LTI Discrete-Time Systems in Transform Domain Ideal Filters Zero Phase Transfer Functions Linear Phase Transfer Functions

Transcription

1 LTI Discrete-Time Systems in Transform Domain Ideal Filters Zero Pase Transfer Functions Linear Pase Transfer Functions Tania Stataki 811b

2 Types of Transfer Functions Te time-domain classification of an LTI digital transfer function sequence is based on te lengt of its impulse response: - Finite impulse response (FIR transfer functions - Infinite impulse response (IIR transfer functions

3 Types of Transfer Functions Several oter classifications are also used In te case of digital transfer functions wit frequency-selective frequency responses (filters, one classification is based on te sape of te magnitude function H ( e j ω or te form of te pase function θ(ω Based on te above, four types of ideal filters are usually defined

4 Ideal Filters Frequency response equal to one at frequencies we wis to keep Frequency response equal to ero at all oter frequencies

5 Ideal Filters Te range of frequencies were te frequency response takes te value of one is called te passband Te range of frequencies were te frequency response takes te value of ero is called te stopband

6 Ideal Filters Frequency responses of te four popular types of ideal digital filters wit real impulse response coefficients are sown below: Lowpass Higpass Bandpass Bandstop

7 Ideal Filters Te inverse DTFT of te frequency response of te ideal lowpass filter is (see previous capter sinωn n = c LP ], < n πn < Te above impulse response is non-causal and is of doubly infinite lengt. Moreover, is not absolutely summable.

8 Ideal Filters Te remaining tree ideal filters are also caracteried by doubly infinite, non-causal impulse responses and are not absolutely summable Tus, te ideal filters wit te ideal brick wall frequency responses cannot be realied wit finite dimensional LTI filters

9 Ideal Filters To develop stable and realiable transfer functions, te ideal frequency response specifications are relaxed by including a transition band between te passband and te stopband Tis permits te magnitude response to decay slowly from its maximum value in te passband to te ero value in te stopband

10 Ideal Filters Moreover, te magnitude response is allowed to vary by a small amount bot in te passband and te stopband Typical magnitude response specifications of a lowpass filter are sown below

11 Zero-Pase and Linear-Pase Transfer Functions A second classification of a transfer function is wit respect to its pase caracteristics In many applications, it is necessary tat te digital filter designed does not distort te pase of te input signal components wit frequencies in te passband

12 Zero-Pase and Linear-Pase Transfer Functions One way to avoid any pase distortion is to make te frequency response of te filter real and nonnegative, i.e., to design te filter wit a ero-pase caracteristic A ero pase cannot be causal

13 Zero-Pase and Linear-Pase Transfer Functions For non-real-time processing of real-valued input signals of finite lengt, ero-pase filtering can be very simply implemented by relaxing te causality requirement One ero pase filtering sceme is sketced below xn] H( vn] un] H( wn] u n] = v n], y n] = w n]

14 Zero-Pase and Linear-Pase Transfer Functions It is easy to verify te above sceme in te frequency domain Let X ( e jω, V ( e j ω, U ( e j ω, W ( e j ω, and Y ( e jω denote te DTFTs ofxn],vn], un],wn], andyn], respectively From te figure sown earlier and making use of te symmetry relations we arrive at te relations between various DTFTs as given on te next slide

15 Zero-Pase and Linear-Pase Transfer Functions xn] H( vn] un] H( wn] V ( ej ω = H ( ejω X ( ejω, U ( ej ω = V*( ejω u n] = v n], y n] = w n] W ( ej ω = H ( ejω U ( ejω Combining te above equations we get Y ejω W*( ejω = H *( ejω U *( e, Y ( ej ω = W*( ejω ( = jω = H* ( ejω V ( ejω = H*( ejω H( ejω X( ejω = H ( ejω X ( ejω

16 Zero-Pase and Linear-Pase Transfer Functions Te functionfftfilt implements te above ero-pase filtering sceme In te case of a causal transfer function wit a non-ero pase response, te pase distortion can be avoided by ensuring tat te transfer function as a unity magnitude and a linear-pase caracteristic in te frequency band of interest

17 Zero-Pase and Linear-Pase Transfer Functions Te most general type of a filter wit a linear pase as a frequency response given by H ( e jω = ejωd wic as a linear pase fromω= 0 toω= π Note also H ( e jω = 1 τ ( ω =D

18 Zero-Pase and Linear-Pase Transfer Functions Te outputyn] of tis filter to an input x n] = Ae j ωn is ten given by y n] = Ae jωdejωn = Aejω ( nd If x a (t and y a (t represent te continuoustime signals wose sampled versions, sampled att= nt, arexn] andyn] given above, ten te delay between x a (t and y a (t is precisely te group delay of amountd

