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Complex Fourier Series and Their Properties

Module by: Don Johnson. E-mail the author

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Summary: This module shows how to find a signal's complex Fourier spectrum. It also lists several properties for that spectrum, including that it obeys Parseval's theorem.

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To aid in finding Fourier coefficients, we note the orthogonality property

∫0Tⅇⅈ2πktTⅇ-ⅈ2πltTdt=Tifk=l0ifk≠l t 0 T 2 k t T 2 l t T T k l 0 k l (1)
We can find a signal's complex Fourier spectrum with
c k =1T∫0Tstⅇ-ⅈ2πktTdt c k 1 T t 0 T s t 2 k t T (2)
The complex Fourier series for the square wave is
sqt=∑k∈…-3-113…2ⅈπkⅇ+ⅈ2πktT sq t k k … -3 -1 1 3 … 2 k 2 k t T (3)

Exercise 1

What is the complex Fourier series for a sinusoid?

Solution

Because of Euler's relation,

sin2πft=12ⅈⅇ+ⅈ2πft-12ⅈⅇ-ⅈ2πft 2 f t 1 2 2 f t 1 2 2 f t (4)
Thus, c 1 =12ⅈ c 1 1 2 , c − 1 =12ⅈ c − 1 1 2 , and the other coefficients are zero.

A signal's Fourier series spectrum c k c k has interesting properties.

property 1

If st s t is real, c − k = c k ¯ c − k c k

Proof

This result follows from the integral that calculates the c k c k from the signal.

Furthermore, this result means that ℜ c k =ℜ c − k c k c − k : The real part of the Fourier coefficients for real-valued signals is even. Similarly, ℑ c k =-ℑ c − k c k c − k : The imaginary parts of the Fourier coefficients have odd symmetry. Consequently, if you are given the Fourier coefficients for positive indices and zero and are told the signal is real-valued, you can find the negative-indexed coefficients, hence the entire spectrum. This kind of symmetry, c − k = c k ¯ c − k c k , is known as conjugate symmetry. We can phrase the property concisely by saying:

property 2

Real-valued periodic signals have a conjugate-symmetric spectrum.

property 3

If s-t=st s t s t , which says the signal has even symmetry about the origin, c − k = c k c − k c k .

Example

Given the previous property for real-valued signals, the Fourier coefficients of even signals are real-valued. A real-valued Fourier expansion amounts to an expansion in terms of only cosines, which is the simplest example of an even signal.

property 4

If s-t=-st s t s t , which says the signal has odd symmetry, c − k =- c k c − k c k . Therefore, the Fourier coefficients are purely imaginary.

Example

The square wave is a great example of an odd-symmetric signal.

property 5

The spectral coefficients for the periodic signal st−τ s t τ are c k ⅇ-ⅈ2πktT c k 2 k t T , where c k c k denotes the spectrum of st s t . Thus, delaying a signal by τ τ seconds results in a spectrum having a linear phase shift of -2πktT 2 k t T in comparison to the spectrum of the undelayed signal. Note that the spectral magnitude is unaffected. Showing this property is easy.

Proof

1T∫0Tst−τⅇ-ⅈ2πktTdt=1T∫-τT−τstⅇ-ⅈ2πkt+τTdt=1Tⅇ-ⅈ2πktT∫-τT−τstⅇ-ⅈ2πktTdt 1 T t 0 T s t τ 2 k t T 1 T t τ T τ s t 2 k t τ T 1 T 2 k t T t τ T τ s t 2 k t T (5)
At this point, the range of integration extends over a period of the integrand. Consequently, it should not matter how we integrate over a period, which means that ∫ − τ T − τ = ∫ 0 T ∫ − τ T − τ= ∫ 0 T , and we have our result.

The Fourier series obeys:

Theorem 1: Parseval's Theorem

Power calculated in the time domain equals the power calculated in the frequency domain.

1T∫0Ts2tdt=∑k=-∞∞| c k |2 1 T t 0 T s t 2 k c k 2 (6)
This result is a (simpler) re-expression of how to calculate a signal's power than with the real-valued Fourier series expression for power.

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