Periodic Waveforms, Fourier Series

A periodic waveform repeats itself exactly at regular time intervals (the period T). It is predictive at any moment in the future we can foresee the exact value. According to Fourier, any periodic waveform can be considered to be the sum of a. fundamental sine wave of frequency fi = 1/T, and sine waves at certain discrete frequencies, the harmonics (2fi, 3fi, 4fi, and so on). A periodic waveform is an idealized concept the waveform is to have lasted and to last forever. At the time we start and stop it, other frequency components than the harmonics appear as transients. 

The sine wave is a very special periodic waveform in the sense fliat it is the only waveform containing just one frequency the fundamental frequency. Why does just the sine wave have such special qualities  

If the sine wave is symmetrical around 0, it has no DC component and is described by the 

The time derivative as well as the time integral of a sine wave is also a sine wave of the same frequency, but phase-shifted 90°. The relationship between a sine wave and the circle is seen more directly in the complex notation of a radius r rotating around the origin in the Wessel diagram  

Because the time derivative of an exponential is the same exponential, then 0(0 )/ dt = jw e . That is why integration and derivation in the equations describing the behavior of electrical circuits can be replaced by algebraic operations with the jw instead of the 0/0t. This is under the assumption that all signals are sine waves of the same frequency. 

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FIGURE 4.2 Nonsinusoidal voltage waveform Fourier series. The Fourier series allows expression of nonsinusoidal periodic waveforms in terms of sinusoidal harmonic frequency components. 

The motivation for the sine-wave representation is that the waveform, when perfectly periodic, can be represented by a Fourier series decomposition in which each harmonic component of this decomposition corresponds to a single sine wave. More generally, the sine waves in the model will be aharmonic as when periodicity is not exact or turbulence and transients are present, and is given by... 

The Fourier expression is an infinite series. In this equation, V, represents the constant or the DC component of the waveform. Vu V2, V3,..., Vn are the peak values of the successive terms of the expression. The terms are known as the harmonics of the periodic waveform. The fundamental (or first harmonic) frequency has a... 

We saw in Section 4.2 that the plucked string supports certain spatial vibrations, called modes. These modes have a very special relationship in the case of the plucked string (and some other limited systems) in that their frequencies are all integer multiples of one basic sinusoid, called thefundamental. This special series of sinusoids is called a harmonic series, and lies at the basis of the Fourier series representation of shapes, waveforms, oscillations, etc. The Fourier series solves many types of problems, including physical problems with boundary constraints, but is also applicable to any shape or function. Any periodic waveform (repeating over and over again), can be transformed into a Fourier series, written as ... 

Any periodic waveform can be made from a sum of suitably scaled and phase shifted sinusoids. This is the Fourier series. 

According to the Fourier series Eq. 8.25, any periodic waveform is the sum of a fundamental sinusoid and a series of its harmonics. Notice that, in general, each harmonic component consists of a sine and cosine component. Of course, either of them may be zero for a given waveform in the time domain. Such a waveform synthesis (summation) is done in the time domain, but each wave is a component in the frequency domain. The frequency spectrum of a periodic function of time f(t) is therefore a line spectrum. The amplitudes of each discrete harmonic frequency component is ... 

Fourier Analysis The Fourier Series for Continuous-Time Periodic Functions The Fourier Transform for Continuous-Time Aperiodic Functions The Fourier Series for Discrete-Time Periodic Functions The Fourier Transform for Discrete-Time Aperiodic Functions Example Apphcations of Fourier Waveform Techniques... 

When the input to the hard limiter is the FM signal, a t) cos(pit)), the output of the hard limiter is the square waveform shown in Fig. 20.53(b). This periodic square waveform has a Fourier series expression given by... 

Most if not all periodic and aperiodic continuous-time waveforms x t), of practical interest, have Fourier series or Fourier transform counterparts. However, to aid in the existence question, the following sufficient technical constraints on the function x t), known generally as the Dirichlet conditions, guarantee convergence of the Fourier technique ... 

The techniques of Fourier analysis are perhaps best appreciated initially by an example. Consider the ideal continuous-time square wave of period T s, having a maximum value of A and a minimum value of 0 as shown at the top of Fig. 20.68. Because of the periodic nature of the waveform, the technique of Fourier series (FS) analysis applies, and the resulting decomposition into an infinite sum of simple trigonometric... 

The discrete-time Fourier series (DTPS) for discrete-time waveforms x n) of period N can also be given in three forms however, the complex exponential form is by far the most common. 

According to Fourier theory, any complicated periodic function can be approximated by this series, by putting the proper values of h, Fh, and ah in each term. Think of the cosine terms as basic wave forms that can be used to build any other waveform. Also according to Fourier theory, we can use the sine function or, for that matter, any periodic function in the same way as the basic wave for building any other periodic function. 

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