Signal processing Part 10 pot

264 Signal Processing Ohsumi & Yamaguchi (2006) to estimate the time-delay of signals in nonstationary random noise, incorporated with the Wigner distribution-based maximum likelihood estimation In this paper the signal detection problem is investigated using the stationarization approach to nonstationary data The model of the corrupting noise is given by an ARMA(p, q) model with unknown time-varying coefficients These coefficient parameters are estimated from the (original) observation data by the Kalman filter Problem Statement Let {y(k)} be the (scalar) observation data taken at sampling time instant tk (k = 1, 2, · · · ), and assume that it can be expressed as y(k) = s(k) + n(k) (k = 1, 2, · · · ), (1) where s(·) is a signal to be detected, whose form is surely known, and is assumed to exist in a brief interval if it exists; and n(·) is the nonstationary random noise In consequence, the observation data {y(k)} becomes nonstationary, but its trend time series is assumed to be removed by the process y(k) = ∆d Y (k), (2) where Y (k) is the original data received by the receiver; ∆Y (k) = Y (k) − Y (k − 1); and d indicates the order In this paper the random noise n(k) is assumed to be given as the output of ARMA(p, q) model with time-varying coefficient parameters: p n(k) + q ∑ α i ( k ) n ( k − i ) = ∑ β j ( k ) w ( k − j ) + w ( k ), i =1 (3) j =1 where w(·) is the white Gaussian noise with zero-mean and variance parameter σ2 ; {αi (·)} and { β j (·)} are slowly and smoothly varying parameters to be specified Then our purpose is to propose a method of detecting the signal s(k) from the noisy observation data {y(k )} The procedure taken in this paper is as follows: (i) First, based on the noise model (3), coefficient functions {αi (·)} and { β j (·)} are estimated using Kalman filter from the observation data {y(k)} ˆ ˆ (ii) Using the estimates {αi (·)} and { β j (·)} obtained in (i), the observation data y(k) is modified to become stationary This procedure is called the stationarization of observation data ˆ (iii) Using the stationarized observation data y(k), the signal detection is based on the model ˆ ˆ y ( k ) = s ( k ) + w ( k ), (4) ˆ where s(k) is the modified signal Equation (4) is familiar in the conventional signal detection problem where the noise is stationary Detection of Signals in Nonstationary Noise via Kalman Filter-Based Stationarization Approach 265 Stationarization of Observation Data Recalling the assumption that the duration of the signal s(k) is short, neglect the signal in the observation data and consider the signal-free case, i.e., y(k) = n(k), then the observation data y(k) is expressed by (1) and (3) as follows: p q y ( k ) = − ∑ αi ( k ) y ( k − i ) + i =1 ∑ β j ( k ) w ( k − j ) + w ( k ) (5) j =1 In order to estimate the time-varying parameters {αi (k)} and { β j (k)} in (5), suppose that they change from step k − to k under random effects {e· (k)} Define vectors      x (k) =     − α1 ( k ) −α p (k) β (k) β q (k)            , v(k) =        − e1 ( k ) −e p (k) e p +1 ( k ) e p+q (k ) Then, {αi (k)} and { β j (k)} are subject to the dynamics,          x ( k + 1) = x ( k ) + v ( k ), (6) (7) 2 where {e· (k)} are assumed to be Gaussian with zero-means and variances τ1 , · · · , τp+q Then, Eq (5) is expressed formally as y(k) = H (k) x (k) + w(k) (8) in which H (k) is given by H (k) = [ y(k − 1), · · · , y(k − p), w(k − 1), · · · , w(k − q)] (9) At this stage it should be noted that the matrix H (k ) consists of the (unmeasurable) past noise sequence {w(·)} To remedy this inadequate situation, we resort to replace it by ˆ H (k) = [ y(k − 1), · · · , y(k − p), νm (k − 1), · · · , νm (k − q)] (10) in which {νm (·)} is the sequence modified from the innovation sequence ν(·) as νm ( ) = c( ) ν( ) ( = k − q, k − q + 1, · · ·, k − 1) , where and (11) ˆ ˆ ν ( ) = y ( ) − H ( ) x ( | − 1) (12) c( ) = + −1 ˆ ˆ H ( ) P ( | − 1) H T ( ) σ (13) ˆ Here, x ( | − 1) and P( | − 1) are the one-step prediction and its covariance matrix computed by Kalman filter for the past interval 266 Signal Processing It is a simple exercise to show that the statistical properties of νm (·) is the same as that of w(·), i.e., E{νm (k)} = and E{|νm (k)|2 } = σ2 (for proof, see Appendix) Then, instead of (8) we have the expression, ˆ y(k) = H (k) x (k) + w(k) (14) ˆ (k) is stated as follows: The procedure for computing H (i) Preliminaries: Assume for the past k (< 0) that {νm (−1), νm (−2), · · ·, νm (−q)} are set approˆ ˆ ˆ priately (may be set all zero), and preassign x (0| − 1), P(0| − 1) and H (0) as initial values Then, at time k (k = 0, 1, 2, · · · ) (ii) Computation of ν( ) and c( ): Compute the innovation ν( ) and coefficient c( ) by (12) and ˆ (13) using H ( ) = [ y( − 1), · · ·, y( − p), νm ( − 1), · · ·, νm ( − q)] (iii) Computation of νm ( ): Compute νm ( ) by (11) using ν( ) and c( ) obtained in the previous step ˆ Repeat Steps (ii) and (iii) for = k − q, k − q + 1, · · ·, k − to obtain H (k) In computing (12) ˆ and (13), x ( | − 1) and P( | − 1) are computed by the Kalman filter (e.g., Jazwinski, 1970): ˆ ˆ x ( + 1| ) = x ( | ) (15) ˆ ˆ x ( | ) = x ( | − 1) + K ( ) ν ( ), ˆ K( ) = P ( | − 1) H T ( ) ˆ ˆ H ( ) P ( | − 1) H T ( ) + σ (16) P ( + 1| ) = P ( | ) + Q (18) ˆ P ( | ) = P ( | − 1) − K ( ) H ( ) P ( | − 1), (19) (17) 2 where Q = diag {τ1 , · · · , τp+q } Thus, the estimates of the coefficient parameters {αi (k)} and { β j (k)} are obtained by the Kalman filter constructed for (7) and (14) (whose form is the same as (15)-(19) replacing by the present k) Under the basic assumption that the coefficient parameters vary slowly and smoothly, they can be treated like constants in an interval Ik around the current time k Write ˆ ˆ them as αik and β jk in Ik Replacing the past {w(k − j)} in (5) by the statistically equivalent ˆ sequence {νm (k − j)}, define the sequence y(k) by p q i =1 ˆ y(k) := y(k) + j =1 ˆ ˆ ∑ αik y(k − i) − ∑ β jk νm (k − j) (20) Then, we have the following adequate approximation for (5), ˆ y(k) = w(k) (21) ˆ which implies that the sequence {y(k)} is stationary because w(k) is the stationary white noise Signal Detection After obtained the estimates of coefficient parameters, the observation process (14) may be written using estimates as ˆ ˆ y(k) = H (k) x (k|k) + w(k) (22) Detection of Signals in Nonstationary Noise via Kalman Filter-Based Stationarization Approach or p q i =1 y(k) + 267 j =1 ˆ ˆ ∑ αik y(k − i) = ∑ β jk νm (k − j) + w(k) (23) Now, let us revive the signal s(k) in the observation data To this, replace {y(k)} formally by {y(k) − s(k)} in (23) to obtain p y(k) + p ˆ ∑ αik y(k − i) = i =1 q ˆ ∑ αik s(k − i) s(k) + i =1 or + ˆ ∑ β jk νm (k − j) + w(k) (24) j =1 ˆ ˆ y ( k ) = s ( k ) + w ( k ), (4)bis ˆ where y(k) has the same form as (20) and p ˆ s(k) = s(k) + ˆ ∑ αik s(k − i) (25) i =1 Note that (4)bis is familiar to us as the mathematical model for the detection problem of signals in stationary noise (e.g., Van Trees, 1968) ˆ ˆ ˆ ˆ Now, consider the binary hypotheses: H 1: y(k) = s(k) + w(k), and H 0: y(k) = w(k), and let Yk ˆ ˆ be the stationarized observation data taken up to k, Yk = {y( ), = 1, 2, · · · , k } Since the additive noise w(k) is white Gaussian sequence with zero-mean and variance σ2 , the likelihoodˆ ˆ ratio function Λ(k) = p{Yk | H }/Yk | H } is evaluated as follows: k ∏ (2π )− Λ(k) = =1 k exp − ∏ (2π ) =1 −1 ˆ ˆ {y( ) − s( )}2 2σ2 ˆ y2 ( ) exp − 2σ2 (26) We use rather its logarithmic form, L(k) := ln Λ(k) = k k ˆ ˆ ˆ s ( ) y ( ) − ∑ s2 ( ) ∑ σ =1 2σ =1 (27) as the signal detector Simulation Studies In this section, we provide a typical set of several simulation results to demonstrate the proposed method (i) Experiment 268 Signal Processing The top of Fig.1 depicts a sample path of the observation data {Y (k)} generated by calculating the output of the ARMA(4, 1)-model: n ( k ) = − ∑ α i ( k ) n ( k − i ) + β ( k ) w ( k − 1) + w ( k ) i =1 Time-varying coefficients {αi (k)} and β(k) are set as α1 (k) = −1.24 sin(0.002k − 0.95), α2 (k) = 0.38 − cos(0.004k − 1.89) α3 ( k ) = α1 ( k ), α4 (k) = 1, β(k) = 1.5 The bottom of Fig.1 shows a signal embedded in the observation data around k = 300 given by s( ) = 12 e−2.78 sin(1.26 ), OBSAERVATION DATA Y(k) ˆ where = k − 300 Figure depicts trend-removed data and stationarized data y(k) The trend was removed by setting d = For the Kalman filter (15)∼(19), the parameters are set 150 100 50 -50 -100 -150 100 200 300 400 500 600 700 800 900 1000 600 700 800 900 1000 EMBEDDED SIGNAL s(k) k step 150 100 50 -50 -100 -150 100 200 300 400 500 k step Fig A sample path of the observation data Y (k) (top) and the embedded signal s(k ) (bottom) TREND-REMOVED DATA y(k) Detection of Signals in Nonstationary Noise via Kalman Filter-Based Stationarization Approach 60 40 20 -20 -40 -60 STATIONARIZED OBSERVATION DATA 269 100 200 300 400 500 600 700 800 900 1000 600 700 800 900 1000 k step 20 10 -10 -20 100 200 300 400 500 k step ˆ Fig The trend-removed data y(k) (top) and the stationarized observation data y(k) (bottom) 270 Signal Processing 1500 Log-likelihood ratio L(k) 1000 500 -500 -1000 100 200 300 400 500 600 700 800 900 1000 k step Fig Log-likelihood function L(k) as Q = diag {0.05, 0.05, 0.05, 0.05, 0.05} and σ2 = 40 It should be noted that from Fig the observation data is well stationarized and that even in this figure the signal emerges from the background noise Figure shows the result of signal detection by the current log-likelihood ratio function L(k) Clearly, it exhibits a salient peak around the true time instant k = 300 and this shows the existence of the signal (ii) Experiment Efficacy of the signal detector proposed in this paper is also tested for the pulse