WO1996024194A1

WO1996024194A1 - Encoding arrangement for encoding a sequence of (n-1)-bit information words into a sequence of n-bit channel words, and a decoding arrangement for decoding a sequence of n-bit channel words into a sequence of (n-1) bit information words

Info

Publication number: WO1996024194A1
Application number: PCT/IB1996/000045
Authority: WO
Inventors: Kornelis Antonie Schouhamer Immink
Original assignee: Philips Electronics N.V.; Philips Norden Ab
Priority date: 1995-02-03
Filing date: 1996-01-19
Publication date: 1996-08-08
Also published as: EP0754373B1; DE69613181T2; EP0754373A1; JPH09511375A; JP3425152B2; TW290764B; US5760718A; CN1110903C; CN1148449A; ATE201943T1; KR970702622A; KR100434207B1; DE69613181D1

Abstract

An encoding arrangement for encoding a sequence of (n-1)-bit information words into a sequence of n-bit channel words is disclosed, comprising input means (1) for receiving the information words, converter means (2) for converting the (n-1)-bit information words into n-bit channel words and output means (3) for supplying the channel words. The converter means (2) comprises weight vector coefficient supply means (7) for supplying a weight vector w, the weight vector having n weight vector coefficients wi, where i is an integer running from 1 to n and the weight vector coefficients being in the form of (n-1)-bit binary words, and calculation means (5) for carrying out a calculation using an information word so as to obtain a channel word, which calculation is based on the following steps: (a) set a running parameter j equal to n; (b) determine whether the value of the information word is larger than or equal to the weight vector coefficient wj, if so, set the binary value in the bit position j of the channel word to 'zero' and subtract the value of the weight coefficient wj from the value of the information word so as to obtain a new value for the information word, if not, set the said binary value to 'one'; (c) repeat the step (b) n-1 times for each next lower value for j, wn being the weight vector coefficient having the largest value and weight vector coefficients having a next lower subscript j having a smaller value, and n is an integer constant. In accordance with the invention, the weight vector coefficient supply means (7) is adapted to supply only p bits of a weight vector coefficient wi, the remaining n-1-p bits of the (n-1)-bit weight vector coefficient word being 'zeroes' to be added before or after the p-bit binary word or before and after the p-bit binary word so as to obtain said weight vector coefficient, and that p is an integer smaller than n-1.

Description

Encoding arrangement for encoding a sequence of (n-l)-bit information words into a sequence of n-bit channel words, and a decoding arrangement for decoding a sequence of n- bit channel words into a sequence of (n-l)-bit information words.

The invention relates to an encoding arrangement for encoding a sequence of (n-l)-bit information words into a sequence of n-bit channel words, the encoding arrangement comprises

- input means for receiving the information words, - converter means for converting the (n-l)-bit information words into n-bit channel words,

- output means for supplying the channel words, the converter means comprising

- weight vector coefficient supply means for supplying a weight vector w, the weight vector having n weight vector coefficients W_j, where i is an integer running from 1 to n and the weight vector coefficients being in the form of (n-l)-bit binary words,

- calculation means for carrying out a calculation using an information word so as to obtain a channel word, which calculation is based on the following steps

(a) set a running parameter j equal to n,

(b) determine whether the value of the information word is larger than or equal to the weight vector coefficient W , if so, set the binary value in the bit position j of the channel word to

'zero' and subtract the value of the weight coefficient W: from the value of the information word so as to obtain a new value for the information word, if not, set the said binary value to 'one',

(c) repeat the step (b) n-l times for each next lower value for j, w_n being the weight vector coefficient having the largest value and weight vector coefficients having a next lower subscript j having a smaller value, to a decoding arrangement for decoding a sequence of n- bit channel words into a sequence of (n-l)-bit information words, the decoding arrangement comprises

- input means for receiving the channel words, - converter means for converting the n-bit channel words into (n-l)-bit information words,

- output means for supplying the information words, the converter means comprising

- inverter means for inverting the channel words so as to obtain inverted channel words,

- weight vector supply means for supplying a weight vector w, the weight vector having n weight vector coefficients w,-, where i is an integer running from 1 to n, each weight vector coefficient W_j being represented as a binary word,

- calculation means for carrying out a calculation using an inverted channel word so as to obtain an information word, which calculation is based on calculating an inner product n

X_j . w_{

where x, is the i-th bit in the inverted channel word and I is the information word obtained from the calculation, and n and p are integer constants, and to an encoding and decoding method.

