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Unfolding
• Use when T∞ is not an integer and/or computation time of a node in DFG > T∞ .
• T∞ increase to J T∞.• Critical path may or may not change.• Clock period is still limited by critical path.• Throughput rate increase by a factor of J
(i.e. Tclk/J).
0D-C4-0B4
0D-C3-0B3
0D-C2-0B2
0D0DC1C10B1
0D0DC0C00B0
1D-B0-1A4
0D-B4-1A3
0D-B3-1A2
0D1DB2B01A1
0D0DB1B11A0
J=5J=2J=5J=2
⎣(i+w)/J⎦(i+w)%Jwi
0D-A4-0D4
0D-A3-0D3
0D-A2-0D2
0D0DA1A10D1
0D0DA0A00D0
2D-E1-7C4
2D-E0-7C3
1D-E4-7C2
1D4DE3E07C1
1D3DE2E17C0
J=5J=2J=5J=2
⎣(i+w)/J⎦(i+w)%Jwi
1D-D1-2E4
1D-D0-2E3
0D-D4-2E2
0D1DD3D12E1
0D1DD2D02E0
1D-B2-3E4
1D-B1-3E3
1D-B0-3E2
0D2DB4B13E1
0D1DB3B03E0
J=5J=2J=5J=2
⎣(i+w)/J⎦(i+w)%Jwi
2)
(a) What is iteration bound and critical path. Assume addition takes 2 u.t. and multiplication takes 4 u.t.
(b) Retime it to minimize the critical path. (c) Unfold it to achieve the throughput rate = T∞ .
Look Ahead Transformation
1) Consider linear time invariant recursion equation: y(n) = ay(n-1)+bu(n)
The input sequence u(n) has sampling rate of 1 sample/µs. Assume TA = 1µs and TM= 2µs. Design hardware that can execute this algorithm at the giving sample rate.
Unfolding after 2 order look ahead
1D-D2-3G21D2DD1D03G11D1DD0D13G01D-C1-2I21D1DC0C12I10D1DC2C02I01D-B0-1I20D1DB2B01I10D0DB1B11I0J=3J=2J=3J=2
⎣(i+w)/J⎦(i+w)%Jwi
2) Rewriting equationy(n) = 0.6y(n-1) − 0.25y(n-2) + x(n)into a state-space formulation. Then apply 2nd
order look-ahead transform.
For 2nd order IIR filter, we havey(n) = 2rcos θy(n-1)-r2y(n-2)+x(n)
Hence r = 0.5, cos θ = 0.6, and sin θ = 0.8.
where bT = [1/(2 sin θ) 0] , cT = [2r cos 2θ r sin 2θ], and d = 1.Apply look-ahead transformation m−1 times and note that [R(θ)]m = R(mθ), we have
1
0( 1) (( ) ) ( 1) ( ) ( )
(( ) ) ( 1) ( )( ) ( ) ( )
mm k
km
T
n r R m n m r R k x n k
r R m n m ny n n d x n
θ θ
θ
−
=
+ = ⋅ − + + −
= ⋅ − + +
= + ⋅
∑v v b
v uc v
[ ]
1 1 1
2 2 2
11 2
2
( 1) ( )cos sin( ) ( ) ( ) ( )
( 1) ( )sin cos
( )( ) ( ) ( ) ( )
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v n v n br x n r R n x n
v n v n b
v ny n c c d x n n d x n
v n
θ θθ
θ θ+ −⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤
= ⋅ + = ⋅ +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥+ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎡ ⎤
= + ⋅ = + ⋅⎢ ⎥⎣ ⎦
v b
c v