EX 2.1 Non-Periodic Sequences x2[小]=cos(n) Suppose it is periodic sequence with period N x2n]=x2[n+n cos( n ) =cos(n+N) for n+2n k=n+N, k: integer there is no integer N 27 2/6/2021 Zhongguo Liu_Biomedical Engineering_shandong Univ
27 2/6/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. EX. 2.1 Non-Periodic Sequences ◆Suppose it is periodic sequence with period N [ ] [ ] x2 n = x2 n + N x [n] cos( n) 2 = cos( n) = cos(n+ N) 2 , : integer, integer for n k n N k there is no N + = +
High and low Frequencies in Discrete-time signal xn= a cos(won (a)W=0or2π Frequency: The rate at which a repeating event occurs (b)W=π8or15/8 (c)=πor74 Zhongguo Liu_Biomedical Engineering_shandong Univ
28 2/6/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. High and Low Frequencies in Discrete-time signal 0 x n A w n [ ] cos( ) = (b) w0 = /8 or 15/8 (c) w0 = /4 or 7/4 (d) w0 = Frequency: The rate at which a repeating event occurs. (a) w0 = 0 or 2
2.2 Discrete-Time System Discrete-Time System is a trasformation or operator that maps input sequence X[n] into an output sequence y[n] ◆y[n]=T{Xn]}; Xn, yn]: discrete-time signal Xn yr Discrete-Time System 29 2/6/2021 Zhongguo Liu_Biomedical Engineering_shandong Univ
29 2/6/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2.2 Discrete-Time System ◆Discrete-Time System is a trasformation or operator that maps input sequence x[n] into an output sequence y[n]. ◆y[n]=T{x[n]}; x[n], y[n]: discrete-time signal T{‧} x[n] y[n] Discrete-Time System
EX 2.2 The Ideal Delay System yn]=xn-n],-00<n<0 If n, is a positive integer the delay of the system, Shift the input sequence to the right by n, samples to form the output oIf nd is a negative integer: the system will shift the input to the left by nal samples, corresponding to a time advance 30 2/6/2021 Zhongguo Liu_Biomedical Engineering_shandong Univ
30 2/6/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. EX. 2.2 The Ideal Delay System y[n] = x[n − nd ], − n ◆If is a positive integer: the delay of the system, Shift the input sequence to the right by samples to form the output . d n d n ◆If is a negative integer: the system will shift the input to the left by samples, corresponding to a time advance. d n d n
EX 2.3 Moving average M, xIn- M1+M2+1k=M xIn+M,+xIn+M-1++xIn +xIn-1++xln-M M1+M,+1 2 forn=7,M1=0,M2=5 [m n-5 0 dummy index m
31 2/6/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. EX. 2.3 Moving Average 2 1 2 1 1 1 2 1 2 1 1 1 1 ... 1 ... 1 M k M y n x n k M M x n M x n M x n x n x n M M M = − = − + + = + + + − + + + − + + − + + x[m] n m n-5 dummy index m for n=7, M1=0, M2=5 y[7]