One-to-two dimensional mapping of DFT

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One-to-two dimensionalmapping of DFT

• Let
• so that
• and

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One-to-two dimensionalmapping of DFT

• Modification of
• Can be a 1-D DFT if
• Modification of resulting array
• Can be a 1-D DFT if

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One-to-two dimensionalmapping of DFT

• Thus the conditions below must prevail
• We identify 4 cases.
•  1)
• When
• 2)
• When

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One-to-two dimensionalmapping of DFT

•  3)
• When
• 4)
• When

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One-to-two dimensionalmapping 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

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One-to-two dimensionalmapping of DFT

• .
• Twiddle factors
• Computational Complexity
• Inner DFT
  Outer DFT
• Twiddle factors
  Total

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One-to-two dimensionalmapping of DFT

• Address mappings (Case A) .
•  

0 1 1
0
1
2
3
..
..

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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.

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Good - Thomas mapping

• When there exist
• such that
• Euclids Algorithm

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Euclids Algorithm

• (31,11)
• (1) 312x119
• (2) 111x92
• (3) 94x21
• 4x(2) yields
• 4x114x94x24x9(9-1)5x9-1
• 5x(1) yields
• 5x3110x115x910x11(4x111)
• Thus 5x31-4x111 ie

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One-to-two dimensionalmapping of DFT

• The mapping from n to is obtained
  from as
• Hence
• and

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Euclids Algorithm

• Given to show that
• Assume and write
• Now consider
• And reduce then the result is in some order

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One-to-two dimensionalmapping of DFT

• For each residue of ,
  will be different, else and for
  and we have
• And hence divides r which is untrue as
  or divides which is also untrue as

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One-to-two dimensionalmapping of DFT

• Hence an n such that
• Thus . But and hence
• Or
• Since and it follows that it is also true

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Prime Radix Algorithm

• Finite duration signal
• DFT
• Set so that

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Prime Radix Algorithm

• Computational Complexity
• Reduction in complexity is achievable via
  segmented computations
• for even N
• point DFTs

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Prime Radix Algorithm
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Prime Radix Algorithm

• May be regarded as

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Prime Radix Algorithm

• Recall that for any
  fixed and with
  prime
• is equal to a rearrangement of the integers

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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.

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Prime Radix Algorithm

• Thus for

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Prime Radix Algorithm
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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

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Prime Radix Algorithm

• The signal flow graphs and transfer functions are

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Prime Radix Algorithm

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Prime Radix Algorithm

P Q Cos(2pQ/P) Shift/PROM Scaling
17 4 0.0923
29 7 0.05411
47 1 0.9911
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Prime Radix Algorithm

• 
  when

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Prime Radix Algorithm

• Approximation is in the denominator where ideally
  
• Actually
• with small
• Hence actual operation can be modified to improve
  performance towards ideal

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Prime Radix Algorithm

• Thus
• hence
• where