In an earlier module, we showed that a square wave could be expressed as a superposition of pulses. As useful as this decomposition was in this example, it does not generalize well to other periodic signals: How can a superposition of pulses equal a smooth signal like a sinusoid? Because of the importance of sinusoids to linear systems, you might wonder whether they could be added together to represent a large number of periodic signals. You would be right and in good company as well. Euler and Gauss in particular worried about this problem, and Jean Baptiste Fourier got the credit even though tough mathematical issues were not settled until later. They worked on what is now known as the Fourier series: representing any periodic signal as a superposition of sinusoids.

But the Fourier series goes well beyond being another signal decomposition method. Rather, the Fourier series begins our journey to appreciate how a signal can be described in either the time-domain or the frequency-domain with no compromise. Let s⁢t st be a periodic signal with period TT. We want to show that periodic signals, even those that have constant-valued segments like a square wave, can be expressed as sum of harmonically related sine waves: sinusoids having frequencies that are integer multiples of the fundamental frequency. Because the signal has period TT, the fundamental frequency is 1T 1T. The complex Fourier series expresses the signal as a superposition of complex exponentials having frequencies kT kT, k=…−101… k … 1 0 1 … .

with ck=12⁢(ak −i⁢bk ) ck 1 2 ak bk . The real and imaginary parts of the Fourier coefficients c k c k are written in this unusual way for convenience in defining the classic Fourier series. The zeroth coefficient equals the signal's average value and is real- valued for real-valued signals: c0=a0 c0 a0 . The family of functions ei⁢2⁢π⁢k⁢tT 2 k t T are called basis functions and form the foundation of the Fourier series. No matter what the periodic signal might be, these functions are always present and form the representation's building blocks. They depend on the signal period T T, and are indexed by k k.

To find the Fourier coefficients, we note the orthogonality property

Assuming for the moment that the complex Fourier series "works," we can find a signal's complex Fourier coefficients, its spectrum, by exploiting the orthogonality properties of harmonically related complex exponentials. Simply multiply each side of Equation 1 by e−(i⁢2⁢π⁢l⁢t) 2 l t and integrate over the interval 0 T 0 T .

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 complex Fourier series obeys Parseval's Theorem, one of the most important results in signal analysis. This general mathematical result says you can calculate a signal's power in either the time domain or the frequency domain.

Let's calculate the Fourier coefficients 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 =1T⁢∫0ΔA⁢e−i⁢2⁢π⁢k⁢tTd t =−(Ai⁢2⁢π⁢k⁢(e−i⁢2⁢π⁢k⁢ΔT−1)) c k 1 T t 0 Δ A 2 k t T A 2 k 2 k Δ T 1 At this point, simplifying this expression requires knowing an interesting property. 1−e−(i⁢θ)=e−i⁢θ2⁢(ei⁢θ2−e−i⁢θ2)=e−i⁢θ2⁢2⁢i⁢sinθ2 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 =−π⁢k⁢ΔT+π⁢neg⁢sinπ⁢k⁢ΔTπ⁢k⁢sign⁢k 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 11 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 2⁢π 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 −π (more negative). 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 −(2⁢π) 2 . Because we can add 2⁢π 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 −π π by adding some (possibly negative) multiple of 2⁢π 2 to each phase value.

Content actions

Footer

This work is licensed by Don Johnson under a Creative Commons Attribution License (CC-BY 3.0), and is an Open Educational Resource.

Last edited by Don Johnson on May 16, 2013 10:58 am -0500.