7.2z变换的性质部分和7.2.9若α<<βf(k) F(z)7ZF(i)则F(z)g(k) =max(α,1) <z|<βz-11=-8若 f(k)ε(k) F(z)α<<βZ则 g(k)=f(i)F(z)5max(α,1)<|z|< β7-i=0证: : g(k)= Zf(i)= Zf(i)e(k-i)= f(k)*e(k)i=-0i=-0ZF(z)g(k) z-1
7.2.9 部分和 若 ( ) 1 ( ) ( ) F z z z g k f i k i − = =− 则 若 ( ) 1 ( ) ( ) 0 F z z z g k f i k i − = = 则 f (k) F(z) z max(,1) z f (k) (k) F(z) z max(,1) z 证： g(k) f (i) f (i) (k i) f (k) (k) i k i = = − = =− =− ( ) 1 ( ) F z z z g k − 7.2 z 变换的性质
7.2z变换的性质例：求()7Za'=Za'e(i)且 α*ε(k)<z-ai=00EaH(z-(z-a)0i=07.2.10初值定理α8f(k)e(k) F(z)若因果序列则f (O) = lim F(z)lim F(z) = f()Z→00Z00证.: F(z2)=Zf(k)z-k =f(0)+ f(1)z- + f(2)z- ..k=0
例：求 = k i i Z a 0 z a z a a i a k k k i i k i i − = = = ( ) ( ) 0 0 且 1 ( 1)( ) 2 0 z z a z z a z z z a k i i − − = − − = 7.2.10 初值定理 若 因果序列 f (k) (k) F(z) z 则 f (0) lim F(z) z→ = 1 2 0 ( ) ( ) (0) (1) (2) − − = − F z = f k z = f + f z + f z k 证 k lim F(z) f (0) z = → 7.2 z 变换的性质
7.2Z变换的性质应用：设因果序列f(k)e(k) F(z)由单边左移性质F(z)-(k)z-kf(k +m)e(k)<zmk=0依据初值定理确定序号m的序列值f（m）F(z)-Zf(k)z-kf(m) = lim z"Z>0k=0终值定理7.2.11若因果序列α8f(k)e(k) F(z)则f (o) = lim(z -1)F(z)Z1
应用：设因果序列 f (k) (k) F(z) 由单边左移性质 + − − = − 1 0 ( ) ( ) ( ) ( ) m k m k f k m k z F z f k z 依据初值定理确定序号m的序列值f (m) = − − = − → 1 0 ( ) lim ( ) ( ) m k m k z f m z F z f k z 7.2.11 终值定理 若 因果序列 f (k) (k) F(z) z ( ) lim( 1) ( ) 1 f z F z z = − → 则 7.2 z 变换的性质
7.27变换的性质证:z[f(k +1)- f(k)]= z[F(z)- f(O)]- F(z)=(z-1)F(z)-zf(0)(z -1)F(z) = zf(O)+ z[f(k +1)- f(k))Z[f(k +1) - f(k)k-k= z(0)+k=0两边取lim有z-1lim(z-1)F(z) = f(0)+Lf(1)-f(0)l+[f(2)- f(1)-1+[f(3) - f(2)]+...+ f (8)= f(8)
证： ( 1) ( ) (0) ( 1) ( ) ( ) (0) ( ) z F z zf Z f k f k z F z f F z = − − + − = − − k k zf f k f k z z F z zf Z f k f k − = = + + − − = + + − 0 (0) ( 1) ( ) ( 1) ( ) (0) ( 1) ( ) 两边取 lim z→1 有 ( ) [ (3) (2)] ( ) lim( 1) ( ) (0) [ (1) (0)] [ (2) (1)] 1 = + − + + − = + − + − → f f f f z F z f f f f f z 7.2 z 变换的性质
7.2Z变换的性质z变换性质表在page315结束7.2
z 变换性质表在page 315 7.2 结束 7.2 z 变换的性质