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m02 - The Fourier Series

Module by: C. Sidney Burrus

Summary: Here is the definition of the Fourier Series and some properties and examples

The Fourier Series

The problem of expanding a finite length signal in a trigonometric series was posed and studied in the late 1700's by renown mathematicians such as Bernoulli, d'Alembert, Euler, Lagrange, and Gauss. Indeed, what we now call the Fourier series and the formulas for the coefficients were used by Euler in 1780. However, it was the presentation in 1807 and the paper in 1822 by Fourier stating that an arbitrary function could be represented by a series of sines and cosines that brought the problem to everyone's attention and started serious theoretical investigations and practical applications that continue to this day [1] [2] [3] [4] [5] . The theoretical work has been at the center of analysis and the practical applications have been of major significance in virtually every field of quantitative science and technology. For these reasons and others, the Fourier series is worth our serious attention in a study of signal processing.

Definition of the Fourier Series

We assume that the signal x ( t ) x ( t ) to be analyzed is well described by a real or complex valued function of a real variable t t defined over a finite interval { 0 ≤ t ≤ T } { 0 ≤ t ≤ T } . The trigonometric series expansion of x ( t ) x ( t ) is given by

x ( t ) = a ( 0 ) 2 + ∑ k = 1 ∞ a ( k ) cos ( 2 π T k t ) + b ( k ) sin ( 2 π T k t ) . x ( t ) = a ( 0 ) 2 + ∑ k = 1 ∞ a ( k ) cos ( 2 π T k t ) + b ( k ) sin ( 2 π T k t ) . (1)

where the sines and cosines are the basis functions for the expansion. The energy or power in an electrical, mechanical, etc. system is a function of the square of voltage, current, velocity, pressure, etc. For this reason, the natural setting for a representation of signals is the Hilbert space of L 2 [ 0 , T ] L 2 [ 0 , T ] . This modern formulation of the problem is developed in [6] [3] . The sinusoidal basis functions in the trigonometric expansion form a complete orthogonal set in L 2 [ 0 , T ] L 2 [ 0 , T ] . The orthogonality is easily seen from inner products

( cos ( 2 π T k t ) , cos ( 2 π T ℓ t ) ) = ∫ 0 T cos ( 2 π T k t ) cos ( 2 π T ℓ t ) ) ⅆ t = δ ( k − ℓ ) ( cos ( 2 π T k t ) , cos ( 2 π T ℓ t ) ) = ∫ 0 T cos ( 2 π T k t ) cos ( 2 π T ℓ t ) ) ⅆ t = δ ( k − ℓ ) (2)
and
( cos ( 2 π T k t ) , sin ( 2 π T ℓ t ) ) = ∫ 0 T cos ( 2 π T k t ) sin ( 2 π T ℓ t ) ) ⅆ t = 0 ( cos ( 2 π T k t ) , sin ( 2 π T ℓ t ) ) = ∫ 0 T cos ( 2 π T k t ) sin ( 2 π T ℓ t ) ) ⅆ t = 0 (3)
where the Kronecker delta function δ ( 0 ) = 1 δ ( 0 ) = 1 and δ ( k ≠ 0 ) = 0 δ ( k ≠ 0 ) = 0 . Because of this, the k k th coefficients in the series can be found by taking the inner product of x ( t ) x ( t ) with the k k th basis functions. This gives for the coefficients
a ( k ) = 2 T ∫ 0 T x ( t ) cos ( 2 π T k t ) ⅆ t a ( k ) = 2 T ∫ 0 T x ( t ) cos ( 2 π T k t ) ⅆ t (4)
and
b ( k ) = 2 T ∫ 0 T x ( t ) sin ( 2 π T k t ) ⅆ t b ( k ) = 2 T ∫ 0 T x ( t ) sin ( 2 π T k t ) ⅆ t (5)
where T T is the time interval of interest or the period of the periodic signal. Because of the orthogonality of the basis functions, a finite Fourier series formed by truncating the infinite series is an optimal least squared error approximation to x ( t ) x ( t ) . If the finite series is defined by
x ̂ ( t ) = a ( 0 ) 2 + ∑ k = 1 N a ( k ) cos ( 2 π T k t ) + b ( k ) sin ( 2 π T k t ) , x ̂ ( t ) = a ( 0 ) 2 + ∑ k = 1 N a ( k ) cos ( 2 π T k t ) + b ( k ) sin ( 2 π T k t ) , (6)
the squared error is
ɛ = 1 T ∫ 0 T | x ( t ) − x ̂ ( t ) | 2 ⅆ t ɛ = 1 T ∫ 0 T | x ( t ) − x ̂ ( t ) | 2 ⅆ t (7)
which is minimized over all a ( k ) a ( k ) and b ( k ) b ( k ) by (Equation 4) and (Equation 5). This is an extraordinarily important property.

It follows that if x ( t ) ∈ L 2 [ 0 , T ] x ( t ) ∈ L 2 [ 0 , T ] , then the series converges to x ( t ) x ( t ) in the sense that ɛ → 0 ɛ → 0 as N → ∞ N → ∞ [6] [3] . The question of point-wise convergence is more difficult. A sufficient condition that is adequate for most application states: If f ( x ) f ( x ) is bounded, is piece-wise continuous, and has no more than a finite number of maxima over an interval, the Fourier series converges point-wise to f ( x ) f ( x ) at all points of continuity and to the arithmetic mean at points of discontinuities. If f ( x ) f ( x ) is continuous, the series converges uniformly at all points [3] [1] [2] .

A useful condition [6] [3] states that if x ( t ) x ( t ) and its derivatives through the q q th derivative are defined and have bounded variation, the Fourier coefficients a ( k ) a ( k ) and b ( k ) b ( k ) asymptotically drop off at least as fast as 1 k q + 1 1 k q + 1 as k → ∞ k → ∞ . This ties global rates of convergence of the coefficients to local smoothness conditions of the function.

The form of the Fourier series using both sines and cosines makes determination of the peak value or of the location of a particular frequency term difficult. A form that explicitly gives the peak value of the sinusoid of that frequency and the location or phase shift of that sinusoid is given by

x ( t ) = d ( 0 ) 2 + ∑ k = 1 ∞ d ( k ) cos ( 2 π T k t + θ ( k ) ) x ( t ) = d ( 0 ) 2 + ∑ k = 1 ∞ d ( k ) cos ( 2 π T k t + θ ( k ) ) (8)
and, using Euler's relation and the usual electrical engineering notation of j = − 1 j = − 1 ,
e j x = cos ( x ) + j sin ( x ) , e j x = cos ( x ) + j sin ( x ) , (9)
the complex exponential form is obtained as
x ( t ) = ∑ k = − ∞ ∞ c ( k ) e j 2 π T k t x ( t ) = ∑ k = − ∞ ∞ c ( k ) e j 2 π T k t (10)
where
c ( k ) = a ( k ) + j b ( k ) . c ( k ) = a ( k ) + j b ( k ) . (11)
The coefficient equation is
c ( k ) = 1 T ∫ 0 T x ( t ) e − j 2 π T k t ⅆ t c ( k ) = 1 T ∫ 0 T x ( t ) e − j 2 π T k t ⅆ t (12)
The coefficients in these three forms are related by
| d | 2 = | c | 2 = a 2 + b 2 | d | 2 = | c | 2 = a 2 + b 2 (13)
and
θ = a r g { c } = tan − 1 ( b a ) θ = a r g { c } = tan − 1 ( b a ) (14)
It is easier to evaluate a signal in terms of c ( k ) c ( k ) or d ( k ) d ( k ) and θ ( k ) θ ( k ) than in terms of a ( k ) a ( k ) and b ( k ) b ( k ) . The first two are polar representation of a complex value and the last is rectangular. The exponential form is easier to work with mathematically.

Although the function to be expanded is defined only over a specific finite region, the series converges to a function that is defined over the real line and is periodic. It is equal to the original function over the region of definition and is a periodic extension outside of the region. Indeed, one could artificially extend the given function at the outset and then the expansion would converge everywhere.

A Geometric View

It can be very helpful to develop a geometric view of the Fourier series where x ( t ) x ( t ) is considered to be a vector and the basis functions are the coordinate or basis vectors. The coefficients become the projections of x ( t ) x ( t ) on the coordinates. The ideas of a measure of distance, size, and orthogonality are important and the definition of error is easy to picture. This is done in [6] [3] [7] using Hilbert space methods.

Examples

  • An example of the Fourier series is the expansion of a square wave signal with period 2 π 2 π . The expansion is
    x ( t ) = 4 π [ sin ( t ) + 1 3 sin ( 3 t ) + 1 5 sin ( 5 t ) ⋯ ] . x ( t ) = 4 π [ sin ( t ) + 1 3 sin ( 3 t ) + 1 5 sin ( 5 t ) ⋯ ] . (15)
    Because x ( t ) x ( t ) is odd, there are no cosine terms (all a ( k ) = 0 a ( k ) = 0 ) and, because of its symmetries, there are no even harmonics (even k k terms are zero). The function is well defined and bounded; its derivative is not, therefore, the coefficients drop off as 1 k 1 k .
  • A second example is a triangle wave of period 2 π 2 π . This is a continuous function where the square wave was not. The expansion of the triangle wave is
    x ( t ) = 4 π [ sin ( t ) + 1 3 2 sin ( 3 t ) + 1 5 2 sin ( 5 t ) + ⋯ ] . x ( t ) = 4 π [ sin ( t ) + 1 3 2 sin ( 3 t ) + 1 5 2 sin ( 5 t ) + ⋯ ] . (16)
    Here the coefficients drop off as 1 k 2 1 k 2 since the function and its first derivative exist and are bounded.

Note the derivative of a triangle wave is a square wave. Examine the series coefficients to see this. There are many books and web sites on the Fourier series that give insight through examples and demos.

References

  1. E. W. Hobson. (1926). The Theory of Functions of a Real Variable and the Theory of Fourier's Series. (Second, Vol. 2). New York: Dover.
  2. H. S. Carslaw. (1906, 1930). Theory of Fourier's Series and Integrals. (third). New York: Dover.
  3. C. Lanczos. (1956). Applied Analysis. Englewood Cliffs, NJ: Prentice Hall.
  4. T. W. Körner. (1988). Fourier Analysis. Cambridge: Cambridge University Press.
  5. Gerald B. Folland. (1992). Fourier Analysis and its Applications. Pacific Grove: Wadsworth & Brooks/Cole.
  6. H. Dym and H. P. McKean. (1972). Fourier Series and Integrals. New York: Academic Press.
  7. R. M. Young. (1980). An Introduction to Nonharmonic Fourier Series. New York: Academic Press.

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