William Stalling - Slide 03 Nyquist Notes

8/8/2019 William Stalling - Slide 03 Nyquist Notes

1/5

ECE-4603htt p: / / www.csc.gatech.edu/ ~copeland/4 603

Prof. John A. Copeland

[email protected] ech.edu

404 894 -5177

Office: GCATT 5 79

email or call Mrs. KathyCheek, 4-5 696 , for off ice visit t ime

Supplement t o Chapt ers 3 and 5

Data & Computer Comm., p 95-98, 148-155

Analog t o Digital Conversion

Bandwidth versus Capacit y

Nyquist and Shannon equations.

SignalDegradat ion

Basebase and Broadband Encoding

ECE4603 5a

Analog and Digit al SignalsAnalog Signal - v alue varies continuously

Sound - air pressure wave, can be convert ed t o analog

volt age or current w ave by microphone.

Electrical ( analog) voice fr equency signals - can be

convert ed to sound by loudspeaker.

Pict ure - light (c olor, brightness) varies cont inuously wit h

position.

Digital Signals - st rings of d iscret e values which can be

represent ed by binary numbers, st rings of one's and zeros.

Time-varying analog signals are convert ed by averaging t he

value over a t ime increment and assigning a binary value.

Images are divided into small areas (pixels), then color and

bright ness are averaged over each pixel and t ransmitt ed are

stor ed in digit al memory sequent ially.

Conversion of A nalog t o Digit al

Time increment s, dt, must be small enough to capt ure highest -

frequency components of interest.

The number of Amp litude Increments, n, must enough to give

an adequately-accurat e estimat ion of t he analog signal.

The bit rate is n/ dt (bits per second).

7 = 0111

-7 = 1111

0 = 0 0 0 0

1101

= - 50101 0 101 01011101 1101 1101Bit Stream

Reconstruction

Reconstruct ion

after low-pass

filter

4 b i ts -> 16levels

maximum

Sampling time = dt

binary numbers for

average value of

signal during dt

intervals

0101

= +5


2/5

Digital Voice

Voice is recognizable and intelligible when frequency range is

limited t o 300 t o 3400 Hz (Hertz = cycles/ second) and

quantizat ion noise < -20 d B.

Telephone standard is to digit ize 800 0 t imes per second (dt=1 25

ms) and to use 12 8 levels (8 bits including sign).

This results in a bit rat e for a voice channel of 64 ,000 b/ s.

Digit al Music

Music requires a frequency range up t o 18 ,00 0 Hz and

quantizat ion noise < -65 d B.

CD music is digit ized 44,100 times per second (dt=2 3 s) into

32,0 00 levels (16 bits including sign).

This results in a bit rat e for a CD music channel of 71 0,00 0 b/ s

(t imes 2 for stereo).

TelephoneVoice

(Sound Waves)

Modem

A/ D Channel

Bank

Digital

Trunk

6 4 k b/ s

ISDN

Telephone

Voice

(Sound Waves)ISDN

Swit ch

Digital

Trunk

6 4 k b/ s

PCAnalog

Swit ch

0110010101100110

Analog and Digital Signals

RS-232

28 ,800 k b/ s

64,000

k b /s

01100 01100

01100

01100

Channel Bandwidt h versus Capacit y

Capacity = Rate in bits/second (b/ s) t hat b its can be transmit ted.

Bandwidth = Frequency range (widt h)of t he channel (Hertz) .

Maximum Baud Rate (symbols/ sec), B = 2 H

Noise = Unpredictable deviation of r eceived value due t o all causes(probability bounded by acceptable error rate).

- Thermal noise, Crosst alk, Impulse, Interm odulatio n, ...

Nyquist Theorem for a noiseless channel: given a bandwidt h H and a signalwith L dif ferent levels, t he capacity C is given by:

Maximum Data Rate (bit s/ sec), C = 2 H log 2 ( L )

Shannon's Theor em for a noisy channel with signal to noise ratio S/N is

given by:

C = H log 2 ( 1+S/N )


3/5

Maximum Symbols ( Baud) per Second

0

1

1 / R

1 / 2 f

Since Bandwidth (H) = fmax:

Rmax = 2 f max =2 H

The maximum number of independent t ime intervals(symbols) is equal t o t wice the bandwidt h.

Number of Bits per Symbol

10

00

The number of bit s per level, N bL , is log 2(L) .

Levels Bit s

2 1

4 2

8 3

16 4

32 5

64 6

Two bit s per symbol requires 4 levels.

11 01 10

11

01

00

01

1

1 0

1 1

0 0

0 1

1 0

0 0

1 0

0 1

1 1

0 0

1 0

1 1

01

0 1

Think of a "Baud" as a " Bucket " t hat holds a

certain number of bit s.

The number of b it s per second t hat pass a

cert ain point (or arrive at a node) =

t he number of Baud (Buckets) per second

mult iplied by

t he number of bit s carried by each Baud (in each

Bucket).


4/5

The Capacity (maximum bit rat e) is equal to t he maximum number of symbols persecon d (2 x Ba nd widt h ) m ul t ip li ed b y t he nu mb er o f b it s p er sym bo l { l o g2 (L ) } . -

Nyquist: C = 2 H log2(L )

Intuitively, t he maximum number of levels with an maximum signal volt age Vs is

approximately Vs/ Vn, where Vn is the noise voltage (separation voltage betweenlevels needed to reduce errors due to noise t o acceptable value). Claude

Shannon (father of Information theory) found a more exact value as a function ofthe rat io of signal power S to noise power N.

Shannon: C


5/5

Dat a Encoding Schemes

Baseband - frequency range is from 0 Hz to f max

(B = f max )

Broadband - frequency range is from f min t o f max

(B = f max - f min )

With broadband signals, dat a is modulat ed ont o a carrier wave whose

frequency is at t he center of the band. Common types of modulation

are:

Amplit ude Shift Keying (ASK)

Frequency Shift Keying ( FSK)

Phase Shift Keying ( PSK)

Quadrat ure Phase Shift Keying ( QPSK)

Quadrat ure Amplit ude Modulat ion (QAM)

With all commonly- used Broadband modulat ion schemes:

Bandwidt h (Hz) = Baud Rat e (Baud/ second)

Broadband Modulat ion Encoding

A sine wave with arbit rary amplit ude A and phase can be

represented by t he amplitude of t he in-phase and quadrat ure

component s, P & Q:

V(t ) = A sine(wt+) = P sine(wt ) + Q sine(wt+ 90)

P

Q

P

Q

P

Q

P

Q

QPSK

2 b i ts

QAM

4 b i ts

PSK

1 bit

ASK

1 bit

Each stat e is represented by a dot in t he

"constellation"diagram.

Baseband Encod ing Schemes

Problems to overcome

Transitions needed t o synchronize receiver bit clock

DC reference level needs t o be maint ained

0 0 1 1 0 1

RZ - Retu rn

t o Zero

NRZ - Non

RTZ

Good ref.,

doubles

bandwidth

Transitions

not certain

NRZ-I

Inverted

Differential:

Polarity

insensitive

Man-

chesterGood timing,

doubles

bandwidth

30

Bit stream

Documents

William Stalling - Slide 03 Nyquist Notes