signal Figure depicts observation data and embedded three pulses Random noise n(k) is generated by the same manner of previous simulation with same coefficients αi (k) and β(k) As a signals s(k), a train of pulses with same magnitude is considered:   20 for Di ≤ k < Di + (i = 1, 2, 3) s(k) =  otherwise, where D1 = 200, D2 = 500, D3 = 800 Detection of Signals in Nonstationary Noise via Kalman Filter-Based Stationarization Approach 271 OBSAERVATION DATA Y(k) ˆ Figure depicts trend-removed data and stationarized data y(k) The trend was also removed by setting d = The parameters of Kalman filter are set as the same of previous experiment Figure shows the result of signal detection Clearly, log-likelihood ratio function L(k) has large value around each time when each pulse exists Thus the signal detection is well succeeded 200 100 -100 -200 100 200 300 400 500 600 700 800 900 1000 600 700 800 900 1000 k step EMBEDDED SIGNAL s(k) 200 100 -100 -200 100 200 300 400 500 k step Fig A sample path of the observation data Y (k) (top) and the pulse signal s(k) (bottom) 272 Signal Processing TREND-REMOVED DATA y(k) 80 60 40 20 -20 -40 -60 100 200 300 400 500 600 700 800 900 1000 600 700 800 900 1000 STATIONARIZED OBSERVATION DATA k step 30 20 10 -10 -20 100 200 300 400 500 k step ˆ Fig The trend-removed data (top) and the stationarized observation data y(k) (bottom) 278 Signal Processing Description and discussion of the proposed allpole filter design We describe the design procedure based on general equations for the allpole filter proposed in Section The parameters of the design are the constant α, the number L, the corresponding frequency values ω l , l = 1, , L, phase values φD (ω l ), l = 1, , L, group delays τ (ω l ), l = 1, , L, and degrees of flatness Kl , l = 1, , L For the real case, i.e., f In = and α is a real constant, the relations (10a) and (10b) become N ∑ (n + τ )k cos(ωn − φD (ω )) f n = −τ k cos(φD (ω )), k odd, (12a) n =1 N ∑ (n + τ )k sin(ωn − φD (ω )) f n = τ k sin(φD (ω )), k even (12b) n =1 Similarly, the condition (11), for the real case becomes N = ( K + 2) + ( K + 2) + ⋅ ⋅ ⋅ + ( K L + 2) (13) The algorithm is described in the following steps: Step Compute the order of the allpole filter N, using (13) for the real case, and (11) for the complex case Step Substitute the frequencies ω l , l = 1, , L, group delays τ (ω l ) and phases φD (ω l ) into (12), for the real case, or (10), for the complex case Step Calculate the filter coefficients f n solving the resulting set of equations The following example illustrates the design of real allpole filter D (z), (α = 1) using three desired frequency points, L = Example The design parameters are shown in Table l ωl φD ( ω l ) τ ( ωl ) Kl π/5 π/2 4π/5 π/3 π/4 π/5 3 Table Design parameters in Example 1, using L = and α = Step From (13), the estimated value of N is 22 Step We substitute the frequencies ω l , group delays τ (ω l ) and phases φD (ω l ), l = 1, , into (12) Step Solving the resulting linear equations, we get the filter coefficients f n Figure 1a shows the corresponding group delay, while the phase response is presented in Fig 1b The desired phases at ω = π/5, ω = π/2 and ω = 4π/5 are also indicated in Fig 1b The following example illustrates the complex case Example We design the complex allpole filter with characteristics given in Table Step The order N of the allpole filter is 13 (see (11)) Direct Design of Infinite Impulse Response Filters based on Allpole Filters Group delay Phase response Normalized phase Samples 279 −2 0.8 0.6 0.4 φD1 φD2 φD3 0.2 −4 0.1 0.2 0.3 Normalized frequency 0.4 0.5 0.1 0.3 0.2 Normalized frequency (a) 0.5 0.4 (b) Fig Phase response and group delay of the designed real allpole filter in Example l ωl φD ( ω l ) τ ( ωl ) Kl π/3 4π/5 8π/5 π/6 − π/20 −3π/20 1/2 1/2 1/2 6 Table Design parameters in Example The value L is and α = Step Using (10a) and (10b), we obtain the set of linear equations with 26 unknowns coefficients; 13 for f Rn and 13 for f In Step Solving the resulting set of equations, we get the coefficients of the complex allpole filter Figure illustrates the phase response and group delay of the designed allpole filter Group delay Phase response 0.4 0.2 Normalized phase Samples 0.5 φD1 φD3 φD2 −0.2 −0.4 −0.6 −0.5 −0.8 −1 0.25 0.5 0.75 Normalized frequency (a) −1 0.25 0.5 0.75 Normalized frequency (b) Fig Group delay and phase response of the complex allpole filter D (z) in Example 280 Signal Processing 3.1 Relationships between allpole filters and allpass filters We consider the relations between allpole filters of order N and allpass filters An allpass filter A(z) is related with an allpole filter as follows (Selesnick, 1999), ˜ D (z) α F (z) , = z− N ∗ A(z) = z− N ˜ (z) α F (z) D (14) ˜ where D (z) is the paraconjugate of D (z), that is, it is generated by conjugating the coefficients of D (z) and