J.Semicond.2013,34(9) Wang Xin et al. R R =4KTR Fig.5.The noise analysis model 60 60 IIP3 @0 dB+ 50 20 IIP3 18dB+ 40 NF @0dB 30 -20 Frequency range -40 NF 18 dB 20 Fig.7.The band selection and frequency tuning scheme. 60 10 102 10 10 10 109 ferent cutoff frequencies.So,by switching resistors in the re- R(2) sistor array,the bandwidth of the filter could change from one to another.The first flaw in this method is that the number of Fig.6.The optimization of IIP3 and NF trade-off. bands is restricted by the number of resistors in the array,and the second is that it can only realize discrete frequency tuning. According to the definition of noise figure in Ref.[10],we get By adding a capacitor array as shown in Fig.7,a mixed ap- proach is employed in this design.The resistor array is used to NF=1+ select a center frequency,and by increasing or decreasing the 4kTRs number of unit capacitors in the capacitor array,the cutoff fre- quency could change around the center frequency.Thus,the ≈1+ 2+H0+ cutoff frequency of the filter is continuously adjustable.This RsHo QopCRa step is determined by the capacitance of the unit capacitor in the capacitor array.To maintain an elliptic frequency response (1+Ho)2+ (9) shape,Ro,Re,Rd,and Cz should change at the same time ac- RsHo +@pR.C cording to Ra and C. Three indications are conveyed in this expression:(1)the noise Accuracy gain is desired to ensure that the auto gain con- contribution of Ra is individually decided by the filter and ca- trol (AGC)loop is working correctly.As analyzed before,the pacitance parameters;(2)Re contributes more noise at high gain of the bi-quad is determined by the ratio of the two re- gain mode;and (3)the OTA noise could be dominant if its input sistors.Re is chosen to change the gain of the bi-quad due to equivalent noise is large enough. Re having no influence on the cutoff frequency.As shown in To show the trade-off between IIP3 and NF.Figure 6 plots Fig.2.R.is a resistor array with segmented resistors.The array the relationship of NF and IIP3 with Ra.C could be chosen operates synchronously with the switching of Ra to maintain an under chip area consideration.In this example,we set C= elliptic filter frequency response,while segmented resistors are 3 pF.As is shown in the graph,the range of resistor values is used to change the gain of the bi-quad.Figure 8 shows the SFG determined by the specification of IIP3 at low gain and NF at of the bi-quad with a non-ideal integrator model.A(j@)is the high gain.If we choose Ra at the right side of the optimal value, open loop gain of OTA.Using SFG analysis,a more accurate IIP3 could be less sensitive to the variation in Ra.but the noise expression of the bi-quad transfer function is given as will be higher because of the larger value of Ra.Below the optimal value of Ra.IIP3 changes dramatically with Ra while 1+5 s2 十 the NF curve is relatively flat. H6)=Ar“a2aT 1+5 2 (10) 3.Circuit design 0.+ag 3.1.Band selection and gain adjustment Subscriptr indicates the effective filter parameter differing Since the cutoff frequency is determined by the RC prod- from the ideal one.As explained in Ref.[9],gain and O are uct,if C remains constant,then different resistors refer to dif- more dependent on A(j).The ratio of effective gain and O 095007-4J. Semicond. 2013, 34(9) Wang Xin et al. Fig. 5. The noise analysis model. Fig. 6. The optimization of IIP3 and NF trade-off. According to the definition of noise figure in Ref. [10], we get NF D 1 C V 2 n;in 4kTRs  1 C Ra RsH2 0  2 C H0 C 1 Q!pCRa  C Rn RsH2 0 " .1 C H0/ 2 C  1 Q C !pRaC 2 # : (9) Three indications are conveyed in this expression: (1) the noise contribution of Rd is individually decided by the filter and ca￾pacitance parameters; (2) Rc contributes more noise at high gain mode; and (3) the OTA noise could be dominant if its input equivalent noise is large enough. To show the trade-off between IIP3 and NF, Figure 6 plots the relationship of NF and IIP3 with Ra. C could be chosen under chip area consideration. In this example, we set C D 3 pF. As is shown in the graph, the range of resistor values is determined by the specification of IIP3 at low gain and NF at high gain. If we choose Ra at the right side of the optimal value, IIP3 could be less sensitive to the variation in Ra, but the noise will be higher because of the larger value of Ra. Below the optimal value of Ra, IIP3 changes dramatically with Ra while the NF curve is relatively flat. 3. Circuit design 3.1. Band selection and gain adjustment Since the cutoff frequency is determined by the RC prod￾uct, if C remains constant, then different resistors refer to dif￾Fig. 7. The band selection and frequency tuning scheme. ferent cutoff frequencies. So, by switching resistors in the re￾sistor array, the bandwidth of the filter could change from one to another. The first flaw in this method is that the number of bands is restricted by the number of resistors in the array, and the second is that it can only realize discrete frequency tuning. By adding a capacitor array as shown in Fig. 7, a mixed ap￾proach is employed in this design. The resistor array is used to select a center frequency, and by increasing or decreasing the number of unit capacitors in the capacitor array, the cutoff fre￾quency could change around the center frequency. Thus, the cutoff frequency of the filter is continuously adjustable. This step is determined by the capacitance of the unit capacitor in the capacitor array. To maintain an elliptic frequency response shape, Rb, Rc, Rd, and Cz should change at the same time ac￾cording to Ra and C. Accuracy gain is desired to ensure that the auto gain con￾trol (AGC) loop is working correctly. As analyzed before, the gain of the bi-quad is determined by the ratio of the two re￾sistors. Rc is chosen to change the gain of the bi-quad due to Rc having no influence on the cutoff frequency. As shown in Fig. 2, Rc is a resistor array with segmented resistors. The array operates synchronously with the switching of Ra to maintain an elliptic filter frequency response, while segmented resistors are used to change the gain of the bi-quad. Figure 8 shows the SFG of the bi-quad with a non-ideal integrator model. A(j!/ is the open loop gain of OTA. Using SFG analysis, a more accurate expression of the bi-quad transfer function is given as Hr .s/ D H0r 1 C s !zrQzr C s 2 !2 zr 1 C s !prQr C s 2 !2 pr : (10) Subscript r indicates the effective filter parameter differing from the ideal one. As explained in Ref. [9], gain and Q are more dependent on A(j!/. The ratio of effective gain and Q 095007-4