 1 symbol over AWGN channel (XIII) BER for M-PSK (matched filter reception)

Transmission of 1 symbol over AWGN channel (XIV) BER for M-QAM (matched filter reception)

Symbol sequence over AWGN channel (I)

• ISI (intersymbol interference) results if

• ISI results in increased BER

0 )

. ( such that

0 



 m g m T

g(t)=p(t)*f(t)

Symbol sequence over AWGN channel (II)

• No ISI (intersymbol interference) if

• zero ISI -> 1-symbol BER analysis still valid

• design zero-ISI pulses ?

0 )

. ( :

0 



 m g m T

Zero-ISI-forcing pulse design (I)

• No ISI (intersymbol interference) if

• Equivalent frequency-domain criterion:

This is called the `Nyquist criterion for zero-ISI’

Pulses that satisfy this criterion are called `Nyquist pulses’

0 )

. ( :

0 



 m g m T

) 0 ( constant

) 1 (

T g f k

T



G 

 





Zero-ISI-forcing pulse design (II)

• Nyquist Criterion for Bandwidth = 1/2Ts

Nyquist criterion can be fulfilled only when G(f)

is constant for |f|<B, hence ideal lowpass filter.

Zero-ISI-forcing pulse design (III)

• Nyquist Criterion for Bandwidth < 1/2Ts

Nyquist criterion can never be fulfilled

Zero-ISI-forcing pulse design (IV)

• Nyquist Criterion for Bandwidth > 1/2Ts

Infinitely many pulses satisfy Nyquist criterion

Zero-ISI-forcing pulse design (V)

• Nyquist Criterion for Bandwidth > 1/2Ts

practical choices have 1/T>Bandwidth>1/2Ts Example:

Raised Cosine (RC) Pulses

1 0

factor' off

-

`roll :



 



(%) 100

. Bandwidth

Excess

2T 1

Bandwidth







 

s

Zero-ISI-forcing pulse design (VI)

Example:

Raised Cosine Pulses

(time-domain)

Zero-ISI-forcing pulse design (VII)

Procedure:

1. Construct Nyquist pulse G(f) (*) e.g. G(f) = raised cosine pulse

(formulas, see Lee & Messerschmitt p.190)

2. Construct F(f) and P(f), such that (**) F(f)=P*(f) and P(f).F(f)=G(f) -> P(f).P*(f)=G(f) e.g. square-root raised cosine (RRC) pulse

(formulas, see Lee & Messerschmitt p.228)

(*) zero-ISI, hence 1-symbol BER performance

(**) matched filter reception = optimal performance

Zero-ISI-forcing pulse design (VIII)

• PS: Excess BW simplifies implementation -`shorter’ pulses (see time-domain plot)

- sampling instant less critical (see eye diagrams)

`eye diagram’ is `oscilloscope view’ of signal before sampler, when symbol timing serves as a trigger

20 % e xc es s- B W 10 0% e xc es s- B W

Zero-ISI-forcing pulse design (IX)

• PPS: From the eye diagrams, it is seen that selecting a proper sampling instant is crucial (for having zero-ISI)

->requires accurate clock synchronization, a.k.a. `timing recovery’, at the receiver

(clock rate & phase)

->`timing recovery’ not addressed here

see e.g. Lee & Messerschmitt, Chapter 17

Questions….

1. What if channel is frequency-selective, cfr. h(t) ?

- Matched filter reception requires that F(f)=P*(f).H*(f) - Zero-ISI requires that P(f).H(f).F(f)=Nyquist pulse

Is this an optimal design procedure ?

aˆk

f(t)

front-end filter

1/Ts

receiver (see lecture-4) n(t)

+

AWGN

s

k E

a .

transmit pulse

p(t)

transmitter

h(t)

channel

Assignment 2.2

Analyze this design procedure for the case where the channel is given as

H(f) = Ho for |f|<B/2 H(f) = 0.1 Ho for B/2<|f|<B

discover a phenomenon known as `noise enhancement’

(=zero-ISI-forcing approach ignores the additive noise, hence may lead to an excessively noise-amplifying receiver)

aˆk

f(t)

front-end filter

1/Ts

receiver (see lecture-4) n(t)

+

AWGN

s

k E

a .

transmit pulse

p(t)

transmitter

h(t)

channel

Questions….

2. Is the receiver structure (matched filter front-end + symbol- rate sampler + slicer) optimal at all ?

Sampler works at symbol rate. With non-zero excess bandwidth this is below the Nyquist rate.

Didn’t your signal processing teacher tell you never to do sample below the Nyquist rate? Could this be o.k. ????

aˆk

f(t)

front-end filter

1/Ts

receiver (see lecture-4) n(t)

+

AWGN

s

k E

a .

transmit pulse

p(t)

transmitter

h(t)

channel

Conclusion

• Transmitter structure:

symbol constellation + transmit pulse p(t)

• Symbol constellation: PAM/PSK/QAM

BER-analysis for transmission of 1 symbol over AWGN-channel

-> Performance of matched filter receiver is independent of transmit pulse

• Transmit pulse p(t):

-> Zero-ISI-forcing design procedure for transmit pulse p(t) and front-end filter f(t), for AWGN channels (-> RRC pulses) -> Even though for more general channels this is not an optimal

procedure (see Lecture 4), transmit pulses are usually designed as

RRC’s .