U N I T I I Baseband Demod V S H

04/10/20231V. S. Hendre Department of E&TC,

TCOER, Pune

UNIT-II

Baseband Demodulation/

Detection Techniques

04/10/2023 2

UNIT-II: Baseband Demodulation/Detection Techniques

• Signals & noise, • Data formats, • Synchronization • multiplexing, • Intersymbol interference, • Equalization,• Detection of binary signals in presence of

Gaussian noise,• Matched and optimum filters.

V. S. Hendre Department of E&TC, TCOER, Pune

04/10/2023 V. S. Hendre Department of E&TC, TCOER, Pune

3

INTRODUCTION

• In case of baseband signaling, the waveforms at Rx are in pulsed form, but these pulses are not in ideal form.

• Due to such degradation & filtering at the transmitter, the problem of Intersymbol Interference occurred.

• The goal of the demodulator are: 1) Recovered the baseband signal with less degradation2)The SNR should be as high as possible3) The received signal should be free from ISI.

Baseband Signaling Transmitter

Baseband Signaling Receiver

04/10/2023 4

INTRODUCTION


Signal Source

Source encoder

MultiplexerChannel encoder modulator

Communication channel

Unit-III

Line codes

(Unipolar, polar)

-NRZ, RZ, AMI

-Manchester

Digital Mux

- Synchronous

-Asynchronous

-quasi-sync

Synchronization-bit, frame

Scrambling & unscrambling

UNIT-II Baseband

Demod/Detection Tech

Line Coding/ Data Formats


5

Binary line codes• Line coding: waveform pattern of voltage or

current used to represent the 1s & 0s on a transmission link Line coding

• Because of the ac coupling in the transformers & repeaters it is desirable to have a ‘0’ dc in the waveform generated by PCM

Line


6

How DC component is generated???Reason 1


7

Reason 2• Digital data representation

• Well suited inside the machines (computers)• Not suitable for long distance-due to presence

of stray capacitance in the transmission medium

• For sufficient capacitance on line-adds DC component to data stream

0 V

5 V


8

Conti…

• Problem of synchronization

• Receivers clock oscillator locks on the signal level

• for long string of 1s & 0s-no level shift• receiver oscillator frequency drifts• Unsynchronise


9

Figure 4.3 Effect of lack of synchronization

04/10/2023 10

Line Coding Formats/ Data Formats/

Transmission Coding Formats• Binary signaling in the original form suffers degradation

& ISI occurs.• To avoid these problems we are converting this digital

pulses into another form of digital pulses which make this data suitable for line or channel.

• Hence this digital to digital conversion is called as Line Coding or Data Formats.

• If baseband data is itself in digital form, then it is necessary to convert that data into a PAM suitable format i.e. Data format

• There are variety of Line coding or Data formats available but they are selected based upon different properties.



11

Requirements• 1) Small BW: to send more signals in a

communication channel• 2) Enough Timing content: for receiver to

extract the clock information & decode the signal

• 3) small probability of error: increases reliability of line codes

• 4) Good power efficiency: for a specific BW; transmitted power should be small

• 5)Transparency: the coded signal should be received correctly

04/10/2023 12

Properties of Line Coding Formats


1) Transmission Bandwidth: It should be as small as possible.

2) Power Efficiency: For a given bandwidth and specified detection error probability, it should be as small as possible

3) Error Detection & Correction Capability: eg in bipolar case single bit error will be indicated by polarity variation

4) Favorable PSD: PSD should be zero at w=0, i.e. D.C. component should be zero.

