2. 2. 4 Frequency representation of periodic signals O The plots of the amplitude An and the phase Pn versus signals angular frequency wo are called amplitude spectrum plot and phase spectrum plot respectively UThe frequency spectrum is displayed graphically by a number of discrete vertical lines representing the amplitude An and the phase n of the analyzed signal respectively The frequency spectrum of a periodic signal is a discrete one
❑The plots of the amplitude An and the phase φn versus signal’s angular frequency ω0 are called amplitude spectrum plot and phase spectrum plot respectively. ❑The frequency spectrum is displayed graphically by a number of discrete vertical lines representing the amplitude An and the phase φn of the analyzed signal respectively. ❖The frequency spectrum of a periodic signal is a discrete one. 2.2.4 Frequency representation of periodic signals
2. 2. 4 Frequency representation of periodic signals Example 1 Find the Fourier series of the periodic square wave signal x(t)shown in Fig. 2. 11 (t) Fig. 2. 11 Periodic square wave signal
Example 1. Find the Fourier series of the periodic square wave signal x(t) shown in Fig. 2.11. 2.2.4 Frequency representation of periodic signals Fig. 2.11 Periodic square wave signal
2. 2. 4 Frequency representation of periodic signals Solution: Within one period, signal x(t)can be expressed as <t<0 x(t) 0<t< According to Eqs.(2.13)and(2.14) x(tcosnootdt=0 T/2 x(tsin noo tdt (1)sin n@o tdt+ sin noo tdt 2|1 T/2 cos n@ot T/2 (cos nooD no 2 cOS nT n=2.4.6
Solution: Within one period, signal x(t) can be expressed as According to Eqs. (2.13) and (2.14) 2.2.4 Frequency representation of periodic signals − − = 2 1, 0 0 2 1, ( ) T t t T x t − = = / 2 / 2 ( ) cos 0 0 2 T T n x t n tdt T a = = = = − = + − = − + = − − − 0, 2,4,6 , 1,3,5, 4 1 cos 2 ( cos ) 1 cos 2 1 ( 1)sin sin 2 ( )sin 2 / 2 0 0 0 0 0 / 2 0 / 2 0 0 0 / 2 0 / 2 / 2 0 n n n n n n t n n t T n n tdt n tdt T x t n tdt T b T T T T T T n
2. 2. 4 Frequency representation of periodic signals The Fourier series expression of the square wave sIgnal x(t)=-(siot+si300t+-sn5001+…) A 7a0 na Fig. 2. 12 Frequency spectrum of periodic square wave signal
The Fourier series expression of the square wave signal 2.2.4 Frequency representation of periodic signals sin 5 ) 5 1 sin 3 3 1 (sin 4 ( ) x t = 0 t + 0 t + 0 t + Fig. 2.12 Frequency spectrum of periodic square wave signal
2. 2. 4 Frequency representation of periodic signals FOurier series can be used to approximate a signal sin t (sint+asin3t) 0丌 3t+-sin5t) Fig. 2. 13 Approximations of a square wave signal using sums of partial terms of Fourier series
❑Fourier series can be used to approximate a signal. 2.2.4 Frequency representation of periodic signals Fig. 2.13 Approximations of a square wave signal using sums of partial terms of Fourier series