4.1 Complex fourier series  (Page 2/2)

A signal's Fourier series spectrum $c_k$ has interesting properties.

If $s(t)$ is real, $c_k=\overline{c_{-k}}$ (real-valued periodic signals have conjugate-symmetric spectra).

This result follows from the integral that calculates the $c_k$ from the signal. Furthermore, this result means that $\Re (c_k)=\Re (c_{-k})$ : The real part of the Fourier coefficients for real-valuedsignals is even. Similarly, $\Im (c_k)=-\Im (c_{-k})$ : The imaginary parts of the Fourier coefficients have oddsymmetry. Consequently, if you are given the Fourier coefficients for positive indices and zero and are told thesignal is real-valued, you can find the negative-indexed coefficients, hence the entire spectrum. This kind of symmetry, $c_k=\overline{c_{-k}}$ , is known as conjugate symmetry .

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-valuedFourier 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 a periodic signal delayed by $\tau$ , $s(t-\tau )$ , are $c_{k}e^{-\left(\frac{i\times 2\pi k\tau }{T}\right)}$ , where $c_k$ denotes the spectrum of $s(t)$ . Delaying a signal by $\tau$ seconds results in a spectrum having a linear phase shift of $-\left(\frac{2\pi k\tau }{T}\right)$ in comparison to the spectrum of the undelayed signal. Notethat the spectral magnitude is unaffected. Showing this property is easy.

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 frequencydomain.

Parseval's theorem

Let's calculate the Fourier coefficients of the periodic pulse signalshown here .

Periodic pulse sequence

Also note the presence of a linear phase term (the first term in $\mathop{\arg }(c_k)$ is proportional to frequency $\frac{k}{T}$ ). Comparing this term with that predicted from delaying a signal,a delay of $\frac{\Delta }{2}$ is present in our signal. Advancing the signal by this amountcenters the pulse about the origin, leaving an even signal, which in turn means that its spectrum is real-valued. Thus, ourcalculated spectrum is consistent with the properties of the Fourier spectrum.

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