Complex Fourier Series and Their Properties 2.6 2000/08/11 2004/08/04 15:43:56.459 GMT-5 Don Johnson dhj@rice.edu Don Johnson dhj@rice.edu Doug Daniels rainking@alumni.rice.edu Fourier orthogonality coefficients series power spectrum Complex spectral square wave conjugate symmetry Parseval Parseval's 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. To aid in finding Fourier coefficients, we note the orthogonality property t 0 T 2 k t T 2 l t T T k l 0 k l We can find a signal's complex Fourier spectrum with c k 1 T t 0 T s t 2 k t T The complex Fourier series for the square wave is sq t k k … -3 -1 1 3 … 2 k 2 k t T What is the complex Fourier series for a sinusoid? Because of Euler's relation, 2 f t 1 2 2 f t 1 2 2 f t Thus, c 1 1 2 , c − 1 1 2 , and the other coefficients are zero. A signal's Fourier series spectrum c k has interesting properties. If s t is real, c − k c k This result follows from the integral that calculates the c k from the signal. Furthermore, this result means that c k c − k : The real part of the Fourier coefficients for real-valued signals is even. Similarly, 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 , is known as conjugate symmetry. We can phrase the property concisely by saying: Real-valued periodic signals have a conjugate-symmetric spectrum. If s t s t , which says the signal has even symmetry about the origin, c − k c k . 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. If s t s t , which says the signal has odd symmetry, c − k c k . Therefore, the Fourier coefficients are purely imaginary. The square wave is a great example of an odd-symmetric signal. The spectral coefficients for the periodic signal s t τ are c k 2 k t T , where c k denotes the spectrum of s t . Thus, delaying a signal by τ seconds results in a spectrum having a linear phase shift of 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. 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 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 , and we have our result. The Fourier series obeys: Parseval's Theorem Power calculated in the time domain equals the power calculated in the frequency domain. 1 T t 0 T s t 2 k c k 2 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.