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m03 - Properties of the Fourier Series

Module by: C. Sidney Burrus

Summary: The Fourier series is linear, supports convolution, and describes modulation easily. Many of its properties are a result of orthogonal basis functions.

Properties of the Fourier Series

The properties of the Fourier series are important in applying it to signal analysis and to interpreting it. The main properties are given here using the notation that the Fourier series of a real valued function x ( t ) x ( t ) over { 0 ≤ t ≤ T } { 0 ≤ t ≤ T } is given by ℱ { x ( t ) } = c ( k ) ℱ { x ( t ) } = c ( k ) and x ˜ ( t ) x ˜ ( t ) denotes the periodic extensions of x ( t ) x ( t ) .

  1. Linear: ℱ { x + y } = ℱ { x } + ℱ { y } ℱ { x + y } = ℱ { x } + ℱ { y } Idea of superposition. Also scalability: ℱ { a x } = a ℱ { x } ℱ { a x } = a ℱ { x }
  2. Extensions of x ( t ) x ( t ) : x ˜ ( t ) = x ˜ ( t + T ) x ˜ ( t ) = x ˜ ( t + T ) x ˜ ( t ) x ˜ ( t ) is periodic.
  3. Even and Odd Parts: x ( t ) = u ( t ) + j v ( t ) x ( t ) = u ( t ) + j v ( t ) and C ( k ) = A ( k ) + j B ( k ) = | C ( k ) | e j θ ( k ) C ( k ) = A ( k ) + j B ( k ) = | C ( k ) | e j θ ( k ) u u v v A A B B | C | | C | 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 continuous cyclic convolution is defined by y ( t ) = h ( t ) ∘ x ( t ) = ∫ 0 T h ˜ ( t − τ ) x ˜ ( τ ) ⅆ τ y ( t ) = h ( t ) ∘ x ( t ) = ∫ 0 T h ˜ ( t − τ ) x ˜ ( τ ) ⅆ τ then ℱ { h ( t ) ∘ x ( t ) } = ℱ { h ( t ) } ℱ { x ( t ) } ℱ { h ( t ) ∘ x ( t ) } = ℱ { h ( t ) } ℱ { x ( t ) }
  5. Multiplication: If discrete convolution is defined by e ( n ) = d ( n ) * c ( n ) = ∑ m = − ∞ ∞ d ( m ) c ( n − m ) e ( n ) = d ( n ) * c ( n ) = ∑ m = − ∞ ∞ d ( m ) c ( n − m ) then ℱ { h ( t ) x ( t ) } = ℱ { h ( t ) } * ℱ { x ( t ) } ℱ { h ( t ) x ( t ) } = ℱ { h ( t ) } * ℱ { x ( t ) } This property is the inverse of property 4 and vice versa.
  6. Parseval: 1 T ∫ 0 T | x ( t ) | 2 ⅆ t = ∑ k = − ∞ ∞ | C ( k ) | 2 1 T ∫ 0 T | x ( t ) | 2 ⅆ t = ∑ k = − ∞ ∞ | C ( k ) | 2 This property says the energy calculated in the time domain is the same as that calculated in the frequency (or Fourier) domain.
  7. Shift: ℱ { x ˜ ( t − t 0 ) } = C ( k ) e − j 2 π t 0 k / T ℱ { x ˜ ( t − t 0 ) } = C ( k ) e − j 2 π t 0 k / T A shift in the time domain results in a linear phase shift in the frequency domain.
  8. Modulate: ℱ { x ( t ) e j 2 π K t / T } = C ( k − K ) ℱ { x ( t ) e j 2 π K t / T } = C ( k − K ) Modulation in the time domain results in a shift in the frequency domain. This property is the inverse of property 7.
  9. Orthogonality of basis functions:
    ∫ 0 T e − j 2 π m t / T e j 2 π n t / T ⅆ t = T δ ( n − m ) = { T if  n = m 0 if  n ≠ m . ∫ 0 T e − j 2 π m t / T e j 2 π n t / T ⅆ t = T δ ( n − m ) = { T if  n = m 0 if  n ≠ m . (1)
    Orthogonality allows the calculation of coefficients using inner products in ((Reference)) and ((Reference)). It also allows Parseval's Theorem in property 6. A relaxed version of orthogonality is called ``tight frames" and is important in over-specified systems, especially in wavelets.

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