1. Professor A G Constantinides 1 One-to-two dimensional mapping of DFT Let so that and

2. Professor A G Constantinides 2 One-to-two dimensional mapping of DFT »Modification of »Can be a 1-D DFT if »Modification of resulting array »Can be a 1-D DFT if

3. Professor A G Constantinides 3 One-to-two dimensional mapping of DFT Thus the conditions below must prevail We identify 4 cases. 1) When 2) When

4. Professor A G Constantinides 4 One-to-two dimensional mapping of DFT 3) When 4) When

5. Professor A G Constantinides 5 One-to-two dimensional mapping of DFT Since. And n must range from 0 to and k must range from 0 to Only two cases are viable Case A and Case B

6. Professor A G Constantinides 6 One-to-two dimensional mapping of DFT. Twiddle factors Computational Complexity: Inner DFT: Outer DFT: Twiddle factors: Total:

7. Professor A G Constantinides 7 One-to-two dimensional mapping of DFT Address mappings (Case A). 011… 0 1… 2… 3….. … …

8. Professor A G Constantinides 8 Good - Thomas mapping Consider the index in the exponential of a DFT For the elimination of cross terms and for proper DFTs we need:- Thus and hence Similarly All conditions w.r.t.

9. Professor A G Constantinides 9 Good - Thomas mapping When there exist such that Euclid’s Algorithm …

10. Professor A G Constantinides 10 Euclid’s Algorithm (31,11) (1) 31=2x11+9 (2) 11=1x9+2 (3) 9=4x2+1 4x(2) yields 4x11=4x9+4x2=4x9+(9-1)=5x9-1 5x(1) yields 5x31=10x11+5x9=10x11+(4x11+1) Thus 5x31-4x11=1 ie

11. Professor A G Constantinides 11 One-to-two dimensional mapping of DFT The mapping from n to is obtained from as Hence and

12. Professor A G Constantinides 12 Euclid’s Algorithm Given to show that Assume and write Now consider And reducethen the result isin some order

13. Professor A G Constantinides 13 One-to-two dimensional mapping of DFT For each residue of, will be different, else and for and we have And hence divides r which is untrue as ordivideswhich is also untrue as

14. Professor A G Constantinides 14 One-to-two dimensional mapping of DFT Hence an n such that Thus. But and hence Or Sinceandit follows that it is also true

15. Professor A G Constantinides 15 Prime Radix Algorithm Finite duration signal DFT Setso that

16. Professor A G Constantinides 16 Prime Radix Algorithm Computational Complexity Reduction in complexity is achievable via segmented computations ;for even N point DFTs

17. Professor A G Constantinides 17 Prime Radix Algorithm

18. Professor A G Constantinides 18 Prime Radix Algorithm May be regarded as

19. Professor A G Constantinides 19 Prime Radix Algorithm Recall that for any fixed and with prime is equal to a rearrangement of the integers

20. Professor A G Constantinides 20 Prime Radix Algorithm Since (i) and there are no common factors between n.k and P (ii) For every multiple of P we have It follows that all powers of W from 0 to P will exist in each but not in the same order.

21. Professor A G Constantinides 21 Prime Radix Algorithm Thus for

22. Professor A G Constantinides 22 Prime Radix Algorithm

23. Professor A G Constantinides 23 Prime Radix Algorithm In general and where Note that in view of the Number Theoretic result we can also rearrange w.r.t. any number Q i.e. Reduce {n.k.Q} mod P

24. Professor A G Constantinides 24 Prime Radix Algorithm The signal flow graphs and transfer functions are + W

25. Professor A G Constantinides 25 Prime Radix Algorithm + + + High speed Imag Real Low speed

26. Professor A G Constantinides 26 Prime Radix Algorithm PQ Cos(2πQ/P) Shift/PROM Scaling 174 0.0923 2970.05411 4710.9911

27. Professor A G Constantinides 27 Prime Radix Algorithm when + + + High speed Imag Real Low speed

28. Professor A G Constantinides 28 Prime Radix Algorithm Approximation is in the denominator where ideally Actually with small Hence actual operation can be modified to improve performance towards ideal

29. Professor A G Constantinides 29 Prime Radix Algorithm Thus hence where