Principles of Communications Lecture 8: Baseband...

Principles of CommunicationsLecture 8: Baseband Communication Systems

Chih-Wei Liu 劉志尉National Chiao Tung [email protected]

Commun.-Lec8 [email protected] 2

Outlines

Introduction

Line codes

Effects of filtering

Pulse shaping toward zero ISI

Zero-forcing equalization

Eye diagrams

Synchronization


Introduction

Digital vs. analog signals

Analog sampling quantization digital

Baseband vs. passband

Channel distortion – ISI

(Channel) bandwidth limitation


Line Codes

Baseband data format used to represent digital data (for transmission purpose).

Examples are given on the next page

Operation: time or frequency shaping

Purposes: usually to cope with the channel limitations (or provide extra function such as synchronization)

Needed for certain applications


Non-return-to-zero (NRZ) change

NRZ mark (data1-> change in level; data0 -> no change)

Unipolar return-to-zero (URZ)

Polar RZ

Bipolar RZ (“0”-> 0 level; “1”-> alternate sign

Split phase (Manchester) (“1”-> level A to level –A at 1/2 interval; “0” -> level –A to level A at 1/2 interval)


Purposes of Line Codes

Self synchronizationProper power spectrumTransmission bandwidthTransparency

Error detection capability

Good error probability performance


Power Spectra (I)The transmitted signal is a pulse train:

( ) ( ).

The amplitudes can be viewed as random variables with 0, 1, 2,

The autocorrelation function of the waveform is

( ) (

kk

m k k m

X m

X t a p t kT

R a a m

R R rτ

∞

=−∞

+

= − − Δ

= = ± ±

=

∑

K

),

1in which ( ) ( ) ( ) .

m

mT

r p t p t dtT

τ

τ τ

∞

=−∞

∞

−∞

−

= +

∑

∫


Power Spectra (II)

2

2

The power spectral density is the Fourier transofrm of ( ):

( ) [ ( )] [ ( )]

[ ( )] ( )

( ) .

1Note that ( ) [ ( )] [ ( ) (

X

X X mm

j mTfm m r

m m

j mTfr m

m

r

R

S f R R r mT

R r mT R S f e

S f R e

S f r p t p tT

π

π

τ

τ τ

τ

τ

∞

=−∞

∞ ∞−

=−∞ =−∞

∞−

=−∞

= = −

= − =

=

= = − ∗

∑

∑ ∑

∑

FT FT

FT

FT FT2| ( ) |)] .P f

T=


Example 1: NRZ

Assume the message m[n] is random (white noise) with equally “0”and “1” values.

Step 1: Compute Rm based on the above assumption and “format”

Step 2: Compute r(t) based on the pulse shape.

2 2 21 12 2

2 21 1 1 14 4 4 2

2 2 2

1. NRZ, 50% "0" and 50% "1".= ( ) , 0;

= ( ) ( ) ( ) 0 , 0. ( ) ( / ) ( ) sinc( )

Therefore, ( ) ( ) sinc ( ).

m

m

NRZ r

R A A A m

R A A A A A A mp t t T P f T Tf

S f A S f A T Tf

+ − = =

+ − + − + − = ≠

= Π → =

= =


Example 2: Unipolar RZ

2 2 2

2

2 2

2 2 2

2. Unipolar RZ, 50% 1-level and 50% 0-level. 1 1 1(0) , 02 2 21 1 1 1 10 0 0 0 , 04 4 4 4 4

( ) (2 / ) ( ) sinc( )

