sampling theoremihayee/teaching/ee5765/ece5765_chapter_6_7.pdf · pulse code modulation (pcm)...

Department of Electrical and Computer Engineering

Sampling Theorem

tsT

t

)(tg t

][ snG

)(G

s

sT

f1

Sampling Frequency

)(tg )(sG

n

sT nTtts

)()(


Sampling Theorem

Let’s g(t) is band limited to B Hz, then samples of g(t) could be written as

)()()()()( ss

n

T nTtTngttgtgs

...]3cos22cos2cos21[1

)( tttT

t sss

s

Ts

s

s

s fT

22

where

We know that


)()()( ttgtgsT

...]3cos)(22cos)(2cos)(2)([1

ttgttgttgtgT

sss

s

By taking Fourier Transform on both sides

1

cos)(2)(

n

s

ss

tntgTT

tg

)]()([1)(

)(1

s

n

s

ss

nGnGTT

GG

n

s

s

nGT

)(1

Bfs 2B

Ts2

1or

Nyquist Criterion


Sampling frequency should be at least equal to or greater

than twice the bandwidth of the message signal for

successful recovery of the signal from it’s samples

Bfs 2

Sampling Theorem

f

sT

1

BBsf sf2 sf3 sf4

sfsf2sf3

sf4

Lowpass Filter

bandwidth of lowpass filter = B Hz


t

)(H)(th

Ideal Reconstruction from Samples

B2

1

B2

1

B2 B2

B2

2

B2

3

)2(sin2)( BtcBTth s

sT

)4

()(B

rectTH s

)2(sin)( Btcth )12( sBT


)()()( ss nTthnTgtg

)](2[sin)( ss nTtBcnTg

)2(sin)( nBtcnTg s

)(H)(t )(th

)(H)(tg )(tg

)()()()()( ss

n

T nTtTngttgtgs

Recall


t

)(tgRecovered


Practical Considerations in Nyquist Sampling

• Gradual roll-off lowpass filter

• Aliasing

• Practical Pulses vs. Ideal Impulses


Sampling Theorem

tsT

t

)(tg t

][ snG

)(G

s

sT

f1

Sampling Frequency

)(tg )(sG

n

sT nTtts

)()(

Not practical to generate!


Let’s Try Square Wave

t

)(tx

sT

t

)(tg t

)(tg

][ snX

)(G

)(G

s

sT

f1

Sampling Frequency

Problem Resolved1


Sampling frequency should be at least equal to or greater

than twice the bandwidth of the message signal for

successful recovery of the signal from it’s samples

Bfs 2

Sampling Theorem

f

sT

1

BBsf sf2 sf3 sf4

sfsf2sf3

sf4

Lowpass Filter

bandwidth of lowpass filter = B Hz


Maximum Information Rate

Noiseless Channel of Bandwidth Bt

Signal of bandwidth B

2B samples per second

2B Independent pieces of information per second

Two independent pieces of information per second per hertz of bandwidth


Sampling Theorem and Digital Communication

• Pulse Amplitude Modulation (M-ary)

• Pulse Width Modulation (M-ary)

• Pulse Position Modulation (M-ary)

• Pulse Code Modulation (Binary)


Pulse Code Modulation (PCM)