5) Adequate Timing Content: possible to extract timing or clock information from signal

6) Transparency: Possible to transmit digital signal correctly regardless of continuous ‘1’ & ‘0’


13

• I) Level codes--They are independent on past data. -They carry information on their voltage levels-two common formats-RZ & NRZ-NRZ:Pulse level remains constant during the bit duration-RZ:Pulse level zero for a portion of bit duration

• II) Transition codes--Current bit level depends on the previous levels-codes have memory

Ex: Miller code, Split phase (mark), Bi-phase (mark), Code mark inversion (CMI), Dou-Binary, Dicode

Line codes: category

04/10/2023 14V. S. Hendre Department of E&TC, TCOER, Pune

Line Coding Formats

Split phase manchester


15

Mapping of Data Symbols into Signal Levels


16

Unipolar• All signal levels are on one side of the time

axis - either above or below• NRZ - Non Return to Zero scheme is an

example of this code. The signal level does not return to zero during a symbol transmission.

• Scheme is prone to baseline wandering and DC components. It has no synchronization or any error detection. It is simple but costly in power consumption.


17

Unipolar NRZ scheme

Advantage:1) only one power supply2) Easy to generateDisadvantage: 1) more power consumption2) prone to DC component3)spectrum is not approaching zero near DC


18

Spectrum: Unipolar NRZ

22

)sin

(4

)(b

bbNRZunipolar fT

fTTAfP

BW= R (Hz), data rate

Application: magnetic tape recording


19

Unipolar RZ

• Advantage:1)relatively simple to implement2)DC level is lower than NRZ

• Disadvantage:1)BW=2R (Hz)2) needs 3 dB more power than polar signaling for the same probability of error

+A

0


20

Spectrum: Unipolar RZ

22

)2/

)2/sin((

16)(

b

bbRZunipolar fT

fTTAfP

Bandwidth: 2R

Application: In baseband data transmission & magnetic tape recording


21

Polar - NRZ

• The voltages are on both sides of the time axis.

• Polar NRZ scheme can be implemented with two voltages. E.g. +V for 1 and -V for 0.

• There are two versions: – NRZ - Level (NRZ-L) - positive voltage for

one symbol and negative for the other– NRZ - Inversion (NRZ-I) - the change or lack

of change in polarity determines the value of a symbol. E.g. a “1” symbol inverts the polarity a “0” does not.


22

Polar NRZ-L and NRZ-I schemes

Advt.:1)relatively easy to generate

2) better error probability

Disadvt.:1)requires two different voltages

2) Large PSD near zero


23

Spectrum: Polar NRZ

• Bandwidth=R (Hz)

22 )sin

()(b

bbNRZpolar fT

fTTAfP


4.24

In NRZ-L the level of the voltage determines the value of the bit.

In NRZ-I the inversion or the lack of inversion

determines the value of the bit.

Note


4.25

NRZ-L and NRZ-I both have an average signal rate of N/2 Bd.

Note


4.26

NRZ-L and NRZ-I both have a DC component problem and baseline

wandering, it is worse for NRZ-L. Both have no self synchronization &no error detection. Both are relatively simple to

implement.

Note


27

Polar - RZ• The Return to Zero (RZ) scheme uses three

voltage values. +, 0, -. • Each symbol has a transition in the middle.

Either from high to zero or from low to zero.• This scheme has more signal transitions

(two per symbol) and therefore requires a wider bandwidth.

• No DC components or baseline wandering.• Self synchronization - transition indicates

symbol value.• More complex as it uses three voltage level.

It has no error detection capability.


4.28

Polar - RZ


4.29

Polar - Biphase: Manchester and Differential Manchester

• Manchester coding consists of combining the NRZ-L and RZ schemes.– Every symbol has a level transition in the

middle: from high to low or low to high. Uses only two voltage levels.

• Differential Manchester coding consists of combining the NRZ-I and RZ schemes.– Every symbol has a level transition in the

middle. But the level at the beginning of the symbol is determined by the symbol value. One symbol causes a level change the other does not.


