離散時間信號處理DSP

1、Digital Signal Processing Chapter 6 Structures for Discrete-Time Systems 2 6.0 Introduction Several ways to describe discrete-time systems: Impulse responses in time domain. Difference equations in time domain. The z transforms in complex frequency domain (transfer functions). Fourier transforms in

2、frequency domain (Frequency response). H(ej) = |H(ej)|ej() N i i i M l l l za zb zX zY zH 1 0 1 )( )( )( 3 6.1 Description of the Digital Filter Structures Its difference equations in time domain is 10 ( )()() NM il il y na y nib x nl 將輸入加以延時，組成 M 節(jié)的延時網(wǎng)絡(luò)，把每 個延時抽頭后加權(quán)，然 后把結(jié)果相加。 將輸出加以延時，組成 N 節(jié)的延時網(wǎng)絡(luò)，把每

3、個延時抽頭后加權(quán)，然 后把結(jié)果相加。 因此，網(wǎng)絡(luò)結(jié)構(gòu)表示一定的運算結(jié)構(gòu)，不同結(jié)構(gòu)所需要因此，網(wǎng)絡(luò)結(jié)構(gòu)表示一定的運算結(jié)構(gòu)，不同結(jié)構(gòu)所需要 的存儲單元以及運算次數(shù)不同，前者影響結(jié)構(gòu)復(fù)雜性，的存儲單元以及運算次數(shù)不同，前者影響結(jié)構(gòu)復(fù)雜性， 后者影響運算速度。后者影響運算速度。 4 6.1 Description of the Digital Filter Structures Three basic elements to implement digital filters: Delay Multiplier Adder Block diagram（方框圖）representation of thr

4、ee basic elements. z1 X(z) x(n)x(n 1) z1X(z) x(n) X(z) k k X(z) k x(n) x1(n) X1(z) x2(n)X2(z) x1(n) + x2(n) X1(z) + X2(z) 5 6.1 Description of the Digital Filter Structures Signal flowgraph（信號流圖）representation of three basic elements. X(z) x(n)x(n 1) z1X(z) z1 x1(n) X1(z) x2(n)X2(z) x1(n) + x2(n) X1

5、(z) + X2(z) X(z) x(n)k x(n) k X(z) k 6 6.1 Description of the Digital Filter Structures Two classes of digital filters: Finite-duration impulse response filters or nonrecursive filters. Its transfer functions are of the polynomial form. Infinite-duration impulse response filters or recursive filters

6、. Its transfer functions are of the rational polynomial form. 7 6.3 Basic structures for IIR digital filters 6.3.1 Direct form I The transfer function of a recursive filter is given by And the difference equations in time domain is In general, M N. 0 1 ( ) ( ) ( ) 1 M i i i N i i i b z N z H z D z a

7、 z M i i N i i inxbinyany 01 )()()( 8 6.3.1 Direct forms I x(n)y(n) z1 z1 b0 b1 b2 z1 bM bM1 z1 z1 z1 a1 a2 aN1 aN y(n1) y(n2) y(nN) x(n1) x(n2) x(nM) M i i N i i inxbinyany 01 )()()( 0 1 ( ) 1 M i i i N i i i b z H z a z 9 6.3.1 Direct forms I x(n)y(n) b0 b1 b2 bM bM1 z1 z1 z1 a1 a2 aN1 aN Direct f

8、orms I structure for IIR digital filters 10 6.3.2 Direct forms II x(n)y(n) z1 z1 b0 b1 b2 z1 bM bM1 z1 z1 z1 a1 a2 aN1 aN 1 ( )( ) ( ) H zN z D z 11 6.3.2 Direct forms II x(n)y(n) b0 b1 b2 bM bM1 z1 z1 z1 a1 a2 aN1 aN Direct forms II structure for IIR digital filters 12 Comparison of the two types x

9、(n)y(n) b0 b1 b2 bM bM1 z1 z1 z1 a1 a2 aN1 aN Direct forms I x(n)y(n) b0 b1 b2 bM bM1 z1 z1 z1 a1 a2 aN1 aN Direct forms II 13 Example 1 Compute H(z) from the following signal flowgraph. Solution: x(n)y(n) 1/4 z1 z1 1/4 -3/8 2 1 1 12 12 1 2 162 4 ( ) 13 823 1 48 z z H z zz zz 14 6.3.3 Cascade form W

