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m11 - Properties of the DFT

Module by: C. Sidney Burrus

Summary: The DFT has many powerful properties, many of which are a result of the orthogonal basis function. If calculated using the fast Fourier transform (FFT), it is a powerful computational tool.

Properties of the DFT

The properties of the DFT are extremely important in applying it to signal analysis and to interpreting it. The main properties are given here using the notation that the DFT of a length- N N complex sequence x ( n ) x ( n ) is ℱ { x ( n ) } = C ( k ) ℱ { x ( n ) } = C ( k ) .

  1. Linear Operator: ℱ { x ( n ) + y ( n ) } = ℱ { x ( n ) } + ℱ { y ( n ) } ℱ { x ( n ) + y ( n ) } = ℱ { x ( n ) } + ℱ { y ( n ) }
  2. Unitary Operator: F − 1 = 1 N F T F − 1 = 1 N F T
  3. Periodic Spectrum: C ( k ) = C ( k + N ) C ( k ) = C ( k + N )
  4. Periodic Extensions of x ( n ) x ( n ) : x ( n ) = x ( n + N ) x ( n ) = x ( n + N )
  5. Properties of Even and Odd Parts: x ( n ) = u ( n ) + j v ( n ) x ( n ) = u ( n ) + j v ( n ) and C ( k ) = A ( k ) + j B ( k ) C ( k ) = A ( k ) + j B ( k ) u u v v A A B B | C | | C | θ θ even 0 even 0 even 0 odd 0 0 odd even π / 2 π / 2 0 even 0 even even π / 2 π / 2 0 odd odd 0 even 0
  6. Cyclic Convolution: ℱ { h ( n ) ∘ x ( n ) } = ℱ { h ( n ) } ℱ { x ( n ) } ℱ { h ( n ) ∘ x ( n ) } = ℱ { h ( n ) } ℱ { x ( n ) }
  7. Multiplication: ℱ { h ( n ) x ( n ) } = ℱ { h ( n ) } ∘ ℱ { x ( n ) } ℱ { h ( n ) x ( n ) } = ℱ { h ( n ) } ∘ ℱ { x ( n ) }
  8. Parseval: ∑ n = 0 N − 1 | x ( n ) | 2 = 1 N ∑ k = 0 N − 1 | C ( k ) | 2 ∑ n = 0 N − 1 | x ( n ) | 2 = 1 N ∑ k = 0 N − 1 | C ( k ) | 2
  9. Shift: ℱ { x ( n − M ) } = C ( k ) e − j 2 π M k / N ℱ { x ( n − M ) } = C ( k ) e − j 2 π M k / N
  10. Modulate: ℱ { x ( n ) e j 2 π K n / N } = C ( k − K ) ℱ { x ( n ) e j 2 π K n / N } = C ( k − K )
  11. Down Sample or Decimate: ℱ { x ( K n ) } = 1 K ∑ m = 0 K − 1 C ( k + L m ) ℱ { x ( K n ) } = 1 K ∑ m = 0 K − 1 C ( k + L m ) where N = L K N = L K
  12. Up Sample or Stretch: If x s ( 2 n ) = x ( n ) x s ( 2 n ) = x ( n ) for integer n n and zero otherwise,then ℱ { x s ( n ) } = C ( k ) ℱ { x s ( n ) } = C ( k ) , for k = 0 , 1 , 2 , … , 2 N − 1 k = 0 , 1 , 2 , … , 2 N − 1
  13. N Roots of Unity: ( W N k ) N = 1 ( W N k ) N = 1 for k = 0 , 1 , 2 , … , N − 1 k = 0 , 1 , 2 , … , N − 1
  14. Orthogonality:
    ∑ k = 0 N − 1 e − j 2 π m k / N e j 2 π n k / N = { N if  n = m 0 if  n ≠ m . ∑ k = 0 N − 1 e − j 2 π m k / N e j 2 π n k / N = { N if  n = m 0 if  n ≠ m . (1)
  15. Diagonalization of Convolution: If cyclic convolution is expressed as a matrix operation by y = H x y = H x with H H given by ((Reference)), the DFT operator diagonalizes the convolution operator H H , or F T H F = H d F T H F = H d where H d H d is a diagonal matrix with the N N values of the DFT of h ( n ) h ( n ) on the diagonal. This is a matrix statement of Property 6. Note the columns of F F are the N N eigenvectors of H H , independent of the values of h ( n ) h ( n ) .

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