Sampling Quantization Binary Coding

Speech Signal from 15Hz to 15kHz

Maximum

Frequency

Sampling

Rate

Quantization

Levels

Number of

Bits/code

Bit Rate

Telephone 3.4 kHz 8 kHz 256 8 64 kb/s

Compact

Disk

15 kHz 44.1 kHz 65,536 16 705.6 kb/s


Quantization

)2(sin)()( nBtcnTgtg s

)2(sin)()( nBtcnTgtg s

)()()( tgtgtq

)2(sin)]()([)( nBtcnTgnTgtq ss

)2(sin)()( nBtcnTqtq s

Reconstruction of actual samples after sampling

Reconstruction of quantized samples after sampling

Distortion in reconstructed signal

where q(nTs ) is the quantization error in nth sample


Power or mean square value of q(t) (quantization noise) is given by

2/

2/

22

)(1

lim)(

T

TT

dttqT

tq

2/

2/

2])2(sin)([1

lim

T

T n

sT

dtnBtcnTqT

Now by using the fact that sinc() are orthogonal functions i.e.,

0

2

1

)2(sin)2(sin

B

dtmBtcnBtc m≠n

m=n

2/

2/

222

)2(sin)(1

lim)(

T

T n

sT

dtnBtcnTqT

tq


2/

2/

222

)2(sin)(1

lim)(

T

T n

sT

dtnBtcnTqT

tq

2/

2/

22 )2(sin)(1

lim

T

Tn

sT

dtnBtcnTqT

)2

1()(

1lim 2

BnTq

T n

sT

n

sT

nTqBT

)(2

1lim 2

=> quantization noise is average of square of quantization error

Let’s calculate this


quantization error lies in the range of -Δv/2 and Δv/2

whereL

mv

p2

Assuming quantization error is equally likely in the range of -Δv/2 and Δv/2

dqqv

q

v

v

2/

2/

22 1

2/

2/

3

3

1v

v

q

v

12

)(1 3v

v

2

22

312

)(

L

mv p


2

2

2

3)(

L

mtqN

p

q

)()()( tqtgtg

)(2

0 tgS

2

2

03L

mNNand

p

q

2

22

0

0 )(3

pm

tmL

N

SSNR Therefore

Desired signal Unwanted noise


Nonuniform Quantization

Uniform Quantization Nonuniform Quantizationpm

pm

pm

pm

t t


Nonuniform Quantization = Compression + Uniform Quantization

(Usually compression increases the bandwidth but not applicable here)

mpm

m

y

y

nonuniform

uniform


)1ln()1ln(

1

pm

my

Example

-law

1000

100

10

1

0 0.15.0

5.0

0.1

pm

m

y

10 20 30 40 50 60

10

20

30

40

50

1

255

256L

)(0

0 dBN

S

Relative Signal Power (dB)


Transmission Bandwidth for PCM

BBandwidthSignal

BRateSampling 2

nLlevelsQuantized 2

nbitsencodedofNumber

sec/2 bitsnBRateBit

HznBBandwidthonTransmissiquired Re


Digital Communication and Time Division Multiplexing

• T1 Carrier System (DS1) is an example

• 1.544Mbits/sec total data rate for 24 channels

• Each channel is encoded with 8 bits

Ch 1

Ch 2

Ch 3

Ch 24

Ch 23

CoderDigital

Processor

Transmission Medium

Digital

ProcessorDecoder

Ch 2

Ch 1

Ch 24

LPF

LPF

LPF


What does Digital Processor do?

1. Synchronization (Framing)

2. Signaling

3. Parity Insertion and Check

4. Scrambling


Energy of a Signal and Parseval’s Theorem

dttgtgEg )(*)(

dtdeGtgE tj

g

)(*2

1)(

ddtetgGE tj

g )()(*2

1

dGGEg )()(*2

1

dGEg

2)(

2

1


Energy Spectral Density

2)()( Gg

dGEg

2)(

2

1

energy per unit bandwidth

ESD of the Input and the Output

)(ty)(tg )(th

)()()( GHY

222)()()( GHY

)()()(2

gy H


Autocorrelation function of a signal

dttgtgg )()()(

dxxgxgdxxgxgg )()()()()(

txlet

dttgtg )()(

)()( gg


ddttgtgeF j

g ])()([)]([

dtdetgtg j ])([)(

dtdetgtg j ])([)(

dteGtg tj ])([)(

dtetgG tj

)()(

2)()()( GGG

)()( gg


Signal Power and Power Spectral Density

dttgT

PT

g )(1

lim 2

2/)(

2/0)(

Tttg

TtT tg

T

dttg

P

T

Tg

)(

lim

2

T

ETg

T lim


])(2

1[

1limlim

2

dG

TT

EP T

T

g

Tg

T

dGdttgE TTgT

22 )(2

1)(

d

T

GP

T

Tg

2)(

lim2

1

T

GS

T

Tg

2)(

lim)(

We define PSD as

So that

0

)(1

)(2

1

dSdSP ggg


Autocorrelation function of a power signal

dttgtgTT

g )()(1

lim)(

)()( gg

Same argument as in energy signals will prove that

)()( gg Sand

also )()()(2

gy SHS

)()()( GHY if


Various Digital Line Codes

1 1 0 1 0 0 1 1 1 0

On-Off

Polar

Bi-Polar

Pulse shape p(t) could be arbitrary

t

t

t


Some Characteristics of Good Line Codes

1. Bandwidth Efficient

2. Power Efficient

3. Error Detection or Correction Capability

4. Favorable PSD

5. Adequate Timing Content

6. Transparency


PSD of various Line Codes

)(tp

t

t

)(ty bTk )1(

bkT

bTk )1(


t

)(tx bTk )1(

bkT

bTk )1(

)(ty)(tx )()( tpth

)()()(2

xy SPS

t

)(ty bTk )1(

bkT

bTk )1(

ka

1ka1ka


t

)(tx bTk )1(

bkT

bTk )1(

PSD of x(t)