4.30

Figure 4.8 Polar biphase: Manchester and differential Manchester schemes


31

Manchester NRZ/Bi--L/split phase/self clocking line code

• Binary 1positive half bit period pulse followed by negative half bit period pulse

• Binary 0 negative half bit period pulse followed by positive half bit period pulse

• Advt:1)always 0 DC value regardless of data sequence

• 2)a string of 0’s will not cause a loss of clocking signal

• 3) In built ‘single error detection’ capacity• Disadvt: needs twice the BW of Unipolar

NRZ/Polar NRZ codes


32

spectrum

)(sin]2/

)2/sin([)( 222

bb

bbNRZmanchester fT

fT

fTTAfP

Applications:1) LAN like Ethernet & cheaper net 2) IEEE 802.3 baseband coaxial & twisted pair CSMA/CD bus LANS


4.33

In Manchester and differential Manchester encoding, the transition

at the middle of the bit is used for synchronization.

Note


4.34

The minimum bandwidth of Manchester and differential Manchester is 2 times that of NRZ. The is no DC component and no baseline wandering. None of

these codes has error detection.

Note


4.35

Bipolar - AMI and Pseudo ternary

• Code uses 3 voltage levels: - +, 0, -, to represent the symbols (note not transitions to zero as in RZ).

• Voltage level for one symbol is at “0” and the other alternates between + & -.

• Bipolar Alternate Mark Inversion (AMI) - the “0” symbol is represented by zero voltage and the “1” symbol alternates between +V and -V.

• Pseudoternary is the reverse of AMI.


36

Alternate Mark Inversion (AMI)/Bipolar RZ/Polar RZ

BW=R (Hz)


37

• Pseudo Ternary: 3 encoded signal levels for two level data

• Advt: 1) Low BW, 2) zero DC value3) In built ‘single error detection’ capacity4) capable of recording clock information

• Disadvt: 1) a long string of successive 0s will adversely affect the precision of synchronization2)The receiver has to distinguish 3 different levels instead of just 23) Needs @ 3dB more signal power than polar signal for same probability of error

• Application: Telephone systems


Unipolar NRZ

Polar NRZ

Unipolar RZ

Bipolar RZ

Split-phase or Manchester code

All Line Codes


39


40

PSD of All Line codes


41

Comparison of all PSD’s


42

MULTIPLEXING

• Whenever the bandwidth of a medium linking two devices is greater than the bandwidth needs of the devices, the link can be shared.

• Multiplexing is the set of techniques that allows the simultaneous transmission of multiple signals across a single data link.

• As data and telecommunications use increases, so does traffic.


43


44

Digital Multiplexing

• Many base-band signals common channel

Analog system: 1) TDM, FDM. Digital system: interleaving ~TDM


45

Digital Multiplexing

• The MUX has to perform four functional operations:

1) Establish a frame as the smallest time interval containing at least one bit from every input.

2) Assign to each input a number of unique bit slots within the frame

3) Insert control bit for frame identification & synchronization

4) Make allowance for any variations of the bit rate.

Analog system: 1) TDM, FDM. Digital system: interleaving ~TDM


46

• Interleaving ProcessEx. TRINITY COE PUNE………..

T R I N I

T Y C O

E P U N

E . . . .

Data Entry Row wise Shift

Register Bank

T R I N I

T Y C O

E P U N

E . . . .

Read Column wise

Interleaved Sequence : TTEERY . I P.NCU.ION.


47

DATA INTERLEAVING/INTERLEAVED CODES/INTERLACED CODE

• Impulse noise-lightening & switching transients- burst of error.• Burst of error of length b -sequence of b bit error (1st & last bits-1)• -in between (b-2) digits-either errornous or correct• Bursty channels-1.channel causing multipath & feding 2.magnetic recording channel (tape/disks)


48

C1 C2 C3 I4 I5 I6 I7

C8 C9 C10 I11 I12 I13 I14

C15 C16 C17 I18 I19 I20 I21

C22 C23 C24 I25 I26 I27 I28

(n-k)

Parity bits

K data bits

m

rows

Read out bits to modulator

data

entry


49

Advantages over TDM

• 1) Free from compulsion of periodic sampling• 2)waveform preservation-not necessary while

multiplexing • Multiplexer used-”Binary Multiplexer”• Multiplexed signal-source digits interleaved • -bit by bit/characters/words• For demultiplexing –constant bit rate at Tx


50

Problem of bit rate variation-solved• A) Synchronous multiplexer

-master clock governs all sources & eliminates bit rate variations-highest efficiency

Disadvantage: needs provision of distributing the master clock, design becomes complex.

