Optimal FIR Filters for DTMF Applications

Transcription

1 ICN 13 : The Twelth International Conerence on Networks Optimal FIR Filters or DTMF Applications Pavel Zahradnik and Boris Šimák Telecommunication Engineering Department Czech Technical University in Prague Prague, Czech Republic {zahradni, simak}@el.cvut.cz Miroslav Vlček Applied Mathematics Department Czech Technical University in Prague Prague, Czech Republic vlcek@d.cvut.cz Abstract A ast and robust analytical procedure or the design o high perormance digital optimal band-pass inite impulse response ilters or dual-tone multi-requency applications is introduced. The ilters exhibit equiripple behavior o the requency response. The approximating unction is based on Zolotarev polynomials. The presented closed orm solution provides ormulas or the ilter degree and or the impulse response coeicients. Several examples are presented. Keywords-FIR ilter; narrow band ilter; dual-tone multirequency; iso-extremal approximation. I. INTRODUCTION There are two basic tasks in the processing o dual-tone multi-requency (DTMF signals, namely the detection o DTMF requencies and the removal o the DTMF requencies in a broad band signal. The DTMF requencies orm two groups with our requencies each. The lower group consists o requencies 697, 77, 85 and 941 Hz while the higher group comprises sinusoids o 19, 1336, 1477 and 1633 Hz. For the processing o DTMF signals, the ininite impulse response (IIR ilters are usually applied because o their lower number o coeicients. The IIR ilters are usually part o the amous Goertzel procedure [1]. In the removal o DTMF requencies in a broad band signal, the IIR ilters produce substantial distortions o the output signal which appear near its lat region due to the group delay variation. This behavior is especially apparent, i pulse like components are present in the signal as demonstrated in []. In order to minimize these distortions in the processing o DTMF signals we propose the application o inite impulse response (FIR ilters which inherit a constant group delay. In order to maximize the discrimination o the DTMF sinusoids, the selective bands o the FIR ilters should be as narrow as possible. In this paper we are ocused upon the design o narrow optimal equiripple (ER band-pass (BP FIR ilters or the DTMF decoding. They are optimal in terms o the shortest possible ilter length related to the requency speciication. Note that the proposed ilter design is based on ormulas, i.e. no numerical procedures are involved. The presented closed orm solution includes the degree equation and ormulas or the robust evaluation o the impulse response coeicients o the ilter. II. TERMINOLOGY We assume a general FIR ilter o type I represented by its impulse response h(k with odd length o N = n + 1 coeicients and with even symmetry (1. In our urther considerations we use the a-vector a(k, which is related to the impulse response h(k a( = h(n, a(k = h(n+k = h(n k, k = 1...n. (1 Further, we introduce an auxiliary real variable w w = 1 (z +z 1 z=e jωt = cos(ωt = cos (π s, ( where s is the sampling requency. The transer unction o the ilter is H(z = k= h(kz k = z n [h(n + h(n±k 1 ( z k +z k] k=1 n = z n a(kt k (w = z n Q(w (3 k= where T k (w = cos(karccos(w is Chebyshev polynomial o the irst kind and Q(w is the real valued zero phase transer unction which we express using the a-vector in orm o the expansion o Chebyshev polynomials Q(w = a(kt k (w. (4 k= The zero phase transer unction o an ER BP FIR ilter is Q(w = Z p,q(w,κ+1 y m +1, (5 where Z p,q (w,κ represents the Zolotarev polynomial [8]. For illustration, the shape o a Zolotarev polynomial is shown in Fig. 1 and the corresponding requency response is shown in Fig.. Copyright (c IARIA, 13. ISBN:

2 ICN 13 : The Twelth International Conerence on Networks Z 1,6 (w, log H(e jωt [db] y w 1 w w w Fig. 1. as[db] Zolotarev polynomial Z 1,6 (w, ωt/π ω T π ωt/π Fig.. Amplitude requency response log H(e jωt [db] corresponding to the Zolotarev polynomial rom Fig. 1. III. OPTIMAL BAND-PASS FIR FILTER An optimal ER BP FIR ilter (Fig. is speciied by the pass band requency ω T (or = ω T s /π, width o the pass band ωt (or = ωt s /π, attenuation in the stop-bands a s [db] and by the sampling requency s. An approximation o the requency response o a ilter is based on the generating unction. The generating unction o an ER BP FIR ilter is the Zolotarev polynomial Z p,q (w,κ which approximates a constant value in equiripple Chebyshev sense in two disjoint intervals 1,w 1 and w,1 as shown in Fig. 1. The main lobe with the maximal value y = Z p,q (w,κ is located inside the interval (w 1,w. The notation Z p,q (w,κ emphasizes the act that the integer value p counts the number o zeros right rom the maximum w and the integer value q corresponds to the number o zeros let rom the maximum w. The real value κ 1 is in act the Jacobi elliptical modulus. It aects the maximum value y and the width w w 1 o the main lobe (Fig. 1. For increasing κ the value y increases and the main lobe broadens. The Zolotarev polynomial is usually expressed in terms o Jacobi elliptic unctions [6]-[8] Z p,q (w,κ = ( 1p H(u p n K(κ H(u+ p n K(κ n (6 + H(u+ p n K(κ n H(u p n K(κ. The actor ( 1 p / appears in (6 as the Zolotarev polynomial alternates (p+1 times in the interval (w,1. The variable u is expressed by the incomplete elliptical integral o the irst kind F(x κ, namely ( p u =F sn n K(κ κ 1+w ( p κ. w+ sn 1 n K(κ κ (7 The unction H (u±(p/n K(κ is the Jacobi Eta unction, sn(u κ, cn(u κ, dn(u κ are Jacobi elliptic unctions and K(κ is the quarter-period given by the complete elliptic integral o the irst kind. The degree o the Zolotarev polynomial is n = p + q. A comprehensive treatise o Zolotarev polynomials was published in [8]. It includes the analytical solution o the coeicients o Zolotarev polynomials, the algebraic evaluation o the Jacobi Zeta unction Z nk(κ κ and o the elliptic integral o the third kind Π(σ m, p n K(κ κ. The position w o the maximum value y = Z p,q (w,κ is sn w = w 1 + n K(κ κ cn n K(κ κ Z dn n K(κ κ n K(κ κ (8 where the edges o the main lobe are w 1 = 1 sn n K(κ κ (9 w = sn ( q n K(κ κ 1. (1 The relation or the maximum value y y = coshn (σ m Z n K(κ κ Π(σ m, p n K(κ κ (11 is useul in the normalization o Zolotarev polynomials. The degree o the Zolotarev polynomialz p,q (w,κ is expressed by the degree ormula ln(y + y n 1 σ m Z n K(κ κ Π(σ m, p. (1 nk(κ κ The auxiliary value σ m in (11, (1 is given by the ormula ( ( 1 σ m = F arcsin κ sn ( w w s p n K(κ κ κ (13 w +1 ( cos π [ ( π cos + ] = F arcsin s s p κ sn( ( n K(κ κ 1+cos π κ. s Copyright (c IARIA, 13. ISBN:

3 ICN 13 : The Twelth International Conerence on Networks The Zolotarev polynomial Z p,q (w,κ satisies the dierential equation ( (1 w dzp,q (w,κ (w w 1 (w w (14 dw = n ( 1 Zp,q(w,κ (w w. Based on the dierential equation (14 we have developed an algorithm or the evaluation o the a-vector and o the impulse response h(k corresponding to the Zolotarev polynomial Z p,q (w,κ in orm o its expansion into Chebyshev polynomials (4. This algorithm is summarized in Table I. There are two goals in the ilter design. The irst one is to obtain the minimal ilter length o N coeicients satisying the ilter speciication. The second one is to evaluate its impulse response h(k. In the standard design o an ER BP FIR ilter, which is represented by the numerical Parks-McClellan procedure (e.g. the unction irpm in Matlab, the exact ilter length is not available because o no approximating unction. Consequently, the ilter length is not the result o the design, it is in act an input argument in the Parks-McClellan procedure. The ilter length is either estimated or successively adjusted in order to meet the ilter speciication, or it is obtained rom empirical approximating ormulas. Moreover, a successul design is not guaranteed in the Parks-McClellan procedure. The alternative design approach that we use here is based on the analytical design which we have developed or ER notch FIR ilters and introduced in [3]. The available pass band requencies which we denote Q are quantized because there is always an integer number o ripples (Fig.. That is why we additionally tune the actual pass band requency Q o the initial ilter to the speciied value. This tuning consists in multiplying the a- vector o the initial ilter by a transormation matrix, resulting in the a-vector o the tuned ilter (16 which exactly meets the speciied pass band requency. For the tuning, we present an eicient algebraic procedure which is a simpliied version o that one introduced in [4]. The tuning procedure results in the zero phase transer unction o the tuned ilter, which is Q t (w = a(kt k (λw±λ = k= a(k k= k α k (mt m (w. m= (15 Based on (15, the a-vector a t (k o the tuned ilter and the a-vector a(k o the initial ilter are related by a triangular transormation matrix A a t (k = [a t ( a t (1 a t (n] = [a( a(1 a(n] α ( α 1 ( α 1 (1 α ( α (1 α ( α 3 ( α 3 (1 α 3 ( α 3 (3 = a(ka... α n ( α n (1 α n ( α n (3 α n (n (16 A ast algorithm or the evaluation o the coeicients α k (m is summarized in Tab. II. The presented tuning o the pass band requency preserves the attenuation a s [db] in the stop bands. IV. DESIGN OF THE OPTIMAL BAND-PASS FIR FILTER Let us speciy the optimal ER BP FIR ilter by the pass band requency [Hz], width o the pass band [Hz], sampling requency s [Hz] and by the attenuation in the stop bands a s [db]. The design procedure reads as ollows: Calculate the Jacobi elliptic modulus κ κ = 1 or the auxiliary values ϕ 1 and ϕ ϕ 1 = π ( + s 1 tan (ϕ 1 tan (ϕ Calculate the rational values p n and q n p n = F(ϕ 1,κ K(κ, ϕ = π ( s s +,. (17 (18 q n = F(ϕ,κ, (19 K(κ where K(κ is a complete elliptic integral o the irst kind, which here represents the elliptic quarter-period. Determine the value y y = 1.5a. ( s[db] Calculate the auxiliary value σ m (13. Calculate and round up the value n (1 which represents the degree n = p + q o the Zolotarev polynomial Z p,q (w,κ. Calculate the integer indices p and q o the Zolotarev polynomial Z p,q (w,κ p = nf(ϕ 1,κ K(κ, q = n F(ϕ,κ K(κ. (1 The arrow brackets in (1 stand or rounding. For values p, q (1, κ (17 and y ( evaluate the a-vector a(k and the related impulse response h(k o the ilter using the algebraical procedure summarized in Tab. I. Check the actual pass band requency Q o the initial BP FIR ilter [ Q = s π arccos 1 sn n K(κ,κ ( sn + n K(κ,κ cn n K(κ,κ p Z( ] dn n K(κ,κ n K(κ,κ. Because o the inherent quantization o the available pass band requencies Q mentioned above, the actual pass band requency Q usually slightly diers rom the speciied pass band requency. That is why we tune the quantized pass band requency Q o the initial ilter to the speciied value using (16 and Tab. II. In our calculations, the Jacobi elliptic Zeta unction Z(x,κ in (8, (11, (1 and the incomplete elliptic integral o the irst kind F(x,κ in (13 are evaluated Copyright (c IARIA, 13. ISBN:

4 ICN 13 : The Twelth International Conerence on Networks given p, q, κ, y initialization n = p+q, w 1 = 1 sn n K(κ,κ ( q, w = sn n K(κ,κ 1, w a = w 1 +w n K(κ,κ ( p Z n K(κ,κ ( p sn n K(κ,κ cn w m = w 1 + ( p dn n K(κ,κ α(n = 1, α(n +1 = α(n+ = α(n+3 = α(n +4 = α(n+5 = body (or m = n+ to 3 8c(1 = n (m+3, 4c( = (m +5(m +(w m w a+3w m[n (m + ] c(3 = 3 4 [n (m +1 ]+3w m[n w m (m+1 w a] (m+1(m +(w 1 w w mw a c(4 = 3 (n m +m (w m w a+w m(n w m m w 1 w c(5 = 3 4 [n (m 1 ]+3w m[n w m (m 1 w a] (m 1(m (w 1 w w mw a 4c(6 = (m 5(m (w m w a+3w m[n (m ], 8c(7 = n (m 3 6 α(m 3 = 1 c(µα(m +4 µ c(7 µ=1 (end loop on m normalization s(n = α( + α(m m=1 a-vector a( = ( 1 p α(, (or m = 1 to n, a(m = ( 1pα(m, (end loop on m s(n s(n a( +1 impulse response h(n = y +1, (or m = 1 to n, h(n±m = a(m, (end loop on m (y +1 TABLE I FAST ALGORITHM FOR EVALUATING THE a-vector a(k AND THE IMPULSE RESPONSE h(k. given Q,, s, k ( cos π ( Q 1 cos π Q +1 initialization i Q < : λ = ( s cos π, s = 1, else : λ = ( s 1 cos π, s = 1 +1 s s λ = 1 λ, α k (k +1 = α k (k + = α k (k +3 =, α k (k = λ k body (or µ = 3... k 4 α k (k µ 4 = { s [(µ+3(k µ 3 ]α λ λ (k µ 3(k µ 7 k (k µ 3 (end loop on µ + λ λ (k µ α k(k µ +s [(µ+1(k µ 1 ]α λ λ (k µ 1(k µ 1 k (k µ 1 } / (µ+4(k µ 4 +µ(k µ α k (k µ TABLE II FAST ALGORITHM FOR EVALUATING THE COEFFICIENTSα k (m OF TRANSFORMATION MATRIXA. Copyright (c IARIA, 13. ISBN:

5 ICN 13 : The Twelth International Conerence on Networks by the arithmetic-geometric mean [7]. The Jacobi elliptic integral o the third kind Π(x, y, κ in (1 is evaluated by a ast procedure proposed in [3]. log H(ejπ /s [db] 1 log H(ejπ /s [db] Fig. 4. Amplitude requency responses log H(ejπ /s [db] o the ilters with N = coeicients log H(ejπ /s [db] V. E XAMPLES OF D ESIGN Let us design three sets o band pass FIR ilters speciied by the DTMF requencies = 697, 77, 85, 941, 19, 1336, 1477, 1633 Hz, width o the pass bands = 5Hz, sampling requency s = 8Hz and with the attenuations in the stop bands as = 8dB, 1dB and 16dB. Using the presented design procedure, we get ilter lengths N = 183 coeicients or as = 8dB, 1551 coeicients or as = 1dB and 19 coeicients or as = 16dB. The corresponding amplitude requency responses are shown in Fig. 3. In order to demonstrate the remarkable selectivity o the ER BP FIR ilters and the robustness o the presented design procedure, let us design the DTMF ilters with very narrow pass band o = 5Hz, sampling requency s = 8Hz and with the attenuation in the stop bands as = 1dB. The ilter length is N = coeicients. The amplitude requency responses are shown in Fig. 4. VI. C ONCLUSION log H(ejπ /s [db] AND F UTURE W ORK We have presented a ast and robust procedure or the design o optimal equiripple narrow band-pass FIR ilters or DTMF applications. In contrast to the established numerical design procedure the proposed methodology solves the approximation problem and provides a ormula or the degree o the ilter and ormulas or the evaluation o the coeicients o the impulse response o the ilter. Our uture activity will include an eicient implementation o the DTMF ilters using digital signal processors Fig. 3. Amplitude requency responses log H(ejπ /s [db]. Copyright (c IARIA, 13. ISBN: ACKNOWLEDGEMENTS This work was thankully supported by the grant MPO No. FR-TI/

6 ICN 13 : The Twelth International Conerence on Networks REFERENCES [1] G. Goertzel, An Algorithm or the evaluation o inite trigonometric Series, Amer. Math. Monthly. Vol. 65, January 1958, pp [] M. Vlček, P. Zahradnik, Digital Multiple Notch Filters Perormance, Proceedings o the 15th European Conerence on Circuit Theory and Design ECCTD 1. Helsinky, August 1, pp [3] P. Zahradnik, M. Vlček, Fast Analytical Design Algorithms or FIR Notch Filters, IEEE Transactions on Circuits and Systems. March 4, Vol. 51, No. 3, pp [4] P. Zahradnik, M. Vlček, An Analytical Procedure or Critical Frequency Tuning o FIR Filters. IEEE Transactions on Circuits and Systems II. January 6, Vol. 53, No. 1, pp [5] N. I. Achieser, Über einige Funktionen, die in gegebenen Intervallen am wenigsten von Null abweichen, Bull. de la Soc. Phys. Math. de Kazan, Vol. 3, pp. 1-69, 198. [6] D. F. Lawden Elliptic Functions and Applications Springer-Verlag, New York Inc., [7] M. Abramowitz, I. Stegun, Handbook o Mathematical Function, Dover Publication, New York Inc., 197. [8] M. Vlček, R. Unbehauen, Zolotarev Polynomials and Optimal FIR Filters, IEEE Transactions on Signal Processing, Vol. 47, No. 3, pp , March Copyright (c IARIA, 13. ISBN: