Optimum Ordering and Pole-Zero Pairing of the Cascade Form IIR. Digital Filter

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1 Optimum Ordering and Pole-Zero Pairing of the Cascade Form IIR Digital Filter There are many possible cascade realiations of a higher order IIR transfer function obtained by different pole-ero pairings and ordering Each one of these realiations will have different output noise power due to product round-offs It is of interest to determine the cascade realiation with the lowest output noise power Copyright 00, S. K. Mitra

2 Optimum Ordering and Pole-Zero Pairing Consider the scaled cascade structure shown below x[n] K e [ n ] a a b 0 b b G G R e e R[ n R [n] ] a R a R b 0 R b R b R y[n] γl [n] l Copyright 00, S. K. Mitra

3 Copyright 00, S. K. Mitra 3 Optimum Ordering and Optimum Ordering and Pole-Zero Pairing Pole-Zero Pairing The scaled noise transfer functions are given by where R G H G R i i R i i,,,, K l l l l l β G R R i H i G l l 0 a a b b b H l l l l l l

4 Copyright 00, S. K. Mitra 4 Optimum Ordering and Optimum Ordering and Pole-Zero Pairing Pole-Zero Pairing Output noise power spectrum due to product round-off is given by The output noise variance is thus σ ω ω γγ R j o e G k P l l l ω π σ σ π π ω γ R j o d e G k l l l R o G k l l l σ

5 Optimum Ordering and Pole-Zero Pairing Note: k l is the total number of multipliers connected to the l-th adder If products are rounded before summation k k R 3 k 5, l,3, K, R l If products are rounded after summation k l, l,, K, R 5 Copyright 00, S. K. Mitra

6 Copyright 00, S. K. Mitra 6 Optimum Ordering and Optimum Ordering and Pole-Zero Pairing Pole-Zero Pairing Recall Thus, Substituting the above in, 0 α β R r r r r,,,, K α α β p p R R i i H F l l l α α β σ ω ω γγ R j o e G k P l l l

7 Copyright 00, S. K. Mitra 7 Optimum Ordering and Optimum Ordering and Pole-Zero Pairing Pole-Zero Pairing we get The output noise variance is now given by σ ω ω γγ R j p p R p o e G F k H k H P l l l l σ σ γ R p p R p o G F k H k H l l l l

8 Optimum Ordering and Pole-Zero Pairing The scaling transfer function F l contains a product of section transfer functions, H i, i,, K, l whereas, the noise transfer function G l contains the product of section transfer functions H i, i l, l, K, R 8 Copyright 00, S. K. Mitra

9 Optimum Ordering and Pole-Zero Pairing Thus every term in the expressions for P γγ ω and σ γ includes the transfer functions of all R sections in the cascade realiation To minimie the output noise power, the norms of H i should be minimied for all values of i by appropriately pairing the poles and eros 9 Copyright 00, S. K. Mitra

10 0 Optimum Ordering and Pole-Zero Pairing Pole-Zero Pairing Rule - First, the complex pole-pair closest to the unit circle should be paired with the nearest complex ero-pair Next, the complex pole-pair that is closest to the previous set of poles should be matched with its nearest complex ero-pair Continue this process until all poles and eros have been paired Copyright 00, S. K. Mitra

11 Optimum Ordering and Pole-Zero Pairing The suggested pole-ero pairing is likely to lower the peak gain of the section characteried by the paired poles and eros Lowering of the peak gain in turn reduces the possibility of overflow and attenuates the round-off noise Copyright 00, S. K. Mitra

12 Optimum Ordering and Pole-Zero Pairing Illustration of pole-ero pairing 3 x x x x x Copyright 00, S. K. Mitra

13 Optimum Ordering and Pole-Zero Pairing After the appropriate pole-ero pairings have been made, the sections need to be ordered to minimie the output round-off noise A section in the front part of the cascade has its transfer function H i appear more frequently in the scaling transfer expressions for P γγ ω and σ γ 3 Copyright 00, S. K. Mitra

14 Optimum Ordering and Pole-Zero Pairing On the other hand, a section near the output end of the cascade has its transfer function H i appear more frequently in the noise transfer function expressions The best locations for H i obviously depends on the type of norms being applied to the scaling and noise transfer functions 4 Copyright 00, S. K. Mitra

15 Optimum Ordering and Pole-Zero Pairing A careful examination of the expressions for P γγ ω and σ γ reveals that if the L -scaling is used, then ordering of paired sections does not affect too much the output noise power since all norms in the expressions are L -norms This fact is evident from the results of the two examples presented earlier 5 Copyright 00, S. K. Mitra

