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Shadow filters generalisation to nth-class
Y. Lakys and A. Fabre
The theory of shadow filters, a new family of second-order filtersrecently introduced, is extended to class n filters, where n is a positiveinteger. It is shown that their centre frequency f0An is given by f0An ¼
f0(1 2 A)n/2 where f0 is the centre frequency of the starting unitygain filter and A the gain of amplifiers. Simulation results of a practicalexample implemented for n ¼ 2 are given. They entirely confirm thetheoretical analysis.
Introduction: A new family of second-order filters, shadow filters, hasbeen introduced in [1]. They can be either pasive or active. Byanother way, a practical implementation of a fully active reconfigurablefilter for cognitive radio communications has been given in [2]. Thisbelongs in fact to a subfamily of shadow filters. The general diagramof a shadow filter is shown in Fig. 1 [1]. A feedback voltage amplifierwith gain A is added externally to a second-order basic cell (whichcould be passive or active) that exhibits a lowpass output VLP and abandpass output VBP. Its centre frequency is f0 and its quality factor isQ. We suppose for this generalisation that the magnitudes of both trans-fer functions of the basic cell are equal to unity. We denote the class-0filter the starting second-order filter and the class-1 filter the shadowfilter in Fig. 1.
inputVE VIn
VBP
VLPf0, Q
A.VLP
second-order filter
A
++
amplifier
bandpassoutput
lowpassoutput
Fig. 1 Class-1, second-order shadow filter
The study in [1] showed that the transfer functions of class-0 andclass-1 filters remain of the same type. The gains of the bandpassoutputs are identical. The magnitude of the gain of the lowpass outputof the class-1 filter is (1 − A) times lower than the gain of the lowpassof the class-0 filter. The centre frequency of the class-1 filter which istunable by A (A , 1) is then given by: f0A1 = f0A
�������
1 − A√
. The 23 dBbandwidths of both bandpass outputs remain unchanged too.
inputVE
inputVE
f0An–1
VBP
VLP
A.VLP
1–A
1–A
++
a
b
outputs
VBP
VLP
outputs
(n–1) class filter
f0An–1
f0An
(n–1) class filter
Aamplifier
Fig. 2 Necessary modifications of (n 2 1) class filter, and class-n shadowfilter
a Necessary modifications of (n 2 1) class filterb Class-n shadow filter
ELECTRONICS LETTERS 8th July 2010 Vol. 46 N
We introduce in this Letter the generalisation of the above theory inorder to obtain the class-n shadow filter (n positive integer) and weinvestigate its properties. Simulation results of a class-2 (n ¼ 2)shadow filter operating in current mode are given to confirm thisgeneralisation.
Theoretical approach: The value of A being always unchanged, wesuppose that the previous transformation is applied as indicated inFig. 1 up to the class-(n 2 1) filter. The latter has then the same proper-ties as the shadow filter in Fig. 1 (class 1) if we consider here the shadowfilter of class-(n 2 2) as the starting filter. Its centre frequency is thengiven by: f0A(n−1) = f0A(n−2)
�������
1 − A√
. As mentioned above, the gain ofthe bandpass output remains unchanged while the gain of the lowpassoutput is divided by (1 − A).
Fig. 2a shows the class-(n 2 1) filter the lowpass output of which isamplified by a factor (1 − A) in order to get the gain back to its initialvalue (unity), therefore the resulting filter will be exactly in the sameconditions as the starting class-0 filter. Fig. 2b shows the class-nshadow filter obtained by applying the same transformations as thosepresented in Fig. 1. Its properties are easily deduced from those of theprevious class-1 filter. Its centre frequency is f0An = f0A(n−1)
�������
1 − A√
=f0A(1 − A)n/2 and its 23 dB bandwidth of the bandpass output isalways identical to that of class-0. Fig. 3 shows the evolution of theratio f0An/f0 as a function of gain A for different values of n. When Ais negative (region 3) f0An is greater than f0 and its evolution is evenfaster as n is greater. Note that for n ¼ 2 the evolution is a linear functionof A.
n = 2
n = 3 f0An/fn
1
0 1
n = 1
A < 0
region 3A < 0
region 2 0 < A < 1
region 1 A > 1
A > 0
shadow filterunfeasible
Fig. 3 Variation of f0An/f0 against A for various values of n
The analysis of different sensitivities shows that Sf0An
A = SQAn
A =−(n/2)SGLPAn
A = −nA/2(1 − A); therefore, for example for n ¼ 2 andA negative, all the values of these sensitivities are less than or equalto unity and an accidental variation on A will not have any impact onthe parameters of the shadow filter.
