2. 2. 4 Frequency representation of periodic signals OFor an odd function x(t) 04 x(tsin notat (2m+1) (n=1,2,…) (2.19) 丌,( m is nteger
❑For an odd function x(t): 2.2.4 Frequency representation of periodic signals x(−t) = −x(t) ( 1,2, ) ( )sin 4 0 / 2 0 0 = = = n x t n tdt T b a T n n (2.18) ( 1,2, ) , integer 2 (2 1) = + = = n m is m A b n n n （ ） (2.19)
2. 2. 4 Frequency representation ofperiodic signals OFor an even function x(t) b.=0 4c7/2 (n=0,1,2…) (220) (t)cos notat A,=an Pn=mI, m is integer) (n=1.2 (2.21) x(t) Fig. 2, 14 Examples of even functions, the functions are symmetrical about the ordinate
❑For an even function x(t) 2.2.4 Frequency representation of periodic signals x(−t) = x(t) ( 0,1,2 ) ( ) cos 4 0 / 2 0 0 = = = n x t n tdt T a b T n n (2.20) ( 1,2 ) , integer = = = n m m is A a n n n （ ） (2.21) Fig. 2,14 Examples of even functions, the functions are symmetrical about the ordinate
2. 2. 4 Frequency representation of periodic signals Eulers formula cos at=-( -JOI (2.22) sin at=(e o-e/o) Substitute Eq. (2.22) into Eg (2.12) x(t)=20 ∑|(an+jbn) -In@o I Jn 2 n=1 Let (an+jbn)n=1,2,3 223)
Euler’s formula: Substitute Eq. (2.22) into Eq. (2.12), Let 2.2.4 Frequency representation of periodic signals = − = + − − ( ) 2 sin ( ) 2 1 cos j t j t j t j t e e j t t e e (2.22) = − = + + + − 1 0 0 0 ( ) 2 1 ( ) 2 1 2 ( ) n j n t n n j n t n n a j b e a j b e a x t 1,2,3 2 ( ) 2 1 ( ) 2 1 0 0 = = = + = − − n a C C a j b C a j b n n n n n n (2.23)
2. 2. 4 Frequency representation of periodic signals x(t)=Co+2C-n e n@o'+2Cn e mof n=1, 2, 3 (224) Or x()=∑C1e0n=0±1+2,… 2.25 T/2 T/2 x(tcos nott x(tsm notat T/2 T/2 1「r712 八2( cos n@ot-/ sin noot)dt (n=0±1,+2…) 少/2x( e Jnoo' dt (2.26
Or 2.2.4 Frequency representation of periodic signals ( ) 1,2,3 1 1 0 0 0 = + + = = = − x t C C− e C e n n j n t n n j n t n (2.24) ( ) = 0 = 0,1,2, =− x t C e n n j n t n (2.25) ( 0, 1, 2, ) ( ) 1 ( )(cos sin ) 1 ( ) cos ( )sin 1 / 2 / 2 / 2 / 2 0 0 / 2 / 2 0 / 2 / 2 0 0 = = = − = − − − − − − n x t e dt T x t n t j n t dt T x t n tdt j x t n tdt T C T T j n t T T T T T T n (2.26)
2. 2. 4 Frequency representation of periodic signals Writing Cn=Cnle%n=Re Cn+j Cn where c,l and n are the amplitude and phase of the complex coefficient respectivexe-C +Im? C (228) Imc P= arct (2.29) Reo
Writing where |Cn | and φn are the amplitude and phase of the complex coefficient respectively. 2.2.4 Frequency representation of periodic signals n n j Cn = Cn e n = ReC + jImC (2.27) Cn Cn Cn 2 2 = Re + Im (2.28) n n n C C arctg Re Im = (2.29)