Let H(z) = P(z)/Q(z) be the system function of a rational form. Let us represent both P(z) and Q(z) as polynomials of z (not z -1 )

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1 Review: Poles and Zeros of Fractional Form Let H() = P()/Q() be the system function of a rational form. Let us represent both P() and Q() as polynomials of (not - ) Then Poles: the roots of Q()=0 Zeros: the roots of P()=0

2 Review: Physical Meanings Physical meanings Pole: The pole of a -transform H() are the values of for which H()=. Zero: The ero of a -transform H() are the values of for which H()=0.

3 This Wee

4 Poles and eros of negative powers of

5 Example of Poles and Zeros

6 Pole-Zero Plot It is useful to display the eros and poles of H() on the complex -plane. Knowing the poles and eros is equivalent to nowing the LTI system. poles eros

7 Significance of the Zeros of H() for The poles a FIR system are always in the origin (or a multi-order ero). The eros of a FIR filter thus play a major role in the frequency response. FIR System

8 Significance of the Zeros of H() for Recall that DTFT is evaluated on the unit-circle on the - plane. If the eros are placed close to the unit circle, we can image that the nearby frequencies will be lowered. FIR System

9 Example: The L-point Running- Sum Filter The system function is We have to represent it as rational form in. From geometric series,

10 Example: The L-point Running- Sum Filter The eros of H() will be determined by the roots of the numerator polynomial, The poles are determined by the denominator. There are (L-)-th roots at =0 and a root at =. So, the is a common location at = for both pole and ero; they are canceled. Hence, finally, the L-point runing-sum filter has L- eros and an (L-)th-order pole at the origin.

11 Example: The L-point Running- For instance, L=0 Sum Filter

12 Example: The L-point Running- Sum Filter The magnitude response: note that there are a total of 9 ero responses from [-, )

13 Recall: Ideal Low-pass Filter s frequency response

14 Passband In the above example of 0-point running sum filter, the passband is roughly centered at w=0. It acts lie a low-pass filter. Can we move the passband to other locations by re-placing eros, so that the filter acts lie some other frequency-selection filters?

15 Frequency-selection Filters: Ideal High-pass Filter

16 Ideal Band-pass Filter

17 Ideal Band-stop Filter

18 Bandpass Filter by Placing the Zeros of an FIR system We can move the passband to a frequency away from w=0 by changing the ero locations: The obvious way is to use all but one of the roots of unity as the eros of an FIR filter. Note that by specifying poles and eros, we completely describe the LTI system. L=0

19 Bandpass Filter by Placing the Zeros of an FIR system Frequency magnitude response:

20 Bandpass Filter with Real Coefficients However, the above filter has complex coefficients. By the symmetry property of DTFT, we now that the magnitude response should be an even function if we hope the coefficients are all real.

21 Practical Band Pass Filtering The above examples enable us a process of manipulating (adding, moving, removing) the eros interactively by using a simulation system (eg. Matlab) for filter design. We show a filter obtained by such an interactive design below, although much better filters can be deigned by more sophisticated methods. It is a useful illustration of the power of the - transform to simplify the analysis of such problems.

22 Example: A Practical Band Pass Filtering

23 Stability of Difference Equations Since the difference equations could be recursive (eg., IIR filter), we have to investigate whether it is stable or not. Stability Bounded input, bounded output (BIBO): For all bounded input, x[n] B x < for all n, the output is also bounded, i.e., there exists a positive value B y s.t. y[n] B y < for all n. Eg., the accumulated system is unstable; easily verified by setting x[n] = u[n], the unit step signal. h n n 0 0 otherwise

24 Convergence Region of Z-transform Stability of an LTI system is highly related to the region of convergence (ROC) of its system function (represented by -transform) Region of convergence (ROC) X n x n xn n n n In fact, convergence of the power series X() depends only on. n n n x

25 ROC of Z-transform n n n x If some value of, say =, is in the ROC, then all values of on the circle defined by = will also be in the ROC. Thus the ROC will consist of a ring in the -plane.

26 ROC of Z-transform Ring Shape

27 Some properties of the ROC The ROC is a ring or dis in the -plane centered at the origin; i.e., 0 r R < r L. The Fourier transform of x[n] converges absolutely iff the ROC includes the unit circle. The ROC cannot contain any poles. If x[n] is a finite-length (finite-duration) sequence, then the ROC is the entire -plane except possible = 0 or =. If x[n] is a right-sided sequence, the ROC extends outward from the outermost (i.e., largest magnitude) finite pole in X() to (and possibly include) =.

