Fundamentals of Measurement Technology Prof Wang Boxiong
Fundamentals of Measurement Technology (2) Prof. Wang Boxiong
2. 2. 4 Frequency representation of periodic signals In a finite interval of time, a periodic signal x(t)can be represented by its fourier series when it complies with the Dirichlet conditions a (t)=+2(a, cosnoot+ bm, sin noot) (2.12) Where 2c7/2 x(tcos nootdi (213) T/2 b x(tsin n@tdt (214) TJ7/2 n=0,1,2,3, 7= the period Wo= the angular frequency or circular frequency Wo=2TT/T an (including ao and bn)are called Fourier coefficients
In a finite interval of time, a periodic signal x(t) can be represented by its Fourier series when it complies with the Dirichlet conditions: where n＝0,1,2,3,…… T= the period ω0= the angular frequency or circular frequency ω0= 2π/T an(including a0 and bn ) are called Fourier coefficients. 2.2.4 Frequency representation of periodic signals = = + + 1 0 0 0 ( cos sin ) 2 ( ) n n n a n t b n t a x t (2.12) − = / 2 / 2 0 ( ) cos 2 T T n x t n tdt T a (2.13) − = / 2 / 2 0 ( )sin 2 T T n x t n tdt T b (2.14)
2. 2. 4 Frequency representation of periodic signals Fourier coefficients an and bn(functions of nwo) a an: even function of n or nwo, a-n=an bn: odd function of n or nwo, b-n=-bn Dirichlet conditions x <OO X(t must be absolutely integrable, X(t possesses a finite number of maxima and minima and finite number of discontinuities in any finite interval
Fourier coefficients an and bn (functions of nω0 ): ▪ an : even function of n or nω0 , a-n = an . ▪ bn : odd function of n or nω0 , b-n = -bn . Dirichlet conditions: ▪ x(t) must be absolutely integrable, ▪ x(t) possesses a finite number of maxima and minima and finite number of discontinuities in any finite interval. 2.2.4 Frequency representation of periodic signals − x(t) dt
2. 2. 4 Frequency representation of periodic signals Rewrite Eq(2. 12) x()=0+∑A,cos(mot+q) (215) Where An=van+b b.、n=1,2 (2.16 arc An: amplitude of signal's frequency component Pn: phaseshift 1,2 (2.17) bn=-A, sin
Rewrite Eq. (2.12): where An : amplitude of signal’s frequency component φn : phase-shift 2.2.4 Frequency representation of periodic signals = = + + 1 0 0 cos( ) 2 ( ) n n n A n t a x t (2.15) 1,2, ( ) 2 2 = = − = + n a b arctg A a b n n n n n n (2.16) 1,2, sin cos = = − = n b A a A n n n n n n (2.17) A−n = An −n =n
2. 2. 4 Frequency representation of periodic signals u a2 is the constant-value or the d. c component of a periodic signal o The term for na 1 is referred to as the fundamenta (component), or as the first harmonic component a the component for n=N is referred to as the Nth harmonic component u The representation of a periodic signal in the form of Eq( 2.15)is referred to as the Fourier series representation An: amplitude of the nth harmonic component Pn: phase shift of the nth harmonic component
❑ a0 /2 is the constant-value or the d.c. component of a periodic signal. ❑ The term for n=1 is referred to as the fundamental (component), or as the first harmonic component. ❑ The component for n=N is referred to as the Nth harmonic component. ❑ The representation of a periodic signal in the form of Eq. (2.15) is referred to as the Fourier series representation: ▪ An : amplitude of the nth harmonic component ▪ φn : phase shift of the nth harmonic component 2.2.4 Frequency representation of periodic signals
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