by replacing z by z−1 The phase φ A (ω ) of A(z) can be expressed as φ A (ω ) = − ωN + 2φD (ω ), (15) where the desired phase φD (ω ) is given by φD ( ω ) = φ A (ω ) + ωN (16) From (15), the group delay of the complex allpass filter τA (ω ) is given by τA (ω ) = N + 2τ (ω ), (17) where τ (ω ) is the group delay of D (z) Using (17), it follows τA (ω ) − N (18) It is well known that the structures based on allpass filters exhibit a low sensitivity to the filter quantization and a low noise level (Mitra, 2005) Therefore, the relationship (14), between allpass and allpole filters, gives the possibility to use efficient allpass structures in the proposed design τ (ω ) = Promising special cases The proposed allpole filters have desired phases, group delays and degrees of flatness at a specified set of frequency points In this section we introduce some new special cases of the proposed design (10), which are used for the design of complex allpole filters, complex wavelet filters, and linear-phase IIR filters 4.1 First order allpole filters Using (12), the filter coefficient f R1 is computed as follows: f R1 = sin(φD1 ) , sin(ω1 − φD1 ) (19) where φD1 is the desired phase at ω = ω1 To ensure the stability of the allpole filter, we have tan (2φD1 ) > − cos(2ω1 ) sin(2ω1 ) (20) Direct Design of Infinite Impulse Response Filters based on Allpole Filters 281 Similarly for the complex case, the filter coefficient f is f1 = sin(φα − φD2 )e j( ω1 +φα −φD1 ) − sin(φα − φD1 )e j( ω2 +φα −φD2 ) , sin(ω1 − ω2 + φD2 − φD1 ) (21) where φD1 and φD2 are the phases of the allpole filter at the desired frequency points ω = ω1 and ω = ω2 , respectively The stability of the allpole filter is satisfied if the following equation holds cos(ω1 − ω2 + φα − φD1 ) − ∣ cos(φD1 − φα )∣ tan(φD2 − φα ) < (22) sin(ω1 − ω2 + φα − φD1 ) + sin(φD1 − φα ) 4.2 Second order allpole filter We consider the following two cases Case For ω = ω1 , we specify the desired phase φD1 and group delay τ Substituting these conditions into the general equations (12), the resulting filter coefficients are (τ + 1) sin(2ω1 ) − sin(2ω1 − 2φD1 ) , (τ + 1) sin ω1 − sin(ω1 − φD1 ) cos(2ω1 − φD1 ) τ sin ω1 + sin(φD1 ) cos(ω1 − φD1 ) = (τ + 1) sin ω1 − sin(ω1 − φD1 ) cos(2ω1 − φD1 ) f R1 = − (23) f R2 (24) Additionally, the condition for the stability of the allpole filter is τ > −1 + ∣ sin(2ω1 − 2φD1 )∣ sin ω1 (25) Case For two phases φD1 and φD2 at the frequencies ω1 and ω2 , the filter coefficients are f R1 = f R2 = sin(2ω1 − φD1 ) sin(φD2 ) − sin(φD1 ) sin(2ω2 − φD2 ) , sin(ω2 − φD2 ) sin(2ω1 − φD1 ) − sin(ω1 − φD1 ) sin(2ω2 − φD2 ) sin(φD1 ) sin(ω2 − φD2 ) − sin(ω1 − φD1 ) sin(φD2 ) sin(ω2 − φD2 ) sin(2ω1 − φD1 ) − sin(ω1 − φD1 ) sin(2ω2 − φD2 ) (26) (27) Furthermore, the stability of the allpole filter is guaranteed if the equation tan(ω1 − φD1 ) < − sin ω1 sin ω2 tan (ω2 − φD2 ) cos ω1 cos ω2 − + ∣ cos ω1 − cos ω2 ∣ (28) is satisfied 4.3 Complex Thiran allpole filters We generalize the result proposed by Thiran (Thiran, 1971), for the design of real allpole filters that are maximally flat at ω = 0, to include both the real and complex cases The required design specifications are the order of the allpole filter N, group delay τ (ω ) at ω = 0, τ0 , degree of flatness K, and the phase value φα Consequently, the allpole filter must satisfy: 𝒜.1 The degree of flatness at ω = is K, where K can be either 2N − or 2N − 𝒜.2 The phase value φD (ω ) is equal to zero at ω = 282 Signal Processing 4.3.1 Degree of flatness K = 2N − Substituting conditions 𝒜.1 and 𝒜.2 into the set of equations (10), we compute the complex coefficients as follows ( ) ) N 2(2τ0 + 1)n−1 ( f n = (−1)n τ0 + ne j( φα−π/2) sin φα , (29) n (2τ0 + N + 1)n where n = 1, , N, the binomial coefficient is given by ( ) N! N , = n n!( N − n )! (30) and the Pochhammer symbol ( x )m indicates the rising factorial of x, which is defined as (Andrews, 1998), { ( x )( x + 1)( x + 2) ⋅ ⋅ ⋅ ( x + m − 1) m > 0, ( x )m = (31) m = The expression in (29) is the extension of the result proposed in (Thiran, 1971), which includes both real and complex cases If φα is or π, the imaginary coefficients are zero, and the result is a real allpole filter, consistent with (Thiran, 1971) For φα = ± π/2, the filter is a real allpole filter (this case is not included in (Thiran, 1971)) For all other phase values, the imaginary coefficients are strictly non-zero, i.e., the filter is complex 4.3.2 Degree of flatness K = 2N − In this case, in order to get a degree of flatness K = 2N − 3, we set f IN = Consequently, the filter coefficients are ( ) ( ) N 2(2τ0 + 1)n−1 (n − N )e jφα cos φα f n = (−1)n τ0 + n + n , (32) n (2τ0 + N + 1)n 2τ0 + N where n = 0, , N In contrast with (32), to obtain a different solution, we now set f RN = Therefore, we have ( ( ( ) )) N 2(2τ0 + 1)n−1 ne jφα ( N − n )(τ0 + N cos2 φα ) f n = (−1)n τ0 + n − τ0 + n + , n (2τ0 + N + 1)n N cos φα 2τ0 + N (33) where n = 0, , N We illustrate the design with one example Example The desired phase φα , and the group delay τ0 at ω = 0, are − π/6, and 7/3, respectively The order N of the filter is We compute the corresponding filter coefficients using (29), (32), and (33) The resulting group delays of D (z) are shown in Fig 3a, while the phase responses of the designed filters are shown in Fig 3b 4.4 Complex allpole filter with flatness at ω = and ω = π Now, we present the design of complex allpole filters of order N (any positive integer) with flatness at ω = and ω = π The design conditions are: (More detailed explanation is given in Section 5.1.) ℬ The phase response of D (z) is flat at the frequency points ω = and ω = π with group delays τ (0) = τ (π ) = − N/2 Direct Design of Infinite Impulse Response Filters based on Allpole Filters Group delays Phase responses K = using (29) Nomalized phase Sampes K = using (29) 0.5 K = using (32) K = using (33) K = using (32) K = using (33) −0.5 −1 −1 −2 283 0.25 0.5 0.75 Normalized frequency (a) −1.5 0.25 0.5 0.75 Normalized frequency (b) Fig Group delays and phase responses of the complex allpole filters in Example ℬ The degree of flatness at these frequency points is the same, i.e., K = N − ℬ The phase values of the allpole filter φD (ω ) at ω = and ω = π, are and π (2N + (2l + 1))/4, respectively, where l is an integer ℬ The desired phase value φD (ω ) at the given frequency ω = ω p is φp , i.e., φp = φD (ω p ) Substituting conditions ℬ 1–ℬ into (10a) and (10b) and solving the resulting set of linear equations, we arrive at ⎧� �  N  n even, ⎨ n f n = � �� (34) �  N √ j(2φα + π )  ⎩ −j n odd, 2e n where � � � � � � ω �� ωp N p φα = ∠ −j − − (−1)⌈ N/2⌉ cot φp − , − tan N 2 (35) and ∠{⋅} indicates the angle of {⋅}, while ⌈⋅⌉ stands for the floor function Next example illustrates the proposed design where the parameters of the design are the filter order N and the phase value φp at the frequency point ω p Example We design a complex allpole filter using the following specifications: the order of the allpole filter is N = and the phase value φD (ω ) at ω p is 1.2π, where ω p = 0.3π The group delay and phase response of the designed filter are presented in Fig 4a and 4b, respectively 4.4.1 Closed form equations for the singularities of the allpole filter In the following, we consider the computation of the poles of D (z) Using (34), we obtain the z-transform of the denominator of D (z) defined in (1) as, � � � � √ N −n N −n F (z) = ∑ z + ( 2e j(2φα+π/4) − j) ∑ z n n n even n odd (36) 284 Signal Processing Magnitude response Phase response 1.2 1.1 0.125 Normalized phase 10 −20 Gain, dB −40 −60 0.15 0.175 −80 −100 1.3 0.1 0.2 0.3 Normalized frequency 0.4 0.5 0.25 0.5 0.75 Normalized frequency (a) (b) Fig Group delay and phase response and of the complex allpole filter in Example After some computations, we get ] e jφα [ F (z) = √ (cos φα − sin φα )(1 + z−1 ) N − (j − 1) sin φα (1 − z−1 ) N (37) Therefore, the corresponding poles are pk = where k = 0, , N − 1, and γk = ( √ γk + , γk − − cot φα )1 (38) N e−j 8k + 4N π (39) 4.5 Complex allpole filters with flatness at ω = 0, and ω = ± ωr In this section, we design a complex allpole filter with the following characteristics: 𝒞 The order N is even 𝒞 The allpole filter has flat group delay at the frequency points ω = 0, ω = − ωr , and ω = ωr The degrees of flatness are K1 (ω = 0) = N − 2, K2 (ω = ± ω r ) = N/2 − The group delay at those frequency points is τ (0) = τ (± ω r ) = − N/2 𝒞 The desired allpole phase value φD (ω ) at the given frequency ω = ω p is φp , i.e., φp = φD ( ω p ) 𝒞 The phase values of the allpole filter φD (ω ) at ω = 0, ω = − ω r, and ω = ω r are 0, π/3 + ω r N/2, and π/3 − ω r N/2, respectively Substituting conditions 𝒞 1– 𝒞 into (10a) and (10b) and solving the resulting set of linear equations, we have [( ) ] ( ) N 4e jφα N/2 − √ (40) f n = (−1)n c N,n (ω r ) cos (φα + π/6) , n n Direct Design of Infinite Impulse Response Filters based on Allpole Filters 285 where n = 0, , N/2, φα = ∠ and {√ } √ 3Rp cot(φp − ω p N/2) + + j 3( Rp + 1) , Rp = where CN (ω r , ω p ) = −2 N −1 sin N ωp ) , (42) ) ( ) N/2 c N,n (ω r ) cos ( N/2 − n )ω p n (43) c N,N/2 (ω r ) + 2CN (ω r , ω p ) N/2−1 ∑ ( (−1) N/2+n n =1 ( (41) The function c N,n (ω r ) for different values of N is given in Table Moreover, we have c N,0 (ω r ) = and f n = f N −n Example The desired design specification is as follows: the allpole filter order is equal to 8, ω p = 0.35π, ω r = 0.75π, and φp = 1.5π The resulting group delay and phase response of the designed filter are shown in Fig Group delay Phase response −1 10 Normalized phase Samples −2 −3 −4 −5 1.5 0.15 0.175 0.2 −6 −7 0.5 0.25 0.75 Normalized frequency 0 (a) 0.25 0.5 0.75 Normalized frequency (b) Fig Group delay and phase response and of the designed complex allpole filter in Example 5 Design of IIR filters based on allpole filters 5.1 Direct design of linear-phase IIR Butterworth filters A filter H (z) has linear-phase if, ˜ H (z) = cz−k H (z), z−k (44) is the delay, the complex constant c has unit magniwhere H (z) is not necessary causal, ˜ tude and H (z) is the paraconjugate of H (z), that is, it is generated by conjugating the coefficients of H (z) and by replacing z by z−1 It has been shown that causal Finite Impulse Response (FIR) filters can be designed to have linear-phase However, Infinite Impulse Response (IIR) filters can have linear-phase property only in the noncausal case (Vaidyanathan & Chen, 1998), (the phase response is either zero or π) It has been recently shown that filters with the linear-phase property are useful in the filter 286 Signal Processing c N,n ( ω r ) = c N,N −n ( ω r ) N n 1 − cos( ω r ) − cos( ω r ) − cos(2ω r ) − cos( ω r ) − cos(2ω r ) 10 − cos( ω r ) − cos(3ω r ) − cos( ω r ) − cos(2ω r ) − cos( ω r ) − cos(3ω r ) 17 − 16 cos(2ω r ) − cos(4ω r ) 10 − cos( ω r ) − cos(2ω r ) − cos( ω r ) − cos(3ω r ) 11 − 10 cos(2ω r ) − cos(4ω r ) 126 − 100 cos( ω r ) − 25 cos(3ω r ) − cos(5ω r ) 12 − cos( ω r ) − cos(2ω r ) 11/2 − 9/2 cos( ω r ) − cos(3ω r ) − cos(2ω r ) − cos(4ω r ) 66 − 50 cos( ω r ) − 15 cos(3ω r ) − cos(5ω r ) 262 − 225 cos(2ω r ) − 36 cos(4ω r ) − cos(6ω r ) 14 − cos( ω r ) − cos(2ω r ) 26/5 − 21/5 cos( ω r ) − cos(3ω r ) − cos(2ω r ) − cos(4ω r ) 143/3 − 35 cos( ω r ) − 35/3 cos(3ω r ) − cos(5ω r ) 127 − 105 cos(2ω r ) − 21 cos(4ω r ) − cos(6ω r ) 1761 − 1225 cos( ω r ) − 441 cos(3ω r ) − 49 cos(5ω r ) − cos(7ω r ) 16 − cos( ω r ) − cos(2ω r ) − cos( ω r ) − cos(3ω r ) 37/5 − 32/5 cos(2ω r ) − cos(4ω r ) 39 − 28 cos( ω r ) − 10 cos(3ω r ) − cos(5ω r ) 87 − 70 cos(2ω r ) − 16 cos(4ω r ) − cos(6ω r ) 715 − 490 cos( ω r ) − 196 cos(3ω r ) − 28 cos(5ω r ) − cos(7ω r ) 3985 − 3136 cos(2ω r ) − 784 cos(4ω r ) − 64 cos(6ω r ) − cos(8ω r ) Table Function c N,n (ω r ) for different values of N bank design and the Nyquist filter design and different methods have been proposed for this design (Djokic et al., 1998; Powell & Chau, 1991; Surma-aho & Saramaki, 1999) A linear-phase lowpass IIR filter H (z) can be expressed in terms of complex allpass filters as (Zhang et al., 2001), ] 1[ ˜ H (z) = (45) A ( z ) + A( z ) , where A(z) is a complex allpass of order N (see (14)) We can note that the filter defined in (45) satisfies the relation (44) if k = and c = Direct Design of Infinite Impulse Response Filters based on Allpole Filters 287 The main goal is to propose a new technique to design real and complex IIR filters with linearphase, based on general design of Section 3, where the design specification is same as in traditional IIR filters design based on analog filters, i.e., the passband and stopband frequencies, ω p and ω s , the passband droop Ap , and the stopband attenuation As , shown in Fig H e jω Ap As ωp ωs π Frequency Fig Design parameters for low pass filter We relate the design of linear-phase IIR filter with allpass filter and in the next section we use the general approach to design the corresponding allpole filter First, we establish the conditions which the auxiliary complex allpass filters from (45) has to satisfy From (45), the magnitude response of H (z) can be expressed as, ( ) ∣ H (e jω )∣ = cos φ A (ω ) , for all ω (46) The magnitude responses of ∣ H (e jω )∣ at ω = 0, and ω = π are and 0, respectively (see Fig 6) Therefore, the values of φ A (ω ) at these frequency points are and (2l + 1)π/2, respectively, where l is an integer Since the magnitude response of H (z) decreases monotonically, relation (46) can be rewritten as, ) ( ∣ H (e jω )∣ = cos φ A (ω ) , ≤ ω ≤ π (47) Note that ∣ H (e jω )∣ has a flat magnitude response at ω = and ω = π, and that the filter A(z) has a flat phase response at the same frequency points As a consequence, the corresponding group delays τA (0) and τA (π ) are equal to Considering the value Ap in dB we write 20 log10 ∣ H (e jω )∣ ω =ωp = − Ap From (47) it follows, (48) ( ) φpA = φ A (ω p ) = arccos 10− Ap /20 (49) In summary, the conditions that the auxiliary complex allpass filter in (45) needs to satisfy are the following: 𝒟 The phase values of φ A (ω ) at ω = and ω = π are and (2l + 1)π/2, respectively 𝒟 The phase response of A(z) is flat at ω = and ω = π Therefore, τA (0) = τA (π ) = 288 Signal Processing 𝒟 The phase value φpA is controlled by Ap (see (49)) In the following, we use the results from Section 3.1 and the Conditions 𝒟 1– 𝒟 in order to obtain the corresponding conditions for the allpole filter D (z) 5.1.1 Design of flat linear-phase IIR filters based on complex allpole filters We relate the allpass filter from (45) with the corresponding allpole filter Using (16) and the phase values φ A (ω ) at ω = and ω = π (see Condition 𝒟 1), we get φ(0) = and φ(π ) = π (2N + (2l + 1))/4 Now, from (18) and Condition 𝒟 2, we have τ (0) = τ (π ) = − N/2 Finally, the following relation is obtained using Condition 𝒟 and (16), ( ) arccos 10− Ap /20 + ω p N (50) φD ( ω p ) = φp = As a consequence, the corresponding conditions that the allpole filter D (z) has to satisfy are: ℰ The phase values of D (z) at ω = and ω = π are and π (2N + (2l + 1))/4, respectively ℰ The group delay τ (ω ) of D (z) at ω = and ω = π are − N/2 ℰ The phase value of D (z) at ω p , φD (ω p ), is given by (50) For a filter having coefficients given in (34) the Conditions ℰ and ℰ are satisfied From the Condition ℰ and (35), the corresponding value of φα ( N, ω p , Ap ) is equal to where { ( ω p )} φα ( N, ω p , Ap ) = ∠ −j − − (−1)⌈ N/2⌉ Ap ′ tan N , Ap ′ = √ (51) 10 Ap /20 + − (52) 10 Ap /20 − We note that the resulting allpole filter has a causal and an anticausal parts The causal part can be implemented with the well known structures for allpass filters while the anticausal part can be implemented with the structures proposed in (Vaidyanathan & Chen, 1998) The degree of flatness of the allpass filter A(z) at ω = and ω = π is equal to N − Based on this result it can be shown that we have 2N − null derivatives in the square magnitude response ∣ H (e jω )∣2 at ω = and ω = π 5.1.2 Closed form equations for the singularities of H (z) It follows from (37) and (45) that the transfer function H (z) is given as, H (z) = where (1 + z −1 ) N E ( z ) , ˜ 2z− N F (z) F(z) (53) E (z) = (1 − sin(2φα ))(1 + z−1 ) N + ((j + 1) − (j − 1)) sin φα (cos(φα ) − sin(φα ))(1 − z−1 ) N (54) Direct Design of Infinite Impulse Response Filters based on Allpole Filters 289 We note that the transfer function H (z) has N zeros at z = −1 and the other zeros are at (see (54)), β +1 , (55) zk = k βk − where k = 0, , N − 1, and the parameter β k is given by, ⎧� �1  1−cos(2φα ) 2N e j 2π k N  N even,  1−sin(2φα ) ⎨ βk = (56) �   1−cos(2φα ) � 2N j 4k −1 π ⎩ e 2N N odd 1−sin(2φ ) α It is easily shown that the absolute values of zk in (55), for even values of N, are always different than However, there also exists one absolute value of zk , for N odd, which is equal to 1, i.e., there is a zero on the unit circle The corresponding frequency ω0 is expressed as, �1 � − cos(2φα ) 2N (57) ω0 = π + arctan − sin(2φα ) As a consequence, the frequency at which H (e jω ) is equal to −1 is given by � �1 1 − cos(2φα ) 2N ω1 = π + arctan − sin(2φα ) (58) Finally, the transfer function H (z) has 2N poles which are poles of the corresponding complex � allpole filters D (z) and D (z) (see Section 4.4.1) 5.1.3 Description of the algorithm The proposed algorithm is described in the following steps: Step Estimate the order of the allpole filter using the following equation, which can be obtained by solving φα ( N, ω p , Ap ) = φα ( N, ω s , As ), � ′ �⎥ ⎢ � � Ap ⎥ ⎢ log ′ ⎢ 10 As ⎥ 10 Ap /20 + 10 As /20 + ′ ′ � ′ � ⎦ , Ap = − 1, As = − 1, (59) N=⎣ Ap /20 − ωs 10 As /20 − 10 log ′ 10 ωp where ⌊⋅⌊ is the ceiling function Step From the values N, ω p and Ap , compute the phase value φα ( N, ω s , As ), using (51) Step Using (34), compute the filter coefficients f n Step Calculate the filter coefficients of H (z) using (45) We illustrate the procedure with the following example Example We design the IIR linear-phase lowpass filter with the passband and stopband frequencies ω p = 0.25π and ω s = 0.5π, respectively The passband droop is Ap = dB, while the stopband attenuation is As = 65 dB Step Using (59), we estimate N = 10 As a consequence, the filter H (z) is real Step We calculate the phase value φα ( N, ω s , As ), to be φα ( N, ω s , As ) = −0.749925π Step The filter coefficients f n are computed from (34) Step We compute the coefficients of the designed filter H (z) The magnitude response of the designed filter is given in Fig 290 Signal Processing Passband detail Magnitude response 0 −0.5 Gain, dB −20 −1 −1.5 −40 −60 0.025 0.05 0.075 0.1 Stopband detail 0.125 −65 −80 −100 −60 0.1 0.2 0.3 Normalized frequency 0.4 0.5 −70 0.225 (a) 0.25 Normalized frequency 0.275 (b) Fig Example 5.1.4 Linear-phase IIR highpass filter design Now, we extend the proposed algorithm for lowpass filter to highpass filter design Using the power-complementary property (Vaidyanathan et al., 1987), it can be shown that the corresponding complementary filter of H (z), defined in (45), is given by H1 (z) = ] [ ˜ A(z) − A(z) , 2j (60) where H1 (z) is a highpass filter Using (60), the phase value φpA is expressed as, ( ) φpA = arcsin 10− Ap /20 Similarly, the phase value φα ( N, ω p , Ap ) is given by, { } ( ) 2(−1)⌊ N/2⌋ N ωp − φα ( N, ω p , Ap ) = ∠ −(j + 1) cot Ap ′ (61) (62) Finally, the filter coefficients of H1 (z) are computed using (60) The following example illustrates the procedure Example 7.The parameters of the design of the highpass filter are: the