VLSI Digital Signal Processing Systems: Design and Implementation

8.8    PROBLEMS

1. Compute the product of 3 complex numbers

   

   using the fewest number of multiplication operations.

2. Calculate f₁ = ax + by and f₂ = bx + ay using 2 multiplications.
3. Calculate f₁ and f₂ given by

   

   using 6 real multiplication operations.

4. Construct a 2 × 2 convolution algorithm using the Cook-Toom algorithm with β = 0, 1, 2.
5. Construct a 3 × 3 convolution algorithm using the Cook-Toom algorithm with β_i's as 0, 1, −1, 2, −2.
6. Construct a 3 × 3 convolution algorithm using the modified Cook-Toom algorithm with β_i's as 0, 1, −1, 2.
7. Construct a 3 × 3 convolution algorithm using the modified Cook-Toom algorithm with β_i's as 0, 1, 2, −2.
8. Use the Cook-Toom algorithm to construct a convolution algorithm for filtering a sequence of 3 data inputs with a 4-tap FIR filter, i.e., 4 × 3 linear convolution.
9. Use the Cook-Toom algorithm to construct a 3 × 4 convolution algorithm.
10. Use the modified Cook-Toom algorithm to construct a convolution algorithm for 4 × 3 linear convolution.
11. Construct a 2 × 2 convolution algorithm using the Winograd algorithm with m(p) = p(p – 1)(p + 1).
12. Construct a 2 × 2 convolution algorithm using the modified Winograd algorithm with m(p) = p(p – 1).
13. Design a Winograd convolution for 4-point by 3-point linear convolution using m(p) = p²(p + 1)(p – 1)(p² + 1).
14. Design ...