![]() ![]() Without affecting the value of the spectrum at index 6, the result is a slightly negative number as shown. Thus, the phase value predicted by the formula is a little less than At index 6, the formula suggests that the phase of the linear term should be less than The phase at index 5 is undefined because the magnitude is zero in this example. We see that at frequency index 4 the phase is nearly We must realize that any integer multiple ofĬan be added to a phase at each frequency without affecting the value of the complex spectrum. There, the phase has a linear component, with a jump ofĮvery time the sinusoidal term changes sign. ![]() The phase plot shown in Figure 2 requires some explanation as it does not seem to agree with what Equation suggests. The complex Fourier series expresses the signal as a superposition of complex exponentials having frequencies 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. 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. They worked on what is now known as the Fourier series: representing any periodic signal as a superposition of sinusoids.īut the Fourier series goes well beyond being another signal decomposition method. 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. You would be right and in good company as well. 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. ![]() In an earlier module, we showed that a square wave could be expressed as a superposition of pulses. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |