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

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Vivien Newton
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1 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 Consider the scaled cascade structure shown below G ( ) K e [ n G ] ) R ( e e R+ [ n] R [n] [n] b y[n] b R + γ [n] a a b R b R a b a R b R The scaled noise transfer functions are given by R R G ( Hi ( βi G (,,,, R i i G R+ ( ) where b ( ) + b + b H + a + a G R ( H i ( i Output noise power spectrum due to product round-off is given by R + Pγγ ( ω) σ o k G( e ) and the output noise variance is thus R+ π σ σ ω γ ( o k G e ) dω π π + R σ o k G Note: k is the total number of multipliers connected to the -th adder If products are rounded before summation k k R + 3 k 5,,3,, R If products are rounded after summation k,,,, R + Recall β, α Thus, R β i α r r α,, r+ α βi α R+ r,, R F p Substituting the above in R + Pγγ ( ω) σ o k G ( e ) H p

2 we get σ R o Pγγ ( ω) kr+ H + p k F G e p ( ) H p The output noise variance is now given by σ o σ γ kr+ H + p k H p R F p G The scaling transfer function F ( contains a product of section transfer functions, H i (, i,,, whereas, the noise transfer function G ( contains the product of section transfer functions H i (, i, +,, R Thus every term in the epressions 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 Rule - First, the comple pole-pair closest to the unit circle should be paired with the nearest comple ero-pair Net, the comple pole-pair that is closest to the previous set of poles should be matched with its nearest comple ero-pair Continue this process until all poles and eros have been paired 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 Illustration of pole-ero pairing 3

3 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 epressions for P γγ (ω) and σ γ 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 epressions The best locations for H i ( obviously depends on the type of norms being applied to the scaling and noise transfer functions A careful eamination of the epressions 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 epressions are L -norms This fact is evident from the results of the two eamples presented earlier If L -scaling is being employed, the sections with the poles closest to the unit circle ehibit 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 The ordering is eactly opposite if the obective 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 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 3

4 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 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 comple structures Methods followed here can be easily etended to the general case Consider the causal unscaled first-order filter shown below + γ [n] + y[ n] [ n] α Its output round-off noise variance was computed earlier and is given by σ σ γ o α Assume the input [n] to be a WSS random signal with a uniform probability density function and a variance σ The variance σ y of the output signal y[n] generated by this input is then σ σ σ y h [ n] α n The SNR of the unscaled filter is then σ y σ SNR σγ σ o SNR is independent of α However, this is not a valid result since the adder is likely to overflow in an unscaled structure The scaled structure is shown below along with its round-off error analysis model assuming quantiation after addition of all products e[n] [n] K + + y[ n] + γ [ n] α 4

5 With the scaling multiplier present, the output signal power becomes K σ σ y α The signal-to-noise ratio is then K SNR σ σ o 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] To guarantee no overflow, we choose K f [ n] n To evaluate the SNR we need to know the type of input [n] being applied If [n] is uniformly distributed with [ n] its variance is given by σ 3 The SNR is then given by ( α ) SNR 3σ o For a (b+)-bit signed fraction with roundoff or two s-complement truncation b σ o / The SNR in db is given by SNR db log ( α ) b 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 ( σ / 9) Sinusoid, known frequency SNR ( α ) 3σ o ( α ) 9σ o ( α ) σ o Typical SNR, db (b, α.95) Consider the causal unscaled second-order IIR filter given below [n] + y[n] α + β 5

6 Its scaled version along with the round-off noise analysis model, assuming quantiation after addition of all products, is shown below e[n] [n] K + + y[ n] + γ [ n] α + β For a WSS input with a uniform probability density function and a variance σ, the signal power at the output is given by σ y σ K h [ n] n The round-off noise power at the output is given by σ γ σ o h [ n] n Thus, the signal-to-noise ratio of the scaled structure is given by σ y K σ SNR σ γ σ γ The scaling transfer function F( is given by F( H ( + α + β If the poles of H( are at re θ, then ± F( H ( r cosθ + r Corresponding impulse response is given by n r sin( n + ) θ f [ n] h[ n] µ [ n] sinθ The overflow is completely eliminated if K h[ n] n n 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 The magnitude response of the unscaled second-order section to a sinusoidal input is given by / H ( e θ ) ( ) ( cos ) r r θ + r But, H ( e θ ) cannot be greater than n h[ n] as the latter is the largest possible value of the output y[n] for an input [n] with [ n] 6

7 Moreover, h[ n] rn sin( n + ) θ n sinθ n rn sinθ ( ) sinθ n r A tighter upper bound on n h[ n] is given by 4 n h[ n] π ( r) sinθ Therefore θ ( r) ( r cosθ + r ) K π ( r ) sin 6 Bounds on the SNR for the all-pole secondorder section can also be derived for various types of inputs 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 fied-point digital filters 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 ( e ), for all valuesof ω Bounded Real Transfer Function If the input and the output to the digital filter are [n] and y[n], respectively with X ( e ) and Y( e ) denoting their DTFTs, then the equivalent condition for the BR property is given by Y ( e ) X ( e ) Integrating the above from π to π, dividing by π, and applying Parseval s relation we get y [ n] [ n] n n 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 H ( e ) for all values of ω then it is a lossless system A causal stable transfer function H( with H ( e ) is called a lossless bounded real (LBR) transfer function 7

