TRANSMISSION FILTER WITH BIQUADRATIC FILTER STAGES

Abstract: The transmission filter according to the invention in­ cludes a plurality of biquadratic filter stages coupled in a leapfrog configuration. It is a sampled filter with characteris­ tics equivalent to those of a filter comprising the cascade connection of a first filter having a transfer function propor­ tonal to and of a second flter comprising a leapfrog arrangement of biquadratic filter stages the first and last ones of which are dissipative for reactive pseudo-power and each have a transfer function proportional to (Z ∓ 1)², the other biqua­ dratic filter stages being non-dissipative for reactive pseudo-power. Various embodiments of the biquadratic filter stages are disclosed.

Description: TRANSMISSION FILTER WITH BIQUADRATIC FILTER STAGES The present invention relates to a leapfrog filter including a plurality of biquadratic filter stages coupled in a leapfrog configuration. Such a filter is generally known in the art e.g. from the article "Synthesis of a leapfrog configuration equivalent to an LC-ladder filter between generalized terminations" by D. Dubois and J. Neirynck, published in the IEEE Transactions on Circuits and Systems, Vol. CAS-24, No 11, November 1977, pp. 590-597. The numerator of the transfer function of such a filter is even and can only have pairs of real transmission zeros at z = 1 and z = -1 in the z-plane. However, it may be necessary in some circumstances to have single real transmission zeros at z = +1 and/or at z = -1, e.g. for the realization of a bandpass filter. A first object of the present invention is to provide a leapfrog filter of the above type but having a transfer function with an odd numerator and more particularly with at least three real transmission zeros at z = + 1 and at least one real transmission zero at z = + 1 in. the z-plane. Reference is now made to the above article and to that by M. Hasler published in the IEEE Transactions on Circuits and Systems, Vol. CAS-27, No 7, July 1980, pp.636-64L, and entitled "A new class of RC-active filters using FDNR elements and generalized terminations". From these it follows that a leapfrog filter comprising a real reciprocal non-dissipative 2-port connected between real non-dissipative terminations has a minimum sensitivity to component variations at the frequencies where the frequency dependent filter polynomial h defined below is equal to zero, i.e. at the frequencies (attenuation zeros where the maximum power transfer occurs (Orchard-Fettweis theorem). The scattering matrixes of such a 2-port defined with respect to real termination impedances W1 and W2 has the form EMI2.1 where : N, D are coprime polynomials such that W1 = N ; W2 D f, g and h are frequency dependent polynomials subject to the following conditions f is even or odd and divisible by D and the upper sign of f corresponds to the upper sign of g is a strictly Hurwitz polynomial ,i.e. its zeros are in the open left half of the s-plane; gg = hh + D ff (1) wherein f =f (s) = f(-s) g =g (s) = g(-s) h =h (s) = h(-s) a function with a - as subscript being the so-called Hurwitz conjugate of the function, i.e. a function of s wherein -s has been substituted for the complex frequency s. In the filter passband and when it is substantially flat, the voltage transfer function f/2g is substantially ff equal to unity and gg is substantially a constant. From the above relation(l) which is a related form of the Feldtkeller relation, it follows that in order to have a maximum number of zeros of h in the passband, the function should be as constant (or flat) as possible. However, D N according to the article by M. Hasler, this choice of has merely permitted to place one pair of complex conjugate zeros of h onto the imaginary axis in the s-plane. A further object of the present invention is to provide a leapfrog filter of the above type, but which does not require D to be substantially constant (or flat) in the passband so as to reach therein a minimum sensitivity to filter component variations. According to the invention the above objects are achieved due to the fact that the present leapfrog filter is a sampled filter with characteristics equivalent to those of a filter comprising the cascade connection of a first filter part having a transfer function proportional z # 1 to and of a second filter part comprising a plurality z # 1 of biquadric filter stages coupled in leapfrog configuration the first and the last stages of which are dissipative for reactive pseudo-power and each have a transfer function proportional to a same factor (z # 1)2 and the other stages being non-dissipative for reactive pseudo-power, wherein z = e T with s being the complex frequency and T being the- sampling period of the filter. Because the total transfer function of the filter is hence proportional z # 1 (z # 1)4 = (z # 1) (z # 1) @ z+l the filter has