information 5 mod

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Encoder Representation (1) VectorForm


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2Polynomial representation

Transfer Domain Representation

Convolution in time domain= Multiplication in

transfer domain.

The convolution operation is replaced by polynomial

multiplication

V1(D)= u(D).g1(D)

V2(D)= u(D).g2(D)

Code word V(D)=V1(D2)+DV2(D2) for a (2,1,m) code

V(D)=V1(Dn)+DV2(Dn)++Dn-1 Vn(Dn) foran (n,k,m) code


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STRUCTURAL PROPERTIES OF CONVOLUTIONAL

CODES*Code Tree* Code Trellis* State Diagram

Tree Diagram (1)


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Tree Diagram (2)

State Diagram

A state diagram is simply a graph of the possible

states of the encoder and the possible transitions fromone state to another. It can be usedtoshow therelationship between the encoder state, input, andoutput.

The stage diagram has2 (K-1) k nodes, each nodestanding for one encoder state.


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Nodes are connected by branches Every node has2k

branches entering it and 2k branches leaving it.

The branches are labelled with c, where c is theoutput. When k=1

The solid branch indicates that the input bit is 0

* The dotted branch indicates that the input bit is 1.

Example of State Diagram


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Distance Properties of Convolutional

Code

* The minimum free distance dfreedenotes The minimum

weight of all the paths in the modified state diagram thatdiverge from and re- emerge with the all-zero state a.

* The lowest power of the transfer function T(X)

Trellis Diagram

Trellis diagram is an extension of statediagram which explicitly shows the passage of time.

* All the possible states are shown for each instant of time.* Time is indicated by a movement to the right.* The input data bits and output code bits are represented

by a unique path through the trellis.* The lines are labelled with the output.

* After the second stage,each node in the trellis has 2kincoming paths and2k outgoing paths.

* When k=1

The solid line indicates that the input bit is 0.

The dotted line indicates that the input bit is 1.


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Example of Trellis Diagram

Maximum Likelihood Decoding

Given the received code word r, determine the mostlikely

path through the trellis. (maximizingp(r|c')) Compare r

with the code bits associated with each path

Pick the path whose code bits are closest to r

Measure distance using either Hamming distance for

Harddecision decoding or Euclidean distance for soft-decision decoding

Once the most likely path has been selected,


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the estimated data bits can be read from the trellis diagramExample of Maximum Likelihood Decoding

The Viterbi AlgorithmA breakthrough in communications

in the late 60s Gu Guarnteed to find the ML solution

However the complexity is only O(2K)Complexity does not depend onthe number of original data bits


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Is easily implemented in hardware

Used in satellites, cell phones, modems, etcExample: Qualcomm Q1900

It used to find the minimum path from one source to

another source.The viterbi algorithm when applied to the

received vector r from a DMC finds the path through the

trellis with the largest metric.At each step ,it compares the

metrics of all paths entering each state and stores the pathwith the largest metric called the survivor together with its

metric.

Algorithm:

Step 1:

Starting at level j=m,compute the partial metric for

the single path entering each node (state). Store the

path(survivor) and its metric for each state.Step:2

Increment the level j by 1.compute the partial metric

for all the paths entering a state by adding the branch metric

entering that state to the metric of the connecting Survivor

at the preceeding time unit.For each state ,store the path

with the largest metric (the survivor) together with its

metric and eliminate all other paths.Step:3

If j < (L + M) repeat step:2 ,otherwise stop.

Disadvantages:

1.Smaller error probabilities cannot be achieved in practise.

2.Algorithm requires 2k fixed number of computations to

be performed per decoded information block. Ie, the


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algorithm is not adaptable to the noise level and thus

results in unnecessary computations and waste of effort.

The ViterbiAlgorithm

Takes advantages of the structure of the trellis:* Goes through the trellis one stage at a time

*At each stage, finds the most likely path leading into each state

(surviving path) and discards all other paths leading into the state(non-surviving paths)

* Continues until the end of trellis is reached* Atthe end of the trellis, traces the most probable path

from right to left and reads the data bits from the trellis

Note that in principle whole transmitted sequence must be

received before decision. However, in practice storing of stages for

input length of 5K is quite adequate

The ViterbiAlgorithm

Implementation:

1. Initialization:

LetMt(i) be the path metric at the i-thnode, the t-thstage in trellisLarge metrics corresponding to likely paths;

small metrics corresponding to unlikely paths

Initialize the trellis, sett=0 and M0(0)=0;

2. At stage (t+1),

Branch metric calculationCompute the metric for each

branch connecting the states at time t to states at time (t+1)

The metric is the likelihood probability between the

received bits and the code bitscorresponding to that branch:p(r(t+1)|c'(t+1))


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The Viterbi Algorithm

3. Sett=t+1; go to step 2 until the end of trellis is reached

4. Trace backAssume that the encoder

ended in the all-zero state

The most probable path leading into the last all-zero

state in the trellis has the largest metricTrace the path from right to left

Read the data bits from the trellis

Examples of the Hard-Decision

ViterbiDecoding


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Examples of the Hard-Decision Viterbi Decoding


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SEQUENTIAL DECODING

A Decoding effort that is adaptable to the noise level

cod ing effort of sequential decoding is

independentof K , so that large constraint lengths can be used.

Arbitrarily low error probabilities can be achieved.

The major draw back of Sequential decoding is

that noisy frames take large amounts of computation,

decoding times may exceed some upper limit, causinginformation to be lost or erased.

SEQUENTIAL DECODING

The various algorithms are

Stack algorithm

Fano algorithm

SEQUENTIAL DECODING

THE PURPOSE OF SEQUENTIAL DECODING IS To search through

the nodes of the code tree in an efficient way in an attempt

to find the maximum likelihood path.

Each node examined represents a path through part of the tree.

Whether or not a particular path is likely to be a part of the

maximum likelihood path depends on the metric value associated with that path.

The metric is a measure of the closeness of a path to the received sequence.

convolutional interleaver

A convolutional interleaver consists of T parallel lines,

with different delay in each line that represents interleaver

period [5]. In general, each successive line has a delay which is M

symbols durations higher than the previous line. The structure for

M=1 and T=3 is illustrated in Figure 1.


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Figure 1: Convolutional interleaver structure with T=3 and M=1.

In a similar manner to the trellis termination technique

(which emits stored data from memories), an insertion of acertain number of zeros at the end of a data block can isolate the

interleaved data block. This guarantees a data block of a specific

length and generates independent blocks of data for the RSC

encoder.

For the data length L=6 and the convolutional interleaver

with M=1 and T=3, the interleaved data would be X1 0 0 X4 X2 0 0

X5 X3 0 0 X6. Since the application of convolutional interleaver in

turbo codes requires an insertion of stuff bits that reduceschannel bandwidth usage, an optimisation should be performed

on the interleaver to control the number of those bits, which can

be equal to the number of applied memories. For this purpose one

block is added after the interleaver to control the data at the

interleaver output deleting extra zero stuff bits that are inserted

at the end of each block.

In this case, the memory contents at the end of each block

have zero value that stays till the beginning of the next block.

Figure 3 shows the structure of the optimised convolutional

interleaver.

Input data Convolul Zer interleaved data

deletion

Interleaver

Convolution

al

interleaver

Zero

deletion


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Figure 2- Optimised Convolutional Interleaver

Following the previous example, the optimised interleaver output is:X1 0 0 X4 X2 0 X5 X3 X6. Figure 3- Optimised Convolutional

Interleaver. The same procedure can be repeated to obtain

a second set of suitably interleaved data for the second RSC encoder.

More optimisation can be done to the zero stuff bits insertion for

the systematic and the first parity data after the trellis termination

procedure of the first RSC encoder, as it is not necessary to transmit

zero stuff bits in the mentioned data parts. However, these bits need

to be considered for the second RSC because of their influence onthe performance of the interleaver

ARQ:

The general idea for achieving error detection and

correction is to add someredundancy(i.e., some extra data) to a

message, which receivers can use to check consistency of the

delivered message, and to recover data determined to beerroneous. Error-detection and correction schemes can be either

systematicor non-systematic: In a systematic scheme, the

transmitter sends the original data, and attaches a fixed number

ofcheck bits (orparity data), which are derived from the data bits

by somedeterministic algorithm. If only error detection is

required, a receiver can simply apply the same algorithm to the

received data bits and compare its output with the received check

bits; if the values do not match, an error has occurred at somepoint during the transmission. In a system that uses a non-

systematic code, the original message is transformed into an

encoded message that has at least as many bits as the original

message.

Good error control performance requires the scheme to be

selected based on the characteristics of the communication

channel. Commonchannel modelsinclude memory-less modelswhere errors occur randomly and with a certain probability, and
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dynamic models where errors occur primarily inbursts.

Consequently, error-detecting and correcting codes can be

generally distinguished between random-error-

detecting/correcting and burst-error-detecting/correcting. Some

codes can also be suitable for a mixture of random errors and

burst errors.

If thechannel capacitycannot be determined, or is highly

varying, an error-detection scheme may be combined with a

system for retransmissions of erroneous data. This is known as

automatic repeat request(ARQ), and is most notably used in theInternet. An alternate approach for error control ishybrid

automatic repeat request(HARQ), which is a combination of ARQ

and error-correction coding.

Error correction may generally be realized in two different ways:

Automatic repeat request(ARQ) (sometimes also referred to

as backward error correction): This is an error controltechnique whereby an error detection scheme is combined

with requests for retransmission of erroneous data. Every

block of data received is checked using the error detection

code used, and if the check fails, retransmission of the data is

requested this may be done repeatedly, until the data can

be verified.

Forward error correction(FEC): The sender encodes the data

using an error-correcting code (ECC) prior to transmission.The additional information (redundancy) added by the code

is used by the receiver to recover the original data. In

general, the reconstructed data is what is deemed the "most

likely" original data.

ARQ and FEC may be combined, such that minor errors are

corrected without retransmission, and major errors are corrected

via a request for retransmission: this is calledhybrid automaticrepeat-request(HARQ).
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Automatic Repeat reQuest(ARQ) is an error control

method for data transmission that makes use of error-detection

codes, acknowledgment and/or negative acknowledgment

messages, andtimeoutsto achieve reliable data transmission. An

acknowledgmentis a message sent by the receiver to indicate that

it has correctly received adata frame.

Usually, when the transmitter does not receive the

acknowledgment before the timeout occurs (i.e., within a

reasonable amount of time after sending the data frame), it

retransmits the frame until it is either correctly received or theerror persists beyond a predetermined number of

retransmissions.Automatic Repeat reQuest(ARQ), also known asAutomatic Repeat

Query, is anerror-controlmethod fordata transmissionthat

usesacknowledgements(messages sent by the receiver indicating that it

has correctly received adata frameorpacket) andtimeouts(specified

periods of time allowed to elapse before an acknowledgment is to be

received) to achieve reliable data transmission over an unreliable service.

If the sender does not receive an acknowledgment before the timeout, it

usuallyre-transmitsthe frame/packet until the sender receives an

acknowledgment or exceeds a predefined number of re-transmissions.

The types of ARQ protocols include

Stop-and-wait ARQ

Go-Back-N ARQ

Selective Repeat ARQ

ARQ is appropriate if the communication channel has

varying or unknowncapacity, such as is the case on the Internet.

However, ARQ requires the availability of aback channel, results

in possibly increasedlatencydue to retransmissions, and requires

the maintenance of buffers and timers for retransmissions, which

in the case ofnetwork congestioncan put a strain on the server

and overall network capacity
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Error-correcting code

Forward error correction

Anerror-correcting code(ECC) or forward error correction

(FEC) code is a system of addingredundantdata, orparity data, to

a message, such that it can be recovered by a receiver even when

a number of errors (up to the capability of the code being used)

were introduced, either during the process of transmission, or on

storage. Since the receiver does not have to ask the sender for

retransmission of the data, aback-channelis not required inforward error correction, and it is therefore suitable forsimplex

communicationsuch asbroadcasting. Error-correcting codes are

frequently used inlower-layercommunication, as well as for

reliable storage in media such asCDs,DVDs,hard disks, andRAM.