16 Optimum Ordering and Pole-Zero Pairing If L -scaling is being employed, the sections with the poles closest to the unit circle exhibit a peaking magnitude response and should be placed closer to the output end The ordering rule in this case is to place the least peaked section to the most peaked section starting at the input end 6 Copyright 00, S. K. Mitra

17 Optimum Ordering and Pole-Zero Pairing 7 The ordering is exactly opposite if the objective is to minimie the peak noise and an L -scaling is used Ordering has no effect on the peak noise if L -scaling is used The M-file psos can be used to determine the optimum pole-ero pairing and ordering according the above discussed rule Copyright 00, S. K. Mitra

18 Signal-to-Noise Ratio in Low-Order IIR Filters The output round-off noise variances of unscaled digital filters do not provide a realistic picture of the performances of these structures in practice This is due to the fact that introduction of scaling multipliers can increase the number of error sources and the gain for the noise transfer functions 8 Copyright 00, S. K. Mitra

19 Signal-to-Noise Ratio in Low-Order IIR Filters Therefore the digital filter structure should first be scaled before its round-off noise performance is analyed In many applications, the round-off noise variance by itself is not sufficient, and a more meaningful result is obtained by computing instead the signal-to-noise ratio SNR for performance evaluation 9 Copyright 00, S. K. Mitra

20 Signal-to-Noise Ratio in Low-Order IIR Filters The computation of the SNR for the firstand second-order IIR structures are considered here Most conclusions derived from the detailed analysis of these simple structures are also valid for more complex structures Methods followed here can be easily extended to the general case 0 Copyright 00, S. K. Mitra

21 Signal-to-Noise Ratio in Low-Order IIR Filters Consider the causal unscaled first-order filter shown below x[n] α y[ n] γ[ n] Its output round-off noise variance was computed earlier and is given by σ σ o γ α Copyright 00, S. K. Mitra

22 Signal-to-Noise Ratio in Low-Order IIR Filters Assume the input x[n] to be a WSS random signal with a uniform probability density function and a variance σ x The variance σ y of the output signal y[n] generated by this input is then σ σ y x n 0 h [ n] σ x α Copyright 00, S. K. Mitra

23 Signal-to-Noise Ratio in Low-Order IIR Filters The SNR of the unscaled filter is then σ y σ SNR x σγ σo SNR is independent of α However, this is not a valid result since the adder is likely to overflow in an unscaled structure 3 Copyright 00, S. K. Mitra

24 Signal-to-Noise Ratio in Low-Order IIR Filters The scaled structure is shown below along with its round-off error analysis model assuming quantiation after addition of all products x[n] K e[n] α y[ n] γ[ n] 4 Copyright 00, S. K. Mitra

25 Signal-to-Noise Ratio in Low-Order IIR Filters With the scaling multiplier present, the output signal power becomes K σ σ x y α The signal-to-noise ratio is then SNR K σ σ o x 5 Copyright 00, S. K. Mitra

26 Signal-to-Noise Ratio in Low-Order IIR Filters 6 Since the scaling multiplier coefficient K depends on the pole location and the type of scaling being followed, the SNR will thus reflect this dependence The scaling transfer function is given by F H α with a corresponding impulse response f [ n] Z { F } α n µ [ n] Copyright 00, S. K. Mitra

27 7 Signal-to-Noise Ratio in Low-Order IIR Filters To guarantee no overflow, we choose K f [ n] n 0 To evaluate the SNR we need to know the type of input x[n] being applied If x[n] is uniformly distributed with x[ n] its variance is given by σx 3 Copyright 00, S. K. Mitra

28 Signal-to-Noise Ratio in Low-Order IIR Filters The SNR is then given by α SNR 3σo For a b-bit signed fraction with roundoff or two s-complement truncation b σo / The SNR in db is given by SNR db 0log0 α b 8 Copyright 00, S. K. Mitra

29 Signal-to-Noise Ratio in 9 First-Order IIR Filters SNR of first-order IIR filters for different inputs with no overflow scaling Input type WSS, white uniform density WSS, white Gaussian density / 9 σ x Sinusoid, known frequency SNR α 3 σ o α 9 σ o α σ o Typical SNR, db b, α Copyright 00, S. K. Mitra

