基于 fft 的频域 rs 译码
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基于 FFT的频域 RS译码季文洁
2014.3.21
频域译码
1 1 2 2( ...... )l t l t l t v l t vE E E E
D D D
1 2 v
… …
… …
… …
l tE
Prime factor Cyclostome FFT
FFT
Prime factor e.g. GF(24)
N=15=3×5
CFFT1
1
( ) , [0, 1]n
j ijj i
i
F f f j n
2( ) , (2 )i m
i ii
L x l x l GF
CFFT
63
0
( ) , (2 )ii i
i
f x f x f GF
polynomial
e.g.GF(23)
F=ALf
CFFT F=ALf
Winograd Algorithm
选择一组正规基 (γ, γ2, γ4) ,此处取 γ = α3 ,满足 γ+γ2+γ4 = 1 )
CFFTGF(23) 3-point cyclic convolution
0a1a2a
124
0b 1b 2b
CFFT
1 2 4 8
2 4 8 1
4 8 1 2
8 1 2 4
1 2 4 8
2 4 8 1
4 8 1 2
8 1 2 415 17
1 2 4 8
2 4 8 1
4 8 1 2
8 1 2 4
1 2 4 8
2 4 8 1
4 8 1 2
8 1 2 4
1 ... 0...
...0 ...
F A
0124836
1295
10007
141311
fffffffffff
ffff
GF(23) n=7 Lr=2
GF(24) n=15 Lr=4
GF(25) n=31 Lr=6
GF(26) n=63 Lr=12
GF(28) n=255 Lr=34
……
预估结果N=7 Mult 346 Add 80 Reg 210 Area LatencyHorner 7 7 7 4452 7TF 6 25 4076 1复用 4 13+14 3 4174 2
N=15 Mult 609 Add 106 Reg 279 Area LatencyHorner 15 15 15 14910 15TF 16 77 17906 1复用 9 19+54 12 16567 4
N=31 Mult 935 Add 133 Reg 349 Area LatencyHorner 31 31 31 43927 31TF 54 315 92385 1复用 10 40+219 25 52522 6
N=63 Mult 1407 Add 159 Reg 419 Area LatencyHorner 63 63 63 125055 63TF 97 805 264474 1复用 12 46+583 66 144549 12
F=ALf计算 n 个校验子的硬件需求
TF
e.g. GF(26) m = 6=2×3 m1 = 2, m2 = 3
CFFT
0 1 2 3 4 50 3 0 3 0 30 4 2 6 4 80 4 2 3 1 5
-0 -3
0 0 4 2 3 1 5 0
4 4 2 0 1 5 3 4
2 2 0 4 5 3 1 2
3 3 1 5 0 4 2 3
1 1 5 3 4 2 0 1
5 5 3 1 2 0 4 5
y a a a a a a xy a a a a a a xy a a a a a a xy a a a a a a xy a a a a a a xy a a a a a a x
1 2 4 8 16 32
2 4 8 16 32 1
4 8 16 32 1 2
8 16 32 1 2 4
16 32 1 2 4 8
32 1 2 4 8 16
1 2 4 8 16 32
2 4 8 16 32 1
4 8 16 32 1 2
8 16 32 1 2 4
16 32 1 2 4 8
32 1 2 4 8 16
1 ... 0...
...
...0 ...
F A
01248
1632...
fffffff
i mod m2 = 0 , 0≤i<m ,→ {0, 3} [y0, y1, y2, y3, y4, y5]T = [a0, a1, a2, a3, a4,a5]C·[y0, y1, y2, y3, y4, y5]T
e.g. GF(26)
预估结果N=63 Mult
1407Add 159 Reg 419 Area Latency
Horner 63 63 63 125055 63TF 97 805 264474 1复用全体 12 46+583 66 144549 12复用 1小块 4 13+583 132 155700 4×12复用 2小块 8 26+583 66 135741 2×12
0 0 4 2 3 1 5 0
4 4 2 0 1 5 3 4
2 2 0 4 5 3 1 2
3 3 1 5 0 4 2 3
1 1 5 3 4 2 0 1
5 5 3 1 2 0 4 5
y a a a a a a xy a a a a a a xy a a a a a a xy a a a a a a xy a a a a a a xy a a a a a a x
相同的例子 如:10=2×515=3×5
常数乘法器
THE END
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