PERIODIC SIGNAL REPRESENTATION BY TRIGONOMETRIC FOURIER SERIES

6.1 PERIODIC SIGNAL REPRESENTATION BY TRIGONOMETRIC FOURIER SERIES

As seen in Section 1.3-3 [ Eq. (1.17) ], a periodic signal x(t) with period T 0 ( Fig. 6.1) has the property

Figure 6.1: A periodic signal of period T 0 .

The smallest value of T 0 that satisfies the periodicity condition (6.1) is the fundamental period of x(t). As argued in Section 1.3-3 , this equation implies that x(t) starts at −∞ and continues to ∞. Moreover, the area under a periodic signal x(t) over any interval of duration T 0 is the same; that is, for any real numbers a and b

This result follows from the fact that a periodic signal takes the same values at the intervals of T 0 . Hence, the values over any segment of duration T 0 are repeated in any other interval of the same duration. For convenience, the area under x(t) over any interval of duration T 0 will be denoted by

The frequency of a sinusoid cos 2 πf 0 t or sin 2 πf 0 t is f 0 and the period is T 0 = 1/f 0 . These sinusoids can also be expressed as cos ω 0 t or sin ω 0 t where ω 0 =2 πf 0 is the radian frequency, although for brevity, it is often referred to as frequency (see Section B.2 ). A

sinusoid of frequency nf 0 is said to be the nth harmonic of the sinusoid of frequency f 0 . Let us consider a signal x(t) made up of a sines and cosines of frequency ω 0 and all of its harmonics (including the zeroth harmonic;

i.e., dc) with arbitrary amplitudes: [ † ]

The frequency ω 0 is called the fundamental frequency. We now prove an extremely important property: x(t) in Eq. (6.3) is a periodic signal with the same period as that of the fundamental,

regardless of the values of the amplitudes a n and b n . Note that the period T 0 of the fundamental is regardless of the values of the amplitudes a n and b n . Note that the period T 0 of the fundamental is

To prove the periodicity of x(t), all we need is to show that x(t) = x(t + T 0 ). From Eq. (6.3)

From Eq. (6.5) , we have n ω 0 T 0 =2 π n, and

We could also infer this result intuitively. In one fundamental period T 0 , the nth harmonic executes n complete cycles. Hence, every sinusoid on the right-hand side of Eq. (6.3) executes a complete number of cycles in one fundamental period T 0 . Therefore, at t = T 0 , every sinusoid starts as if it were the origin and repeats the same drama over the next T 0 seconds, and so on ad infinitum. Hence, the

sum of such harmonics results in a periodic signal of period T 0 .

This result shows that any combination of sinusoids of frequencies 0, f 0 , 2f 0 ,..., kf 0 is a periodic signal of period T 0 = 1/f 0 regardless of the values of amplitudes a k and b k of these sinusoids. By changing the values of a k and b k in Eq. (6.3) , we can construct a variety of

periodic signals, all of the same period T 0 (T 0 = 1/f 0 or 2 π/ω 0 ).

The converse of this result is also true. We shall show in Section 6.5-4 that a periodic signal x(t) with a period T 0 can be expressed as

a sum of a sinusoid of frequency f (f = 1/T ) and all its harmonics, as shown in Eq. (6.3) . [ † ] 0 0 0 The infinite series on the right-hand side of Eq. (6.3) is known as the trigonometric Fourier series of a periodic signal x(t).