• B) Asynchronous Multiplexer-used for data source that operates in start/stop mode with burst of characters with variable spacing betn bursts-principle-buffering & interleaving-Application-computer networks

• C) Quasi-synchronous multiplexer-used when input bit rates have the same nominal value but varies within specified bound


51

Multiplexing hierarchy for digital telecommunications:Increasing BIT RATES

Channel Bank

Digital

DataOnly

MultiplexingOther MUX point to point transmission

• Multiplexing patterns- 1) American Telephone & Telegraph Company (AT &T) hierarchy, 2) International Telegraph and Telephone Consultative Committee (CCIT) hierarchy

-both 64 kbps voice PCM unit Layout


52

Parameters of AT&T and CCITT hierarchies

Out put bit rate>sum of input bit rates

Surplus-control bit+ stuff bits (for steady output rate)

Level AT &T (North America & Japan)

CCIT (Europe & Asia)

No of Inputs

Output Rates (Mbps)

No of Inputs

Output Rates (Mbps)

First 24 1.544 30 2.048

Second 4 6.312 4 8.448

Third 7 44.736 4 34.368

Fourth 6 274.176 4 139.264


53

Illustrative configuration of the AT&T hierarchy


Figure 6.23 Digital hierarchy


55

Example-synchronous multiplexersBasic TDM scheme

Wide Band co-axial cable

Sequencial sampling

Local clock repeaters

Framing &


56

Figure 6.24 T-1 line for multiplexing telephone lines


57

Figure 6.25 T-1 frame structure


58

• Bits/frame:commutator speed-8000 revolution/sec:sampling rate-8000 samples/sec: each sample 8 bits\ no. of output bits=24x8=192 bits*frame synchronisation:synchronisation information-extra bits/frame

193 bits, 125 s

Channel 1Channel 23Channel 24

Frame

bit 8 bits8 bits8 bits


59

Figure 6.22 Framing bits


60

• Bit rate

-frame time Tp=1/8000=125 s-Tp occupies 193 bits-bit rate on T1 channel

fb(T1)=no. of bits/Time=193/125 Mb/s=1.544 Mbps• Signaling information/supervisory information-Dial pulses, busy signal with speech signal-added to voice signal-method “BIT ROBBING”-8th (LSB) bit of 6th sample- Voice transmission + signaling-1st five samples-Eight bit, 6th sample- 7 bits + 8th bit for

signalingNo. of bits in six frames=[5 frames x 8 bits]+[1 frame x 7 bits] =47 bitsAvg. bits/sample=47/6= 7 (5/6) bits


61

• Signalling bit frequecny=(1/6)xframe bit rate =(1/6) x 8000fb(T1) signalling = 1333 Hz• Signalling Technique- -“Channel Associated Signalling”


62

• Types: 1) Symbol / Bit synchronization

2) Frame Synchronization

3) Carrier synchronization• Synchronization in a Binary Receiver

Synchronization Techniques

Bit Synchronization:-Open loop bit synchronization

-closed loop bit synchronization

-Early Late Synchronization

y(t) LPF Regenerator

Bit Sync. CLK.

Frame Sync.