at least three real tensmission zeros at z = + 1 and one transmission zero at z = + 1 and the numerator of its transfer function is odd. It should be z+l @ noted that the factor does not affect the dagree of z # 1 the filter. The invention is based on the insight that when z+l one puts an additional stage with transfer function z+l in cascade with the input of the above mentioned lossless 2-port and thus multiplies the overall transfer function by this factor, it is necessary if the overall (unity) transfer function in the passband should remain unchanged to multiply ff- z # 1 in the passband by about () 2 . gg z + 1 From the above relation (1) it then follows that for sensitivity reasons, i.e. to have h equal to zero, one should take N = (z # l)ê D (z - 1)2 @ instead of D having to be substantially flat as indicated by M. Hasler. This leads to the choice of the bilinear transfer functions for the terminations W1 and W2, i.e. z # 1 1 z # 1 W1 = q1 and W2 = (q1 and q2 are constants) @@ q@ z + 1 N Due to D being no longer flat it has been possible to place a maximum number of zeros of h in the passband. Another characteristic feature of the present leapfrog filter is that the factor (z + 1)2 to which the transfer function of said first stage is proportional is the numerator of a function T1 ¯ (z + 1)2 T1' = = h1' . 1 + W1T1 z2 - 2 z R1 cos tpl' + R12 ztl wherein W1 = q1 z + 1 is the tians@@@ function @@ s@@@ @@@@@ termination stage (W1) and (z # 1)2 T1 = h1 is the transfer function of z2 - 2 z cos tpl + 1 the first biquadratic filter stage of said two-port, and wherein hl, h'1 and q1 are positive constants and tpl is the angle of a transmission pole of T1 in the z-plane defined by z = esT. Still another characteristic feature of the present leapfrog filter is that the factor to which the transfer function of said last stage is proportional, is the numerator of a function T 2 T ' = m = h' (z + 1) m 1 + W2 1T m z2 - 2 z R cos t' + R 2 m m m pm m z # 1 wherein W2 = 1/q2 is the transfer function of said z # 1 second terminal stage (W2) and 2 (z # 1) Tm = hm is the transfer function z2 - 2 z cos t @ + 1 pm of the last biquadratic stage of said two-port and wherein hm, hm' and q2 are positive constants and tpm is the angle of a transmission pole of T in the z-plane defined by z = e m Thus each termination stage and the adjacent stage of the two-port are combined to constitute a new biquadratic filter stage. Yet another characteristic feature of the present leapfrog filter is that it includes a biquadratic filter input stage with a first input constituting the input of the leapfrog filter, a second input and an output and includes a forward circuit between said first input and said output which realizes the transfer function of said first filter part and a feedback circuit between said second input and said output which realizes the transfer function of said first stage of said second filter part. Thus a same biquadratic filter stage is used for realizing, on the one hand, the transfer function partly resulting from the additional stage with transfer function z # l z # l and,on the other hand,of the transfer function of the first stage of said second filter part. The present invention also relates to a switched capacitor biquadratic filter circuit with an input and an output and including a non-overlapping two-phase clock source and first and second operational amplifiers each having an inverting input, a grounded non-inverting input, an output and a feedback capacitance, the output of said first amplifier being coupled to the input of said second amplifier via first capacitance means including a first capacitance the plates of which are connected to first and second change-over switches able, under the control of said clock source, to connect them between a ground and said output of said first amplifier and between said ground and the input of said second amplifier, the output of said second amplifier being coupled to the input of said first amplifier via second capacitance means including a second capacitance the plates of which are connected to third and fourth changeover switches able under the control of said clock source, to connect them between said ground and the output of said second amplifier and between said ground and said input of said first amplifier1 and said filter input being coupled to an input of one of said amplifiers via third capacitance means including at least a third capacitance. Such biquadratic filter stages realized in switched capacitor technique have already been designed and are for instance disclosed in the article "An NMOS analog building block for telecommunication applications " by P.E. Fleischer et al, published in IEEE Transaction on Circuits and Systems, Vol. CAS-27, No 6, June 1980, pp.552-559 and more particularly in Fig. 5 thereof. Another object of the present invention is to provide switched capacitor biquadratic filter stages which are insensitive to parasitic capacitances and particularly adapted for use in leapfrog filters previously defined. According to the invention such an object is achieved by a switched capacitor biquadratic filter wherein said clock source controls said switches in such a way that said second capacitor is connected to the input of said first amplifier and said first- capacitor is connected to said input of said second amplifier for respective different phases of said clock source. The present invention also relates to a switched capacitor biquadratic filter circuit, characterized in that it simultaneously realizes a transfer function proportional 2 to z - 1 between a first input and an output and a second transfer function proportional to (z + 1) between a second input and said outputthe transfer fuations being considered in the z-plane. Such a filter circuit can thus be directly used to realize the above mentioned input stage of a leapfrog filter defined above. The above mentioned and other objects and features of the invention will become more apparent and the invention itself will be best understood by referring to the following description of embodiments taken in conjunction with the accompanying drawings wherein Fig. 1 is a block diagram of a subscriber line circuit SLC for use in a PCM telephone system and incorporating leapfrog filters TBPF and RLPF according to the invention; Fig. 2 is a flowgraph of a classical leapfrog filter; Fig. 3 is a flowgraph of a leapfrog filter according to the invention; Figs. 4, 5 and 6 show transfer function versus frequency characteristics used to explain the invention; Fig. 7 is a flowgraph of a leapfrog filter according to the invention; Fig. 8 is a modified version of the flowgraph of Fig. 7; Figs. 9 to 16 represent various switched capacitor biquadratic filter stages according to the invention; ; Fig. 17 shows the transmit bandpass filter TBPF of Fig. 1 in detail; Fig. 18 represents the attenuation versus frequency characteristic of the filter of Fig. 17; Fig. 19 shows the group delay versus frequency characteristic of the filter of Fig. 17; Fig. 20 represents the lowpass filter RLPF of Fig. 1 in detail; Fig. 21 shows the attenuation versus frequency characteristic of the filter of Fig. 20. The subscriber line circuit SLC for a PCM telephone switching system shown in Fig. 1 includes a subscriber line interface circuit SLIC which is connected to a coder/decoder circuit or codec COD, operating at 8 kHz, via the cascade connection of an anti-aliasing transmit filter TIF, a transmit bandpass filter TBPF and a smoothing transmit filter TOF, whilst the codec COD, is connected to the SLIC via the cascade connection of a receive lowpass filter RLPF and a smoothing receive filter ROF. The filters TIF, TOF, ROF all are of the second order Sallen-Key type. The bandpass filter TBPF is an eighthorder switched capacitor leapfrog filter operating as an anti-alias filter and rejecting 50 Hz and 60 Hz power supply frequencies. The lowpass filter RLPF is a fourth-order switched capacitor leapfrog filter having substantially the same lowpass characteristic as the TBPF and realizing a sin x correction for the sampling operation in the codec. x In both filters the capacitors ratio has a minimum dispersion and the filters have a small sensitivity to component variations and use a single clock frequency of 256 kHz. Switched capacitor filters for use in a subscriber line circuit for a PCM telephone switching system are already known from the following articles, both published in the IEEE Journal of Solid-State Circuits, Vol. SC-15, No 6, December 1980, and entitled "A precision low-power PCM channel filter with on-chip power supply regulation" by H. Ohara et al, pp. 1005-1013, and "A low-power NMOS transmit-receive IC filter for PCM.telephony" by I.A. Young, pp. 997-1005. None of these circuits however includes an eighth order leapfrog bandpass filter and a fourth order leapfrog lowpass filter. A structure comprising a 2-port which is lossless for reactive power and connected to terminations W1 and W2 which are dissipative for reactive. power will now be considered. In fact the power is pseudo power because it cannot be measured. This 2-port is constituted by a plurality of biquadratic filter stages T1 to T which are arranged in m a leapfrog configuration. These stages have an infinite quality factor and transfer functions which are even functions because they are lossless for reactive power. On the contrary, the transfer functions of the termination stages W1 and W2 are odd functions because they are dissipative for reactive power. The termination stage W1 is connected in series with a generator e (having zero impedance ) across the input I1, I2 of the 2-port and thus provides a voltage e - W1I'1 (I'1 being the input current) across this input. In the flowgraph this is