2. 2. 4 Frequency representation of periodic signals 10 100-50 (a)One-sided amplitude spectrum (b) Two-sided amplitude spectrum (c)One-sided phase spectrum (d) Two-sided phase spectrum Fig. 2. 15 Two graphic representations of spectra of periodic signals
2.2.4 Frequency representation of periodic signals Fig. 2.15 Two graphic representations of spectra of periodic signals
2. 2. 4 Frequency representation of periodic signals s The spectrum of a periodic signal has the following three features 1. The spectrum is a discrete spectrum 2. The spectral lines appear only at the fundamental frequency and the harmonic frequencies; 3. The amplitude of harmonic component decreases with the increase in its frequency, the higher the harmonic frequency, the lower the amplitude is
❖ The spectrum of a periodic signal has the following three features: 1. The spectrum is a discrete spectrum; 2. The spectral lines appear only at the fundamental frequency and the harmonic frequencies; 3. The amplitude of harmonic component decreases with the increase in its frequency, the higher the harmonic frequency, the lower the amplitude is. 2.2.4 Frequency representation of periodic signals
2. 2. 4 Frequency representation of periodic signals Example 2. Find the frequency spectrum of the periodic sequence of rectangular pulses(also called periodic gate function) shown in Fig. 2. 16 x(2) 22 Fig. 2. 16 Periodic sequence of rectangular pulses
Example 2. Find the frequency spectrum of the periodic sequence of rectangular pulses (also called periodic gate function) shown in Fig. 2.16. 2.2.4 Frequency representation of periodic signals Fig. 2.16 Periodic sequence of rectangular pulses
2. 2. 4 Frequency representation of periodic signals Solution: From Eq(2.26), We have T/2 x(1)e T/2 e e /2 T -Nao noT sIn 2 noo nOT sIn 2 0.±1.±2
Solution: From Eq. (2.26), we have 2.2.4 Frequency representation of periodic signals 0, 1, 2, 2 2 sin 2 sin 2 1 1 ( ) 1 0 0 0 0 / 2 / 2 0 / 2 / 2 / 2 / 2 0 0 0 = = = − = = = − − − − − − n n n T n n T j n e T e dt T x t e dt T C j n t j n t T T j n t n
2. 2. 4 Frequency representation of periodic signals Substituting wo=2T/ into the above equation, we have sIn ,n=0,±1,±2, (236) 1丌r Defining sin c(r sin x (237) then Eq (2.36 changes to nOT C=-sin =0.±1+2 (238) 2 So x()=∑ ∑ Smcn丌r (239)
Substituting ω0=2π/T into the above equation, we have Defining then Eq.(2.36) changes to So 2.2.4 Frequency representation of periodic signals , 0, 1, 2, sin = = n T n T n T Cn (2.36) x x c x def sin sin ( ) = (2.37) , 0, 1, 2, 2 sin sin 0 = = = n n c T T n c T Cn (2.38) =− =− = = n j n t n j n t n e T n c T x t C e 0 0 ( ) sin (2.39)