28 ROC vs. Linear System Consider the system function H() of a linear system: If the system is stable, the impulse response h(n) is absolutely summable and therefore has a Fourier transform, then the ROC must include the unit circle. If the system is causal, then the impulse response h(n) is right-sided, and thus the ROC extends outward from the outermost (i.e., largest magnitude) finite pole in H() to (and possibly include) =. That is, a causal LTI system is stable if the poles are all inside the unit circle. FIR system is always stable because its poles are always at the origin.

29 Digital Filter Structures We now that there could be multiple ways to realie the system function of a difference equation. Z-transform is also useful in help characterie these solutions, and find an efficient implementation structure.

30 Digital Filter Structures (for IIR Filter) 0 N M y n a y n b x n M n x b n v 0 ] [ ] [ n v n y a n y N Direct Form I implementation

31 On the -domain or equivalently M N b a H H H 0 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 0 X b X H V M ) ( ) ( ) ( ) ( ) ( V a V H Y N

32 By changing the order of H and H, onsider the equivalence on the -domain: where Let Then ) ( ) ( ) ( H H H M b H 0 ) ( ) ( ) ( N a H ) ( ) ( ) ( ) ( ) ( X a X H W N ) ( ) ( ) ( ) ( 0 W b W H Y M

33 In the time domain, We have the following equivalence for implementation: N n x w n a w n ] [ ] [ ] [ M w n b n y 0 ] [ ] [ We assume M=N here

34 Note that the exactly the same signal, w[], is stored in the two chains of delay elements in the bloc diagram. The implementation can be further simplified as follows: Direct Form II (or Canonic Direct Form) implementation

35 By using the direct form II implementation, the number of delay elements is reduced from (M+N) to max(m,n). Example: H ( ) Direct form I implementation Direct form II implementation

36 Representing by signal-flow graph Example: the signal-flow graph of direct form II.

37 Example: ( )( ) H( ) ( 0.5 )( 0.5 ) Cascade structure: direct form I implementation Cascade structure: direct form II implementation

38 Cascade Form From the fundamental theorem of algebra, we now that an n-th order real-coefficient polynomial has n roots that are real or complex-conjugate pairs. From this theorem, it ensures that we can factor the numerator and denominator polynomials. We can express H() in the form: H M M ( f ) ( ) N N ( c ) ( g ( d )( g )( d ) ) where M=M +M and N=N +N, g and g * are a complex conjugate pair of eros, and d and d * are a complex conjugate pair of poles.

39 A general form is where and we assume that MN. The real poles and eros have been combined in pairs. If there are an odd number of eros, one of the coefficients b will be ero. s N a a b b b H 0 ) ( ( ) / N N s

40 It suggests that a difference equation can be implemented via the following structure consisting of a cascade of second-order and first-order systems: cascade form of implementation (with a direct form II realiation of each second-order subsystems)

41 Parallel Form We also now that a rational function can be represented as sum of partial fractions. If we represent H() by additions of low-order rational systems: where N=N +N. If MN, then N p = MN. 0 ) )( ( ) ( ) ( N N N d d e B c A C H p

42 Parallel Form Alternatively, we may group the real poles in pairs, so that p N s N a a e e C H 0 0 ) (

43 Illustration of parallel-form structure for six-order system (M=N=6) with the real and complex poles grouped in pairs.

44 Example: consider still the same system ) ( H

45 another alternation of the same system H ( ) 8 Hence, given a system function, there are many ways to implement it. There are equivalent when infinite-precision arithmetic is used. However, their behavior with finiteprecision arithmetic could be quite different.

46 Remar: While the signal flow graph is an efficient way to represent a difference equation, not all of its instances are realiable: If a system function has poles, a corresponding signal flow graph will have feedbac loops. A signal flow graph is computable if all loops contain at least one unit delay element. Eg. A non-computable system Computable systems

47 Inverse Z-transform Given X(), find the sequence x[n] that has X() as its -transform. We need to specify both algebraic expression and ROC to mae the inverse Z-transform unique. Techniques for finding the inverse -transform: Investigation method: By inspect certain transform pairs. Eg. If we need to find the inverse -transform of X 0. 5 From the transform pair we see that x[n] = 0.5 n u[n].