The encoding and decoding arrangement defined in the opening paragraph is described in the publication 'Fibonacci codes for synchronization control' by W.H. Kautz, in IEEE Trans. Inform. Theory, Vol. IT-11, pp. 284-292, 1965. The encoding method described in the publication can be used for obtaining a k-constrained sequence of channel words and is known under the name of 'enumerative encoding'. Enumerative decoding is done by forming the inner product of the received binary channel word and the weighting vector of n coefficients. The weighting vector is a function of the channel constraints in force and is usually precalculated. Note that multiplications are simple additions as the received channel word is binary.

Encoding is done by a method which is similar to decimal to binary conversion where, instead of the usual powers of two, the weighting vector of n coefficients is used. The binary representation of the weight coefficients in the Kautz method requires n-l bits. As there are n weight coefficients, a memory capacity of n(n-l) storage locations is required for storing the n coefficients. When encoding/decoding 100-bit channel words, a memory capacity of 9.9 K storage locations would be needed. A second drawback of the prior art is that the additions in forming the inner product between the binary channel word and the vector of n coefficients requires a double carry, which complicates the structure of a parallel adder. This renders the conversion of the adder from a parallel to a simple serial form practically impossible.

These and other drawbacks are so serious that enumerative encoding and decoding have been confined to information theory practice. The invention aims at providing a simplified encoding and decoding arrangement. The encoding and decoding arrangement are characterized in that the weight vector coefficient supply means is adapted to supply p bits of a weight vector coefficient w_i? the remaining n-l-p bits of the (n-l)-bit weight vector coefficient word being 'zeroes' to be added before or after the p-bit binary word or before and after the p-bit binary word so as to obtain said weight vector coefficient, and that p is an integer larger than one and smaller than n-l.

The invention is based on the recognition that it is possible to generate the n weight vector coefficients by generating for a weight vector coefficient a p-bit word and adding 'zeroes' before, or after the p-bit binary word, or both before and after the p-bit word so as to obtain the vector coefficient. By doing so, still a sufficient amount of coefficients are available to carry out the encoding. Further, if the weight vector coefficient generator means are adapted to generate a p-bit word for each of the n weight coefficients and if those n p-bit words are stored in the weight vector coefficient generator means, this means that a memory capacity of n.p memory locations are needed now, as the 'zeroes' to be added before and/or after the p-bit word so as to obtain the weight vector, need not be stored.

The encoding arrangement may be further characterized in that for generating a sequence of channel words having the virtue that at most k consecutive 'zeroes' occur between subsequent Ones' in the sequence, the channel words further satisfying the requirement that at most r consecutive 'zeroes' occur at one and the same end of the channel words, the weighting coefficients W_j satisfy the following equation:

, n ,

where W_j = 0 for j ≤ 0, and where the FLOOR of a value equals the largest integer value which is smaller than said value, and where r is smaller than k. This enables the generation of a sequence of channel signals that satisfy the k-constraint, even after concatenation of subsequent channel words. By specifying the value for r, r being smaller than k, it is automatically satisfied that channel words are obtained having at most 1 'zeroes' at the one end of the channel word, more specifically the leading end, of the channel word and at most r 'zeroes' at the other end, more specifically the trailing end, of the channel words, where l+r=k.

The encoding method is especially useful for high rate encodings, such as in a 196 to 197 conversion code.

Further preferred embodiments of the encoding arrangement are the subject of the claims 3, 4 and 5. Preferred embodiments of the decoding arrangement are the subject of the claims 7 to 10. Claims 11 and 12 define corresponding encoding and decoding methods.

These and other objects of the invention will be apparent from and further elucidated with reference to the embodiments described hereafter. In the figure description shows figure 1 an embodiment of an encoding arrangement, figure 2 an embodiment of the weight coefficient generator in the encoding arrangement of figure 1, and figure 3 an embodiment of a decoding arrangement.

The description and explanation of the encoding and decoding arrangement in accordance with the invention starts with an explanation of the Kautz encoding and decoding method. Table 1 shows an example of a 4-to-5 encoding to clarify the operation.

Table 1

decimal value information word channel word

0 0000 11111

1 0001 11110

2 0010 11101

3 0011 11011

4 0100 11010

5 0101 11001

6 0110 10111

7 0111 10110

8 1000 10101

9 1001 10011

10 1010 10010

11 1011 01111

12 1100 oiiio

13 1101 01101

14 1110 01011

15 mi 01010

The table shows an encoding/decoding of 4-bit information words into 5- bit channel words. The resulting channel words satisfy the k=2 constraint, that at most two subsequent 'zeroes' are present between 'ones'.

The left hand column denotes the decimal representation of the information word and the middle column presents the binary representation of the information words. The right hand column displays the corresponding channel words. The channel words start and end with at most one 'zero', so that the k-constraint is also satisfied when concatenating subsequent channel words.

The decoding process is as follows. Define the weight vector {w} for example as {11,6,3,2,1}. The weight vector is in the form of a number of (five) weight coefficients W_j, where i runs from 1 to 5. The weight coefficients are expressed in decimal notation for explanatory purposes, but it will be clear that actually, the weight coefficients are in 4-bit binary form. 6

The decoding arrangement forms the inner product

n

Σ < . W,

where _j is the i-th bit in the inverted channel word and I is the information word obtained from the calculation. It can be verified that for each channel word in the table the inner product so formed equals the integer representation I of the information word associated with the channel word as defined in the table. As an example, the channel word '01010' is decoded as 1.11+0.6+1.3+0.2+1.1 = 15, resulting in the information word '1111'. From this example, it is clear that the inner product simplifies to simple additions. The encoding process translates the information words into the channel words in the following way. A calculation is carried out using an information word so as to obtain a channel word, which calculation is based on the following steps

(a) set a running parameter j equal to n,

(b) determine whether the value of the information word is larger than or equal to the weight vector coefficient W_j, if so, set the binary value in the bit position j of the channel word to

(c) repeat the step (b) n-l times for each next lower value for j, w_n being the weight vector coefficient having the largest value and weight vector coefficients having a next lower subscript j having a smaller value.

In accordance to Kautz, the weight coefficients W; can be obtained using the following formula:

As explained in the foregoing, the Kautz' method requires a weight vector of which the binary value of the weight coefficients are expressed in n-l bits, so that, in the case of high values for n a memory of large memory capacity is needed.

In accordance with the invention, the weighting coefficients w. are chosen such that they can be expressed using p-bit binary words for which p is smaller than n-l, and where 'zeroes' need to be added before or after, or both before and after the p-bit binary word so as to obtain the (n-l)-bit weight vector coefficient word.

The weight coefficients W_j can be obtained using the following equation:

, n ,

where W_j=0 for j ≤O and where the FLOOR of a value equals the largest integer value which is smaller than said value. When using these weighting vector coefficients, channel words are obtained where at most r consecutive 'zeroes' occur at one and the same end, more specifically the trailing end, of the channel words. The trailing end of the channel words are defined as that end where the least significant bits of the channel word are positioned. This is realized by restricting the number of coefficients v/.- being equal to 2¹"¹ to r+ 1 coefficients. As a result, a concatenation of subsequent channel words will not lead to a violation of the k- constraint, as each n-bit channel word generated has at most k-r 'zeroes' at the leading end of the channel word. From the first formula 2^1"1, it is clear that the weight coefficients w_i have only one 'one' bit at a certain bit position in the weight coefficient, whereas the remaining bits of the weight coefficient are all 'zero*. The other weight coefficients are obtained with the second and the third formula, which includes the floor function. The second formula defines the weight coefficients w_r+2 to w_p and the third formula defines the weight coefficients w_p+1 to w_n.

From the second formula it is clear that each subsequent coefficient W_j is obtained by summing the k previous coefficients v/_{i k} to WJ.J, where those previous coefficients w₄ having an index number smaller than or equal to zero are set to zero. As a result, weight coefficients are obtained having a sequence of r+ 1 'ones' in r+ 1 neighbouring bit positions in the weight coefficient, whereas the remaining bits of the weight coefficient are all 'zero'. It is further clear that, for deriving w, using the third formula, including the floor function, the sum term in the floor function is of the order of 2'^"1. Dividing the sum term by 2^1"1*1* will thus result in value which is of the order of 2^P. This value can thus be expressed in p bits. Taking the floor and again multiplying by 2^1~1~p will result in weight coefficients that are expressed using p bit words and adding 'zeroes' before, or after, or both before and after the p-bit words.

If p equals k+2, the above equations for the weighting coefficients W_j change into

2'^" for i=l. ,r+l , - ^•j(2^r+1-l)2'-^r-², for i =r+2 , , k+2 ,

((2^r+1-l)2 j*fc-~r^r+⁺l¹+;c+2-i)2i-k-3 for i=k+3 , ...n.

In an 196-to-197 encoding method, where the sequence of channel words obtained satisfies the requirements k=7 and r=4, the weighting coefficients w. satisfy the following equation:

2'-¹, for i=l, .. ,5, w_{ = . 31.2'^"6, for i=6, ... ,9,

(505-i) .2'^"10, for i=10, ... , 197.

As a result, the values for w- are as follows:

Table 2

i W_j (decimal) W_j (binary)

1 1 0... ....0

2 2 0... ...10

3 4 0... ...100

4 8 0... ...1000

5 16 0... ...10000

6 31 0... ...11111

7 31x2 0... ...111110

8 31x4 0... ...1111100

9 31x8 0... ...11111000

10 495 0... ...111101101

11 494x2 0... ...1111011000

12 493x4 0.. ..11110101100

13 492x8 0.. ..111101010000

194 311x2¹⁸⁴ 000100110111.. ..0

195 310x2¹⁸⁵ 00100110110.. ..0

196 309x2¹⁸⁶ 0100110101.. ..0

197 308x2¹⁸⁷ 100110100.. ..0

The weight vector coefficients w_} in the table 2 are in the form of 196 bits wide binary numbers.

Note that each w₄ can be obtained by generating a 9-bit binary word at the maximum and adding 'zeroes' before or after, or both before and after the 9-bit binary word. This for the reason that the weight coefficients w₁₀ to w₁₉₇ comprise a term (505-i), which equals the decimal value between 495 and 308. Those decimal values can be represented by a binary value of 9 bits, as can be seen in table 2, the right column.

From the table 2 it is clear that the weight coefficients W_j to w₅ can even be obtained by generating 5-bit binary values '00001' to '10000' and adding 191 'zeroes' before these 5-bit binary values so as to obtain the 196-bit wide weight coefficients. It is even possible to generate an i-bit binary number for i equals 1 to 5, having a '1' bit as the most significant bit and having 'zero' bits for the lower significant bits (if present). The weight coefficients w₆ to w₉ can be obtained by generating the decimal number '31', which is the 5-bit binary word '11111' and adding 191 'zeroes' before this word for i=6, adding 190 'zeroes' before and one 'zero' after the word for i=7, adding 189 'zeroes' before and two 'zeroes' after the word for i=8 and adding 188 'zeroes' before and three after the word for i=9. The weight coefficients w₁₀ to w₁₉₇ are obtained by generating the 9-bit binary word corresponding to the decimal word '505-i', as explained above, and adding 197-i 'zeroes' before and i-10 'zeroes' after the 9-bit binary word.

Figure 1 shows an embodiment of an encoding arrangement for encoding (n-l)-bit information words into n-bit channel words. The embodiment of figure 1 comprises an input terminal 1 for receiving a sequence of (n-l)-bit information words, a converter unit 2 for converting the (n-l)-bit information words into the n-bit channel words, and an output terminal 3 for supplying a sequence of n-bit channel words. The converter unit 2 comprises a calculation unit 5 and a weight coefficient generator 7. The weight coefficient generator 7 generates the weight coefficients W; at an output 11 in response to a value i supplied to an input 9, where i runs from 1 to n. The value i is also generated internally in the arrangement.

The conversion of an information word into a corresponding channel using the n weight vectors w- has been explained above, and is moreover well known in the prior art, so that no further discussion of this conversion will be given. The generation of the weight coefficients will be further explained with reference to figure 2, which discloses a further embodiment of the weight coefficient generator 7.

The generator 7 of figure 2 comprises a memory 15 having a memory capacity for storing n p-bit binary words. In the example of table 2, this memory can have 197 9-bit words stored in it. In response to the value i supplied to an address input 17, the memory 15 supplies a p-bit binary word to a p-bit output 19. The p-bit binary words are supplied to a p-bit input 20 of a multiplexer 22, which has a (n-l)-bit output 26. In response to the value i supplied to an input 24, the multiplexer 22 multiplexes the p-bit binary word to p neighbouring output terminals 26.q to 26.p+q and connects the output terminals 26.1 to 26.q and 26.p+q+ l to 26.n-l to a 'zero' value terminal (not shown). In the embodiment of the coefficients of the table 2, it would suffice to have a 'one' terminal for connecting to the terminal 26.i, in the case that i runs from 1 to 5, to have the binary word (11111) stored in the memory 15 and connect the five output terminals of the output 19 of the memory 15 that have the word (11111) available, to the teπninals 26.i-5 to 26.i-l for i running from 6 to 9, and to have 188 9-bit words stored in the memory 15 and to supply those 9-bit words via the multiplexer 22 to the output terminals 26.i-9 to 26.i-l for i running from 10 to 197.

Figure 3 shows an embodiment of a decoding arrangement for decoding n- bit channel words into (n-l)-bit information words. The embodiment of figure 3 comprises an input terminal 30 for receiving a sequence of n-bit channel words, a converter unit 32 for converting the n-bit channel words into the (n-l)-bit information words, and an output terminal 33 for supplying a sequence of (n-l)-bit information words. The converter unit 32 comprises a calculation unit 35 and a weight coefficient generator 7. The weight coefficient generator 7 generates the weight coefficients w- at an output 11 in response to a value i supplied to an input 9, where i runs from 1 to n. The value i is also generated internally in the arrangement. The weight coefficient generator 7 can be identical to the weight coefficient generator in the encoding arrangement of figure 2 so that it generates the same coefficients W] in response to the value i supplied to its input 9.

The conversion of a channel word into a corresponding information using the n weight vectors w_} has been explained above, and is moreover well known in the prior art, so that no further discussion of this conversion will be given. The generation of the weight coefficients has been explained above, so that no further discussion of the generation of the weight coefficients will be given.

It should be noted that the p-bit words needed for generating the weight vector coefficients may be stored in a memory. This is not strictly necessary. It may be possible to generate the p-bit words using a calculation algorithm, such as the formulae given above, each time when a vector coefficient W_j is needed for encoding or decoding.

Claims

1. Encoding arrangement for encoding a sequence of (n-l)-bit information words into a sequence of n-bit channel words, the encoding arrangement comprises

- input means for receiving the information words,

- converter means for converting the (n-l)-bit information words into n-bit channel words, - output means for supplying the channel words, the converter means comprising

- weight vector coefficient supply means for supplying a weight vector w, the weight vector having n weight vector coefficients W_j, where i is an integer running from 1 to n and the weight vector coefficients being in the form of (n-l)-bit binary words, - calculation means for carrying out a calculation using an information word so as to obtain a channel word, which calculation is based on the following steps

(a) set a running parameter j equal to n,

(b) determine whether the value of the information word is larger than or equal to the weight vector coefficient W , if so, set the binary value in the bit position j of the channel word to 'zero' and subtract the value of the weight coefficient w: from the value of the information word so as to obtain a new value for the information word, if not, set the said binary value to 'one',

(c) repeat the step (b) n-l times for each next lower value for j, w_n being the weight vector coefficient having the largest value and weight vector coefficients having a next lower subscript j having a smaller value, characterized in that the weight vector coefficient supply means is adapted to supply p bits of a weight vector coefficient W;, the remaining n-l-p bits of the (n-l)-bit weight vector coefficient word being 'zeroes' to be added before or after the p-bit binary word or before and after the p-bit binary word so as to obtain said weight vector coefficient, and that p is an integer larger than one and smaller than n-l.

2. Encoding arrangement as claimed in claim 1, characterized in that for generating a sequence of channel words having the virtue that at most k consecutive 'zeroes' occur between subsequent 'ones' in the sequence, the channel words further satisfying the requirement that at most r consecutive 'zeroes' occur at one and the same end of the channel words, the weighting coefficients w_; satisfy the following equation:

>i-l for i-l, .. ,r+l i-l fori=r+2, ^>fl j=i-l-J i-l ψ-P^'γ .FLOOR {2P^'*¹. ∑ _j ), for i=p+l, ,Λ, =i-l-7c where W_j = 0 for j ≤ 0, and where the FLOOR of a value equals the largest integer value which is smaller than said value, and where r is smaller than k.

3. Encoding arrangement as claimed in claim 2, characterized in that p=k+2 and that the weighting coefficients W_j satisfy the following equation:

2'^"1, for i-l, ,2"+l

Wf = (2^r+1-l)2'-^r-², for i=r+2, ,k+2,

((2^r*1-l)2*^~r+1+/c+2-i)2'^~*^~3, for i=Jc+3,...n.

4. Encoding arrangement as claimed in claim 3, characterized in that n=197, k=7 and r=4, and that the weighting coefficients w₅ satisfy the following equation:

>i-l for i-l, .. ,5, w_{ = < 31.2^1'"6, for i=6, ... ,9,

(505-i) .2' -10 for i=l0, ... ,197.

5. Encoding arrangement as claimed in claim 2, characterized in that the channel words satisfy the requirement that at most r consecutive 'zeroes' occur at the trailing end of the channel words.

6. Decoding arrangement for decoding a sequence of n-bit channel words into a sequence of (n-l)-bit information words, the decoding arrangement comprises

- input means for receiving the channel words,

- converter means for converting the n-bit channel words into (n-l)-bit information words, - output means for supplying the information words, the converter means comprising

- weight vector supply means for supplying a weight vector w, the weight vector having n weight vector coefficients w_i( where i is an integer running from 1 to n, each weight vector coefficient v/_i being represented as a binary word,

- calculation means for carrying out a calculation using an inverted channel word so as to obtain an information word, which calculation is based on calculating an inner product n ¹ - ∑ ^xi-^wi r=ι

where X_j is the i-th bit in the inverted channel word and I is the information word obtained from the calculation, and n and p are integer constants, characterized in that the weight vector coefficient supply means is adapted to supply p bits of a weight vector coefficient W_j, the remaining bits of the weight vector coefficient word being 'zeroes' to be added before or after the p-bit binary word or before and after the p-bit binary word, and that p is an integer larger than one and smaller than n-l.

15

7. Decoding arrangement as claimed in claim 6, the sequence of channel words having the virtue that at most k consecutive 'zeroes' occur between subsequent 'ones' in the sequence, the channel words further satisfying the requirement that at most r consecutive 'zeroes' occur at one and the same end of the channel words, characterized in that the weighting coefficients w₅ satisfy the following equation:

r for i=l, ,-τ+l i¹-,l

^WJ fori-r+2 , ....p, j-i-l-k i-l ψ-^P-^χ .FLOOR(2*^'i+1. ∑V; ), for i=p+2, ,n_t =i-l-/c

where W_j = 0 for j ≤ 0, and where the FLOOR of a value equals the largest integer value which is smaller than said value, and where r is smaller than k.

8. Decoding arrangement as claimed in claim 7, characterized in that p=k+2 and that the weighting coefficients W_j satisfy the following equation:

2 , for i=l, ,r+l

Wi = (2^r+1- 1)2'--², for i=r+2, ,k+2,

((2^r+1-l)2*^~r+1+/c+2-i)2'^~*^~3, for i=k+3, ...n.

9. Decoding arrangement as claimed in claim 8, characterized in that n=197, k=7 and r=4, and that the weighting coefficients w_i satisfy the following equation:

2'^"1, for i-l, .. ,5,

Wi = 31.2'^"6, for i=6, ... ,9 ,

(505-i) .2'^"10, for i-10, ... ,197

10. Decoding arrangement as claimed in claim 7, characterized in that the channel words satisfy the requirement that at most r consecutive 'zeroes' occur at the trailing end of the channel words.

11. Encoding method for encoding a sequence of (n-l)-bit information words into a sequence of n-bit channel words, that can be carried out in the encoding arrangement as claimed in anyone of the claims 1 to 5.

12. Decoding method for decoding a sequence of n-bit channel words into a sequence of (n-l)-bit information words, that can be carried out in the decoding arrangement as claimed in anyone of the claims 6 to 10.