Output Message

Frame Identification


63

• Carrier Sync.:1)Mth power Law carrier synchronization

2)Costas Loop• Bit Synchronization:-Open loop bit synchronization• Used when y(t)=unipolar RZ format• y(t) from polar to unipolar conversion


64


65

Closed loop bit synchronization


66

• Disadvantages• Sync will suffer from timing jitter when 1)

zero crossing of y(t) are not spaced by integer multiples of Tb

• 2)message includes a long string of 1’s & 0’s

• Problem is solved by message scrambling


67

Early-late bit synchronization (a) waveform (b) block diagram

• Independent of zero crossings• Properly filtered digital signal has

peaks at the optimum sampling times & symmetric on either side

• tksinchronised & <Tb/2

|y(tk-)|~ |y(tk+)|<|y(tk|

• Early synchronisation

|y(tk-)|<|y(tk+)|

v(t)=|y(tk-)|-|y(tk+)|>0

speeds up clock

Filtered digital signal

Late sync

|y(tk-)|>|y(tk+)|


68

Scrambler & message sequence generator

• Coding technique at transmitter—long string of likely bits occure —randomized

• Eliminates periodic bit patterns-undesired discrete frequency components in the power spectrum

• Tapped shift registers are used


69

(a)Binary scrambler (b) un-scrambler at receiver

tap gains 1=2=0,

& 3=4=1

M”k=M’k-3 M’k-4

M’k=Mk M”k

a)

))(

)(""

"""'

bMMMM

MMMMM

kkkk

kkkkk


70

Input sequence:10 11 00 00 00 00 01


71

Frame synchronization

• Receiver should know-when signal is present

• Aspects of frame sync.1) Identify start of frame2) Identify subdivisions/subframes within the message

• For frame sync-special N bit sync word

t

prefix-time for bit-sync acquisation

N bit sync. word

Message bits

Start of message


72

• Start of message-different codeword followed by Prefix

• Frames are labeled by sync words-inserted periodically in the bit stream

• Frame synchronizer

~sync word bits in polar form

o/p bits with polar format

V


73

Carrier Synchronizer-1) Mth power carrier recovery circuit

Received signal

BPF ()M Narrow band F

Divide by M

Phase locked loop

VCO

LPF

PhaseCompa-

rator

Recovered carrier

Removes noise outside the Band

MfcSpectral components + Mfc fc

• Due to jitter-carrier at transmitter is unstable• Oscillator frequency drifts: f & larger• Narrowband filter cannot be made as narrow as our requirements• Replace narrowband filter by PLL• PLL-filter whose bandpass is determined by LPF• PLL will follow oscillator jitter


74

b) The Costas Loop

• Involves two PLL’s with common VCO & loop filter

VCO Loop filter

LPF, fc

LPF, fc

900 phase shift

Phase comparator

Phase comparator

)cos()(2 ttbP cs

)cos( tc

)sin( tc

)cos()(22

1 tbPs

)sin()(22

1 tbPs

Vm

)(2sin4

)cos()sin()()2(4

1 2

sm

sm

PV

tbPV


75

• If VCO frequency differs form carrier frequency progressive change in phase difference -

• This change change in Vm—increases/decreases VCO frequency


76

Signal & Noise• Error Performance Degradation in Digital

Communication System: The task of the detector is to retrieve the bit stream from the received waveform as error free as possible, notwithstanding the impairments to which the signal have been subjected.

• Two Causes of error:1) The effect of filtering at the transmitter,

channel, and receiver which causes ISI.2) Electrical noise and Interference produced by

a variety of sources, such as atmospheric noise switching transients, as well as interfering signals from other sources.


77

• No channel has infinite bandwidth• Most transmission schemes require higher bandwidth than available in the

channel.- Square wave requires infinite bandwidth.- Synch function is not possible due to causality violation.- Modified synch function to satisfy the causality requires higher bandwidth.

• Each symbol may be smeared into adjacent time slots. • Intersymbol Interference (ISI) is the spreading of symbol pulses from one slot into adjacent slots.