t

)(ˆ tx bTk )1(

bkT

bTk )1(

1kh

kh

1kh

kk ah

ka

1ka1ka


when

2/

2/

ˆ )(ˆ)(ˆ1

lim)(

T

TT

x dttxtxT

k

kT

x hT

)(1

lim)( 2

ˆ

k

kT

aT

)(1

lim2

2

)( kk ah

)1(0

bT

R

k

kb

Ta

T

TRwhere 2

0 lim

k

kT

aT

)1(1

lim 2

k

k

b

b

Ta

T

T

T

2)1(1

lim


bT

TNnotice

)1()( 0ˆ

b

xT

R

as

TwhenN

k

kb

Ta

T

TRwhere 2

0 lim

k

kN

aN

R 2

0

1lim

2

ka

)1()( 0ˆ

b

xT

R

)(ˆ x is even function of time


bTat

111

1lim

kkk

k

kN

aaaaN

Rwhere

)1()( 1ˆ

b

xT

R

nkknk

k

kN

n aaaaN

RWhere

1

lim

)1()(ˆ

b

nx

T

R

Similarly, at bnT


)1()(ˆ

b

nx

T

R

)(lim)( ˆ0

xx

)()( b

b

nx nT

T

R

n

bn

b

nTRT

)(1

n

Tjn

n

b

xbeR

TS

1

)(

,bnTat

n

b

b

nx nT

T

R)()(

For all


n

Tjn

n

b

xbeR

TS

1

)(

1

0 )cos2(1

n

bn

b

TnRRT

)()()(2

xy SPS

1

0

2

)cos2()(

)(n

bn

b

y TnRRT

PS

We know that


PSD of Polar Signaling

1

0

2

)cos2()(

)(n

bn

b

y TnRRT

PS

transmission of 1 with p(t) and 0 with –p(t)

k

kN

aN

R 2

0

1lim

1)(1

lim

NNN

11

1lim

k

k

kN

aaN

R

0)]1(2

)1(2

[1

lim

NN

NN

0 nR 0nif

b

yT

PS

2)(

)(


b

yT

PS

2)(

)(

)()(bT

trecttpif

)2

(sin)( bb

TcTPthen

)2

(sin)( 2 bby

TcTS

*Tweaking PSD with Pulse Shaping

having null at zero frequency


Noise and Digital Communication

at Transmittert

1 1 0 1 0 0 1 1 1 0

t

t

at Receiver

(less noise)

at Receiver

(more noise)

pA

pA


Gaussian Noise and Probability Density Function

n

)(np

2

2

2

2

1)( n

n

n

enp

dnednnpnob n

n

n

2

2

2

2

1)()(Pr


Error Probability for Polar Signaling

)]1(Pr)0([Pr2

1)(Pr EobEobEob

pAnthatyprobabilitEob )0(Pr

pAnthatyprobabilitEob )1(Pr


p

n

p A

n

nA

dnednnpEob2

2

2

2

1)()0(Pr

p

n

A

n

n

dne2

2

2

2

1

npA

x

dxe /

2

2

2

1)(

n

nx

y

x

dxeyQ 2

2

2

1)(

Let’s define

)()0(Prn

pAQEob


)(n

pAQ

p

n

p A n

n

A

dnednnpEob2

2

2

2

1)()1(Pr

)]1(Pr)0([Pr2

1)(Pr EobEobEob

Now

)]()([21

n

p

n

p AQ

AQ

)(n

pAQ


Error Probability for ON-Off Signaling

)]1(Pr)0([Pr2

1)(Pr EobEobEob

)2

(2

)0(Prn

pp AQ

AnthatyprobabilitEob

)2

(2

)1(Prn

pp AQ

AnthatyprobabilitEob

)]2

()2

([2

1)(Pr

n

p

n

p AQ

AQEob

)2

(n

pAQ


Error Probability for Bi-Polar Signaling

)]1(Pr)0([Pr2

1)(Pr EobEobEob

)2

()0(PrpA

nprobEob

)2

(2n

pAQ

)2

()2

(pp A

nprobA

nprob

)]2

([2pA

nprob


)2

()1(PrpA

nprobEob

)2

(n

pAQ

)2

(pA

nprobor

when positive pulse is used

when negative pulse is used

)]1(Pr)0([Pr2

1)(Pr EobEobEob

)]2

()2

(2[2

1

n

p

n

p AQ

AQ

)2

(2

3

n

pAQ

Now

sampling theoremihayee/teaching/ee5765/ece5765_chapter_6_7.pdf · pulse code modulation (pcm)...

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