International Symposium on Information Theory and its Applications, ISITA2004
Parma, Italy, October 10–13, 2004
On the Calculation Method of Input-Output Weight Distribution of
Terminated Convolutional Codes
Hideki YOSHIKAWA
Suzuka National College of Technology
Suzuka-shi, Mie, 510-0294 Japan
E-mail: hyoshi@info.suzuka-ct.ac.jp
Abstract
In this paper, first, the calculation method of inputoutput weight distribution of terminated convolutional
codes is shown. Next, it is shown that generalized
weight distribution can be obtained since the constraint
length is two or three. Furthermore, some analytical
results of terminated convolutional codes are shown as
function of the number of information symbols, and
application for error performance evaluation is demonstrated.
1. INTRODUCTION
The weight distribution of error correcting codes
can be used to compute the error performance bounds
for the performance evaluation of any linear block
codes. As any terminated convolutional code can be
represented by an equivalent block code if the convolutional encoders are forced to the all-zero state by zerotail terminating at end of each block. Although it is not
easy to calculate the weight distribution, generally, the
bounding technique can be applied for the performance
evaluation for any convolutional codes. The simple calculation method of the weigh distribution of terminated
convolutional codes based on the state transition matrix of encoder was already shown[1]. But the complexity depends on the codeword length.
In [2], it is shown that the low-term weight distribution of terminated convolutional codes can be obtained,
directly, for d ≤ h ≤ 2d − 1, where h is weight of codeword, and d is minimum free Hamming distance of the
convolutional code. In this case, the low-term weight
distribution can be expressed as a linear function of x
which is the number of information bits. Therefore, we
can apply the bounding technique for the performance
evaluation of convolutional code when any distribution
can be derived directly by x.
In this paper, first, the calculation method of inputoutput weight distribution of terminated convolutional
codes is presented. Next, it is shown that generalized
weight distribution can be obtained since the constraint
length is two or three even if h ≥ 2d. In this case, the
weight distribution can be given by a function of only
input-output weight and x. The input-output weight
distribution expressed as a function of x is useful for
the performance evaluation of serial concatenated code
and parallel concatenated convolutional codes[3, 4].
2. CALCULATION METHOD OF WEIGHT
DISTRIBUTION OF TERMINATED CONVOLUTIONAL CODES
Let us assume a rate 1/2 binary convolutional codes
of constraint length L. We can define a state transition
matrix[1] of the encoder P to be a 2L−1 × 2L−1 matrix
with i − jth element given by Pi,j = W w H h where
w and h are Hamming weights of input information
bits and encoder’s output, respectively. For example,
the matrix P of 4-state non-systematic convolutional
encoder with generating polynomial [1+D+D2 , 1+D2 ]
are given by


0
1 W H2 0
 0
0
H WH 
.
P =
(1)
 H2
W
0
0 
0
0
H WH
Let A(W, H, x) denote the number of codewords
with weight h generated by information bits of weight w
in the (2(x+L−1), x) linear binary block code obtained
by terminating with L − 1 tail zeros after x information
bits. In this case, the the input-output weight enumerating function (IOWEF), A(W, H, x) can be expressed
by
A(W, H, x) = IP x+L−1 I T =
x n
Aw,h W w H h .
w=0 h=0
(2)
where Aw,h represents the number of codewords with
output codeword weight h generated by input information bits of weight w, I = (1, 0, . . . , 0), and the superscript T denotes transpose. The low-weight terms
852
of the weight distribution A(W, h, x)|W =1 can be obtained as a linear function of x[2]. For d ≤ h ≤ 2d − 1,
A(1, h, x) can be expressed by Nh x − ah where Nh is
the number of paths of output weight h in the trellis diagram that remerge with the all-0 path at x+L−1. Although this value can be obtained by a transfer function
of convolutional code, the matrix approach is applied
because it is hard to calculate the transfer function for
longer constraint length.
Here, let us consider the output weight distribution.
