1 Cicuits and Systems, 013, 4, Published Online Januay 013 ( Pefomance Analysis of an Invese Notch Filte and Its Application to F 0 Estimation Yosuke Sugiua, Aata Kawamua, Youji Iiguni Gaduate School of Engineeing Science, Osaka Univesity, Osaka, Japan Received Novembe 1, 01; evised Decembe 1, 01; accepted Decembe 9, 01 ABSTRACT In this pape, we analyze an invese notch filte and pesent its application to F 0 (fundamental fequency) estimation. The invese notch filte is a naow band pass filte and it has an infinite impulse esponse. We deive the explicit foms fo the impulse esponse and the sum of squaed impulse esponse. Based on the analysis esult, we deive a nomalized invese notch filte whose pass band aea is identical to unit. As an application of the nomalized invese notch filte, we popose an F 0 estimation method fo a musical sound. The F 0 estimation method is achieved by connecting the nomalized invese notch filtes in paallel. Estimation esults show that the poposed F 0 estimation method effectively detects F 0 s fo piano sounds in a mid-ange. Keywods: Band Pass Filte; Invese Notch Filte; Impulse Response Analysis 1. Intoduction In speech pocessing, image pocessing, biomedical signal pocessing, and many othe signal pocessing fields, it is impotant to eliminate the naowband signal. The examples of the naowband signal ae a hum noise fom the powe supply, an acoustic feedback, and an intefeeence noise, and so on. A notch filte is useful fo the elimination of the naowband signal [1-7], whee the notch filte passes all fequencies expect of a stop fequency band centeed on a cente fequency, called as the notch fequency. The notch filte has a simple stuctue, and its stop bandwidth and its notch fequency ae individually designed. The notch filte is used in many applications and it has been analyzed in many liteatues [1,4-7]. On the othe hand, an invese notch filte is a band pass filte which has the invese chaacteistics of the notch filte. In contast to the notch filte, thee ae few applications of the invese notch filte. As an example of the applications, an active noise contol system fo educing a sinusoidal noise has been poposed . In this system, the invese notch filte is used to extact the sinusoidal noise. Unfotunately, the system is designed without espect to the impulse esponse of the invese notch filte. Hence, the invese notch filte cannot accuately extact the sinusoidal noise when the filte output is in the tansient state. To utilize the invese notch filte moe effectively fo not only the active noise contol system but also many othe applications, a moe detail analysis of the impulse esponse fo the invese notch filte needs to be equied. In this pape, we deive an explicit fom fo the infinite impulse esponse of the invese notch filte. Additionally, we deive an explicit fom fo the sum of the squaed impulse esponse. Then, we eveal the limit values of these two infinite sequences. Next, based on the analysis esults, we popose a nomalized notch filte whose pass band aea is adjusted to unit. The nomalized invese notch filte is efficient to estimate the output powe in the shot time such as the fame pocessing. Finally, as an application of the nomalized invese notch filte, we pesent an F 0 estimation method fo a musical sound. In the F 0 estimation method, we use multiple nomalized invese notch filtes whose pass fequencies ae identical to F 0 s fo each monophonic sound, espectively. These nomalized invese notch filtes ae connected in paallel. In the estimation pocedue, we detect F 0 fom the invese notch filte whose output powe is lagest among all the invese notch filte output powes. Fom the simulation esults, we see that the poposed F 0 estimation method can effectively detect the F 0 both of fo the monophonic sounds and the polyphonic sounds.. Pefomance Analysis of Invese Notch Filte In this section, we explain both of the notch filte and the invese notch filte, whee the latte filte has an invese chaacteistic of the notch filte. The notch filte passes all fequencies expect of the naow fequency band centeed on the notch fequency. The stop bandwidth and the Copyight 013 SciRes.
2 118 Y. SUGIURA ET AL. notch fequency can be individually designed [1-7]. The seveal stuctues of the notch filte have been poposed and all of them can be tansfomed to the invese notch filte. In this pape, we use the stuctue of the notch filte poposed in [3-5], since the invese notch filte can be simply obtained fom the notch filte s tansfe function. The tansfe function of the notch filte N z is given by 1 1 z z Nz 1 1, (1) 1 z z whee is a paamete to design the notch fequency and 1 1 is the stop bandwidth paamete. The notch fequency paamete is given by F 1 cosπ, () FS F Hz denotes the notch fequency and F Hz whee S denotes the sampling fequency. When we put the stop bandwidth as K Hz, the elational expession of and K is epesented as K FSsin πk FS K F sin πk F 1cos π. (3) 1 cos π S Fom (1), we can deive the invese notch filte epesented as 1 1z Iz1Nz, (4) 1 1 z z whee the I z is the tansfe function of the invese notch filte. We see fom (4) that the invese notch filte is vey easy to implement. Note that the pass bandwidth paamete is also given as (3), whee K denotes the pass bandwidth. Figue 1 shows the stuctue of the invese notch filte, whee xn is the input signal, y n is the output signal, and un is the signal obtained fom the IIR unit within the invese notch filte. We see fom this figue that the invese notch filte equies only thee multiplications and thee additions to calculate the output signal. Figue shows the fequency amplitude esponse of I z when 0F FS with = 0.8, 0.9, 0.99, whee the vetical axis denotes the amplitude and the hoizontal axis denotes the nomalized fequency. We see fom Figue that the amplitude at the notch fequency is 1 egadless of, and the pass bandwidth becomes naow with inceasing Figue 1. Stuctue of invese notch filte. Copyight 013 SciRes. S Figue. Powe spectum of the invese notch filte. towad to 1, i.e., we can accuately extact a single sinusoidal signal by setting extemely close to 1. When filteing an input signal, one of the most impotant factos is the impulse esponse of the filte. We fistly deive the impulse esponse of the invese notch filte as an explicit fomulation. We see fom () o Figue 1 that the signal y n and un ae given as 1 yn unun (5) with 1 u n x n u n u n. (6) To obtain the impulse esponse, we put the input signal as the impulse signal epesented as n x n n, (7) whee is the Konecke s delta. In this case, (6) can be epesented as the following equation u n u n1 u n 0, (8) whee n. Solving the above homogeneous equation with espect to un and intoducing the initial condition that u 0 1 and u 1, we obtain the solution expessed as n1 un sin n1, (9) p p 4, (10) p actan. (11) We assume that 4. Note that this assumption is satisfied when 1. By substituting (9) into (5), we obtain the impulse esponse of the invese notch filte h n n expessed as yn h n xn n n1 n n 1 sin sin. p (1) Fom (1), we see that the impulse esponse becomes n 1 close to 0 with inceasing n due to the tem. When 0, 0, we also have h 0 and h 1 un n
3 Y. SUGIURA ET AL. 119 epesented as h n n h , (13) 1 h 1. (14) Next, we fomulate the sum of squaed impulse esponse to evaluate its convegence popety. Taking squae of (1), the squaed impulse esponse is obtained as p n1 h n p Re p j 1 1 e n j n1 whee we use the following elation 1 e, (15) cos 1 p. (16) The above elation is deived fom (10) and (11). Using (13), (14), and (15), the sum of the squaed impulse esponse J n is epesented as whee 1 1 n J n h m 4 4 m n 1 1 qcn, p 1 q, (18) (17) c n 1 cos n cos n1. (19) Fom (17), we easily obtain the limit value of with n as J n 1 lim Jn. (0) n The sum of the squaed impulse esponse J n conveges to the constant which is depending on the pass bandwidth paamete. Fom Paseval s theoem, we see that (0) is identical to the sum of the squaed fequency esponse. Note that (0) also shows the pass band aea of the invese notch filte, since its fequency esponses ae almost zeo expect of the pass band. Figue 3 shows the actual convegence popeties fo the sum of the squaed impulse esponse with = 0.8, 0.9, 0.99, whee the solid line denotes the sum of the squaed impulse esponse and the dashed line denotes the theoetical limit calculated fom (0). The hoizontal axis denotes sample numbe. We see fom Figue 3 that the sum of the squaed impulse esponse conveged to each theo- etical limit. Also we see that convegence speed becomes fast with deceasing. In the audio signal pocessing, the invese notch filte is often utilized fo measuing a naowband fequency powe which is coesponding to the invese notch filte s output powe. Howeve, the pass band aea of the invese notch filte depends on the paamete as shown in (0), and thus the output powe also depends on. Hence, it is difficult to evaluate the invese notch filte s output powe when thee exist multiple invese notch filtes which have diffeent s. To solve this poblem, we deive a nomalized invese notch filte whose output powe is faily available independently with. Since the output powe is actually calculated in a shot fame length, we have to establish the nomalized invese notch filte by taking into account the fame length. The sum of the squaed output signal is given by L1 V L y n n1 L1 n n n0 m0 l0 h m h l x nm x nl whee L is the fame length. Hee, we conside the case that the obseved signal xn is a white noise whose mean value and vaiance ae 0 and N, especttively. Taking the expectation value of (1), we have L1 n L1 n N N E V L h m J n n0 m0 n0 m0 Substituting (17) into () gives whee L1 EV L N p 1 L q L, dl, (1). () (3) d cos 4 cos L. (4) L The white noise has the same magnitude fo all fequencies. Thus, it is desiable that the sum of the squaed Figue 3. Convegence popety fo sum of squaed impulse esponse. Copyight 013 SciRes.
4 10 Y. SUGIURA ET AL. output signal of the invese notch filte is always constant egadless of the values of,, L. Howeve, as shown in (3), the expectation value of V Lstongly depends on the espective values. To solve this poblem, we popose the following nomalized invese notch filte. I z I z EV L 1 1z. 1 EVL 1 z z (5) The above nomalized invese notch filte adjusts the total pass band aea in L samples to unit. Figue 4 shows the stuctue of the nomalized invese notch filte, whee y n denotes its output signal. Compaing Figue 4 with Figue 1, we see that the diffeence is only one multiplie s value. Hence, the computational complexities of those filtes ae the same. To confim the popety of the nomalized invese notch filte, we caied out a simulation. In this simulation, the capability of the nomalized invese notch filte was compaed with the geneal invese notch filte shown in (4). We pepaed fou filtes which designed by diffeent paametes. The paamete setting is summaized in Table 1. We used white noise as the obseved signal, whee its mean and vaiance ae 0 and N 1. Figue 5 shows EV L fo fame length L, whee denotes the aveage value of V L in 1000 simulations and the solid line denotes the theoetical value calculated by (3). In this figue, the hoizontal axis de- Figue 4. Stuctue of nomalized invese notch filte. notes fame length. We see that the each invese notch filte I gave diffeent cuves of m EV L due to the diffeent paamete setting. In this case, it is not easy to evaluate the elation between filte output powes. Figue 6 shows the esult of the nomalized invese notch filte I m. We see that all the obtained EV L ae unit fo evey fame length. Hence, we can evaluate the elation between the output powe egadless of the values of,, L. 3. Application to F 0 Estimation In this section, as an application of the nomalized invese notch filte, we pesent an F 0 estimation method fo musical signal. Hee, we assume that the music signal consists of the F 0 fequency and its hamonics, and the amplitude of F 0 fequency is geate than othe fequency amplitudes. We epesent the F 0 of the music signal such as P, whee i denotes an octave numbe and j denotes a pitch name numbe, e.g., the pitch 440 Hz is epesented as P 4,10. The estimating pitch ange is set to P3,9 P5,3, whee a piano sound in this fequency ange has the maximum amplitude at its F 0 fequency. We set the notch fequency of the nomalized invese notch filte to coespond to the pitch P. Then, the i, j-th nomalized invese notch filte is epesented as 1 1 z I z, (6) 1 EVL 1 z z 1 cosπp F, (7) S and whee ae the notch fequency paamete and the pass bandwidth paamete fo I z, espectively. To eliminate ovelap with the neighbohood pass bandwidth of I z, we design the pass bandwidth paamete as K FS K FS K FS K FS 1cos π sin π 1 cos π sin π K P Pi 1,1, j 1 Pi, j Pi, j1,othewise, (8) Figue 5. Sum of squaed output signal of invese notch filte output. Table 1. Paametes fo nomalized invese notch filte. Filtes I, I 1 1 I, I I, I 3 3 I, I 4 4 Paametes Figue 6. Sum of squaed output signal of nomalized invese notch filte. Copyight 013 SciRes.
5 Y. SUGIURA ET AL. 11 Hee, we designed the pass bandwidth as the ange fom its notch fequency to one of the lowe neighboing notch fequency. The poposed F 0 estimato is achieved by connecting the designed nomalized invese notch filtes I z in paallel. The F 0 estimato is shown in Figue 7, whee y n denotes the output signal of I z. The estimation pocedue is the follows: Fist, we calculate V L defined in (1) fo each I z. We then detect the nomalized invese notch filte whose V L is lagest among all of filtes. Its filte numbe i and j diectly gives the fist F 0 estimate as P. Next, we emove the nomalized invese notch filte fo Pki, j k which is coesponding to hamonics of P. Repeating the above estimation pocedue gives the second and latte F 0 estimates. The epetition of the estimation pocess is finished when all the esidual V L s ae smalle than the theshold. We caied out simulations to confim the capability of the poposed F 0 estimato. In the simulations, we set the sampling fequency FS 10 khz, and the fame length L = 100 (=10 [ms]). We empiically set the theshold T to As the fist simulation, we caied out the F 0 estimation fo the monophonic sound which was played with a electonic piano. Figue 8 shows the wavefom of the input signal and the estimation esult, whee the tue octave numbe i and pitch numbe j ae displayed on the wavefom as i, j. We plotted the estimated F 0 as the thick black line. Fom the esult, we see that the F 0 estimation method can accuately estimate the F 0 of the obseved signal, although some eos occued in the keystoke. Additionally, we caied out the simulation fo the same monophonic signal with a white noise. The estimation esult shows in Figue 9. We see fom the figue that the F 0 estimation method can obustly estimate the F 0 unde the noisy envionment as accuately as unde the envionment without noise. As the second simulation, we caied out the F 0 estimation fo the polyphonic sound. The polyphonic sound P P P4.3, P5,3 contains the octave note 4.1, 5,1 and which ae known as a difficult combination to sepaately detect. Figue 10 shows the estimation esult. We see fom the figue that the F 0 estimation method can estimate the F 0 although thee also exist some eos at the keystoke. Especially, the poposed method can detect F 0 fo the octave note. Fom these esults, we confimed the nomalized invese notch filte is efficiently fo F 0 estimation. Figue 7. Stuctue of F 0 estimation method. Figue 9. Simulation esult fo noisy monophonic sound. Figue 8. Simulation esult fo monophonic sound. Figue 10. Simulation esult fo polyphonic sound. Copyight 013 SciRes.
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