Intersymbol Interference


78

ISI & EYE Pattern

• Fundamental limitation (digital transmission)relationship betn ISI, BW & signaling rate • “Given an ideal Low pass channel of BW ‘B’,

it is possible to transmit independent symbols at a rate r≤2B boud without Inter- Symbol Interference”

• ‘No transmission at r>2B’• Practically, no channel-ideal freqn response• Linear distortion=amplitude & delay• Solution-channel+equalisationideal freqn

response


79

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

(a) Baseband transmission system (b) signal-plus-noise (ISI) waveform

Actual signal


80

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Baseband binary receiverFigure 11.2-1


81

• Overall T.F.=Hc(f).Heq(f)filtering characteristic

• If Heq(f)-chosen to minimize ISIfilter at Rx ‘Equalizing filter’

• Final o/p - distortion less (minimum ISI) if

Hc(f) Heq(f)x(t)

channel Equalizer

y(t)

dt-jeqc ek(f)(f).HH

Gain factor

Time delay


82

Methods to eliminate ISI• a) Nyquist First Method:Zero ISI• b) Nyquist Second Method:control of ISI• c) Nyquist Third Method• Eye-Pattern:effect of channel filtering & channel noise-seen

by observing received line codes on an Analog Oscilloscope-displayeye-pattern

• Eye pattern provides excellent way of assessing the quality of the received line code & the ability of the receiver to combat bit errors

• General eye pattern


83

• Under normal operating conditions (No detected bit error)

eye –opened

• For noise/ISI eye closedbit error at receiver


84

Eye Pattern Seen in oscilloscopeThe Cleaner, the betterGood indication of transmission quality


85

(a) Distorted polar binary signal (b) eye pattern


86

Binary Baseband Demodulation / Detection


87

Demodulation / Detection• For any binary channel, the transmitted signal

over a symbol interval (0, T) is represented by

• The received signal r(t) degraded by noise n(t) and possibly degraded by the pulse response of the channel hc(t) was described


88

Demodulation & Detection• Demodulation: Demodulation is a recovery of a

waveform (to an undistorted baseband pulse),• Detection: To mean the decision-making

process of selecting the digital meaning of that waveform.

• Frequency down-conversion block: performs frequency translation for band pass signals operating at some radio frequency (RF). It may take place within the front end of the receiver, within the demodulator, shared between the two locations, or not at all.


89

Demodulation & Detection• The Receiving filter: which performs waveform

recovery in preparation of the next important step-detection.

• The goal of the receiving filter is to recover a baseband pulse with the best possible signal-to-noise ratio (SNR), free of any ISI.

• The optimum receiving filter for accomplishing this function is called a matched filter or correlator.

• An optional equalizing filter follows the receiving filter; it is only needed for those systems where channel induced ISI can distort the signals.


90

Baseband Signal Detection: Integrate & Dump Switch


91



92



93


• We will be interested in a quantity called Peak pulse signal-to-noise ratio at the output which is given by,

To find , we need transfer function of integrator which is given by,


94



95


Thus. we can see from above equation that-(1) The signal-to-noise ratio at the output of integrate-and-

dump circuit increases with bit duration T.(2) It also depends on A2Twhich is normalized energy of the

bit (symbol).(3) Since and the signal

voltage increases linearly with T and the noise voltage increases slowly with. Hence we can say that integrator enhances the signal more than the noise.


96

Probability of Error

Probability density function (pdf) of the Gaussian random noise no can be represented as

The conditional PDF:


97

Probability of Error


98

ExampleFind the error probability of a binary baseband receiver with the binary pulse S(t) = +0.5 V and -0.5 V with bit rate 1 kbps. The noise power spectral density Is 10-5 W/Hz. What is the probability of error if the transmitted amplitudes are reduced by50%? Given erf (3.535)=0.999999445


99

Example


100

Optimum FilterThe integrate-and-dump circuit emphasizes signal output in comparison with the noise voltage. The error probability of this circuit is dependent on Eb/No ratio. But then, is it an optimum value of probability that we get?