19 Zero-Pase and Linear-Pase Transfer Functions If D is an integer, ten yn] is identical to xn], but delayed by D samples If D is not an integer, yn], being delayed by a fractional part, is not identical to xn] In te latter case, te waveform of te underlying continuous-time output is identical to te waveform of te underlying continuous-time input and delayed D units of time

20 Zero-Pase and Linear-Pase Transfer Functions If it is desired to pass input signal components in a certain frequency range undistorted in bot magnitude and pase, ten te transfer function sould exibit a unity magnitude response and a linear-pase response in te band of interest

21 Zero-Pase and Linear-Pase Transfer Functions Figure below sows te frequency response of a lowpass filter wit a linear-pase caracteristic in te passband Since te signal components in te stopband are blocked, te pase response in te stopband can be of any sape

22 Example 1: Ideal Lowpass Filter Determine te impulse response of an ideal lowpass filter wit a linear pase response: H LP( ejω = e jωno 0,, 0< ω<ωc ω ω π c

23 Example 1: Ideal Lowpass Filter Applying te frequency-sifting property of te DTFT to te impulse response of an ideal ero-pase lowpass filter we arrive at sinω nn n = c( o LP ], < n< π( nn As before, te above filter is non-causal and of doubly infinite lengt, and ence, unrealiable o

24 Example 1: Ideal Lowpass Filter By truncating te impulse response to a finite number of terms, a realiable FIR approximation to te ideal lowpass filter can be developed Te truncated approximation may or may not exibit linear pase, depending on te value of n o cosen

25 Example 1: Ideal Lowpass Filter If we coose n o = N/ witna positive integer, te truncated and sifted approximation ^ sinω n N n c( / LP ] =, 0 n N π( nn / will be a lengtn1 causal linear-pase FIR filter (symmetric filter: see analysis later

26 Example 1: Ideal Lowpass Filter Figure below sows te filter coefficients obtained using te functionsinc for two different values ofn 0.6 N = N = Amplitude 0. Amplitude Time index n Time index n

27 Zero-Pase and Linear-Pase Transfer Functions Because of te symmetry of te impulse response coefficients as indicated in te two figures, te frequency response of te truncated approximation can be expressed as: ^ H LP N ω ^ LP HLP n= 0 ( ej = n] ejωn = ejωn/ were H LP(ω, called te ero-pase response or amplitude response, is a real function of ω ( ω

28 Linear-Pase FIR Transfer Functions It is impossible to design a linear-pase IIR transfer function It is always possible to design an FIR transfer function wit an exact linear-pase response Consider a causal FIR transfer function H( of lengt N1, i.e., of ordern: = N n n = n H( 0 ]

29 Linear-Pase FIR Transfer Functions Te above transfer function as a linear pase if its impulse responsen] is eiter symmetric, i.e., n] = Nn], 0 or is antisymmetric, i.e., n N n] = Nn], 0 n N

30 Linear-Pase FIR Transfer Functions Since te lengt of te impulse response can be eiter even or odd, we can define four types of linear-pase FIR transfer functions For an antisymmetric FIR filter of odd lengt, i.e.,neven N/] = 0 We examine next eac of te 4 cases

31 Linear-Pase FIR Transfer Functions Type 1: N = 8 Type : N = 7 Type 3: N = 8 Type 4: N = 7

32 Linear Linear-Pase FIR Transfer Functions Pase FIR Transfer Functions Type 1: Symmetric Impulse Response wit Odd Lengt In tis case, te degreenis even AssumeN= 8 for simplicity Te transfer functionh( is given by = H ] ] ] ] ( ] ] ] ] ]

33 Linear Linear-Pase FIR Transfer Functions Pase FIR Transfer Functions Because of symmetry, we ave0] = 8], 1] = 7],] = 6], and3] = 5] Tus, we can write ]( ]( ( = H ] ]( ]( ]( ]( { = ]} ]( ]( 4 3 1

34 Linear-Pase FIR Transfer Functions Te corresponding frequency response is ten given by H ( e jω = e j4ω {0]cos(4ω 1]cos(3ω ]cos(ω 3]cos( ω 4]} Te quantity inside te brackets is a real function ofω, and can assume positive or negative values in te range 0 ω π

35 Linear-Pase FIR Transfer Functions Te pase function ere is given by θ( ω =4ωβ wereβis eiter 0 orπ, and ence, it is a linear function ofωin te generalied sense Te group delay is given by dθ (ω τ ( ω = = dω indicating a constant group delay of 4 samples 4

