复杂,自适应,陷波器

Complex adaptive notch filter structure for tracking multiple complex sinusoidal signalsP.T.Wheeler and J.A.ChambersA new type of complex adaptive notch filter is developed,which is capable of tracking multiple complex sinusoidal signals.The proposed complex structure is developed from an all-pass design based on a structurally lossless prototype,which is canonic in the number of mul-tipliers and delay elements.The learning algorithm is based upon output error calculation,and is better matched for tracking closely located signal frequencies than a recently introduced approach of Regalia.Introduction:Adaptive notch filters have the ability to track time-varying sinusoidal signals;many structures have been proposed for such filters that can be used to track the frequencies of real sinusoidal signals.However,few of these structures have been modi fied so they have the ability to track the frequencies of complex sinusoidal signals.Systems are now commonly functioning with complex signals particu-larly in communications,i.e.in-phase and quadrature (I&Q)sampling in modulators/demodulators or sensors.Hence,tracking these signals and eliminating noise is necessary,particularly as technology advances with smaller geometries,thus increasing the vulnerability of a system to noise related errors.Real designs in the past have been created with constrained poles and zeros usually through all-pass decompositions,with notable relevant designs being [1,2].In this reported work,we extend the real design in [1]to a complex form.This design has the advantage that,being developed from an all-pass design based on a structurally lossless pro-totype,which is canonic in the number of multipliers and delay elements,it can also be easily cascaded for tracking multiple complex sinusoidal signals.We also develop an output error learning algorithm,which facilitates the tracking of complex sinusoidal signals that are close in frequency,which is not possible in the recent work of Regalia shown in [3].Filter realisation:First,consider a standard first-order real all-pass filter,whose z -domain transfer function isA (z )=z −1−a1−a z −1(1)where 0≪a ,1and is a real coef ficient.For our implementation,the all-pass structure is modi fied to the following formA (z )=z −1b −a 1−a z −1b(2)wherein an additional phase parameter βis introduced,this value equates to β=e j θ,where θis the complex phase shift angle which is equivalent to the frequency being tracked.This structure is then utilised to create a notch structure as shown in Fig.1.Thus,the z -domain trans-fer function of the proposed complex notch filter (shown in Fig.1)can be derived asC (z )=E (z )U (z )=12{1−A (z )}=12(1+a )(1−z −1b )1−a z −1b(3)In this equation the parameter αcontrols the notch width of the filter,and for stability reasons αis fixed during the learning process.Parameter βon the other hand,controls the notch frequency,and is adapted during learning.u(n)e(n)Z –11/2grad(n)–jαβ––Fig.1Proposed complex notch filter structure,where the structure within thebox de fined by dashed line is denoted CNF_in Fig.2The z -domain transfer function from the input to the gradient output,later used in developing the learning algorithm,is given byGRAD (z )U (z )=12(1+a )(−jz −1b )1−a z −1b(4)Next,we develop the learning algorithm to control β,or equivalently θ.Learning algorithm development:We derive an output error basedlearning algorithm,where the cost function (J ),is de fined asJ =e (n )2=e (n )e ∗(n )(5)where |.|and (.)*denote,respectively,the modulus and conjugate of a complex number.A least mean square (LMS)type update for the unknown parameter θcan be derived from J .This derivation begins with the update equationu n =u n −1−m2∇J |u =u n −1(6)where μis the adaptation gain.Hence,by applying differentiation by parts to (5)gives∇J =e (n )∇e ∗(n )+∇e (n )e ∗(n )(7)thus (6)becomesu n =u n −1−m Re (e ∗(n )∇e (n ))(8)wherein the signal ∇e (n )=grad (n )in Fig.1.Due to the non-quadratic nature of the cost function J in (5)with respect to θin β=e j θ,we adopt a normalised LMS type update for θ.The normalised (N)LMS algorithm includes a recursive calculation of the gradient energy ψ.This calculation isc n =c n −1g +(1−g )(grad (n ).grad ∗(n ))(9)where the term γis the forgetting factor –which is a value between zero and one,and is chosen to be ≥0.97in the simulations in this Letter.Also,the initialising value used in implementations is ψ0=1.0.Thus,updating (8)with (9)yields the final update equation for θu n =u n −1−m Re (e (n ).grad (n )/c n )(10)This algorithm and new structure is then applied to track the frequen-cies of multiple complex sinusoidal signals.Tracking two complex sinusoids:To track two complex sinusoids a cascade structure as shown in Fig.2is employed,wherein an additional complex notch filter structure CNF_3is required to generate the gradient value for CNF_2.This requires only two additional signi ficant multi-pliers as in Fig.1,i.e.the αand βmultipliers.u(n)e(n)grad(n)grad_1(n)CNF_2CNF_3CNF_1grad_2(n)θ1θ2θ1Fig.2Cascade structure for tracking two complex sinusoidsSuch a cascade structure can be easily expanded to track more than two frequencies.Fixing αas 0.8in the simulations preserves the stability of the notch structures;in our experiments μis chosen small enough to avoid instability in the learning algorithm.Unfortunately,there is not space for formal stability analysis in this Letter,but background in-formation can be found in [4].Simulation results and comparison:We first consider tracking a single frequency ω0,for the complex sinusoidal signalu (n )=Se (v 0n +w )j+W (n )(11)where S is a scale factor which de fines the signal-to-noise ratio (SNR),φis a random phase uniformly distributed over (0,2π),and W (n )is zero mean unity variance complex white noise.The results in this Letter,as in [3],have been achieved with a signal-to-noise ratio of 0dB.Figs.3and 4both include a target signal that is frequency hopping.ELECTRONICS LETTERS 31st January 2013Vol.49No.3This target signal (shown in red)is first initialised,then instantaneously changes or hops every 1000samples in Fig.3,whilst in Fig.4one value hops at 1000samples then the second value hops at 2000samples.The result for the proposed structure in Fig.3is achieved with μ=0.15,γ=0.97and θ0=−1.26;whereas,to achieve the same initial tracking per-formance,for Regalia ’s structure [3],also shown in Fig.3,the par-ameters required are μ=0.05and γ=0.8.3210–1–2–35001000150020002500sample numberthis Letter target signal Regalia [3]n o r m a l i s e d n o t c h f r e q u e n c y300035004000Fig.3Single frequency signal tracking comparison2.52.01.51.00.50–0.5–1.0–1.52.52.01.51.00.50–0.5–1.0n o r m a l i s e d n o t c h f r e q u e n c yn o r m a l i s e dno t c h f r e q u e n c y5001000150020002500sample number3000350040005001000150020002500sample number300035004000Fig.4Two frequency signal tracking comparison for the proposed structure (top),Regalia ’s structure [3](below)The results show that the proposed structure performs as well as,or better than,the previous approach [3],as there are visible convergence improvements in these results.For example,in Fig.3observe the faster convergence to the final frequency following the change at sample number 3000.Note,adding the term ψimproves the performance of Regalia ’s scheme presented in [3],which adopts an LMS-type learning algorithm.The addition of ψto structure [3]also instantiates a fair com-parison with the proposed algorithm,as if ψis omitted the structure pro-posed in this Letter signi ficantly outperforms [3].Next,we compare the ability of both structures to track the frequen-cies of two complex exponential signals.The parameters that facilitated the results in Fig.4(top)are m =0.08and γ=0.99,whereas for Fig.4(below)μ=0.02and γ=0.8;these values have been found empirically to yield best results with equivalent tracking performance.Observe that when the target frequencies are closer together beyond sample number 2000,Regalia ’s structure fails to converge;this effect worsens with lower values of α.This is a consequence of not using an output error based learning algorithm in [3]to reduce the complexity in gradient gen-eration in the learning algorithm.Table 1shows the average estimated frequency errors and the variances in the frequency estimates over the last 200samples for the results presented in Fig.4.The reduction in error and variance con firms the advantage of our proposed structure.Table 1:Dual complex sinusoidal signal tracking resultsConclusion:We have proposed a new complex adaptive notch filter structure based on an output error learning algorithm.We have demon-strated its superior tracking performance particularly for two closely spaced frequencies.The low complexity and robustness of our proposed structure is very attractive for practical applications.©The Institution of Engineering and Technology 20133August 2012doi:10.1049/el.2012.2958One or more of the Figures in this Letter are available in colour online.P.T.Wheeler and J.A.Chambers (Loughborough University,United Kingdom )E-mail:p.wheeler2@ References1Chambers,J.A.,and Constantinides,A.G.:‘Frequency tracking using constrained adaptive notch filters synthesized from allpass sections ’,Proc.IEE F ,1990,137,pp.475–4812Regalia,P.A.:‘An improved lattice-based adaptive IIR notch filter ’,IEEE Trans.Signal Process .,1991,39,pp.2124–21283Regalia,P.A.:‘A complex adaptive notch filter ’,IEEE Signal Process.Lett.,2010,17,pp.937–9404Ljung,L.,and Söderstrom,T.:‘Theory and practice of recursive identi-fication ’(MIT Press,Cambridge,MA,1983)ELECTRONICS LETTERS 31st January 2013Vol.49No.3。