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m06 - Properties of the Fourier Transform

Module by: C. Sidney Burrus

Summary: The Fourier Transform has properties similar to those of the Fourier series.

Properties of the Fourier Transform

The properties of the Fourier transform are important in applying it to signal analysis and to interpreting it. The main properties are given here using the notation that the FT of a real valued function x ( t ) x ( t ) over all time t t is given by ℱ { x } = X ( ω ) ℱ { x } = X ( ω ) .

  1. Linear: ℱ { x + y } = ℱ { x } + ℱ { y } ℱ { x + y } = ℱ { x } + ℱ { y }
  2. Even and Oddness: if x ( t ) = u ( t ) + j v ( t ) x ( t ) = u ( t ) + j v ( t ) and X ( ω ) = A ( ω ) + j B ( ω ) X ( ω ) = A ( ω ) + j B ( ω ) then 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
  3. Convolution: If continuous convolution is defined by: y ( t ) = h ( t ) * x ( t ) = ∫ − ∞ ∞ h ( t − τ ) x ( τ ) ⅆ τ = ∫ − ∞ ∞ h ( λ ) x ( t − λ ) ⅆ λ y ( t ) = h ( t ) * x ( t ) = ∫ − ∞ ∞ h ( t − τ ) x ( τ ) ⅆ τ = ∫ − ∞ ∞ h ( λ ) x ( t − λ ) ⅆ λ then ℱ { h ( t ) * x ( t ) } = ℱ { h ( t ) } ℱ { x ( t ) } ℱ { h ( t ) * x ( t ) } = ℱ { h ( t ) } ℱ { x ( t ) }
  4. Multiplication: ℱ { h ( t ) x ( t ) } = 1 2 π ℱ { h ( t ) } * ℱ { x ( t ) } ℱ { h ( t ) x ( t ) } = 1 2 π ℱ { h ( t ) } * ℱ { x ( t ) }
  5. Parseval: ∫ − ∞ ∞ | x ( t ) | 2 ⅆ t = 1 2 π ∫ − ∞ ∞ | X ( ω ) | 2 ⅆ ω ∫ − ∞ ∞ | x ( t ) | 2 ⅆ t = 1 2 π ∫ − ∞ ∞ | X ( ω ) | 2 ⅆ ω
  6. Shift: ℱ { x ( t − T ) } = X ( ω ) e − j ω T ℱ { x ( t − T ) } = X ( ω ) e − j ω T
  7. Modulate: ℱ { x ( t ) e j 2 π K t } = X ( ω − 2 π K ) ℱ { x ( t ) e j 2 π K t } = X ( ω − 2 π K )
  8. Derivative: ℱ { ⅆ x ⅆ t } = j ω X ( ω ) ℱ { ⅆ x ⅆ t } = j ω X ( ω )
  9. Stretch: ℱ { x ( a t ) } = 1 | a | X ( ω / a ) ℱ { x ( a t ) } = 1 | a | X ( ω / a )
  10. Orthogonality: ∫ − ∞ ∞ e − j ω 1 t e j ω 2 t = 2 π δ ( ω 1 − ω 2 ) ∫ − ∞ ∞ e − j ω 1 t e j ω 2 t = 2 π δ ( ω 1 − ω 2 )

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