101

Error Probability of Optimum FilterIn order to find error probability in this receiver, consider that S2 (t) was transmitted.• Let no(T) be positive and SOl (T) > SO2 (T). If no(T) is larger than error will be made in decision-making i.e. we will be deciding in favor of Sl(t).Thus, error in detection occurs when,


102

Error Probability of Optimum Filter


103

Error Probability of Optimum FilterLet


104

Transfer function of Optimum Filter


105

Matched FilterThe optimum filter is considered with generalized Gaussian noise. An optimum filter which gives a maximum ratio

when input noise is white Gaussian noise, is called a matched filter


106

Properties of Matched Filter

1. The spectrum of output signal of a matched filter with matched signal as input is proportional to energy density of input signal.

Hence, spectrum of output signal [Y(t)] is proportional to its Energy Spectral Density


107

Properties of Matched Filter2. The output signal of a matched filter is proportional to a shifted version of autocorrelation function of input signal to which the filter is matched.

3. The output signal-to-noise ratio of matched filter depends only on the ratio of the signal energy to P.S.D. of white noise at filter input.

04/10/2023 108

Conclusion : Baseband Demodulation/Detection Techniques

• Signals & noise, • Data formats, • Synchronization • multiplexing, • Intersymbol interference, • Equalization,• Detection of binary signals in presence of

Gaussian noise,• Matched and optimum filters.



109

Maximum length /Pseudo noise sequence Generator

• Shift resister-non-zero state & output is fed back to input

• Unit acts-periodic sequence generator• Ex:5 stage shift resister with [5,2] configurations with

initial non zero states

periodic

11111 00 11 0 1 00 1 0000 1 0 1 0 111 0 11 000……


110

• Longest possible sequence for n stage shift resister

L=2n -1, output MLS/PN sequence• Pseudo noise-correlation properties of PN sequence• PN signal –acts like-white noise with small DC

component• Application of PN sequence:• 1) in test instruments• 2)radar ranging• 3)spread spectrum communication• 4) Digital framing


111

Pseudorandom Numbers• Generated by algorithm using initial

seed• Deterministic algorithm

– Not actually random– If algorithm good, results pass

reasonable tests of randomness• Need to know algorithm and seed to

predict sequence


112

Properties of PN sequence• PN sequence is periodic• In each period-number of 1’s >0’s by 1• Among the runs of consecutive 1’s & 0’s

-(1/2)-of the runs of each kind are of length 1

-(1/4)-are of length 2

-(1/8)-are of length 3 etc.

Ex: calculate PN sequence for 4 bit shift resister with [3,2] feedback connections

• Autocorrelation of a PN sequence


113


114

Unit-III-ENDS

[1]

[2,1]

[3,2]

[4,3]

[5,3]

[6,5]

[7,6]

[8,4,3,2]

[9,5]

[10,7]


115

Maximum length shift register codes /PN sequence

• Class of cyclic codes with (n,k)=(2 m-1,m) , where m= +ve integer

• M stage digital shift register –feedback based on parity polynomial

• Ex. 3 stage(m=3) shift register with feedback

Sourcem bits

Flip-flop

12

Out

put


116

• For each code word- m info. Bits ->SR• Switch –from 1 to 2• Shift register –shifted 1 bit left for 2m-1

shifts• Systematic code –length n= 2m-1 • Data undergo cyclical shift for 2m-1

shifts• SR-original state –in 2m-1 shifts


117

Consider example with I/p: 0 0 1

0 0 1

Flip-flop

12

0 0 10


118

0 1 1

1 1 1

1 1 0

1 0 1

0 1 0

1 0 0

0

1

1

1

1

0


119

Possible combination ofK=m= 3 bits

Codeword generated

After 2m-1 shifts= 8 bits

0 0 0 0 0 0 0 0 0 0 initially

0 0 1 0 0 1 1 1 0 1

0 1 0 0 1 0 0 1 1 1

0 1 1 0 1 1 1 0 1 0

1 0 0 1 0 0 1 1 1 0

1 0 1 1 0 1 0 0 1 1

1 1 0 1 1 0 1 0 0 11 1 1 1 1 1 0 1 0 0


120

• Output seqn-periodic & length n= 2m-1

• Length-largest possible period• Hence , 2m-1 codewords –different cyclic-

shift of a single codeword

• Not all f/b arrangements- MLS• To check-polynomial

f(x)= 0+ 1x+ 2x2 +-------+(n-1)x(n-1)+xn

• Check if irreducible.


121

Shift register connection – MLSR code (for 2<=m<=34)

m

2stages1,2

m

8Stages1,5,6,7

m

14

stages1,5,9,14

m

20

Stages1,18

3 1,3 9 1,6 15

1,15 21

1,20

4 1,4 10

1,8 16

1,5,14,16

22

1,22

5 1,4 11

1,10 17

1,15 23

1,19

6 1,6 12

1,7,9,12

18

1,12 24

1,18,23,24

7 1,7 13

1,10,11,13

19

1,5,18,19

25

1,23

m stages

26

1,21,25,26

27

1,23,26,27

28

1,26

29

1,28

30

1,8,29,30

… …..


122

Properties of MLSR code

• Sequence –periodic• Each codeword (except all zero)-

2m-1 ones & 2m-

1 -1 zeros• All codewords-identical weights w =

2m-1 =dmin

• Codeword compared-cyclical shift of itself no of agreements differ from no of

• disagreements - by one


123

• Same shift register arrangements-a periodic binary sequence –

period n= 2m-1• Seqn-periodic autocorrelation (m) =n for m=0,+-

n,+-2n,--- (m) =-1 for all

other shifts• ->impulse like autocorrelation =>power spectrum ~white• Seqn resembles white noise• Maximum length sequence-Pseudo noise sequence

-used-for data scrambling -SS generation

• All zero state-prohibited


124

PN sequence

• mls –long string of likely bits- at receiver -disturbs synchronisation

• Randomized at transmitter• ->PN sequence • Device-scrambler

-eliminates periodic bit pattern

• At receiver- Unscrambler


125

Scrambler & unscrambler –4 stage SRtap gains 1= 2=0 & 3= 4=1

• Scrambler unscrambler

m’k-1

m’k-2

m’k-3

m’k-4

m’k-1

m’k-2

m’k-3

m’k-4m”k

mk m’k m’k mk

m”k


126

• m’’k=m’k-3 m’k-4 & m’k=mk m’’k (a)

• m’k m’’k = [mk m’’k] m’’k

= mk[ m’’k m’’k] = mk 0

= mk


127

I/p seqn: 1011 0000 0000 01*SR-initially- zeros

register m’k-1 0 1 0 1 0 1 1 1 1 0 0 0 1 0

content m’k-2 0 0 1 0 1 0 1 1 1 1 0 0 0 1

m’k-3 0 0 0 1 0 1 0 1 1 1 1 0 0 0

m’k-4 0 0 0 0 1 0 1 0 1 1 1 1 0 0

register o/p m'’k 0 0 0 1 1 1 1 1 0 0 0 1 0 0

I/p sequence mk 1 0 1 1 0 0 0 0 0 0 0 0 0 1

o/p sequence m'k 1 0 1 0 1 1 1 1 0 0 0 1 0 1


128

Application of PN sequence• In Test equipments• Radar ranging• Spread spectrum communication• Digital framing


129

).........12

]88

[3

1

]3

)2/(

3

)2/([

1

]3

[1

).1

(

).()(

.222

233

33

2/2/

32/

2/

2

22

)(.)(

II

d

dfEnoisefor

dxx xfxxEx

12

2

Putting II) into I), mean square value of noise voltage

=

At R=1, noise power is normalized

Normalized noise power/ quantization noise power = 12

2

Mean square value of r.v. x

U N I T I I Baseband Demod V S H

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