We can greatly simplify the expressions for a signal's Fourier series by using Euler's relations. In this way, when we combine cosine and sine terms at the same frequency By defining ∀k,k≠0: c k =12 a k -ⅈ b k k k 0 c k 1 2 a k b k and c 0 = a 0 c 0 a 0 , we can rewrite the Fourier series of a square wave as with ∀k,k≠0: c - k = c k ¯=12 a k +ⅈ b k k k 0 c - k c k 1 2 a k b k . In this way, we have the complex Fourier series. The positive index terms correspond to the first portion of the common-frequency term and the negative indexed terms to the second. Thus, the complex Fourier series and the sine-cosine series are identical, each representing a signal's spectrum in slightly different ways. Manipulations and calculations are more streamlined with complex Fourier series, and it is our representation of choice.

To aid in finding Fourier coefficients, we note the orthogonality property We can find a signal's complex Fourier spectrum with The complex Fourier series for the square wave is

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

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.

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.

Therefore, the Fourier coefficients are purely imaginary. The square wave is a great example of an odd-symmetric signal.

The Fourier series obeys Parseval's Theorem, one of the most important results in signal analysis. This general mathematical result formalizes what we have already seen: you can calculate a signal's power in either the time domain or the frequency domain.

Let's calculate the spectrum of the periodic pulse signal shown here. The pulse width is Δ Δ, the period T T, and the amplitude A A. The complex Fourier spectrum of this signal is given by c k =AT∫0Δⅇ-ⅈ2πkΔTdt=-Aⅈ2πkⅇ-ⅈ2πkΔT-1 c k A T t 0 Δ 2 k Δ T A 2 k 2 k Δ T 1 At this point, simplifying this expression requires knowing an interesting property. 1-ⅇ-ⅈθ=ⅇ-ⅈθ2ⅇ+ⅈθ2-ⅇ-ⅈθ2=ⅇ-ⅈθ22ⅈsinθ2 1 θ θ 2 θ 2 θ 2 θ 2 2 θ 2 Armed with this result, we can simply express the Fourier series coefficients for our pulse sequence. Because this signal is real-valued, we find that the coefficients do indeed have conjugate symmetry: c k = c − k ¯ c k c − k . The periodic pulse signal has neither even nor odd symmetry; consequently, no additional symmetry exists in the spectrum. Because the spectrum is complex valued, to plot it we need to calculate its magnitude and phase. ∠ c k =-πkΔT+πnegsinπkΔTπksignk c k k Δ T neg k Δ T k sign k The function neg· neg · equals -1 if its argument is negative and zero otherwise.

The somewhat complicated expression for the phase results because the sine term can be negative; magnitudes must be positive, leaving the occasional negative values to be accounted for as a phase shift of π .

Also note the presence of a linear phase term (the first term in ∠ c k c k is proportional to frequency kT k T ). Comparing this term with that predicted from delaying a signal, a delay of Δ2 Δ 2 is present in our signal. Advancing the signal by this amount centers the pulse about the origin, leaving an even signal, which in turn means that its spectrum is real-valued. Thus, our calculated spectrum is consistent with the properties of the Fourier spectrum.

The phase plot shown in Figure 2 requires some explanation as it does not seem to agree with what Equation 10 suggests. There, the phase has a linear component, with a jump of π every time the sinusoidal term changes sign. We must realize that any integer multiple of 2π 2 can be added to a phase at each frequency without affecting the value of the complex spectrum. We see that at frequency index 4 the phase is nearly -π . The phase at index 5 is undefined because the magnitude is zero in this example. At index 6, the formula suggests that the phase of the linear term should be less than (more negative) than -π . In addition, we expect a shift of -π in the phase between indices 4 and 6. Thus, the phase value predicted by the formula is a little less than -2π 2 . Because we can add 2π 2 without affecting the value of the spectrum at index 6, the result is a slightly negative number as shown. Thus, the formula and the plot do agree. In phase calculations like those made in MATLAB, values are usually confined to the range -ππ by adding some (possibly negative) multiple of 2π 2 to each phase value.