Lecture 12: Algorithmic Strength 2 Reduction in Filters and Transforms Saeid Nooshabadi

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1 Multimedia Systems Lecture 12: Algorithmic Strength 2 Reduction in Filters and Transforms Saeid Nooshabadi

2 Overview Cascading of Fast FIR Filter Algorithms (FFA) Discrete Cosine Transform and Inverse DCT Algorithm Architecture A hit t Transformation Rotator Operation

3 Review Parallel FIR Filters can be optimized through the use of decomposition and fast FIR filter algorithms (FFA) Any parallel FIR filter structure can be used to derive another parallel equivalent structure by transpose operation (or transposition) Optimum Linear Convolution can be used to construct fast FIR filter from the Linear Convolution algorithm

4 2 phase Fast FIR Algorithms x(n) y(n) H(z) N-tap FIR N multiplications and (N 1) additions 4(N/2)=2N multiplications 3(N/2) multiplications 4(N/2 1)+2=2(N 1) additions 3(N/2 1)+4=3N/2+1 additions

5 3 phase Fast FIR Algorithms 9(N/3)=3N multiplications 6(N/3)=3N multiplications 9(N/3 1)+6=3(N 1) additions 6(N/3 1)+10=2N+4 additions

6 Fast Parallel FIR Algorithms for Large Block Sizes (#1/2) Parallel aa FIR filters eswith long gboc block sizes escan be designed ed by cascading smaller length fast parallel filters Example: an m parallel FFA can be cascaded with an n parallel FFA to produce an (m x n) parallel filtering structure. The set of FIR filters resulting from the application of the m parallel FFA can be further decomposed, one at a time, by the application of the n parallel FFA. The resulting set of filters will be of length N/(m x n). When cascading the FFAs, it is important to keep track of both the number of multiplications and the number of additions i required dfor the filtering i structure

7 Cascading FFA for block size of 4 (#1/3) Thereduced complexity complexity 4 parallel filtering structure is obtained by first applying the 2 parallel FFA to then applying the FFA a second time to each of the filtering operations that result from the first application of the FFA Application 1

8 Cascading FFA for block size of 4 (#2/3) Application 2: Filtering operation {X 0 H 0 } Filtering operation {X 1 H 1 } Filtering operation {(X 0 +X 1 )(H 0 +H 1 )}

9 Cascading FFA for block size of 4 (#3/3) The second application of the 2 parallel FFA leads to the 4 parallel filtering structure, requiring 9 filtering operations withlength N/4 x D y y=z -2 x x 0 y 0 D X 1 y 1 z -4 9 length N/4 filtering operations 9N/4 multiplications and 20+9(N/4-1) additions 44% less hardware than the traditional parallel FIR

10 Cascading FFA for block size of 6 (#1/2) The reduced complexity 6 parallel filtering structure is obtained by first applying the 2 parallel FFA to then applying the 3 parallel FFA a second time to each of the filtering operations that result from the first application of the FFA The reduced complexity 6 parallel filtering structure is obtained by first applying the 3 parallel FFA to then applying the 2 parallel FFA a second timeto to each of the filtering operations that result from the first application of the FFA

11 Cascading FFA for block size of 6 (#2/2) x D y y=z -2 x x 0 y 0 x 1 y 1 x 2 D y 2 z -6 18N/6=3N filtering operations 3N,, multipliers and 42+18(N/6-1) ) adders. 50% less hardware than the traditional parallel FIR

12 Computational Complexity of Cascading FFA L parallel cascaded as L=L 1 x L 2 x x l r Number of multiplications L i the block size of FFA Number of adders M i is the number of filters that result from the application of the i th FFA A i is the number of pre/post- p processing adders required by the i th FFA N is the length of the filter.

13 Computational Complexity of Cascading FFA (L=6) L parallel cascaded as L=L 1 x L 2 x x l r Number of multiplications L i the block size of FFA Number of adders M i is the number of filters that result from the application of the i th FFA A i is the number of pre/post- p processing adders required by the i th FFA N is the length of the filter.