36 Linear-Pase FIR Transfer Functions In te general case for Type 1 FIR filters, te frequency response is of te form j ω jnω/ ~ H ( e = e H ( ω were te amplitude response H ~ (ω, also called te ero-pase response, is of te form ~ H (ω = N ] N / n= 1 N n]cos(ω n

37 Example : A Type 1 Linear Pase Filter Consider wic is seen to be a sligtly modified version of a lengt-7 moving-average FIR filter Te above transfer function as a symmetric impulse response and terefore a linear pase response ] ( = H

38 Example : A Type 1 Linear Pase Filter A plot of te magnitude response of H 0 ( along wit tat of te 7-point moving-average filter is sown below modified filter moving-average Magnitude ω/π

39 Example : A Type 1 Linear Pase Filter Note tat te improved magnitude response obtained by simply canging te first and te last impulse response coefficients of a moving-average (MA filter It can be sown tat we can express ( 1 1 ( H0 = 1 ( 1 6 wic a cascade of a -point MA filter wit a 6- point MA filter Tus, H 0 ( as a double ero at =1, i.e.,ω= π

40 Linear Linear-Pase FIR Transfer Functions Pase FIR Transfer Functions Type : Symmetric Impulse Response wit Even Lengt In tis case, te degreenis odd AssumeN= 7 for simplicity Te transfer function is of te form = H ] ] ] ] ( ] ] ] ]

41 Linear Linear-Pase FIR Transfer Functions Pase FIR Transfer Functions Making use of te symmetry of te impulse response coefficients, te transfer function can be written as ]( ]( ( = H ]( ]( ]( ]( { / / / / / = } ]( ]( / / / /

42 Linear-Pase FIR Transfer Functions Te corresponding frequency response is given by jω j7ω/ 7ω 5 ω H ( e = e {0]cos( 1]cos( ]cos( 3 ω 3]cos( ω} As before, te quantity inside te brackets is a real function ofω, and can assume positive or negative values in te range 0 ω π

43 Linear-Pase FIR Transfer Functions Here te pase function is given by θ( ω = ωβ were againβis eiter 0 orπ As a result, te pase is also a linear function ofωin te generalied sense Te corresponding group delay is τ( ω = indicating a group delay of samples 7 7 7

44 Linear-Pase FIR Transfer Functions Te expression for te frequency response in te general case for Type FIR filters is of te form j ω jnω/ ~ H ( e = e H ( ω were te amplitude response is given by ( 1 / N n= 1 H ~ (ω = N 1 n]cos( ω( n 1

45 Linear-Pase FIR Transfer Functions Type 3: Antiymmetric Impulse Response wit Odd Lengt In tis case, te degreenis even AssumeN= 8 for simplicity Applying te symmetry condition we get H( = { 0]( 1]( ]( 3]( }

46 Linear-Pase FIR Transfer Functions Te corresponding frequency response is given by H ( e jω = e j4ω jπ/ It also exibits a generalied pase response given by θ( ω =4ω β wereβis eiter 0 orπ e {0]sin(4ω ]sin(ω 3]sin( ω} π 1]sin(3ω

47 Linear-Pase FIR Transfer Functions Te group delay ere is τ( ω = 4 indicating a constant group delay of 4 samples In te general case ω H ( e = je j jnω/ ~ H ( ω were te amplitude response is of te form / H ~ (ω = Nn]sin( ωn N n= 1

48 Linear-Pase FIR Transfer Functions Type 4: Antiymmetric Impulse Response wit Even Lengt In tis case, te degreenis even AssumeN= 7 for simplicity Applying te symmetry condition we get 7 / 7 / 7 / 5/ 5/ H( = { 0]( 1]( 3/ 3/ 1/ 1/ ]( 3]( }

49 Linear-Pase FIR Transfer Functions Te corresponding frequency response is given by jω j ω/ jπ/ H e = e e {0]sin( 7ω 1]sin( ( 7 5 ω 3 ω ω} ]sin( 3]sin( It again exibits a generalied pase response given by θ( ω = ω β wereβis eiter 0 orπ 7 π

50 Linear-Pase FIR Transfer Functions Te group delay is constant and is given by τ( ω = 7 In te general case we ave H ( e = je H ( ω were now te amplitude response is of te form j ω jnω/ ~ ( 1 / N n= 1 H ~ (ω = N 1 n]sin( ω( n 1

51 Zero Locations of Linear Zero Locations of Linear-Pase FIR Pase FIR Transfer Functions Transfer Functions Consider first an FIR filter wit a symmetric impulse response: Its transfer function can be written as By making a cange of variable, we can write = = = = N n n N n n n N n H 0 0 ] ] ( ] ] n N n = n N m = = = = = = N m m N N m m N N n n m m n N ] ] ]

