Example 4.1:Sampling and Reconstruction of a sinusoidal signal Compare the continuous-time and discrete-time FTs,if we sample signal x.(t)=cos(4000nt) with sampling period T=16000 Solution:x(-cos-co-co) The highest frequency of x(t):=4000n sampling frequency -12000x-20 T 20
20 Example 4.1: Sampling and Reconstruction of a sinusoidal signal ( ) ( ) ( ) ( 0 ) 4000 2 cos cos cos 6000 3 x n x n n c n nT = = = = 0 2 sampling frequency s 12000 2 T = = ( ) 0 : 4000 The highest frequency of c x t = Solution: Compare the continuous-time and discrete-time FTs, if we sample signal x t t c ( )=cos 4000 , ( ) with sampling period T =1 6000
Example 4.1:Sampling and Reconstruction of a sinusoidal signal continuous-time FT of x.()=cos(4000m)=(e4o0m+e4o0m X.(j2)=πδ(2-4000m)+δ(2+4000m) discrete-time FT of x=cos3πn=cos(@wm -dfm.t-e -4000π 4000π X)X..=12000 21
21 Example 4.1: Sampling and Reconstruction of a sinusoidal signal X j c ( = − + + ) ( 4000 4000 ) ( ) −4000 0 4000 ( ) j X e ( )| ( ( )) 1 | s T T s c k X j X j k T = = =− = = − x (t) ( t) c = cos 4000 ( ) ( 0 ) 2 cos cos 3 x n n n = = continuous-time FT of discrete-time FT of ( ) 1 4000 4000 2 j t j t e e − = + ( ) 2 2 1 3 3 2 j n j n e e − = + , 12000 s = ( ) 0 2 2 0 : j n DTFT r can use e r =− ⎯ + → −