represented by the connection a = e -W1b. The termination stage W2 is connected across the output 1 02 of the 2-port and thus provides a voltage v = W21'2 (I'2 being the output current) across this output. Because -l I'2 = v (-W2) -1 in the flowgraph this is represented by c = (-W2) d. As the stages T1 to T have an infinite quality m factor z - 2 z cos t @ + 1 T. = h2 ol (2) z - 2 z cos t . + 1 pl wherein hi is a constant with i = 1 to m and toi and tpi are the angles in the z-plane of the transmission zero and pole of T. respectively. The circuit comprising -W1 and the stage T13 on the -l one hand, and the circuit comprising -W2 and the stage Tm on the other hand can be realized by the biquadratic stages T1' and Tm' respectively,the transfer functions of which are related to those of W1, T1 and W-12, Tm by T1 T1' = 1 + W1T1 (3) Tm and T'm = 1 + w2-1 Tm (4) These end stages T1' and Tm' are shown is Fig. 3 (tobe considered without the additional stage AS). In order that T1' and Tm' should be biquadratic functions the transmission poles of W1 should be the trans mission zeros of T1 and the transmission poles of W2 -1 should be the transmission zeros of T . Moreover because m W1 and W2 are dissipative for reactive power their transfer functions have to be odd. For this reason, W1 and W2 have been chosen as follows z2 - 1 W1 = q1 (5) 2 @@ z - 2z cos to1 + 1 z2 - 1 W2 = q2 (6) z2 - 2z @@@ @ @@ om By combining the relations (2) to (6) one obtains 2 2 @ @@ @@@ @ol @ 1 1 = h1' (7) z2 - 2zR1 cos tpl' + R12 z2 ¯ 2z cos tom + 1 @@ Tm' = hm' (8) z2 - 2zRm cos tpm' + Rm2 where h1 1 - q1h1 h1' = ; R1 1 + q1h1 1 + q1h1 hm 1 - q2hm hm' = ; Rm = 1 + q2hm 1 + q2hm EMI11.1 T1' and Tm' are damped stages. From the relations (5) and (6) it follows that (z2 - 1)2 N/D = W1/W2 = q1q2 (9) (z2 - 2z cos tol'+1) (z2 - 2z cos tom'+1) From the above it also follows that the transfer function of the leapfrog filter shown in Fig. 3 (and considered without the stage AS) is of even order and has an even numerator. Reference is now made to the latter figure, but now with the additional stage AS having a transfer function equal to z+l z+l As already explained in the introduction, this additional stage leads to the following choice of the function N , i.e. (z # 1)2 N/D = W1/W2 = (10) (z # 1)2 To satisfy this relation, W1 and W2 (see previous relations 5 and 6) have now been chosen as follows z2 - 1 z # 1 W1 = q1 = q1 (11) (z # 1)2 z # 1 i.e. relation (5) with cos tol = # 1 (z # 1)2 z # 1 W2 = 1/q2 = 1/q2 (12) z2 - 1 z # 1 i.e. relation (6) with cos t = + 1. om In this case T1' and Tm' (see previous relations (7) and (8) become 1 = h1' 2 (Z + 1)2 2 (z + 1)2 (13) @ @ z@ - 2zR1 cos tpl' + R1@ P1 (z # 1)2 (z # 1)2 Tm' = h2' = (14) z2 - 2zR cos @' + R2 Pm z m pm m The numerator of the transfer function of the filter can therefore be written z # l. (z + 1)2 (z2 - 2z cos to2 + 1) ... (z + 1)2 z # l Because (z # 1)4 = (z # 1)4 = (z # 1 (z # l) z # 1 this transfer function thus has at least three transmission zeros at z = - 1 and at, least one transmission zero at z = + 1. The structure of Fig. 3 permits, for instance, to realize a PCM bandpass filter operating at a sampling frequency fs and having a transfer function T/frequency characteristic such as shown in Fig. 4 i.e. with a flat passband and with three real zeros at frequency zero (DC) and one real zero at half the sampling frequency fs, i.e. in the z-plane at z = 1 and at z = -1 respectively. The transfer function T shown in Fig. 4 is a combination of the trans z + 1 fer function e.g. of the additional stage AS (Fig. 3) z - 1 and shown in Fig. 5 and of the transfer function represented in Fig. 6 of the filter part T'1, T2, T3, Tm' (Fig. 3). As z + 1 follows from Fig. 5, has a real transmission zero at z - 1 z =-1 and a real transmission pole at z = +1, whilst the filter part T1' - T' has four real transmission zeros at m zero frequency (i.e. z = +1) and one real transmission fs zero at 2 (z = -1). The overall filter has three real transmission zeros at frequency zero and one real trans mission zero at 2s because the transmission pole at z = +1 2 of Fig. 5 compensating a transmission zero at z = +1 of Fig. 4. On the other hand from Figs. 4 to 6 it also follows that in the passband the slope of Fig. 5 compensates that of Fig. 6 to provide the flat passband of Fig. 4. This inclined passband permits to place a maximum of attenuation zeros of the polynome h in this passband. The stages T'1 and AS in Fig. 3 can be realized by a single equivalent biquadratic filter stage T"1 shown in the flowgraph of Fig. 7 (where m = 4). T" has a forward path A1 z2 - 1 1 with transfer function = and a feedback path with p1 p1 transfer function @/@ = -- , P1 being the P1 P1 @ T'1 in the relation 13 given above. Indeed, in the structure of Fig. 3 the transfer function of the forward path from IN to the output OUT1 of T'1 is equal to z + 1 2 - = . Tg or to Z - 1 z + 1 1 P1 when taking relation (13) into account, and the transfer function of the feedback path from point A to output OUT1 is equal to (z # 1)2 T1' = P1 In the structure of Fig. 7 the signals at the summing points S2 and S3 have a different sign because the forward signals are plus signals, whereas the feedback signals are minus signals. For this reason the structure cannot be realized directly. On the other hand it is desirable to have as much negative signals as possible because biquadratic filter stages with a negative output signal (a negative hi) have been designed and are thus available as will be explained later. Therefore the signals in the flowgraph of Fig. 7 have been manipulated to remove a maximum number of positive signals. The result is shown in Fig. 8 but is still not satisfactory because still two positive signals remain which will give rise to some difficulty because they require the use of non-existent negative capacitances. The structure of Fig. 8 will now be used to realize an 8th order bandpass leapfrog filter having a transfer function with an odd numerator and making use of switched capacitor biquadratic filter stages which are insensitive to parasitic capacitmces. As mentioned above, the internal biquadratic filter stages such as T2 and T3 of the above considered leapfrog filter have an infinite quality factor and a small sensitivity to component variations, whereas the biquadratic end stages T" and T'4 are damped stages and should have a sensitivity which is as small as possible (for the quality factor). Such non-damped and damped stages are shown in Figs. 9 to 16 representing more particularly switched capacitor biquadratic stages which are each insensitive to parasitic capacitances. Some of these stages are known from the above mentioned article of P.E. Fleischer et al. Each of these stages makes use of - two operational amplifiers OAl and OA2; - a two-phase clock (not shown) with non-overlapping phases; - MOS switches such as Sll, S12, ... and S21, S22, ... the first indicia indicating the phase of the clock which has to be applied to close the switch; - four capacitor connecting circuits which are insensitive to parasitic capacitors, i.e. = a first or non-inverting type, hereinafter called TP1, such as shown in the upper left part of Fig. 9 and able to connect a capacitor (k3C1) between an input (IN) and ground via switches (S25, S15) during one clock phase (here clock phase 2 or CP2) and able to connect this capacitor between ground and an input (IN1) of an opera tional amplifier (OA1) via switches (S15, S11) during another clock phase (here clock phase 1 or CP1); ; = a second or inverting type, hereinafter called TP2, such as shown in the lower part of Fig. 9 and able to connect a capacitor (klC2) between two grounds via switches (S13, S14) and during one clock phase (CP1) and able to connect this capacitor between an output (of OAl) and an input (IN2) of an operational amplifier (OA2) via switches (S23, S24) during another clock phase (CP2); = a third type, hereinafter called TP3, such as shown in Fig. 9 and connecting a capacitor (k4C2) directly to an input (IN2) of an operational amplifier (OA2); ; = a fourth type, hereinafter called TP4, such as shown in Fig. 11 and which includes a capacitor (K4C2) one plate of which is permanently connected to an input (IN2) of an operational amplifier (OA2) and the other plate of which can be connected to ground via a switch and during one clock phase (CP1) . During the other clock phase the circuit is able to connect capacitor k C between a circuit input (IN) and an input (IN2) of an operational amplifier (OA2). The biquadratic filter stage of Fig. 9 can be characterized as follows - output 02 of OA2 is coupled to input IN1 of OA1 via a capa citor connecting circuit (S11, S21, S12, S22, K2C1) of type TP1 which connects capacitor k C to IN, on clock phase CP1; - input IN is coupled to IN1 via a capacitor connecting circuit (S11, S21, S15, S25) of type TP1 which connects capacitor K3C1 to IN1 on clock phase CP1; - output O1 of OA1 is coupled to input IN2 of OA2 via a capacitor connecting circuit (S13, S23, S14, S24' klC2) of type TP2 which connects capacitor klC2 to IN2 on clock phase CP2 ; - input IN is coupled to IN2 via a capacitor connecting circuit K4C2 of type TP3. Likewise, and in brief, the biquadratic filter stages of Figures 10 to 16 can be characterized as follows Fig. 10 : 02 to IN1 : TP2, CP1 IN to IN1 : TP2, CP1 O1 to IN2 : TP1, CP2 IN to IN2 : TP3 Fig. 11 : 2 to IN1 : TP1 (K2C1), CP1 2 to IN1 : TP2 (k3Cl), CP1 O1 to IN2 : TP1, CP2 IN to IN2 : TP4, CP2 Fig. 12 : O2 to IN1 : TP1 (K2C1), CP2 O2 to IN1 : TP1 (K3C1), CP1 O1 to IN2 : TP2, CP2 IN1 to IN2:, TP4, CP2 Fig. 13 : 02 to IN1 : TP2 (k2C1), CP2 2 to IN1 : TP1 (K3C1), CP2 IN to IN1 : TP4, CP2 1 to IN2 TP1 (K1C2), CP1 O1 to IN2 : TP4 (k5C2), CP2 Fig. 14 : O2 to IN1 : TP1 (K2C1), CP1 O2 to IN1 : TP1 (k3Cl), CP2 IN to IN1 : TP4, CP2 2 to IN2 : TP2 (K1C2), CP2 O2 to IN2 : TP4 (K5C2), CP1 Fig. 15 : 02 to IN1 : TP1, CP1 IN to IN1 : TP3 1 to IN2 : TP2, CP2 IN to IN1 : TP3 Fig. 16 : 02 to IN1 : TP2, CP1 IN to IN1 : TP3 O1 to IN2 : TP1 CP2 IN to IN2 : TP3 All these biquadratic filter stages, except the one of Fig. 12 are characterized by the fact that. the capacitors K1C2 and k C are connected to IN2 and IN1 on different phases of the clock respectively. A further difference with the prior art is that different combinations of circuits TP1 to TP4 are used. The above biquadratic filter stages of Figs. 9 to12 (to be considered without damping capacitor k3C2) have an infinite quality factor and a transfer function of the form 2 z - 2 z cos to + 1 T=-h 2 @ (15) z - 2 z cos t + 1 with h > O and their components satisfy the following relations - Figs. 9 to 10 (without k5C2) K1K2 = 2 (1 - cos tp) K1K3 = 2 (1 - cos to) K4 K4 = h - Fig. 11 (withoutk5C2) K1K2 = 2 (1 - cos to klk3 = 2 (cos t - cos t p o K1K3 = 2 (cos tp - cos to k4 with t < to p o - Fig. 12 (without K5C2) K1K2 = 2 (1 - cos t @@1@@2 @ @@ @@@ @O@ K1K3 = 2 (cos tp - cos tp) K4 = h with tp < tp From these four filter stages one can derive damped filter stages with transfer function 2 z - 2 z cos to + 1 T=-h 2 z - 2 z R cos t + R2 with h > o and o < R < 1 These damped filter stages are also shown in Figs. 9 to 12, now to be considered with the damping capacitor K5C2. Their components satisfy the following equations Fig. 9 (with k5C2) K1K2 = 1 + R-2 - 2 R-1 cos tp K1K3 = 2 (1 - cos to) K4 k4 = hR K5 = R-2 - 1 Fig.10 (with k5C2) K1K2 = 1 + R2 - 2 R cos tp K1K3 = 2 (1 - cos t@) kq = k4 = h K1K5 = 1 - R2 Fig. 11 (with k5C2) K1K2 = 2 (1 - cos to) K1K3 = 1 - R-2 + 2R-1 cos tp - 2 cos to K4 = R-2h K5 = 1 - R-2 with tp < to Fig. 12 (with k5C2) K1K2 = 2 (1 - cos to) K1K3 = R-2 - 1 + 2 cos to - 2 R-1 cos tp K4 = R-2h K5 = R-2 - 1 with to < tp Figs. 13 and 14 show other damped biquadratic stages which realize the transfer function 2 z - 2 z cos to + 1 T = h zê - 2 z R cos tp + Rê where h > o and o < R < 1 Their components satisfy the following relations Fig. 13 K1K2 = 2 (1 - cos t @@ @ K1K3 = 1 - R-2 + 2 R cos tp - 2 cos to k 3k5 = 1 - R2 k4k5 =h with tp < to Fig. 14 K1K2 = 2 (1 - cos to) K1K3 = 1 - R2 + 2 R2 cos to @ 2 R cos tp k 3k5 = 1 - R2 K4K5 = h with to < tp Finally, Figs. 15 and 16 show biquadratic bandpass filter stages realizing the transfer function z2 - 1 T = h (17) z2 - 2 z R cos tp + R2 with o < R < 1 Their components satisfy the following relations Fig. 15 k1k2 = 1 + R-2 - 2 R-1 cos tp k4 = h R-2 klk3 = 2 h R-2 k5 =R -2 1 with h > O Fig. 16 klk2 = 1 + R2 - 2 R cos t p k4 = -h k 1k3 = -2h K1K5 = 1 - R2 with h < O To understand how the above relations between components of the biquadratic filter stages are obtained, reference is made to the above mentioned article by P.E. Fleischer. As an example, the relations for Fig. 10 will be given for both the undamped and damped embodiments. When using the same references as in the article one has C1 = D C2 B K1C2 = A K2C1 = C K3C1 = G k4C2 =K and E = F = H = I = J = L = O (for the undamped version) and E = K5C1 and F = H = I = J = L = O (for the damped version). From the article it follows that the transfer functions are EMI21.1 (undamped version) (damped version) On the other hand, from the above it follows (see relations 15 and 16) that these transfer functions should be 2 2 z cos t + 1 z # Tu = - h @2 @ @ @@@ @ @ @ z p # z - 2 z cos t + 1 Td = - h z2 - 2zR cos tp + R2 p so that the following relations have to be satisfied undamped stage K ¯ B @@ AG - 2 (1 - cos t DK 2 (1 - cos t ) AC = 2 (1 - cos tp) DB Damped stage K B AG = 2 (1 - cos = 2 (1 - cos to) AC + AE = 2 (1 - R cos tp) DB AE DB = l - Rê Because B = C2 and D = C1, solutions for these equations are Undamped stage A = K1C2 or A = K1C1 C = K2C1 C = K2C2 G = K3C1 E = K3C2 K = K4C2 K = K4C2 so that k4 = h k1k3 k4 = 2 (1 - cos to) K@K@ = 2 (1 - cos t p which are the relations given above for Fig. 10 (undamped). Damped stage A = K1C2 or A = K1C1 C = K2C1 C = k2C2, E = K5C1 E = k5C2 G = K3C1 G = K3C2 K = K4C2 R = K4C2 so that k4 = h klk3 = 2 (1 - cos to) K4 K1K5 = 1 - R2 K1K2 = 1 + R2 - 2 R cos tp which are the relations given above for Fig. 10 (damped). For stages T1", T2, T3 and T4' of the bandpass filter of Fig. 17 the following choice has now been made. Biquadratic filter stage T'i is a combination, as explained above, of a forward circuit between IN and OUT A1 with transfer function @/P1 and of a backward circuit between A and OUT1 with transfer function A2/P1@ These transfer functions are respectively provided by the biquadratic filter stages of Fig. 15 and of Fig. 9 (damped and cos tol = 1) having substantially the same topology. Indeed, the transfer functions of the stage according to Fig. 15 is (see relation 17) z2 - 1 h11 2 @ z - 2 z R cos tpl + R11 with h11 > 0, and O < R11 ( 1 A1 and this function is indeed of the type .7, whilst the transfer function of the stage of Fig. 9 (damped and with cos tol = 1) is (see relation 16) (z - 1)2 @ @12 z2 - 2 z R12 cos tp1 + R122 with h12 > O and O < R12 < 1 and this is indeed of the type A2/P2 This merging of the stages of Figs. 15 and 9 in one single stage is important because it ensures a perfect matching of the pole and zero at z = + 1. Biquadratic stages T2 and T3 are of the type shown in Fig. 9 without damping capacitor k5C2. They have the respective transfer functions (see relation 15) 2 z - 2 z cos to2 + 1 T2 = - h2 2 o2 z - 2 z cos t 2 + 1 p with h2 > O z2 - 2 z cos to3 + 1 T3 = -h3 z - 2 z cos t@3 + 1 with h3 > o Finally, biquadratic stage T'4 is of the type shown in Fig. 9 with damping capacitor k5C2 and cos to4 = 1 and having a transfer function (see relation 16) (z - 1)2 T4' = h4 z2 - 2 z R4 cos tp4 Z + R42 with h4 > 0 and 0 < R4 < 1 In Fig. 17, to be considered with the dotted connections but without the dashed line connections, the stages T"1, T2, T3 and T4' may be identified in a relatively easy way with the stages of Figs. 15 and 9 (damped and with cos tol = 1); Fig. 9 (undamped); Fig. 9 (undamped) and Fig. 9 (damped and with cos to4 = 1) respectively. The only problems might be the biquadratic stage T"1 and the leapfrog connections between the stages. Both are therefore explained in detail hereinafter. The stages of Figs. 15 and Fig. 9 only differ in the connection of their entries and therefore it was possible to use in Fig. 17 for T"1 a same circuit with an input IN, constituting the input of the filter and connected as in Fig. 15, and with another input A connected as in Fig. 9. However, because the signal at the input A should be positive (see Fig. 8) but is negative at the output OUT2 (see the transfer function of T2) to which A is coupled a negative capacitor -Ca corresponding to k4C2 in Fig. 9 has to be used. From 8 it also follows that OUT1 of T"1 and the output OUT3 of T3 are both connected to the input of T2 via a summing circuit S2. In Fig. 17 the output OUT1 is coupled to the inputs IN21 and IN22 of T2 via capacitors C8 and Cg corresponding to k3Cl and k4C2 in Fig. 9 and the output OUT3 is coupled to the same inputs IN21 and IN22 via capacitors C10 and C11 respectively also corresponding to k 3C1 and k C in Fig. 9. The interconnections between the capacitors C8 and C10 as well as between the capacitors C2 and -Ca constitute summing point S1 in Eg. 8. From this figure it also follows that OUT2 of T2 and OUT of T'4 are both connected to the input of T3 via a summing circuit S3.In Fig.17 output OUT2 is coupled to the inputs IN31 and IN32 of T 3 via respective capacitors C14 and C15 corresponding to k3C1 at kqC2 in Fig. 9 and the output OUT is coupled to the same inputs IN21 and IN22 via capacitors C16 and C17 corresponding to k3C1 and k4C2 in Fig. 9. The interconnections between the capacitors C14 and C16 as well as between the capacitors C15 and C17 constitute summing point S3 in Fig. 8. The output OUT3 of T3 is also connected to the input IN42 via negative capacitor -Cb. A negative capacitor -Cb is required because the signal at the output OUT3 of T3 is negative, as follows from the above given transfer function, whereas according to Fig. 8 the signal at the input of T'4 should be positive. As follows from the above the bandpass filter of Fig. 17 requires two negative capacitors -Ca and -Cb which do not exist. However, by manipulation of the charges on the various capacitors of the circuit these negative capacitors can be removed. Thus finally the filter circuit of Fig. 17 without the dotted connections, but with the dashed line connections is obtained i.e. with - capacitor C3 between OUT3 and IN12; - capacitor C20 between OUT2 and IN42; - and capacitor C21 between the junction point of sell, s and 5116' S216 This filter has a transfer function versus frequency characteristic such as shown in Fig. 4. Part of the corresponding attenuation versus frequency characteristic is also represented in Fig. 18. The group delay versus frequency characteristic of the filter is shown in Fig. 19 and fully satisfies the CCITT recommendations as indicated by the gauge. The receive lowpass filter RLPF of Fig. 1 is represented in detail in Fig. 20 and includes biquadratic stages T'1 and T'2 of the type shown in Fig. 9 (damped version) and Fig. 10 (damped version) respectively. In Fig. 20 capacitors C1 and C2 correspond to capacitors k C and k C in Fig. 9 and capacitors C8 and Cg correspond to capacitors k3C1 and k4C2 in Fig. 10. The output OUT of T'2 is fed back to IN12 and IN22 of T'1 via connections including capacitors C3 and C4 respectively. Part of the attenuation versus frequency characteristic of the filter of Fig. 20 is shown in Fig. 21. In the lower passband the above mentioned sin x correction is visible.

