Coding and error correction

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Coding and error correction

WS 19/20 Wireless Communications ‐ Coding and

Error Correction 1

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GENERAL CONCEPTS

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Error detection

Wireless Communications - Coding and Error Correction

check bits

Image source : William Stallings, „Data and Computer Communications“, 2004

Recap „Grundlagen der Rechnernetze“:

Parity, Checksum, CRC

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Check bits, error detecting code, error detection function, detectable errors, non detectable errors

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Error detection supports error control

Recall „Grundlagen der Rechnernetze“: Stop-and-Wait, Go-Back-N, Selective-Reject

Use of error detection e.g.

 with wired connection (e.g., HDLC)

 at IP transport level (e.g., TCP) Use in the wireless case? problems:

 High bit error rate (compared to wired communication) leads to frequent retransmissions

 Long latency connections (e.g., in the case of satellite communications) require large transmission windows and, in the event of an error,

retransmission of many frames Solution for wireless networks?

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Error correction procedure

Image source: William Stallings, „Data and Computer Communications“, 2004

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FEC (Forward Error Correction); code word;

detectable and correctable errors; detectable and uncorrectable errors; unrecognized errors;

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Hamming distance

Hamming distance d(v₁, v₂) between two n bit sequences v₁ and v₂

Beispiel: vier 4-Bit-Sequenzen mit einer paarweisen Hamming-Distanz von

mindestens 2

Wieviele Bit-Fehler können erkannt werden?

Example: four 4 bit sequences with a pairwise Hamming distance of at least 2

How many bit errors can be detected?

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Coding gain

Image source : William Stallings, „Data and Computer Communications“, 2004

coding gain

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conflicting (As few redundant bits as possible to use as little bandwidth as possible) goals (As many redundant bits as possible to minimize the bit error rate)

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LINEAR BLOCK CODES

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In general:

Transmission procedure in case of no bit errors

(n,k) block codes: detecting bit errors

data block code word 00 -> 00000

01 -> 00111 10 -> 11001 11 -> 11110

Detection of bit errors: consider code {b₁,...,b_k} and let b be received:

Sender

Empfänger

f : data block  code word

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Receiver

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Correction of bit errors: Consider code {b₁,...,b_k} and let b be received:

(n,k) block codes: correcting bit errors

received next valid CW data data block code word

00 -> 00000 01 -> 00111 10 -> 11001 11 -> 11110

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Facts for general block codes

code distance of d_min >= 2t+1 supports correcting of how many c bits in error?

… and detection of how many d bits in error?

Thus: code distance of d_minsupports correction of how many erronous bits?

… and detection of how many erronous bits?

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A (7,4) code example

William Stallings, Wireless Communications & Networks, 2nd edition, Prentice Hall, 2005

Generator polynomial: P(X) = X³ + X² + 1

Remark:

• With 7 = 2 ^ 3-1, so after previous

consideration, all 1-bit errors are correctable in principle

• Also note d_min of code words is 3, so in fact all 1-bit errors are correctable

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How do you actually implement the polynomial division?

    

Example:

X^n-k + A_n-k-1 x^n-k-1 + ... + A₂ X² + A₁ X + 1

Linear-Feedback-Shift-Register (LFSR)

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Concrete CRC examples

Bose-Chaudhuri-Hocquenhem (BCH)

 Here no exact details how these are constructed

 Only in general: suitable binary (n, k) -BCH codes for given m and t can be constructed with the following parameters

 Block length: n = 2m - 1

 Number of check bits: n - k <= m * t

 Minimum distance of the codewords: dmin> = 2t + 1

 Code can then correct all combinations of t or less erroneous bits

 Examples of BCH generator polynomials

m = log (n + 1)

Reed-Solomon codes (RS) are a BCH subclass (here is just the code name, no further details) William Stallings, Wireless Communications & Networks, 2nd edition, Prentice Hall, 2005

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CONVOLUTIONAL CODES

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Error Correction 51

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Convolutional Codes

Idea of (n, k, K) convolutional codes

• Separate bit stream to be transferred into (very small) k-bit blocks

• Transfer every k-bit block into an n-bit block

• n-bit block is the k-bit block with additional redundancy

• In contrast to block codes:

• The last K-1 k-bit blocks to be transferred are included in the redundancy calculation of the current n-bit block

• At the receiver side k-bit blocks are corrected taking previously received k- bit blocks into account (typically a fixed window of previously received k-bit blocks

b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12

example K=4

c1 c2 c3 c4 c5

…

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Example of an (2,1,3) encoder

Wireless Communications - Coding and Error Correction William Stallings, Wireless Communications & Networks, 2nd edition, Prentice Hall, 2005



• Transmitter sends the output bits

• Receiver must

reconstruct the input bits

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

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Output bits of the sender describe a path in a trellis; example

Zu senden: 1 1 0 1 0 0 0 Erzeugt: 11|01|01|00|10|11|00

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Decoding: initially without errors

Wireless Communications - Coding and Error Correction William Stallings, Wireless Communications & Networks, 2nd edition, Prentice Hall, 2005

Gesendet: 11|01|01|00|10|11|00 Empfangen: 11|01|01|00|10|11|00 also: 1 1 0 1 0 0 0

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Decoding: now with an error

Gesendet: 11|01|01|00|10|11|00 Empfangen: 11|01|01|01|10|11|00 also: 1 1 0 ?

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Viterbi algorithm

Principle idea: find the path in the trellis that differs least from the received bit sequence

 Distance metric allows different variants

 We are looking at the Hamming distance here Algorithm for a window size b

 Step 0: mark trellis start state with 0

 Step i: For each trellis state, find the path(s) minimizing the following equation

 Weight of predecessor state + Hamming distance between the label of the last selected edge and the received edge label

 Step b: if all the paths so found have a first common edge, then the input for that edge is the result; otherwise not correctable error

An example!!!! ...

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Example for window size 7

10 01 01 00 10 11 00

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ERROR CORRECTION PROPERTIES OF

CONVOLUTIONAL CODES

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Convolutional encoder example (n=3, k=1, K=3)

Image source: Andrea Goldsmith, Wireless Communications, Cambridge University Press, 2005, Figure 8.6

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The corresponding trellis diagram

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Distance properties discussed; distances to the all-zero path

The paths to be considered further are Path1, Path3 and Path4.

All other paths are time shifted versions or concatenations.

Hamming distances to the all-zero path:

Path1 = 6, Path3 = 8, Path4 = 6

The minimum free distance d_f(which is equivalent to d_minin block codes) is thus 6 in this case. With maximum likelihood detection (as done by the Viterbi algorithm for sufficiently large windows) we can always detect up to d_f-1 bit errors and correct floor(d_f/2-1) bit errors.

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Thus error correction capabilities of a convolutional code depend on the paths that diverge and remerge with the all-zero path

These form the basis to obtain probabilities of error bounds How to characterize all diverging and remerging paths?

 Transfer functions

 These are determined from the codes state diagram representing all possible transitions from the all-zero state to the all-zero state

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State diagram in our example

solid line = transition due to 0 bit dashed line = transition due to 1 bit

exponent of D corresponds to the Hamming distance between the code word output and the all-zero code word

D⁰

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Representation by state equations

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D⁰

… equations representing the states’ incoming arcs

X_c = D³X_a + DX_b, X_b = DX_c + DX_d, X_d = D²X_c + D²X_d, X_e = D²X_b

… dummy variables X_a, … , X_e are representing the partial paths

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Transfer function

The transfer function

T(D) = X_e / X_a is describing the paths from state a to state e.

Solving the equation yields a form

𝑇 𝐷 𝑎 𝐷

where a_d is the number of paths with Hamming distance d from the all- zero path.

In particular, we have 𝑎 paths with minimum distance.

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In our example we get

𝑇 𝐷 𝐷

1 2𝐷 𝐷 2𝐷 4𝐷 ⋯

Thus, we have

• one path with minimum distance 6

• two paths with distance 8

• four paths with distance 10

• …

Conclusion: the transfer function is a convenient shorthand to enumerate the number and corresponding Hamming distance of all paths diverging and remerging with the all-zero path.

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Extended state diagram

Transfer function captures the number and Hamming distance of the paths to the all-zero path

To derive bit error probabilities we need to extend the diagram to

represent Hamming distance, length, and number of bit errors that correspond to each path diverging and remerging to the all-zero path.

Introduce two additional parameters

• N = multiplied to all branches associated with a 1-bit input  to count the number of bits differing from the all-zero path

• J = multiplied to all branches  to count the number of branches taken by any given path from a to e

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The extended state diagram in our example

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D⁰

X_c = JND³X_a + JNDX_b X_b = JDX_c + JDX_d X_d = JND²X_c + JND²X_d X_e = JD²X_b

State equations:

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The transfer function for the extended state diagram

The transfer function associated with this representation T(D,N,J) = X_e / X_a

in our example yields

𝑇 𝐷, 𝑁, 𝐽 𝐽 𝑁𝐷

1 𝐽𝑁𝐷 1 𝐽

𝐽 𝑁𝐷 𝐽 𝑁 𝐷 𝐽 𝑁 𝐷 𝐽 𝑁 𝐷 ⋯ Thus, we have

• one path with minimum distance 6 of length 3 and a single bit error

• one path with Hamming distance 8 of length 4 and two bit errors

• one path with Hamming distance 8 of length 5 and two bit errors

• …

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The transfer function for the extended state diagram

Conclusion: the extended transfer function is a convenient shorthand to enumerate

• Hamming distance

• path length

• number of bit errors

that correspond to each path diverging and remerging with the all-zero path.

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Error probability for convolutional code (n,k,K)

Convolutional codes are linear codes  probability of error can be

determined by just studying the all-zero code word being transmitted We study the bit error probability under soft decoding (the typical

application for convolutional codes)

We limit our study to studying the case: coded symbols are output from the convolutional encoder and sent over an

 AWGN channel

 using coherent BPSK

We have to consider E_c the energy per bit transmitted by the convolutional encoder. With an energy per bit of E_b and a coding rate of R_c = k/n we get

E_c = R_c ∙ E_b

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Error probability with soft decision decoding

We first consider the pairwise error probability:

 assuming the all-zero sequence was sent

 the probability P₂(d) of receiving a sequence with a Hamming distance d away the all-zero sequence

For AWGN and coherent BPSK it can be shown

𝑃 𝑑 Q 2𝐸

𝑁 𝑑 Q 2𝛾 𝑅 𝑑

The transfer function enumerates all possible paths that diverge and remerge with the all-zero path, thus by the union bound the probability P_e to mistake the all- zero path for another is

𝑃 𝑎 𝑄 2𝛾 𝑅 𝑑

where a_d is the number of paths with distance d from the all-zero path

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Error probability with soft decision decoding

With 𝑄 2𝛾 𝑅 𝑑 𝑒 (Chernoff upper bound) this yields

𝑃 𝑎 𝑒

which can also be expressed in short by 𝑃 𝑇 𝑒

What about the bit error probability? We can use the transfer function of the extended state diagram

𝑇 𝐷, 𝑁 𝑎 𝐷 𝑁

where f(d) = number of bit errors associated with path distance d to the all-zero path (note, we consider sufficient large sequences here, i.e. ignore J)

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Error probability with soft decision decoding

For k=1 the bit error probability P_b can be upper bounded by

𝑃 𝑎 𝑓 𝑑 𝑄 2𝛾 𝑅 𝑑

and

𝑃 𝑑𝑇 𝐷, 𝑁 𝑑𝑁 evaluated at N=1 and D=𝑒

If k>1 division by k (i.e. P_b / k) yields the bit error probability

Remark, this section showed derivation for coherent BPSK. For other modulation schemes the pairwise error probability P₂(d) must be recomputed.

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Error probability with hard decision decoding

Following a similar approach for hard decision decoding we get 𝑃 𝑇 4𝑝 1 𝑝

where p = probability of a bit error on the channel and

𝑃 𝑑𝑇 𝐷, 𝑁 𝑑𝑁 evaluated at N=1 and D= 4𝑝 1 𝑝

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INTERLEAVING

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Coding with interleaving

Block codes and convolutional codes are designed for good performance in AWGN channels with independent bit errors

In fading channels errors tend to occur in bursts (deep fades)!

Codes may have worst performance in fading than an uncoded transmission

Interleaving: a technique to mitigate the

effects of error bursts by spreading error bursts over many codewords

Spreading is done by the interleaver, error correction is done by the code

Interleaving can be applied for block codes and convolutional codes; we consider block codes here

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Block coding with interleaving

d rows and n columns for an (n,k) code codewords are read

into interleaver by rows

and read out into modulator by columns

Thus symbols in the same codeword are separated by d-1 other code words

.. Wireless Communications - Coding and Error Correction

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