ZQC2
feedbackconveyor
C1VC1
QCCCII+
YQ
IQ
I0CR
XQ
X2 2CCCII+
Y2
I02
Z2
ZCR1; Gi = 2
ZCR2; Gi = |A|
CCCII+
Z1 1CCCII+
X1
I01
Y1ZOUT
IOUT(t)
basic filter (class-0)
IIN(t)
CR
XCR
YCR
Fig. 4 Class-2 current mode shadow filter, A , 0
Practical example: The generalisation above is also valid for circuitsimplemented in current mode. Because the latter is preferred for circuitsoperating at high frequencies [3] and signal summing is easier to obtain,the current mode shadow filters will occupy the smallest silicon area. Wehave implemented our circuits in current mode using current controlledconveyors (CCCII+) [3, 4]. Fig. 4 shows the entire class-2 shadow filter.The basic filter (class-0) which comprises conveyors 1, 2 and Q is thesame as in [1] and Iout is its bandpass output. Its centre frequency isf0 = 1/2pRX C for I01 = I02 = I0 and C1 = C2 = C;RX = VT/2I0
being the intrinsic resistance on output X of the CCCII+ [3]. CurrentIQ allows tuning of the quality factor. The voltage VC1 across capacitorC1 is lowpass so that the total feedback current which is added to the
o. 14
input is IZCR = (2 + |A|)VC1/RXCR;RXCR = VT/2I0CR being the intrinsicresistance of the conveyor (CR). Note that the emitter areas of transistorsof the two outputs of the conveyor (CR) are properly dimensioned [3], ina way to have IZCR1 = 2IXCR and IZCR2 = |A|IXCR with A , 0. Thus, con-veyor (CR) allows synthesising all the necessary current amplifiers thatare deduced for n ¼ 2 from Figs 1 and 2b, the bias current of the con-veyor (CR) being in this way chosen as |A| = I0CR/I0.
This filter was implemented in 0.25 mm SiGe BiCMOS technologyfrom STMicroelectronics. The main limitations result from the transitionfrequency for the PNP transistors in this technology, that is about 6 GHz,so the maximal operating frequency does not have to exceed 1 to 2 GHz.With such frequencies, the limitations owing to the variations of A andRX according to the frequency can be neglected. Biased under +2.5 Vwith I0 ¼ 50 mA and C ¼ 5 pF, the centre frequency of the startingbandpass (class-0) filter is 104 MHz. For the class-2 shadow filter weobtained f0A2 ¼ 211 MHz (with A ¼ 21), 374 MHz (with A ¼ 22)and 585 MHz (with A ¼ 25), the corresponding values calculatedtheoretically from the formula above being 208, 312, and 624 MHz,respectively. The value of the Q-factor can be adjusted to the desiredvalue, varying current IQ, without incidence on f0A2 (in fact lowerthan 2% in all cases). All these results entirely validate thegeneralisation.
Conclusions: The theory of shadow filters has been generalised toclass-n shadow filters, thus opening the door to new applications. Theonly necessary amplifiers for this generalisation have either gain A or(1 − A) which are easily synthesised in current mode. A practical
ELECTR
example of a class-2 (n ¼ 2) shadow filter operating in current modehas been given. It has a minimum number of CCCII+ and its centre fre-quency f0A2 is directly proportional to gain A.
# The Institution of Engineering and Technology 201016 February 2010doi: 10.1049/el.2010.0452One or more of the Figures in this Letter are available in colour online.
Y. Lakys and A. Fabre (IMS Laboratory, IPB-ENSEIRB - UMR5218CNRS, Universite Bordeaux 1, 351 Cours de la liberation - 33405,Talence Cedex, France)
E-mail: [email protected]
References
1 Lakys, Y., and Fabre, A.: ‘The shadow filters – a new family of 2nd orderfilters’, Electron. Lett., 2010, 46, (4), pp. 276–277
2 Lakys, Y., Godara, B., and Fabre, A.: ‘Cognitive and encryptedcommunications. Part 2: a new theory for frequency agile active filtersand the validation results for an agile bandpass in SiGe-BiCMOS’.Invited paper, 6th Int. Conf. on Electrical and Electronics Engineering,ELECO 2009, Bursa, Turkey, November 2009, pp. 16–29
3 Fabre, A.: ‘Electronique analogique rapide, circuits et applications’, (inFrench), TechnoSup, Ellipses edit., Paris, France, 2009
4 Fabre, A., Saaid, O., Wiest, F., and Boucheron, C.: ‘High frequencyapplications based on a new current controlled conveyor’, IEEE Trans.Circuits Syst., 1996, 43, pp. 82–91
ONICS LETTERS 8th July 2010 Vol. 46 No. 14