In order to obtain Nh , let us define a matrix Q to be a
matrix which is removed the element of remergence to
all-0 path in P [5]. Consider P in (1) for example, the
matrix Q can be written as


0
1 W H2 0
 0
0
H WH 
.
Q=
(3)
 0
W
0
0 
0
0
H WH
as a function of x. Using this coefficient, the conditional weight enumerating function (CWEF) is given
by
n
Aw,h (x)H h .
(7)
Â(w, H, x) =
The number of paths with output weight h that diverge
from all-0 at the origin x = 0 and remerge only once,
vh,x can be obtained by
as a generalized formula. By substituting above equation in (6) and w = h − 2 for ch,k = 0, the IOWEF
coefficient Aw,h (x) can be expressed as
x − w + 1 if w = h − 2
(2)
Âw,h (x) =
.
0
otherwise
V (H, x) = IQx+L−2 P I T =
n
vh,x H h .
(4)
h=0
Hence, the number of paths with weight h that diverge
from all-0 at the origin x = 0 and remerge only once
at x = k + L − 1, These path is defined as single error
event[4].
For performance evaluation of terminated convolutional codes, the weight distribution, i.e, weightenumerating function (WEF) needs the number of single error event. But for h ≥ 2d, the number of codewords, A(W, H, x) in (2) includes not only single error
events but also multiple error events[2, 4].
Here, a coefficient ch,k is defined as the number
of paths with weight h that the length of single error events is k + L − 1. This can be obtained by
ch,k = vh,k − vh,k−1 . By this coefficient, the number of codewords with weight h by single error events,
Â(W, h, x) can be generalized as follows
lh,max
Â(W, h, x) =
ch,k (x − k + 1)
(5)
k=1
where lh,max is a maximum length of single error events
of weight h. The expression is valid even for h ≥ 2d.
From (5), the IOWEF coefficient Aw,h in (2) is given
by
1 ∂ w Â(W, h, x) Âw,h (x) =
(6)
w!
∂W w
W =0
h=0
3. ANALYTICAL RESULTS
In this section, some analytical weight distributions
are presented as functions of x. At first, it is shown that
the weight distribution can be obtained by generalized
formulas for L = 2, 3.
Example 1 For the convolutional code with L = 2
given by the generating polynomial[1, 1+D], we see that
ch,k = W h−2 only if h − 2 = k, and ch,k = 0 otherwise,
By (5), the output weight distribution is given by
Â(2) (W, h, x) = (x − h + 3)W h−2
Example 2 For the convolutional code with L = 3
given by the generating
+ D + D2 , 1 + D2 ],
h−4
h−5 polynomial[1
we obtain ch,k = k−h+4 W
. From (5), the output
weight distribution is given by
(3)
A
(W, h, x) =
h−5
q=0
h−5
(x − h − q + 5)W h−4
q
as a generalized formula. By substituting above equation in (6) and w = h − 4 for ch,k = 0, the IOWEF
coefficient Aw,h (x) can be expressed as
 w−1

 w−1
(x − w − q + 1) if w = h − 4
(3)
q
Âw,h (x) =
.
q=0


0
otherwise
For L ≥ 4, the input-output weight distribution
can be derived as a function of x from (5)(6) when
cj,k can be obtained. In Table 1-4, the coefficients cj,k
and IOWEF coefficients Âw,h are listed where G means
generator polynomial of convolutional code denoted by
the octal form.
