A statement of the Fourier theorem is that any periodic signal gT(t) of period T, such as shown in Fig. C.1(a), can be expressed as the sum of sinusoids with frequencies at integer multiples (called harmonics) of the fundamental frequency, and with appropriate amplitudes and phases. Thus we have the Fourier series expression for gT(t): 1 X gT tÞ 1/4 Ao þ Ancos2nnfot þ ψnÞ n1/41 1 X Ao þ n1/41 1 Xancos2nnfotÞ þ n1/41 bn sin 2nnfotÞ 1 fo 1/4 C.1Þ T The coefficients in Eq. (C.1) are obtained as follows: Integrating both sides of Eq. (C.1) over an interval of one period Tyields the average value (or DC component) Ao of the periodic signal. First multiplying both sides by cos(2nmfot) before integrating over one period yields the mth cosine coefficient am; and multiplying first by sin(2nmfot) before integrating over one period gives the mth sine coefficient bm: Ao 1/4 an 1/4 T/2 1 ( TJ g tÞ dt -T/2 T/2 2 ( TJ g tÞcos2nnfotÞ dt -T/2 T/2 _ 2 bn T An 1/4 g tÞsin2nnfotÞ dt -T/2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2n þ b2 Amplitude of nth harmonic component n , ψn 1/4 -arctan n/anÞ, Phase of nth harmonic component C.2Þ Employing Euler's formula exp(j2nnfot) 1/4 cos(2nnfot) þjsin(2nnfot) in Eq. (C.1) leads to the exponential form of Fourier series: where 1 X g tÞ 1/4 n1/41 Cnej2nnfot T/2 11 Cn 1/4 2an - jbnÞ 1/4 T gtÞe_j2nnfotdt -T/2