Error-correcting codes are usually distinguished between

convolutional codesandblock codes:

Convolutional codes are processed on a bit-by-bit basis. They

are particularly suitable for implementation in hardware,

and theViterbi decoderallowsoptimal decoding.

Block codes are processed on ablock-by-blockbasis. Early

examples of block codes arerepetition codes,Hamming

codesandmultidimensional parity-check codes. They were

followed by a number of efficient codes,Reed-Solomon

codesbeing the most notable due to their current
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widespread use.Turbo codesandlow-density parity-check

codes(LDPC) are relatively new constructions that can

provide almostoptimal efficiency.

Shannon's theoremis an important theorem in forward

error correction, and describes the maximuminformation rateat

which reliable communication is possible over a channel that has

a certain error probability orsignal-to-noise ratio(SNR). This

strict upper limit is expressed in terms of thechannel capacity.

More specifically, the theorem says that there exist codes such

that with increasing encoding length the probability of error on adiscrete memoryless channelcan be made arbitrarily small,

provided that thecode rateis smaller than the channel capacity.

The code rate is defined as the fraction k/n ofksource symbols

and n encoded symbols.

The actual maximum code rate allowed depends on the

error-correcting code used, and may be lower. This is because

Shannon's proof was only of existential nature, and did not showhow to construct codes which are both optimal and haveefficient

encoding and decoding algorithms.

Hybrid schemes

Hybrid ARQ

Hybrid ARQis a combination of ARQ and forward error correction.There are two basic approaches:

Messages are always transmitted with FEC parity data (and error-

detection redundancy). A receiver decodes a message using the

parity information, and requests retransmission using ARQ only if the

parity data was not sufficient for successful decoding (identified

through a failed integrity check).

Messages are transmitted without parity data (only with error-detection information). If a receiver detects an error, it requests FEC
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information from the transmitter using ARQ, and uses it to

reconstruct the original message.

The latter approach is particularly attractive on anerasure channelwhen

using arateless erasure code.

Applications

Applications that require low latency (such as telephone

conversations) cannot useAutomatic Repeat reQuest(ARQ); they must use

Forward Error Correction(FEC). By the time anARQsystem discovers an

error and re-transmits it, the re-sent data will arrive too late to be anygood.

Applications where the transmitter immediately forgets the

information as soon as it is sent (such as most television cameras) cannot

useARQ; they must useFECbecause when an error occurs, the original

data is no longer available. (This is also whyFECis used in data storage

systems such asRAIDanddistributed data store).

Applications that use ARQ must have areturn channel. Applications

that have no return channel cannot use ARQ.

Applications that require extremely low error rates (such as digital

money transfers) must useARQ.

Stop-and-wait ARQ

Stop-and-wait ARQ is a method used intelecommunicationsto send

information between two connected devices. It ensures that information is

not lost due to dropped packets and that packets are received in the correctorder. It is the simplest kind ofautomatic repeat-request(ARQ) method. A

stop-and-wait ARQ sender sends oneframeat a time; it is a special case of

the generalsliding window protocolwith both transmit and receive

window sizes equal to 1. After sending each frame, the sender doesn't send

any further frames until it receives anacknowledgement(ACK) signal.

After receiving a good frame, the receiver sends an ACK. If the ACK does not

reach the sender before a certain time, known as the timeout, the sender

sends the same frame again.
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The above behavior is the simplest Stop-and-Wait

implementation. However, in a real life implementation there are problems

to be addressed.

Typically the transmitter adds aredundancy checknumber to the

end of each frame. The receiver uses the redundancy check number to

check for possible damage. If the receiver sees that the frame is good, it

sends an ACK. If the receiver sees that the frame is damaged, the receiver

discards it and does not send an ACK -- pretending that the frame was

completely lost, not merely damaged.

One problem is where the ACK sent by the receiver is damagedor lost. In this case, the sender doesn't receive the ACK, times out, and

sends the frame again. Now the receiver has two copies of the same frame,

and doesn't know if the second one is a duplicate frame or the next frame of

the sequence carrying identical data.

Another problem is when the transmission medium has such a

longlatencythat the sender's timeout runs out before the frame reaches

the receiver. In this case the sender resends the same packet. Eventually

the receiver gets two copies of the same frame, and sends an ACK for eachone. The sender, waiting for a single ACK, receives two ACKs, which may

cause problems if it assumes that the second ACK is for the next frame in

the sequence.

To avoid these problems, the most common solution is to define

a 1 bitsequence numberin the header of the frame. This sequence number

alternates (from 0 to 1) in subsequent frames. When the receiver sends an

ACK, it includes the sequence number of the next packet it expects. This

way, the receiver can detect duplicated frames by checking if the framesequence numbers alternate. If two subsequent frames have the same

sequence number, they are duplicates, and the second frame is discarded.

Similarly, if two subsequent ACKs reference the same sequence number,

they are acknowledging the same frame.

Stop-and-wait ARQ is inefficient compared to other ARQs,

because the time between packets, if the ACK and the data are received

successfully, is twice the transit time (assuming the turnaround time can be

zero). The throughput on the channel is a fraction of what it could be. Tosolve this problem, one can send more than one packet at a time with a
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larger sequence number and use one ACK for a set. This is what is done

inGo-Back-N ARQand theSelective Repeat ARQ.

Selective Repeat ARQ

It may be used as a protocol for the delivery and acknowledgement

of message units, or it may be used as a protocol for the delivery of

subdivided message sub-units.

When used as the protocol for the delivery ofmessages, the

sending process continues to send a number offramesspecified by

a window size even after a frame loss. UnlikeGo-Back-N ARQ, the receiving

process will continue to accept andacknowledgeframes sent after an initial

error; this is the general case of thesliding window protocolwith both

transmit and receive window sizes greater than 1.

The receiver process keeps track of the sequence number of the

earliest frame it has not received, and sends that number with

everyacknowledgement(ACK) it sends. If a frame from the sender does not

reach the receiver, the sender continues to send subsequent frames until ithas emptied its window. The receiver continues to fill its receiving window

with the subsequent frames, replying each time with an ACK containing the

sequence number of the earliest missingframe. Once the sender has sent all

the frames in its window, it re-sends the frame number given by the ACKs,

and then continues where it left off.

The size of the sending and receiving windows must be equal,

and half the maximum sequence number (assuming that sequence numbers

are numbered from 0 to n1) to avoid miscommunication in all cases ofpackets being dropped. To understand this, consider the case when all

ACKs are destroyed. If the receiving window is larger than half the

maximum sequence number, some, possibly even all, of the packages that

are resent after timeouts are duplicates that are not recognized as such.

The sender moves its window for every packet that is acknowledged.[1]

When used as the protocol for the delivery ofsubdivided

messages it works somewhat differently. In non-continuous channels

where messages may be variable in length, standard ARQ or Hybrid ARQprotocols may treat the message as a single unit. Alternately selective
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retransmission may be employed in conjunction with the basic ARQ

mechanism where the message is first subdivided into sub-blocks (typically

of fixed length) in a process calledpacket segmentation. The original

variable length message is thus represented as a concatenation of a

variable number of sub-blocks. While in standard ARQ the message as a

whole is either acknowledged (ACKed) or negatively acknowledged

(NAKed), in ARQ with selective transmission the NAKed response would

additionally carry a bit flag indicating the identity of each sub-block

successfully received. In ARQ with selective retransmission of sub-divided

messages each retransmission diminishes in length, needing to only

contain the sub-blocks that were NAKed.

In most channel models with variable length messages, the

probability of error-free reception diminishes in inverse proportion with

increasing message length. In other words it's easier to receive a short

message than a longer message. Therefore standard ARQ techniques

involving variable length messages have increased difficulty delivering

longer messages, as each repeat is the full length. Selective retransmission

applied to variable length messages completely eliminates the difficulty in

delivering longer messages, as successfully delivered sub-blocks areretained after each transmission, and the number of outstanding sub-

blocks in following transmissions diminishes.