30 Signal-to-Noise Ratio in Second-Order IIR Filters Consider the causal unscaled second-order IIR filter given below x [n] y[n] α β 30 Copyright 00, S. K. Mitra

31 Signal-to-Noise Ratio in Second-Order IIR Filters 3 Its scaled version along with the round-off noise analysis model, assuming quantiation after addition of all products, is shown below x[n] K e[n] α β y[ n] γ[ n] Copyright 00, S. K. Mitra

32 Signal-to-Noise Ratio in 3 Second-Order IIR Filters For a WSS input with a uniform probability density function and a variance σ x, the signal power at the output is given by σ σ y x K h [ n] n0 The round-off noise power at the output is given by σ σ γ o h [ n] n 0 Copyright 00, S. K. Mitra

33 Signal-to-Noise Ratio in Second-Order IIR Filters Thus, the signal-to-noise ratio of the scaled structure is given by σ y K σ SNR x σγ σ γ The scaling transfer function F is given by F H α β 33 Copyright 00, S. K. Mitra

34 Signal-to-Noise Ratio in Second-Order IIR Filters If the poles of H are at re, then F H r cos θ r Corresponding impulse response is given by n ± jθ r sin n θ f [ n] h[ n] µ [ n] sin θ 34 Copyright 00, S. K. Mitra

35 Signal-to-Noise Ratio in Second-Order IIR Filters 35 The overflow is completely eliminated if K 0 n h[ n] n0 h[ n] is difficult to compute analytically It is possible to establish some bounds on the summation to provide a reasonable estimate of the value of K Copyright 00, S. K. Mitra

36 Signal-to-Noise Ratio in Second-Order IIR Filters 36 The magnitude response of the unscaled second-order section to a sinusoidal input is given by / θ H e j r r cosθ r But, jθ H e cannot be greater than n0 h[ n] as the latter is the largest possible value of the output y[n] for an input x[n] with x[ n] Copyright 00, S. K. Mitra

37 Signal-to-Noise Ratio in 37 Second-Order IIR Filters Moreover, n 0 h[ n] sin θ sin θ n 0 n sin n θ A tighter upper bound on n0 h[ n] is given by 4 n 0 h[ n] π r sin θ r n n r sin θ 0 r Copyright 00, S. K. Mitra

38 Signal-to-Noise Ratio in Second-Order IIR Filters Therefore π r sin θ r r cosθ r K 6 Bounds on the SNR for the all-pole secondorder section can also be derived for various types of inputs 38 Copyright 00, S. K. Mitra

39 39 Signal-to-Noise Ratio in Second-Order IIR Filters As in the case of the first-order section, as the poles move closer to the unit circle r, the gain of the filter increases The input signal then needs to be scaled down to avoid overflow while boosting the output round-off noise This type of interplay between the round-off noise and dynamic range is a characteristic of all fixed-point digital filters Copyright 00, S. K. Mitra

40 Bounded Real Transfer Function Bounded Real Transfer Function - A causal stable real coefficient transfer function H is defined to be a bounded real BR function if it satisfies the following condition: H ω e j, for all valuesof ω 40 Copyright 00, S. K. Mitra

41 4 Bounded Real Transfer Function If the input and the output to the digital filter are x[n] and y[n], respectively with jω and jω X e Y e denoting their DTFTs, then the equivalent condition for the BR property is given by jω jω Y e X e Integrating the above from π to π, dividing by π, and applying Parseval s relation we get y [ n] x [ n] n n Copyright 00, S. K. Mitra

42 4 Bounded Real Transfer Function Thus, a digital filter characteried by a BR transfer function is passive as, for all finiteenergy inputs, Output energy Input energy If ω H e j for all values of ω then it is a lossless system A causal stable transfer function H with jω H e is called a lossless bounded real LBR transfer function Copyright 00, S. K. Mitra

43 Requirements for Low Coefficient Sensitivity A causal stable digital filter with a transfer function H has low coefficient sensitivity in the passband if it satisfies the following conditions: H is a bounded-real transfer function There exists a set of frequencies ω k at which H e jω k 3 The transfer function of the filter with quantied coefficients remains bounded real 43 Copyright 00, S. K. Mitra

44 Requirements for Low Coefficient Sensitivity jω Since H e is bounded above by unity, the frequencies must be in the passband H e ω k jω ω k ω Copyright 00, S. K. Mitra