1 1( ) sinc ( )4 2 2 4

m

T T

URZ

A A mR

A A A A A m

p t t T P f f

T TS f f A A

⎧ + = =⎪⎪= ⎨⎪ ⋅ + ⋅ + ⋅ + ⋅ = ≠⎪⎩= Π → =

= + 2, 0

2 2 2 2

22 2 2

1 1sinc ( )4 2 4 4

1 1 1sinc ( ) ( ) . ( )4 2 4 4

mj mTf

m m

mj mTf

m

n m nj mTf

n m n

e

T T f A A e

T T A n nf A f e fT T T T

π

π

πδ δ

=∞−

=−∞ ≠

=∞−

=−∞

=∞ =∞ =∞−

=−∞ =−∞ =−∞

⎡ ⎤⎢ ⎥⎣ ⎦

⎡ ⎤= +⎢ ⎥⎣ ⎦⎡ ⎤

= + − = −⎢ ⎥⎣ ⎦

∑

∑

∑ ∑ ∑Q


Inter-symbol Interference (ISI)Intersymbol interference refers to one specific type of distortion. It is typically due to insufficient bandwidth of the channel. Note: Narrow BW Long time-duration

Example: Rectangular waveforms through a lowpass RC filter. The neighboring pulses smear out and interfere with each other.


BW= 1bit/sec


BW= 0.5bit/sec


ISI Reduction

Pulse shaping: Assume the channel is ideal (flat) with the minimum BW (W). What shape of “pulse” warrants ISI-free transmission?

Equalization: If the channel is non-ideal (not ideally flat LPF), an equalizer helps in reducing ISI.


Pulse Shaping

Consider 2W independent samples per second are transmitted through a channel with bandwidth W Hz. The output is as follows.

( ) ( ) sinc 2 .2

The samples at / 2 are ISI-free, sinc( ) 0.

n nn n

m

ny t y t a W tW

t m W m n

∞ ∞

=−∞ =−∞

⎡ ⎤⎛ ⎞= = −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦= − =

∑ ∑Q

Are there other pulse shapes warrant ISI-free? -T 0 T 2T t

t

Sampling of p(t-2T)

)2(2 Ttpa −

)3(3 Ttpa −

)(1 Ttpa −

)(tp


Raised Cosine FamilyRaised Cosine: a example of ISI-free pulses – raised cosine spectra at BW edges.

Roll-off factor β: β=0 rectangular pulse (sinc)β=1 “cosine” in freq; time pulse has narrow main lobe with very low sidelobes

1, | |2

1 1 1( ) 1 cos | | | |2 2 2 2

10 | |2

RC

T fT

T TP f f fT T T

fT

β

π β β ββ

β

−⎧ ≤⎪⎪

⎧ ⎫⎡ ⎤− − +⎪ ⎛ ⎞= + − < ≤⎨ ⎨ ⎬⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎩ ⎭⎪⎪ +

>⎪⎩


2

Time pulse of the raised cosine:cos( / )( ) sinc( ), is called the roll-off factor.

1 (2 / )RCt T tp tt T T

πβ ββ

=−

cosine curves


Nyquist’s Pulse Shaping CriterionA pulse shape ( ) with a Fourier transform ( ) satisfying

1( ) , | |2

1, 0then its sampled values ( ) .

0, 0.

k

p t P fkP f T fT T

np nT

n

∞

=−∞

+ = ≤

=⎧= ⎨ ≠⎩

∑


( )

2

2

2 12 2

2 12

122

12

2

Proof of Nyquist's Pulse Shaping Criterion

( ) ( )

Sampling at , ( ) ( )

( ) ( )

( ) ;

( )

j ft

j fnT

kT j fnT

kkT

j f unT knTk

T

j funT

k

p t P f e df

nT p nT P f e df

p nT P f e df

k kP u e du u fT T

kP u eT

π

π

π

π

π

∞

−∞

∞

−∞

+∞

−=−∞

∞ +

−=−∞

=−

=

=

=

= + = −

= +

∫∫

∑ ∫

∑ ∫1

21

2

12 2

12

1, 00, 0

T

T

T j funT

T

du

nTe du

nπ

∞

−∞

−

=⎧= = ⎨ ≠⎩

∑∫

∫


Pulse Transmission System

( ) ( ).

