山东大学：《生物医学信号处理 Biomedical Signal Processing》精品课程教学资源（PPT课件讲稿）Chapter 02 Discrete-Time Signals and Systems

Example of Linear System KEX. 2. 6 Accumulator system yin =∑x for arbitrary x,n]and x2n] 团=∑x]y2=∑x k=-∞ When a3=an小+bxl =∑x]=∑(axk+b2少 k ]+b∑xk]=cz]+by k k 27 2/2/2021 Zhongguo Liu_ Biomedical Engineering_shandong Univ

27 2/2/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Example of Linear System ◆Ex. 2.6 Accumulator system      =− = n k y n x k     (    ) a x k b x k ay n by n y n x k ax k bx k n k n k n k n k 1 2 1 2 3 3 1 2 = + = + = = +     =− =− =− =−      =− = n k y n x k 1 1      =− = n k y n x k 2 2 x nand x n 1 2 for arbitrary x n ax n bx n when 3 = 1 + 2

Example 2.7 Nonlinear Systems Method: find one counterexample ◆Fory{z]=(xn]) ◆ counterexample12+12≠(1+1) ◆Fory]=logn(x]) ◆ counterexample 10×b90(kg(0x) 28 2/2/2021 Zhongguo Liu_ Biomedical Engineering_shandong Univ

28 2/2/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Example 2.7 Nonlinear Systems ◆Method: find one counterexample ( ) 2 2 2 ◆ counterexample 1 +1  1+1   ( ) 2 ◆ For y n = x[n]   log ( [ ]) y n = 10 x n 10 log (1) log (10 1)  10  10  ◆ counterexample ◆ For

Properties of Discrete-time systems 2.2.3 Time-Invariant Systems Shift-Invariant Systems X,In X, In 2[m]=n{n 29 2/2/2021 Zhongguo Liu_ Biomedical Engineering_shandong Univ

29 2/2/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Properties of Discrete-time systems 2.2.3 Time-Invariant Systems ◆Shift-Invariant Systems     2 1 n n0 x n = x −     2 1 n n0 y n = y − y n x1 n T{‧} 1 T{‧}

Example of Time-Invariant System EX 2.8 Accumulator system y小=∑] k X=rIn-n y=∑x=∑-m=∑]=-m k k= 30 2/2/2021 Zhongguo Liu_ Biomedical Engineering_shandong Univ

30 2/2/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Example of Time-Invariant System ◆Ex. 2.8 Accumulator system      =− = n k y n x k   1 n n0 x = x −           1 1 0 1 0 0 1 y n x k x k n x k y n n n n k n k n k =  =  − =  = − − =− =− =−

Example of Time-varying System Ex. 2.9 The compressor system 团T份}y=Mh 0o<n<oo -{T 6|2 0 0 =x1b[m=[T1127- yIn=x Mn=xMn-nyn-n]=xM(n-no) 31 2/2/2021 Zhongguo Liu_ Biomedical Engineering_shandong Univ

31 2/2/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Example of Time-varying System ◆Ex. 2.9 The compressor system yn= xMn, −  n            ( ) 1 1 0 0 M n n0 y n = x Mn = x Mn − n  y n − n = x − x n  T{‧} 0 T{‧} 0  n  2n x n x n n 1 0   = −   0  n −1 0 T{‧}  2 1 n − 