45 45 Requirements for Low Coefficient Sensitivity Any causal stable transfer function can be scaled to satisfy the first two conditions Let the digital filter structure N realiing the BR transfer function H be characteried by R multipliers with coefficients m i Let the nominal values of these multiplier coefficients assuming infinite precision realiation be m i 0 Copyright 00, S. K. Mitra

46 Requirements for Low Coefficient Sensitivity 46 Because of the third condition, regardless of the actual values of m i in the immediate neighborhood of their design values m i0, the actual transfer function remains BR Consider H e k which for multiplier values is equal to m i0 j ω The third condition implies that if the coefficient m i is quantied, then H e can only become less than j ω k Copyright 00, S. K. Mitra

47 Requirements for Low Coefficient Sensitivity j ω A plot of H e k will thus appear as indicated below 47 Copyright 00, S. K. Mitra

48 Requirements for Low Coefficient Sensitivity Thus the plot of H e ω k will have a erovalued slope at m i m i 0 i.e. jω H e k 0 m i The first-order sensitivity of H e with respect to each multiplier coefficient is ero at all frequencies where jω ω k H e assumes its maximum value of unity j m i m i0 jω 48 Copyright 00, S. K. Mitra

49 Requirements for Low Coefficient Sensitivity Since all frequencies ω k, where the magnitude function is exactly equal to unity, are in the passband of the filter and if these frequencies are closely spaced, it is expected that the sensitivity of the magnitude function to be very small at other frequencies in the passband 49 Copyright 00, S. K. Mitra

50 Requirements for Low Coefficient Sensitivity A digital filter structure satisfying the conditions for low coefficient sensitivity is called a structurally bounded system Since the output energy of such a structure is also less than or equal to the input energy for all finite energy input signals, it is also called a structurally passive system 50 Copyright 00, S. K. Mitra

51 Requirements for Low Coefficient Sensitivity jω If H e, the transfer function H is called a lossless bounded real LBR function, i.e., a stable allpass function An allpass realiation satisfying the LBR condition is called a structurally lossless or LBR system implying that the structure remains allpass under coefficient quantiation 5 Copyright 00, S. K. Mitra

52 Copyright 00, S. K. Mitra 5 Low Low Passband Passband Sensitivity Sensitivity IIR Digital Filter IIR Digital Filter Let G be an N-th order causal BR IIR transfer function given by with a power-complementary transfer function H given by N N N N d d p p p D P G L L 0 N N N N d d q q q D Q H L L 0

53 Low Passband Sensitivity 53 IIR Digital Filter Power-complementary property implies that jω jω G e H e Thus, H is also a BR transfer function We determine the conditions under which G and H can be expressed in the form G { A A } 0 H { A0 A } where A 0 and A are stable allpass functions Copyright 00, S. K. Mitra

54 Low Passband Sensitivity IIR Digital Filter G { A0 A } G must have a symmetric numerator, i.e., p n p N n H { A0 A } H must have an anti-symmetric numerator, i.e., q n q N n 54 Copyright 00, S. K. Mitra

55 Low Passband Sensitivity IIR Digital Filter Symmetric property of the numerator of G implies P N P Likewise, antisymmetric property of the numerator of H implies N Q Q 55 Copyright 00, S. K. Mitra

56 Copyright 00, S. K. Mitra 56 Low Low Passband Passband Sensitivity Sensitivity IIR Digital Filter IIR Digital Filter By analytic continuation, the powercomplementary condition can be rewritten as Substituting in the above we get H H G G D D Q Q P P, D Q D P H G

57 Copyright 00, S. K. Mitra 57 Low Low Passband Passband Sensitivity Sensitivity IIR Digital Filter IIR Digital Filter Using the relations and in the previous equation we get or, equivalently P P N Q Q N ] [ D D Q Q P P N ] ][ [ Q P Q P Q P D N D

58 Low Passband Sensitivity IIR Digital Filter N Using the relations P P and N Q Q we can write P Q N[ P Q ] Let ξk, k N, denote the eros of [P Q] Then, k N, are the eros of ξk [ P Q ] 58 Copyright 00, S. K. Mitra

59 Low Passband Sensitivity IIR Digital Filter 59 [ P Q ] is the mirror image of the polynomial [P Q] From P Q N[ P Q ] it follows that the eros of [P Q] inside the unit circle are also eros of D, and the eros of [P Q] outside the unit circle are eros of D, since G and H are assumed to be stable functions Copyright 00, S. K. Mitra