passband and stopband frequencies are ω p = 0.75π and ω s = 0.4π, respectively The stopband attenuation and passband droop are 50 dB and dB, respectively The resulting filter order is equal to and φα ( N, ω p , Ap ) = −0.002569π The magnitude response, the passband and stopband details of the designed filter are shown in Fig 5.2 Direct design of linear-phase IIR filter banks The modified two-band filter bank (Galand & Nussbaumer, 1984), is shown in Fig The analysis filter H0 (z) and the synthesis filter G0 (z) are lowpass filters, while the analysis filter H1 (z) and the synthesis filter G1 (z) are highpass filters However, both the analysis and the synthesis filters are not causal As a difference with traditional structure, in this structure Direct Design of Infinite Impulse Response Filters based on Allpole Filters 291 Passband detail Magnitude response 0 −0.5 Gain, dB −20 −1 −40 −1.5 0.35 −60 0.4 0.425 0.45 Stopband detail 0.475 0.5 −50 −80 −100 0.375 −45 0.1 0.2 0.3 Normalized frequency 0.5 0.4 −55 0.175 0.2 Normalized frequency (a) (b) Fig Magnitude response of H1 (z) in Example X (z) H0 (z) 2 G0 (z) z −1 z −1 H1 (z) 2 Y (z) G1 (z) Fig Modified two-band filter bank there are two extra delays, one in the highpass analysis filter and another one in the lowpass synthesis filter (see Fig 9) The output Y (z) is obtained using some multirate computations (Jovanovic-Dolecek, 2002), i.e., ( ) ( ) ( ) z −1 X (z) G0 (z) H0 (z) + G1 (z) H1 (z) + X (− z) G0 (z) H0 (− z) − G1 (z) H1 (− z) Y (z) = (63) The output of the filter bank (63) suffers from three types of errors, i.e., aliasing, amplitude distortion and phase distortion To avoid aliasing, the synthesis filters are related to the analysis filter H0 (z) in the following form (Vaidyanathan et al., 1987), ˜ G0 (z) = H0 (z), G1 (z) = H0 (− z), (64) ˜ ˜ where H0 (z) is the paraconjugate of H0 (z) and H1 (z) = H0 (− z) The amplitude and phase distortions are eliminated if the analysis filters are chosen to satisfy ˜ ˜ H0 (z) H0 (z) + H0 (− z) H0 (− z) = (65) ∣ H0 (e jω )∣2 + ∣ H0 (e j( ω −π ) )∣2 = (66) From (65), the following relation holds, 292 Signal Processing The relationship between the passband frequency ω p and stopband frequency ω s of H0 (z), is given by, (67) ω p + ω s = π Additionally, using (66) and (67) we have 10− Ap /10 + 10− As /10 = 1, (68) where Ap and As are the passband droop and the stopband attenuation in dB According to (Zhang et al., 2001), the analysis filters are given by, H1 (z) = 2j H0 (z) = ] ˜ A(z) + A(z) , [ ] ˜ A(z) − A(z) , [ (69) (70) ˜ where A(z) is a complex allpass filter and A(z) is its paraconjugate From (69) and (70), we can see that the design of perfect reconstruction filter banks is reduced to the design of the complex allpass filter A(z) In the following, we present one method for the modified two-band filter bank design based on the results obtained in Section 5.1 The perfect reconstruction condition for the modified two-band IIR filter banks is established in (Vaidyanathan et al., 1987; Zhang et al., 2001), which implies that the poles of H0 (z) and H1 (z) must appear on the imaginary axis and in pairs jp and 1/jp, where p is a pole From this condition, it follows that the filter coefficients given in (34) must be imaginary for even values of n (Vaidyanathan et al., 1987) Consequently, the values of φα in (34) for an even N, must be { − π for N even, (71) φα = − π for N odd Similarly, the values of φα when N is odd must be, { + − π for N2 even, φα = N +1 − π for odd (72) 5.2.1 Description of the algorithm In the following, we describe the proposed algorithm for a linear-phase IIR filter banks The IIR filters are real if N is even, otherwise they are complex The steps of the algorithm are described in the following Step Calculate the order N of the allpole filter using (68), (67) and (59) (Note that the filter H0 (z) has order 2N.) Step If N is even compute the filter coefficients (34) using (71), otherwise use (72) We illustrate the method with the following examples Example Stopband frequency ωs of the analysis filter H0 (z) is 0.65π, while the stopband attenuation As is 45 dB Step From (68) and (67), it follows that Ap = 1.373381 × 10−4 and ωp = 0.35π Using (59), the order of the complex allpole filter is 12 From (71), φα = − π ... the parameters are set 150 100 50 -50 -100 -150 100 200 300 400 500 600 700 800 900 100 0 600 700 800 900 100 0 EMBEDDED SIGNAL s(k) k step 150 100 50 -50 -100 -150 100 200 300 400 500 k step Fig... 272 Signal Processing TREND-REMOVED DATA y(k) 80 60 40 20 -20 -40 -60 100 200 300 400 500 600 700 800 900 100 0 600 700 800 900 100 0 STATIONARIZED OBSERVATION DATA k step 30 20 10 -10 -20 100 ... 700 800 900 100 0 600 700 800 900 100 0 k step EMBEDDED SIGNAL s(k) 200 100 -100 -200 100 200 300 400 500 k step Fig A sample path of the observation data Y (k) (top) and the pulse signal s(k)