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m13 - Properties of the DTFT

Module by: C. Sidney Burrus

Summary: The properties of the DTFT are very similar to those of the DFT and other Fourier transforms.

Properties

The properties of the DTFT are similar to those for the DFT and are important in the analysis and interpretation of long signals. The main properties are given here using the notation that the DTFT of a complex sequence x ( n ) x ( n ) is ℱ { x ( n ) } = X ( ω ) ℱ { x ( n ) } = X ( ω ) .

  1. Linear Operator: ℱ { x + y } = ℱ { x } + ℱ { y } ℱ { x + y } = ℱ { x } + ℱ { y }
  2. Periodic Spectrum: X ( ω ) = X ( ω + 2 π ) X ( ω ) = X ( ω + 2 π )
  3. Properties of Even and Odd Parts: x ( n ) = u ( n ) + j v ( n ) x ( n ) = u ( n ) + j v ( n ) and X ( ω ) = A ( ω ) + j B ( ω ) X ( ω ) = A ( ω ) + j B ( ω ) u u v v A A B B | X | | X | θ θ even 0 even 0 even 0 odd 0 0 odd even 0 0 even 0 even even π / 2 π / 2 0 odd odd 0 even π / 2 π / 2
  4. Convolution: If non-cyclic or linear convolution is defined by: y ( n ) = h ( n ) * x ( n ) = ∑ m = − ∞ ∞ h ( n − m ) x ( m ) = ∑ k = − ∞ ∞ h ( k ) x ( n − k ) y ( n ) = h ( n ) * x ( n ) = ∑ m = − ∞ ∞ h ( n − m ) x ( m ) = ∑ k = − ∞ ∞ h ( k ) x ( n − k ) then ℱ { h ( n ) * x ( n ) } = ℱ { h ( n ) } ℱ { x ( n ) } ℱ { h ( n ) * x ( n ) } = ℱ { h ( n ) } ℱ { x ( n ) }
  5. Multiplication: If cyclic convolution is defined by: Y ( ω ) = H ( ω ) ∘ X ( ω ) = ∫ 0 T H ˜ ( ω − Ω ) X ˜ ( Ω ) ⅆ Ω Y ( ω ) = H ( ω ) ∘ X ( ω ) = ∫ 0 T H ˜ ( ω − Ω ) X ˜ ( Ω ) ⅆ Ω ℱ { h ( n ) x ( n ) } = 1 2 π ℱ { h ( n ) } ∘ ℱ { x ( n ) } ℱ { h ( n ) x ( n ) } = 1 2 π ℱ { h ( n ) } ∘ ℱ { x ( n ) }
  6. Parseval: ∑ n = − ∞ ∞ | x ( n ) | 2 = 1 2 π ∫ − π π | X ( ω ) | 2 ⅆ ω ∑ n = − ∞ ∞ | x ( n ) | 2 = 1 2 π ∫ − π π | X ( ω ) | 2 ⅆ ω
  7. Shift: ℱ { x ( n − M ) } = X ( ω ) e − j ω M ℱ { x ( n − M ) } = X ( ω ) e − j ω M
  8. Modulate: ℱ { x ( n ) e j ω 0 n } = X ( ω − ω 0 ) ℱ { x ( n ) e j ω 0 n } = X ( ω − ω 0 )
  9. Sample: ℱ { x ( K n ) } = 1 K ∑ m = 0 K − 1 X ( ω + L m ) ℱ { x ( K n ) } = 1 K ∑ m = 0 K − 1 X ( ω + L m ) where N = L K N = L K
  10. Stretch: ℱ { x s ( n ) } = X ( ω ) ℱ { x s ( n ) } = X ( ω ) , for − K π ≤ ω ≤ K π − K π ≤ ω ≤ K π where x s ( K n ) = x ( n ) x s ( K n ) = x ( n ) for integer n n and zero otherwise.
  11. Orthogonality: ∑ n = − ∞ ∞ e − j ω 1 n e − j ω 2 n = 2 π δ ( ω 1 − ω 2 ) ∑ n = − ∞ ∞ e − j ω 1 n e − j ω 2 n = 2 π δ ( ω 1 − ω 2 )

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