Claims: CLAIMS 1. Leapfrog filter including a plurality of biquadratic filter stages coupled in a leapfrog configuration, characterized in that it is a sampled filter with characteristics equivalent to those of a filter comprising the cascade connection of a first filter part having a transfer function proportional to + 1, and of a second filter part comprising a plurality of Z 1 1 biquadratic filter stages coupled in leapfrog configuration the first and the last stages of which are dissipative for reactive pseudo-power and each have a transfer function proportional to a same factor (z + 1)2 and the other stages being non-dissipative for reactive pseudo-power, wherein z = e eST with s being the complex frequency and T being the sampling period of the filter. 2, Leapfrog filter according to claim 1, characterized in that the characteristics of said second filter part are equivalent to those of a two-port non-dissipative for reactive pseudo-power and coupled between first (W1) and second (W2) termination stages dissipative for reactive pseudo-power and having transfer functions W1 and z # 1 W2 proportional to and the reverse thereofrespectively. z # 1 3. Leapfrog filter according to claim 2, characterized in that the factor (z + 1)2 to which the transfer function of said first stage is proportional is the numerator of a function T1 (z # 1)2 T1' = = h1' 1 + W1T1 z2 - 2 z R1 cos tp1' + R12 z # 1 wherein W1 = q1 is the transfer function of said first z # 1 termination stage (W1) and (z # 1)2 T1 = h1 is the transfer function of z2 - 2 z cos tp1 + 1 the first biquadratic filter stage of said two-port, and wherein hl, h1' and q1 are positive constants and tpl is the angle of a transmission pole of T1 in the z-plane defined by z = esT. 4. Leapfrog filter according to claim 2, characterized in that the factor to which the transfer function of said last stage is proportional, is the numerator of a function Tm (z # 1)2 Tm' = = hm'. 1 + W2-1Tm z2 - 2 z Rm cos tpm' + Rm2 z # 1 wherein W2 = 1/q2 is the transfer function of said z # 1 second termination stage (w2) and T = h (z + 1)2 is the transfer function m @m z2 - 2 z cos tpm + 1 of the last biquadratic stage of said two-port and wherein hm, h'm and q2 are positive constants and tpm is the angle of a transmission pole of Tm in the z-plane defined by z = esT. 5. Leapfrog filter according to claim 4, characterized in that it includes a biquadratic filter input stage (T"1) with a first input (IN) constituting the input of the leapfrog filter, a second input (A) and an output (OUT1) and includes a forward circuit between said first input and said output (OUT1) which realizes the transfer function of said first filter part and a feedback circuit between said second input (A) and said output (OUT1) which realizes the transfer function of said first stage (T'1) of said second filter part. 6, Leapfrog filter according to claim 5, characterized in that said forward and backward circuits are substantially constituted by a same biquadratic filter circuit. 7 Switched capacitor biquadratic filter circuit with an input and an output and including a non-overlapping two-phase clock source and first and second operational amplifiers each having an inverting input, a grounded noninverting input, an output and a feedback capacitance, the output of said first amplifier being coupled to the input of said second amplifier via first capacitance means including a first capacitance the plates of which are connected to first and second change-over switches able, under the control of said clock source, to connect them between a ground and said output of said first amplifier and between said ground and the input of said second amplifier, the output of said second amplifier being coupled to the input of said first amplifier via second capacitance means including a second capacitance the plates of which are connected to third and fourth changeover switches able under the control of said clock source, to connect them between said ground and the output of said second amplifier and between said ground and said input of said first amplifier, and said filter input being coupled to an input of one of said amplifiers via third capacitance means including at least a third capacitance, characterized in that sad clock source controls said switches (S13, S23; S14, S24; S12, S22; S11, S21) in such a way that said second capacitor (k2C1) is connected to the input (IN1) of said first amplifier (OA1) and said first capacitor (klC2) is connected to said input (IN2) of said second amplifier (OA2) for respective different phases of said clock source (Figs. 9-11, 12-16). 8. Switched capacitor biquadratic filter circuit with an (Fig. 16) input and an output and including a nonoverlapping two-phase clock source and first and second amplifiers each having an inverting input, a grounded noninverting input, an output and a feedback capacitance, the output of said first amplifier being coupled to the input of said second amplifier via first capacitance means including a first capacitance the plates of which are connected to first and second change-over switches able,under the control of said clock source, to alternately ground the one and other plates of said first capacitance for different phases said clock source respectively, the output of said second amplifier being coupled to the input of said first amplifier via second capacitance means including a second capacitance the plates of which are connected to third and fourth change-over switches able, under the control of said clock source, to simulaneously ground both plates of said second capacitance for a same phase of said clock source, said filter input being directly connected to the input of said first amplifier via a third capacitance and to the input of said second amplifier via a fourth capacitance, a damping fifth capacitor being connected between the output of said second amplifier and the input of said first amplifier, characterized in that said clock source controls said switches in such a way that, on the one hand, said second capacitance (k2Cl) is connected to said input (IN1) of said first amplifier (OA1) and, on the other hand, said first capacitance (klC2) is connected to said input (IN2) of said second amplifier (OA2) for respective different phases of said clock source. 9. Switched capacitor biquadratic filter circuit (Fig. 16) according to claim 8, characterized in that it has a transfer function, considered in the 2-plane, z2 - 1 T = h @ @ 2 - 2 z R cos tlz + R with klk2 = 1 + R2 - 2 R cos t p K4 = - h K1K3 = - 2 h K1K5 = 1 - R2 wherein tp is the angle of the pole of T; klC2, k2Cl, k3C1, k4C2 and k5C2 are said first, second third, fourth and fifth capacitances and C1 and C2 are said feedback capacitances of said first (OA1) and second (OA2) amplifiers respectively; kl to k5 and R are positive constants and h < C and O < R < l. 10. Switched capacitor biquadratic filter circuit, characterized in that it simultaneously realizes a transfer function proportional to z2 - 1 between a first input (IN) and an output (OUT1) a n d a second transfer function propor tional to (z + 1)2 between a second input (A) and said output the transfer functions being considered in the z-plane.

Inventors: GUEBELS, PIERRE-PAUL FRANCOIS M.M., Guebels, Pierre-Paul François M.M.
Applicants: ALCATEL N.V.
Application year: 1983
Main classification: H03H19
IPC codes: H03H19