60 Low Passband Sensitivity IIR Digital Filter 60 Let ξk, k r, be the r eros of [P Q] inside the unit circle, and the remaining N r eros, ξk, r k N, be outside the unit circle Then the N eros of D are given by ξ ξ k k, k r, r k N Copyright 00, S. K. Mitra

61 Low Passband Sensitivity IIR Digital Filter To identify the above eros of D with the appropriate allpass transfer functions A 0 and A we observe that these allpass functions can be expressed as A A 0 P Q G H D P Q G H D 6 Copyright 00, S. K. Mitra

62 Copyright 00, S. K. Mitra 6 Low Low Passband Passband Sensitivity Sensitivity IIR Digital Filter IIR Digital Filter Therefore, the two allpass transfer functions can be expressed as ξ ξ N r k k k A * 0 ξ ξ r k k k A *

63 Low Passband Sensitivity IIR Digital Filter 63 In order to arrive at the above expressions, it is necessary to determine the transfer function H that is power-complementary to G Let U denote the N-th degree polynomial U P N D D N n 0 u n n Copyright 00, S. K. Mitra

64 Low Passband Sensitivity IIR Digital Filter 64 Then Q N Solving the above equation for the coefficients q k of Q we get q 0 u 0 u q q0 u k q q q q k l l n l k N k, k q 0 n 0 u n n Copyright 00, S. K. Mitra

65 Low Passband Sensitivity IIR Digital Filter After Q has been determined, we form the polynomial [P Q], find its eros ξ k, and then determine the two allpass functions A 0 and A It can be shown that IIR digital transfer functions derived from analog Butterworth, Chebyshev and elliptic filters via the bilinear transformation can be decomposed into the sum of allpass functions 65 Copyright 00, S. K. Mitra

66 Low Passband Sensitivity IIR Digital Filter For lowpass-highpass filter pairs, the order N of the transfer function must be odd with the orders of A 0 and A differing by For bandpass-bandstop filter pairs, the order N of the transfer function must be even with the orders of A and A differing by 0 66 Copyright 00, S. K. Mitra

67 Low Passband Sensitivity IIR Digital Filter A simple approach to identify the poles of the two allpass functions for odd-order digital Butterworth, Chebyshev, and elliptic lowpass or highpass transfer functions is as follows Let λk, 0 k N denote the poles of G or H 67 Copyright 00, S. K. Mitra

68 Low Passband Sensitivity IIR Digital Filter λ k Let θ k denote the angle of the pole Assume that the poles are numbered such that θk < θk Then the poles of A 0 are given by λ k and the poles of A are given by λ k 68 Copyright 00, S. K. Mitra

69 Low Passband Sensitivity IIR Digital Filter The pole interlacing property is illustrated below Im 0.5 θλ 6 λ θ 6 θ 0 λ Re Copyright 00, S. K. Mitra

70 Low Passband Sensitivity IIR Digital Filter 70 Example- Consider the parallel allpass realiation of a 5-th order elliptic lowpass filter with the specifications: ωp 0. 4π, α p 0.5dB, and 40 db α s The transfer function obtained using MATLAB is given by G Copyright 00, S. K. Mitra

71 Low Passband Sensitivity IIR Digital Filter Its parallel allpass decomposition is given by G [ ] 7 Copyright 00, S. K. Mitra

72 Low Passband Sensitivity IIR Digital Filter A 5-multiplier realiation using a signed 7- bit fractional sign-magnitude representation of each multiplier coefficient is shown below 7 Copyright 00, S. K. Mitra

73 Low Passband Sensitivity IIR Digital Filter The gain response of the filter with infinite precision multiplier coefficients and that with quantied coefficients are shown below 0 ideal - solid line, quantied - dashed line Gain, db ω/π Copyright 00, S. K. Mitra

74 Low Passband Sensitivity IIR Digital Filter Passband details of the parallel allpass realiation and a direct form realiation Gain, db Parallel Allpass Realiation ideal - solid line, quantied - dashed line ω/π Gain, db Direct Form Realiation ideal - solid line, quantied - dashed line ω/π 74 Copyright 00, S. K. Mitra

Optimum Ordering and Pole-Zero Pairing. Optimum Ordering and Pole-Zero Pairing Consider the scaled cascade structure shown below

Optimum Ordering and Pole-Zero Pairing. Optimum Ordering and Pole-Zero Pairing Consider the scaled cascade structure shown below Pole-Zero Pairing of the Cascade Form IIR Digital Filter There are many possible cascade realiations of a higher order IIR transfer function obtained by different pole-ero pairings and ordering Each one