ece 4710: lecture #10 1 digital signaling what is appropriate way to mathematically represent the...

ECE 4710: Lecture #10 1

Digital Signaling

What is appropriate way to mathematically represent the waveform of a digital signal?

What is the bandwidth of the digital signal? BW depends on pulse shape used to represent digital data

Only indirectly related to bandwidth of analog signal bandwidth via

sampling frequency fs Digital waveform can be represented by series summation of

N orthogonal functions

N is dimension of orthogonal function set = # of (t) functions required to represent all possible waveforms for digital signal

wk represents the digital data (e.g. 101 w1 = 1, w2 = 0, w3 = 1)

N

kkk Tttwtw

100,)()(


Orthogonal Functions

What is orthogonal? satisfies mathematical condition

Example: sin(t) and cos(t)

mndttt m

b

a

n ere wh0)()(

0)(4

1

)()(4

1)(cos)sin(

)(2

1)cos( soand

)(2

1cos)(

2

1sin

0220

dteeeej

dteeeej

tt

eetjj

eeteej

t

b

a

jtjt

jtjtb

a

jtjtb

a

jtjt

jtjtjtjt


Orthogonal Functions

Orthogonal Another word is “perpindicular”

» Sine and cosine are 90° out of phase

In complex domain» Orthogonal characteristic of sine/cosine» Cosine Real axis» Sine Imaginary axis

» Used to represent vector R

Uniqueness of orthogonal characteristic

enables the vector representation Many other types of orthogonal function sets

|R|cos

|R|sin

|| RR

Real

Im


Symbol & Bit Rate

For N dimension waveform set transmitted over T0 seconds: Symbol Rate = D = N / T0 (symbols/sec or sps)

» Also called baud rate outdated» Please use symbol rate (sps) in this class

Bit Rate or Data Rate = R = n / T0 (bits/s or bps)

If wk’s have binary values then n = N and D = R

» 2 states only per symbol binary signaling

If wk’s have more than 2 possible states and D R


Vector Representation

Orthogonal function space

can be represented in vector space by

where

w is an N dimensional vector and

the set {j } is orthogonal set of N directional vectors Shorthand notation for w is row vector

Note book uses bold w for vector representation

N

kkk Tttwtw

100,)()(

N

jjjww

1

Nwwww ,...,, 21



Three-bit binary signal s(t) represented by 3-bit waveform

Let p(t)

So

t

5 V

T

T0 = 3T

t

5 VT

(1,0,1)),,( e wher)()( 321

3

1

ddddtpdtsN

jjj

])([)( and 21 Tjtptp j

Bit-Shape Waveform

Functional Space



Orthogonal Function Set

t

5 VT

p(t) t5 V

T 2T 3T

p1(t)

t5 V

T 2T 3T

p2(t)

t5 V

T 2T 3T

p3(t)



Orthogonal Vector Space

3

1

)(N

jjjsts

N = 3 dimensions

2N = 8 possible

messages for each symbol


Digital Signal BW

Lower bound for digital signal BW

Lower bound only achieved for sin(x)/x pulse shape Other real pulse shapes will have larger BW

Binary Signal Example M = 256 possible message & n = 8-bit binary words

T = 1 msec so T0 = 8 msec

Example message = 01001110 so

)Hz( 2

1

2 0

DT

NB

)...,,,(0,1,1,1,0,0,1,0 821 wwww


Binary Signal Example

Bit Rate = R

= n / T0 = 8 / 8 ms

= 1 kbps

Symbol Rate = D = R = 1 ksps since it is a binary signal

Rectangular Pulse Shape, Tb = 1 ms

sin(x)/x Pulse Shape, Tb = 1 ms

0 1 0 0 1 1 1 0

0 1 0 0 1 1 1 0


Binary Signal BW

Rectangular Pulse FNBW = 1 / T = 1 / 1 msec = 1 kHz Digital source info transmitted with digital waveform

Sin(x) / x Pulse Smooth rounded corners have much less frequency content Digital source information transmitted with analog waveform Pulse shape has no ISI if sampled exactly at midpoint of bit period

see sampling points in previous figure

Absolute BW = minimum BW = 0.5 D = 500 Hz

Other Pulse Shapes Filter rectangular pulses to reduce BW Studied next


Multi-Level Signaling

Multi-level signaling Binary signals have L = 2 states/symbol

» “0” = 0 V and “1” = +5V

Multi-level signaling has L > 2 states/symbol» # bits / symbol = log2 (L)

Two possible benefits:1) For same symbol period, if L then # bits per unit time

data rate is increased OR

2) If L we can increase the symbol period to maintain the same data rate BW 1 / Ts so BW will be reduced



Binary to Multi-Level Conversion for L = 4

Binary Input Output Level

11 +3

10 +1

00 -1

01 -3

Example message = 01001110 -3, -1, +3, +1 Same message as previous binary signal example



Bit Rate = R = n / T0

= 8 / 8 ms = 1 kbps

Symbol Rate = D = N / T0

= 4 / 8 ms = 500 sps

N = 4 Dimensions

{w1, w2, w3, w4} =

{-3, -1, +3, +1}

Multi-Level Rectangular Pulse Shape

Multi-Level sin(x)/x Pulse Shape

Ts = 2 msec

01 00 11 10


Multi-Level Signal BW

Rectangular Pulse FNBW = 1 / Ts = 1 / 2 msec = 500 Hz

Sin(x) / x Pulse Absolute BW = minimum BW = 0.5 D = 250 Hz

BW’s are 2 smaller than same message for binary signal Data rate kept the same Symbol period increased by factor of 2 BW Alternate approach would be to keep same BW and

increase data rate by factor of 2

ece 4710: lecture #10 1 digital signaling what is appropriate way to mathematically represent the...

Documents