10、riting the numerator and denominator polynomials of H(z) as products of second-order factors, respectively, we have that 12 012 0 12 1 12 1 1( ) ( ) ( )1 1 M i im ikk N i k kk i i b z zzN z H zH D zm zm z a z x(n)y(n) 11 z1 z1 m11 m2121 1m z1 z1 m1m m2m2m H0 15 6.3.4 Parallel form H(z) can also be e

11、xpressed as an addition of second- order partial-fractions, such that 0 1 1 01 12 1 12 ( ) ( ) ( ) 1 1 M i i i N i i i m kk k kk N z H z D z b z a z z m zm z x(n)y(n) 11 z1 z1 m11 m21 z1 z1 m1m m2m 0m 01 1m 16 Example 2 Figure the signal flowgraph of the following system by the direct form (type I a

12、nd II), cascade form and parallel form. Solution: 311 ( )(1)(2)( )(1) 483 y ny ny nx nx n x(n)y(n) 1/3 z1 z1 3/4 -1/8 Type II x(n) y(n) 3/4 -1/8 z1 z1 1/3 Type I 17 Example 2 x(n)y(n) 1/3 z1 1/4 z1 1/2 Cascade form 121 111 1211 11 311 ( )(1)(2)( )(1) 483 311 ( )( )( )( )( ) 483 111 111 1 333 ( ) 311

13、111 11111 484242 y ny ny nx nx n Y zz Y zz Y zX zz X z zzz H z zzzzzz 18 Example 2 11 1211 11 11710 11 3333 ( ) 311111 11111 484242 zz H z zzzzzz x(n)y(n) z1 1/4 z1 1/2 10/3 -7/3 Parallel form 19 Example 3 Determine the transfer function of the system below: x(n)y(n) z1 1/3 z1 1/5 -15/2 -3 1 11 1511

14、 3 ( ) 11 2 11 35 z H z zz 1 15111 ( )( )( )( )3(1) 2355 nnn h nnu nu nu n 20 6.5 Basic structures for FIR digital filters The difference equation of FIR filters 0 0 0 0 ( )() ( )( ) () ( )( )( ) ( ) ( )( ) ( ) M l l M l M l l M l l y nb x nl y nh l x nl Y zh l z X z Y z H zh l z X z 21 6.5.1 Direct

15、 form x(n) y(n) z1z1z1 h(0)h(1)h(2)h(M1) h(M) 0 0 12 ( )( ) () ( ) ( )( ) ( ) (0)(1)(2)() M l M l l M y nh l x nl Y z H zh l z X z hhzhzh M z 22 6.5.1 Direct form Transposed direct form（直接型結(jié)構(gòu)的轉(zhuǎn)置） 0 ( )( ) M l l H zh l z x(n) y(n) z1z1z1 h(0)h(1)h(2)h(M1)h(M) 23 Example 4 Compute the transfer functio

16、n given by the signal flowgraph and the direct form of H(z). x(n) y(n) h(0) h(1) h(2) h(3) h(4) h(5) h(6) h(7) h(8) 3 z 3 z 1 z 1 z 24 Example 4 1234 5678 8 0 ( ) (0)(1)(2)(3)(4) (5)(6)(7)(8)( ) ( )( ) n n Y zhhzhzhzhz hzhzhzhzX z X zh n z x(n) y(n) z1z1z1 h(0)h(1)h(2)h(3)h(4)h(5)h(6)h(7)h(8) z1z1z1

17、z1z1 25 6.5.2 Cascade form Writing H(z) as a product of second-order factors, we get that 112 012 01 is even 2 ( )( )(), 1 is odd 2 NM kkk lk M M H zh l zzzN M M x(n)y(n) z1 z1 01 11 21 z1 z1 02 12 22 z1 z1 2N 1N 0N 26 6.5.3 Linear-phase forms（線性相位型） An important subclass of FIR digital filters is t