( ) ( ) ( ) [ ( ) ( )] ( ).Consider the combined effect of three filters, we would liketo have ( ) to be ISI-free.

k Tk

R C R

x t a h t kT

v t y t h t x t h t h t

v t

∞

=−∞

= −

= ∗ = ∗ ∗

∑


Transmitter and Receiver Filters

Practically, Hc(f) is unknown of time-varying

2

Let ( ) ( )

where represents a scale factor and a possible delay.( ) ( ) ( ) ( ), or in frequency domain,

( ) ( ) ( ) ( ).If ( ) is given, then

d

k RC dk

d

RC d T C Rj ft

RC T C R

C T

v t A a p t kT t

A tAp t t h t h t h t

AP f e H f H f H fH f H

π

∞

=−∞

−

= − −

− = ∗ ∗

=

∑

( ) ( ) 1 ( )The overall spectrum should satisfy the Nyquist criterion to avoid ISI. A filter design problem.

R Cf H f H f=

⇒


1/2 Let ( ) ( ) 1 ( )T R CH f H f H f= =


Zero-Forcing Equalization

Now, assume digital processing at the receiver.Purpose: Design an FIR filter that compensates for the channel distortion zero-ISI.Show an example of zero-ISI equalizer design.


Let the combined impulse response after the channel be ( ),Pass such a pulse through our finite-length equalizer, then

( ) ( ).

If the resulting pulse satisfies the zero-ISI condition

c

n

eq n cn N

p t

p t p t nα=−

= − Δ∑, ISI-free

transmission is achieved. Assume that , the condistion is1, 0

( ) [( ) ] 0, 1, , .0, 0

Notice that we only enforce the zero-ISI condition on the 2 1 samples

n

eq n cn N

Tm

p mT p m n T m Nm

N

α=−

Δ =

=⎧= − = = ± ±⎨ ≠⎩

+

∑ K

. Why? We only have 2 1 coefficients (variables)and can only achieve this much. Solve the 2 1 equations andyou obtain the zero-forcing equalizer.

NN

++


1

0

(0) ( ) ( 2 )0

( ) (0) (( 2 1) )10

(2 ) (0)

0

c c c N

c c c N

c c N

p p T p NT ap T p p N T a

p NT p a

−

− +

⎡ ⎤⎢ ⎥⎢ ⎥ − −⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥− +⎢ ⎥ ⎢ ⎥ ⎢ ⎥= ⋅⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦

ML

L

M M M

M

Example: A 3-tap equalizer The equalizer cannot

eliminate ISI beyond its span.


Eye Diagrams

Eye diagram: Constructed by overlapping a number ofsegments of the base-band signals

A qualitative measure of the system performance.

(Lee et al., Digital Comm., 1994)


The best place tosample the signal.

Amplitude jitter:ICI causes the amplitudeof each symbol fluctuate.

Timing jitter:ICI causes the timing ofeach symbol fluctuate.

Increasing bandwidth may mitigate ICI, butwill also allows more noise to enter. Anotherexample of trade-offs in commu systems design

eye eye


Symbol, Bit, Word, and Frame

Bits Symbol (a single pulse in transmission) ---carrier sync., symbol timing

Bits Word --- word sync.

Words Frame ---frame sync.

(Benedetto et al., Digital TansmissionTheory, 1987)


Synchronization

Methods : (1) external sync signals;

(2) self-synchronization: derivation from the modulated signals

Example 1: squaring the received NRZ signals and PLL

Sync signal


Example 2

4

Sync signal


Carrier Modulation

Simple modulation applied to baseband digital signals

Example: Let ( ) be the NZR waveform. Amplitude Shift-Keying (ASK): ( ) [1 ( )]cos(2 )

Phase Shift-Keying (PSK): ( ) cos[2 ( )]2

Frequency Shift-Keying (FSK): ( ) cos[2

ASK c c

PSK c c

FSK c c

d tx t A d t f t

x t A f t d t

x t A f t

πππ

π

= +

= +

= + ( ) ]t

fk d dα α∫

Principles of Communications Lecture 8: Baseband...

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