52 Zero Locations of Linear-Pase But, FIR Transfer Functions N m = 0 m ] = H( 1 Hence for an FIR filter wit a symmetric impulse response of lengt N1 we ave N 1 H( = H( A real-coefficient polynomialh( satisfying te above condition is called a mirror-image polynomial (MIP m

53 Zero Locations of Linear Zero Locations of Linear-Pase Pase FIR Transfer Functions FIR Transfer Functions Now consider first an FIR filter wit an antisymmetric impulse response: Its transfer function can be written as By making a cange of variable, we get ] ] n N n = = = = = N n n N n n n N n H 0 0 ] ] ( ( ] ] = = = = H m n N N N m m N N n n n N m =

54 Zero Locations of Linear-Pase FIR Transfer Functions Hence, te transfer functionh( of an FIR filter wit an antisymmetric impulse response satisfies te condition N H( = H( A real-coefficient polynomialh( satisfying te above condition is called a antimirror-image polynomial (AIP 1

55 Zero Locations of Linear-Pase FIR Transfer Functions N It follows from te relation H( = ± H( tat if = ξ o is a ero ofh(, so is =1/ ξ o Moreover, for an FIR filter wit a real impulse response, te eros ofh( occur in complex conjugate pairs Hence, a ero at = ξ o is associated wit a ero at = ξ * o 1

56 Zero Locations of Linear-Pase FIR Transfer Functions Tus, a complex ero tat is not on te unit circle is associated wit a set of 4 eros given by ± jφ = re, jφ r e ± = 1 A ero on te unit circle appears as a pair ± jφ = e as its reciprocal is also its complex conjugate

57 Zero Locations of Linear-Pase FIR Transfer Functions Since a ero at =±1 is its own reciprocal, it can appear only singly Now a Type FIR filter satisfies N H( = H( wit degreenodd N HenceH( 1 = ( 1 H( 1 =H( 1 implying H( 1 = 0, i.e.,h( must ave a ero at =1 1

58 Zero Locations of Linear-Pase FIR Transfer Functions Likewise, a Type 3 or 4 FIR filter satisfies Tus N H( = H( N H( 1 =( 1 H( 1 =H( 1 implying tath( must ave a ero at= 1 On te oter and, only te Type 3 FIR filter is restricted to ave a ero at =1 since ere te degreenis even and ence, N 1 H( 1 =( 1 H( 1 =H( 1

59 Zero Locations of Linear-Pase FIR Transfer Functions Typical ero locations sown below Type 1 Type Type 3 Type

60 Zero Locations of Linear-Pase FIR Transfer Functions Summariing (1 Type 1 FIR filter: Eiter an even number or no eros at= 1 and=1 ( Type FIR filter: Eiter an even number or no eros at= 1, and an odd number of eros at =1 (3 Type 3 FIR filter: An odd number of eros at= 1 and=1

61 Zero Locations of Linear-Pase FIR Transfer Functions (4 Type 4 FIR filter: An odd number of eros at= 1, and eiter an even number or no eros at=1 Te presence of eros at =±1 leads to te following limitations on te use of tese linear-pase transfer functions for designing frequency-selective filters

62 Zero Locations of Linear-Pase FIR Transfer Functions A Type FIR filter cannot be used to design a igpass filter since it always as a ero =1 A Type 3 FIR filter as eros at bot = 1 and =1, and ence cannot be used to design eiter a lowpass or a igpass or a bandstop filter

63 Zero Locations of Linear-Pase FIR Transfer Functions A Type 4 FIR filter is not appropriate to design a lowpass filter due to te presence of a ero at= 1 Type 1 FIR filter as no suc restrictions and can be used to design almost any type of filter

64 Bounded Real Transfer Functions A causal stable real-coefficient transfer functionh( is defined as a bounded real (BR transfer function if jω H ( e 1 for all values ofω Let xn] and yn] denote, respectively, te input and output of a digital filter caracteried by a BR transfer function H( jω jω wit X ( e and Y ( e denoting teir DTFTs

65 Bounded Real Transfer Functions Ten te condition H ( e 1implies tat Y ( e jω Integrating te above from π toπ, and applying Parseval s relation we get n= y n] jω X ( e n= jω x n]

66 Bounded Real Transfer Functions Tus, for all finite-energy inputs, te output energy is less tan or equal to te input energy implying tat a digital filter caracteried by a BR transfer function can be viewed as a passive structure jω If H ( e = 1, ten te output energy is equal to te input energy, and suc a digital filter is terefore a lossless system

67 Bounded Real Transfer Functions A causal stable real-coefficient transfer jω functionh( wit H ( e = 1 is tus called a lossless bounded real (LBR transfer function Te BR and LBR transfer functions are te keys to te realiation of digital filters wit low coefficient sensitivity