18、he one that includes linear-phase filters, that is and the frequency response has the following form )( () ()() () ( ) jjj jjj jj H eH ee H ee Be 27 6.5.3 Linear-phase forms where b(n) is the inverse Fourier transform of B(), and Since B() is real, So 11 ( )()( ) 22 ( ) 2 () jj njjj n j jj n j h nH

19、eedBeed e Beed e b n * ( )()b nbn * ()() jj eh ne hn *2 ()() j h nhne ( )() j b neh n 28 6.5.3 Linear-phase forms In the common case where all the filter coefficients are real, so If h(n) is causal, that is h(n) = 0, for n 2. So, This equation shows that the h(n) of a linear-phase filter is symmetri

20、c or antisymmetric about M/2. MnnMhnh M Mnnhnh 0),()( 2 0),2()( 2 ()()()( )(2 ) j h nhnehnh nhn * ()() and , 2 k hnhnkZ 29 6.5.3 Linear-phase forms n h(n) 2103 4 5 6 7 8n h(n) 2103 4 5 6 7 8 9 n h(n) 2103 4 5 6 7 8 n h(n) 2103 4 5 6 7 8 9 symmetric antisymmetric M even M odd Type IType II Type IIITy

21、pe IV 30 6.5.3 Linear-phase forms: type I ( )(), is evenh nh MnM 1 2 2 0 1 2 11 22 () 2 00 1 2 () 2 0 ( )( )()( ) 2 ( )()() 2 ( )() 2 M M M nn M n n MM M nMn nn M M nMn n M H zh n zhzh n z M h n zhzh Mn z M h nzzhz 31 6.5.3 Linear-phase forms: type I x(n) y(n) z1z1z1 h(0)h(1)h(2)h(M/2) z1z1z1 h(M/21

22、) 1 2 () 2 0 ( )( )() 2 M M nMn n M H zh nzzhz 32 6.5.3 Linear-phase forms: type II ( )(), is oddh nh MnM 1 2 1 0 2 11 22 () 00 1 2 () 0 ( )( )( ) ( )() ( ) M M nn M n n MM nMn nn M nMn n H zh n zh n z h n zh Mn z h nzz 33 6.5.3 Linear-phase forms: type II x(n) y(n) z1z1z1 h(0)h(1)h(2) z1z1 z1 z1 2

23、3M h 2 1M h 1 2 () 0 ( )( ) M nMn n H zh nzz 34 6.5.3 Linear-phase forms: type III ( )(), is evenh nh MnM 1 2 0 1 2 11 22 () 00 1 2 () 0 ( )( )( ) ( )() ( ) M M nn M n n MM nMn nn M nMn n H zh n zh n z h n zh Mn z h nzz 35 6.5.3 Linear-phase forms: type III x(n) y(n) z1z1z1 h(0)h(1)h(2) z1z1z1 h(M/2

24、1) 1111 1 2 () 0 ( )( ) M nMn n H zh nzz 36 6.5.3 Linear-phase forms: type IV ( )(), is oddh nh MnM 1 2 1 0 2 1 1 22 () 00 1 2 () 0 ( )( )( ) ( )() ( ) M M nn M n n MM nMn nn M nMn n H zh n zh n z h n zh Mn z h nzz 37 6.5.3 Linear-phase forms: type IV x(n) y(n) z1z1z1 h(0)h(1)h(2) z1z1 z1 z1 2 3M h

25、2 1M h 11111 1 2 () 0 ( )( ) M nMn n H zh nzz 38 Example 5 Draw the signal flow-gragh of the direct form and linear-phase form for the FIR system. Solution: direct form ( )( )2 (1)3 (2)4 (3) 3 (4)2 (5)(6) h nnnnn nnn 123456 ( )1 23432H zzzzzzz x(n) y(n) z1z1z1 1-23-21 z1z1 3-4 z1 39 Example 5 Linear

26、-phase form 123456 615243 ( )1 23432 (1)2()3()4 H zzzzzzz zzzzzz x(n) y(n) z1z1z1 1-23-4 z1z1z1 40 6.5.3 Linear-phase forms Clearly, the linear-phase form structure requires about 50% fewer multiplications than that of the direct forms. 41 Digital network analysis The analysis of digital networks is realized through the signal flow graph representation. A digital network