4. APPLICATION
FOR
THE
UNION
BOUND OF BIT ERROR PROBABILITY
For binary transmission with maximum likelihood
decoding over additive white Gaussian noise(AWGN)
853
Table 1: The coefficients ch,k for L = 4, G = [17, 13]
k\h
1
2
3
4
5
6
7
8
9
10
6
0
7
W
0
W3
W3
0
0
0
0
0
0
4
W2
0
0
0
0
0
0
0
0
2
lh,max
8
0
0
9
0
0
0
0
2
W
W4
W4
2W 4
0
0
0
0
6
3W 3 + W 5
W5
3W 5
3W 5
0
0
8
Table 2: The coefficients ch,k for L = 5, G = [35, 23]
k\h
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
10
0
0
0
W2
W4
4
2W + W 6
5W 4 + W 6
4W 6
5W 6
5W 6
10
0.1
Upper Bound (L=2)
Upper Bound (L=3)
Upper Bound (L=4)
Upper Bound (L=5)
Simulation (L=2)
Simulation (L=3)
Simulation (L=4)
Simulation (L=5)
Bit Error Probability
0.01
lh,max
7
W
0
0
W3
0
0
0
0
0
0
0
0
0
0
0
4
8
0
W2
0
0
W4
0
0
W6
0
0
0
0
0
0
0
8
9
0
0
W3
0
0
2W 5
0
0
W7
0
0
0
0
0
0
9
10
0
0
W2
W2 + W4
W2
2W 4
4
2W + 2W 6
2W 4
W6
6
W + W8
W6
0
0
0
0
11
11
0
0
0
W3
2W 3 + W 5
2W 3
4W 5
5
4W + 2W 7
4W 5
3W 7
7
4W + W 9
4W 7
W9
2W 9
2W 9
15
0.001
eight distribution are independent for medium to high
signal-to-noise power ratio per information bits, Eb /No .
0.0001
5. CONCLUSIONS
1e-05
1e-06
0
1
2
3
4
5
6
7
Eb/No (dB)
Figure 1: Analytical bounds for terminated convolutional codes.
channel, the bit error probability of (n, x) linear block
codes can be upper-bounded as
x
n w
hREb
Aw,k erfc
(8)
Pb ≤
x
No
w=1
8
In this paper, the calculation method of inputoutput weight distribution of terminated convolutional
codes is shown. As the results, it is shown that the
weight distribution of terminated convolutional codes
can be given by function of only the input-output
weight, w, h, and the number of information bits x.
Furthermore, generalized weight distribution of terminated convolutional codes with constraint length two or
three are shown. Because of the computational complexity not depend on codeword length, it is useful for
application of error performance evaluation. It is expected that this analytical technique can be applied for
concatenated codes, or a class of turbo codes based on
terminated convolutional codes.
h=d
where erfc(x) is the complementary error function, R
is the code rate, and Eb /No is the signal-to-noise power
ratio per information bit[3]. Above bound can be applied for the error performance evaluation of terminated convolutional codes.
The obtained upper bound by analytical inputoutput distribution is plotted in Fig.1. In this figure,
x is set to 1024, and curves come from the theoretical
approach in (8) and the points come from the computer simulations, respectively. In these results, it is
concluded clearly that high-weight coefficients in the w
References
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854
Table 3:
h\w 1
6
0
7
x
8
0
0
9
10
0
11
0
0
12
13
0
14
0
0
15
16
0
0
17
18
0
19
0
IOWEF
2
x−1
0
x−2
0
x−3
0
0
0
0
0
0
0
0
0
coefficients Âw,k (x)
3
4
0
0
2x − 5
0
0
4x − 17
3x − 12
0
0
8x − 44
3x − 13
0
0
9x − 56
x−6
0
0
5x − 36
0
0
0
x−9
0
0
0
0
0
0
for L = 4.
5
0
0
0
8x − 48
0
20x − 142
0
26x − 207
0
19x − 168
0
7x − 71
0
x − 12
Table 4: IOWEF coefficients Âw,k (x) for L = 5.
h\w
7
8
9
10
11
12
13
14
15
16
17
18
19
1
x
0
0
0
0
0
0
0
0
0
0
0
0
2
0
x−1
0
3x − 9
0
0
0
0
0
0
0
0
0
3
x−3
0
x−2
0
5x − 21
0
9x − 54
0
0
0
0
0
0
4
0
x−4
0
7x − 39
0
8x − 41
0
21x − 153
0
27x − 243
0
0
0
5
0
0
2x − 10
0
13x − 88
0
36x − 300
0
42x − 348
0
82x − 842
0
81x − 972
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