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