Examples

TheTransmission Control Protocoluses a variant ofGo-Back-N

ARQto ensure reliable transmission of data over theInternet Protocol,

which does not provide guaranteed delivery of packets; with SelectiveAcknowledgement (SACK), it uses Selective Repeat ARQ.

TheITU-TG.hnstandard, which provides a way to create a high-

speed (up to 1 Gigabit/s)Local area networkusing existing home wiring

(power lines, phone lines andcoaxial cables), uses Selective Repeat ARQ to

ensure reliable transmission over noisy media.G.hnemployspacket

segmentationto sub-divide messages into smaller units, to increase the

probability that each one is received correctly.
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Go-Back-N ARQ

Go-Back-N ARQ is a specific instance of theautomatic repeat

request(ARQ) protocol, in which the sending process continues to send a

number offramesspecified by a window size even without receiving

anacknowledgement(ACK) packet from the receiver. It is a special case of

the generalsliding window protocolwith the transmit window size of N

and receive window size of 1.

The receiver process keeps track of the sequence number of the next frame

it expects to receive, and sends that number with every ACK it sends. Thereceiver will ignore any frame that does not have the exact sequence

number it expects whether that frame is a "past" duplicate of a frame it

has already ACK'ed or whether that frame is a "future" frame past the last

packet it is waiting for. Once the sender has sent all of the frames in

its window, it will detect that all of the frames since the first lost frame

are outstanding, and will go back to sequence number of the last ACK it

received from the receiver process and fill its window starting with that

frame and continue the process over again.Go-Back-N ARQ is a more efficient use of a connection thanStop-and-wait

ARQ, since unlike waiting for an acknowledgement for each packet, the

connection is still being utilized as packets are being sent. In other words,

during the time that would otherwise be spent waiting, more packets are

being sent. However, this method also results in sending frames multiple

times if any frame was lost or damaged, or the ACK acknowledging them

was lost or damaged, then that frame and all following frames in the

window (even if they were received without error) will be re-sent. To avoidthis,Selective Repeat ARQcan be used.

Convolutional Interleavers

A convolutional interleaver consists of a set of shift registers, each with a fixed

delay. In a typical convolutional interleaver, the delays are nonnegative integermultiples of a fixed integer (although a general multiplexed interleaver allows
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array row by row. To configure this interleaver, use the Number of columns of

helical array parameter to set the width of the array, and use the Group

size andHelical array step size parameters to determine how symbols are placedin the array. See the reference page for theHelical Interleaverblock for more

details and an example.

Delays of Convolutional Interleavers

After a sequence of symbols passes through a convolutional interleaver and a

corresponding convolutional deinterleaver, the restored sequence lags behind the

original sequence. The delay, measured in symbols, between the original and

restored sequences is

(Number of shift registers) * (Maximum delay among all shift registers)

for the most general multiplexed interleaver. If your model incurs an additional

delay between the interleaver output and the deinterleaver input, then the restored

sequence lags behind the original sequence by the sum of the additional delay and

the amount in the preceding formula.

Note For proper synchronization, the delay in your model between the interleaver output

and the deinterleaver input must be an integer multiple of the number of shift registers. Youcan use theInteger Delayblock in the DSP Blockset to adjust delays manually, if necessary.

Convolutional Interleaver block. In the special case implemented by the

Convolutional Interleaver/Convolutional Deinterleaver pair, note that the number

of shift registers is the Rows of shift registers parameter, while the maximum

delay among all shift registers is

(Register length step)*(Rows of shift registers-1)

Helical Interleaver block. In the special case implemented by the Helical

Interleaver/Helical Deinterleaver pair, the delay between the restored sequence and

the original sequence is

where C is the Number of columns in helical array parameter, N is the Group

size parameter, and s is the Helical array step size parameter.
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Example: Convolutional Interleavers

The example below illustrates convolutional interleaving and deinterleaving usinga sequence of consecutive integers. It also illustrates the inherent delay and the

effect of the interleaving blocks' initial conditions.

information 5 mod

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