48 Some Common Z-transform Pairs a n u[ n ]

49 Some Common Z-transform Pairs (continue) 0. ROC : 0 0. ROC :. ROC :. ROC :. ROC :. ROC : a a otherwise N n a r r w r w r n n u w r r r w r w r n n u w r w w n n u w w w n n u w a a a n u na N N n n n n cos sin sin cos cos cos cos sin sin cos cos cos

50 Inverse Z-transform by Partial Fraction Expansion If X() is the rational form with An equivalent expression is N m M m m a b X 0 0 N N M m M M m m N N N N m M M m m M a b a b X

51 Inverse Z-transform by Partial Fraction Expansion (continue) There will be M eros and N poles at nonero locations in the -plane. Note that X() could be expressed in the form X b a 0 0 m N where c s and d s are the eros and poles, respectively. M d m c m

52 Inverse Z-transform by Partial Fraction Expansion (continue) Then X() can be expressed as N A X d Obviously, the common denominators of the fractions in the above two equations are the same. Multiplying both sides of the above equation by d and evaluating for = d shows that A d X d

53 Example Find the inverse -transform of X() can be decomposed as Then 4 X / / 4 4 / / / / X A X A 4 A A X / /

54 Example (continue) Thus From the ROC if we have a right-hand sequence, 4 X / / n u n u n x n n 4

55 Find the inverse -transform of Since both the numerator and denominator are of degree, a constant term exists. B 0 can be determined by the fraction of the coefficients of, B 0 = /(/) =. Another Example X / 0 A A B X /

56 From the ROC, the solution is right-handed. So Another Example (continue) A A A A X / / / / / n u n u n n x X n / /

57 Power Series Expansion X n x n n x x 0 x x x We can determine any particular value of the sequence by finding the coefficient of the appropriate power of.

58 Example: Finite-length Sequence Find the inverse -transform of By directly expand X(), we have Thus, 0 5 X. x X. n n 0. 5 n n 0. 5 n

59 Suddenly Applied Complex Exponential Inputs In practice, to get the frequency response, we may not apply the complex exponential inputs e jwn to a LTI system, but the more practical-appearing inputs of the form x[n] = e jwn u[n] i.e., complex exponentials that are suddenly applied at an arbitrary time, which for convenience we choose n=0. Consider its output to a causal LTI system: y n h xn

60 y n h xn n h h h n 0 jw( n ) e un jw e un h e jw 0 e e jw jwn e jwn e jwn Consider h is a causal LTI system: h()=0 when <0 n n 0 0 Consider u[n-] is ero for >n

61 Suddenly Applied Complex Exponential Inputs (continue) We consider the output for n 0. jw jwn y n h e e 0 h H e jw jwn jw jwn e h e e n n jw Hence, the output can be written as y[n] = y ss [n] + y t [n], where jw jwn y n H e e Steady-state response ss e e jwn y t jw jwn n h e e n Transient response

62 Suddenly Applied Complex Exponential Inputs (continue) If h[n] = 0 except for 0 n M (i.e., a FIR system), then the transient response y t [n] = 0 for n+ > M. That is, the transient response becomes ero since the time n = M. For n M, only the steady-state response exists. For infinite-duration impulse response (i.e., IIR) y t jw jwn n h e e h Qn n n For stable system, Q n must become increasingly smaller as n, and so is the transient response.

63 Discrete Fourier Transform (DFT) Currently, we have investigated three cases of Fourier transform, Fourier series (for continuous periodic signal) Continuous Fourier transform (for continuous signal) Discrete-time Fourier transform (for discrete-time signal) All of them have infinite integral or summation in either time or frequency domains.

64 There is still another type of Fourier transform Consider a discrete sequence that is periodic in the time domain. Eg., it can be obtained by a periodic expansion of a finite-duration sequence, ie., we image that a finitelength sequence repeats, over and over again, in the time domain). Then, in the frequency domain, the spectrum shall be both periodic and discrete, ie, the frequency sequence is also made up of a finite-length sequence, which repeats over and over again in the frequency domain.

65 Discrete Fourier Transform Considering both the finite-length (or finite-duration) sequences in one period of the time and frequency domains, leads to a transform called discrete Fourier transform (DFT). In principle, DFT acts lie performing multiple (a ban of) FIR filters simultaneously at the same time.

66 DFT Definition The DFT of x[n], denoted by X() is as follows: Inverse DFT Inverse DFT j / N where W e, and W n are the roots of W N =.

67 Relation to DTFT for Finite-length sequence X() is the uniform samples of the DTFT X(e jw ) at the discrete frequency w = (/N), when the frequency range [0, ] is divided into N equally spaced points. Note that the above explains only the relation between DFT and DTFT for the simple case that x[n] is a finite-length sequence. We will investigate more in-depth relationships between DFT continuous FT (and also between DFT and DTFT) for length-unlimited sequences in the future.

68 From Kuhn 005

69 From Kuhn 005

70 Four types of Fourier Transforms Frequency domain nonperiodic Frequency domain periodic Time domain nonperiodic Continuous Fourier transform (both domains are continuous) DTFT (time domain discrete, frequency domain continuous and band-limited or periodic) Time domain periodic Fourier series (time domain continuous and periodic, frequency domain discrete) DFT/DTFS (time domain discrete, frequency domain discrete, and both finite-duration or periodic)