8 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 eists a set of frequencies ω k at which H ( e ω k ) (3) The transfer function of the filter with quantied coefficients remains bounded real Since H ( e ) is bounded above by unity, the frequencies must be in the passband H ( e ) ω k..4 ω k.6.8 ω 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 Because of the third condition, regardless of the actual values of m i in the immediate neighborhood of their design values m i, the actual transfer function remains BR Consider H ( e ω k) which for multiplier values m i is equal to The third condition implies that if the coefficient is quantied, then ( k m H e ω i ) can only become less than A plot of H ( e ω k) will thus appear as indicated below Thus the plot of H ( e ω k) will have a erovalued slope at m i m i i.e. H ( e ω k) m i The first-order sensitivity of H ( e ) with respect to each multiplier coefficient is ero at all frequencies ω k where H ( e ) assumes its maimum value of unity m i m i 8

9 Since all frequencies ω k, where the magnitude function is eactly equal to unity, are in the passband of the filter and if these frequencies are closely spaced, it is epected that the sensitivity of the magnitude function to be very small at other frequencies in the passband 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 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 Let G( be an N-th order causal BR IIR transfer function given by P( p + p + + p N N G( + d + + d N with a power-complementary transfer function H( given by Q( q + q H ( + d N + + q + + d N N N N Power-complementary property implies that G( e ) + H ( e ) Thus, H( is also a BR transfer function We determine the conditions under which G( and H( can be epressed in the form G { A ( + A ( )} ( H ( { A ( A ( )} where A ( and A ( are stable allpass functions G ( { A ( + A ( )} G( must have a symmetric numerator, i.e., pn p N n H ( { A ( A ( )} H( must have an anti-symmetric numerator, i.e., q n q N n 9

10 Symmetric property of the numerator of G( implies P( ) N P( Likewise, antisymmetric property of the numerator of H( implies N Q( ) Q( By analytic continuation, the powercomplementary condition can be rewritten as G( ) G( + H ( ) H ( which is equivalent to P N [ ( + Q( ][ P( ) Q( )] ) ) N Using the relations P( ) P( and Q( ) NQ( 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 ξ k [ P( Q( ] [ P( Q( ] is the mirror image of the polynomial [P( + Q(] FromP( Q( N [ P( ) + Q( )] it follows that the eros of [P( + Q(] inside the unit circle are also eros of, and the eros of [P( + Q(] outside the unit circle are eros of ), since G( and H( are assumed to be stable functions 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 are given by ξk,, ξk k r r + k N To identify the above eros of with the appropriate allpass transfer functions A ( and A ( ) we observe that these allpass functions can be epressed as P( + Q( A( G( + H ( ( ) ( ) ( P Q A G( H (

11 Therefore, the two allpass transfer functions can be epressed as A ( N k r+ ( ξ * k ) ξ k r ξ A k ( k ξk * In order to arrive at the above epressions, 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 ) ) N n n u n Then ) N n n Q ( u n Solving the above equation for the coefficients q k of Q( we get q u u q q u k q q q q k n k N k, k q After Q( has been determined, we form the polynomial [P( + Q(], find its eros ξ k, and then determine the two allpass functions A ( and A ( ) It can be shown 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 For lowpass-highpass filter pairs, the order N of the transfer function must be odd with the orders of A ( 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 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, k N denote the poles of G( or H(

12 λ k Let θ k denote the angle of the pole Assume that the poles are numbered such that θ k < θ k+ λ k + < θ k + Then the poles of A ( are given by λ k and the poles of A ( ) are given by λ k+ The pole interlacing property is illustrated below Im λ6 θ6 7 θ 6 θ λ 6 λ Re Eample- Consider the parallel allpass realiation of a 5-th order elliptic lowpass filter with the specifications: ω p. 4π, α p.5 db, and α s 4 db The transfer function obtained using MATLAB is given by G( (.49 )( ) ( ) Its parallel allpass decomposition is given by (.49 )( G( [ (.49 )( ] ) ) A 5-multiplier realiation using a signed 7- bit fractional sign-magnitude representation of each multiplier coefficient is shown below The gain response of the filter with infinite precision multiplier coefficients and that with quantied coefficients are shown below ideal - solid line, quantied - dashed line Gain, db ω/π

13 Passband details of the parallel allpass realiation and a direct form realiation Parallel Allpass Realiation ideal - solid line, quantied - dashed line Gain, db ω/π Gain, db Direct Form Realiation ideal - solid line, quantied - dashed line ω/π 3

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

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