[6.8 SUMMARY](#page-12-0)

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6.8 SUMMARY

In this chapter we showed how a periodic signal can be represented as a sum of sinusoids or exponentials. If the frequency of a periodic signal is f0, then it can be expressed as a weighted sum of a sinusoid of frequency f⁰ and its harmonics (the trigonometric Fourier series). We can reconstruct the periodic signal from a knowledge of the amplitudes and phases of these sinusoidal components (amplitude and phase spectra).

If a periodic signal x(t) has an even symmetry, its Fourier series contains only cosine terms (including dc). In contrast, if x(t) has an odd symmetry, its Fourier series contains only sine terms. If x(t) has neither type of symmetry, its Fourier series contains both sine and cosine terms.

At points of discontinuity, the Fourier series for x(t) converges to the mean of the values of x(t) on either side of the discontinuity. For signals with discontinuities, the Fourier series converges in the mean and exhibits Gibbs phenomenon at the points of discontinuity. The amplitude spectrum of the Fourier series for a periodic signal x(t) with jump discontinuities decays slowly (as 1/n) with frequency. We need a large number of terms in the Fourier series to approximate x(t) within

668 CHAPTER 6 CONTINUOUS-TIME SIGNAL ANALYSIS: THE FOURIER SERIES

a given error. In contrast, the amplitude spectrum of a smoother periodic signal decays faster with frequency and we require a smaller number of terms in the series to approximate x(t) within a given error.

A sinusoid can be expressed in terms of exponentials. Therefore, the Fourier series of a periodic signal can also be expressed as a sum of exponentials (the exponential Fourier series). The exponential form of the Fourier series and the expressions for the series coefficients are more compact than those of the trigonometric Fourier series. Also, the response of LTIC systems to an exponential input is much simpler than that for a sinusoidal input. Moreover, the exponential form of representation lends itself better to mathematical manipulations than does the trigonometric form. This includes the establishment of useful Fourier series properties that simplify work and help provide a more intuitive understanding of signals. For these reasons, the exponential form of the series is preferred in modern practice in the areas of signals and systems.

The plots of amplitudes and angles of various exponential components of the Fourier series as functions of the frequency are the exponential Fourier spectra (amplitude and angle spectra) of the signal. Because a sinusoid cosω0t can be represented as a sum of two exponentials, ej^ω0*^t* and e−jω0*^t* , the frequencies in the exponential spectra range from ω = −∞ to ∞. By definition, frequency of a signal is always a positive quantity. Presence of a spectral component of a negative frequency −nω⁰ merely indicates that the Fourier series contains terms of the form e−jnω0*^t* . The spectra of the trigonometric and exponential Fourier series are closely related, and one can be found by the inspection of the other.

In Sec. 6.5 we discuss a method of representing signals by the generalized Fourier series, of which the trigonometric and exponential Fourier series are special cases. Signals are vectors in every sense. Just as a vector can be represented as a sum of its components in a variety of ways, depending on the choice of the coordinate system, a signal can be represented as a sum of its components in a variety of ways, of which the trigonometric and exponential Fourier series are only two examples. Just as we have vector coordinate systems formed by mutually orthogonal vectors, we also have signal coordinate systems (basis signals) formed by mutually orthogonal signals. Any signal in this signal space can be represented as a sum of the basis signals. Each set of basis signals yields a particular Fourier series representation of the signal. The signal is equal to its Fourier series, not in the ordinary sense, but in the special sense that the energy of the difference between the signal and its Fourier series approaches zero. This allows for the signal to differ from its Fourier series at some isolated points.