14 Computational Complexity of N=24, L=6 {L 1 =2, L 2 =3} Cascading FFA (L=6) {L 1 =3, L 2 =2}

15 FFA Example (#1/3) x(n) y(n) y(2n) H(z) 2 N-tap FIR N tap p FIR requires N multiplications and (N 1) additions per input sample Since odd output samples are not retained computation can be reduced dto N/2 multiplications li li i and (N 1)/2 additions i Requires 2(N/2) muls and 2(N/2-1)+1 Requires 2(N/2) muls and 2(N/2-1)+1 adds for 2 input samples N/2 muls and (N-1)/2 adds per input sample

16 FFA Example (#2/3) This structure requires 6 x (N/4) multiplications and 6 x (N/4 1) +10 =3N/2 + 4 additions per 4 input samples. Thus, 3N/8 multiplications and 3N/8 +1 additions per input samples

17 FFA Example (#3/3) X 00 x(4k) + y(4k) Y 00 + y(4k+2) Y 01 X 01 x(4k+2) D X 10 x(4k+1) X 11 x(4k+3)

18 Discrete Cosine Transform and Inverse DCT (#1/2) The discrete cosine transform (DCT) is a frequency transform used in still or moving video compression. We discuss the fast implementations of DCT based on algorithm architecture transformationsandthe and the decimation in frequencyin approach Denote the DCT of the data sequence x(n), n=0, 1,, N 1, by X(k), k=0, 1,, N 1. The DCT and inverse DCT (IDCT) are described by the following equations:

19 Discrete Cosine Transform and Inverse DCT (#2/2) DCT is an orthogonal transform, i.e., the transformation matrix for IDCT is a scaled version of the transpose of that for the DCT and vice versa. Therefore, the DCT architecture can be obtained by transposing the IDCT, i.e., reversing the direction of the arrows in the flow graph of IDCT, and the IDCT can be obtained by transposing the DCT Direct implementation of DCT or IDCT requires N(N 1) multiplication operations, i.e., O(N 2 ), which is hardware expensive. Strength reduction can reduce the multiplication complexity of a 8 point DCT from 56 to 13.

20 8 point DCT It can be written in matrix ti form as follows: c i =cos(iπ/16)

21 Strength Reduction for 8 point DCT (#1/8) Step 1:Using trigonometricproperties the 8 point DCTcan Step 1:Using trigonometric properties, the 8 point DCT can be written as:

22 Strength Reduction for 8 point DCT (#2/9) Using trigonometric properties, the 8 point DCT can be g g p p, p written as:

23 Strength Reduction for 8 point DCT (#3/8) x(0) P 0 X(1) x(7) M 0 x(3) P 1 x(4) M 1 x(1) P 2 x(6) M 2 x(2) P 3 x(5) M 3 P 10 M 10 P 11 P 100 M 11 M 100 X(7) X(5) X(3) X(2) X(6) X(0) X(4)

24 Strength Reduction for 8 point DCT (#4/8) Step 2: the DCT structure is grouped into different functional units represented by blocks and then the whole DCT structure is transformed into a block diagram

25 Strength Reduction for 8 point DCT (#5/8) x(0) P 0 x(7) M 0 x(3) P 1 x(4) M 1 x(1) P 2 x(6) M 2 x(2) P 3 x(5) M 3 X(1) X(7) X(5) X(3) X(2) X(6) X(0) X(4)

26 Strength Reduction for 8 point DCT (#6/8) Step 3: Reduced complexity complexity implementations of various blocks are exploited The block can be realized using 3 multiplications and 3 additions instead of using 4 multiplications and 2 additions, as shown in follows Define the block with {a=sin(θ), b=cos(θ)} and reversed outputs as a rotator block that performs the following computation:

27 Strength Reduction for 8 point DCT (#7/8) The angles of cascaded rotators can be simply added, as shown in the transformation block as follows: Based on the fact that a rotator with θ=π/4 is just like the block, we modify it as the following structure:

28 Strength Reduction for 8 point DCT (#8/8) From the three steps, we obtain the finalstructure where only 13 multiplications are required

29 Conclusion Parallel Fast FIR Filter Algorithms (FFA) can be cascaded together to realize large block size filters Discrete Cosine Transform (DCT) can be reduced in computational complexity through algorithm strength reduction.

SOME CONCEPTS IN DISCRETE COSINE TRANSFORMS ~ Jennie G. Abraham Fall 2009, EE5355

SOME CONCEPTS IN DISCRETE COSINE TRANSFORMS ~ Jennie G. Abraham Fall 009, EE5355 Under Digital Image and Video Processing files by Dr. Min Wu Please see lecture10 - Unitary Transform lecture11 - Transform