IMPLEMENTATION ASPECTS OF GENERALIZED BANDPASS SAMPLING

Transcription

1 15th European Signal Processing Conerence (EUSIPCO 27), Poznan, Poland, September 3-7, 27, copyright by EURASIP IMPLEMENTATION ASPECTS OF GENERALIZED BANDPASS SAMPLING Yi-Ran Sun and Svante Signell, Senior Member IEEE Department o Electronic, Computer and Sotware Systems, School o ICT KTH - Royal Institute o Technology, Stockholm, Sweden yiran@imitkthse, srs@kthse ABSTRACT BandPass Sampling (BPS) realizes requency downconversion in radio receiver ront-ends by a sampling rate that can be slightly larger than twice the inormation bandwidth compared to twice the highest requency or traditional LowPass Sampling (LPS) However, some implementation problems, harmul signal spectral olding, noise aliasing and sampling jitter, are unavoidably present in conventional BPS systems Under recent research, generalized bandpass sampling combined with iltering has been proposed or dealing with these problems In this paper, the novel sampling architectures with intrinsic FIR and IIR iltering are implemented using a sampled-data technique called Switched-pacitor (SC) circuit technique Speciically complex SC iltering is designed and analyzed Both the analysis and simulation results show that the designed complex SC ilter ulils the expectations o generalized bandpass sampling 1 INTRODUCTION Under the concept o Sotware Deined Radio (SDR) [1], the A/D converter is placed as close as possible to the antenna in radio receiver ront ends BPS realizes down-conversion by using a sampling rate that can be slightly larger than twice the inormation bandwidth, which is similar to down-conversion mixing By using ideal BPS, the receiver architecture is simpliied and the requirements on the ollowing A/D converter can be relaxed compared to using conventional LPS BPS may be a very easible solution or SDR In addition to the advantage o lower sampling rate, BPS has serious drawbacks in real implementations The BPS rate has to be careully chosen in order to avoid harmul signal spectral aliasing [2] It is well known that a resistor charging a capacitor gives rise to a total thermal noise with power o kt/c [3], k is Boltzmann constant, T is the absolute temperature and C is the capacitance By using BPS, the wideband RC-iltered white noise will be olded due to subsampling resulting in a serious degradation o the Signal-to- Noise Ratio (SNR) compared to the equivalent LPS system Normally or wireless communication the carrier requency is much higher than the bandwidth, or example 8211a use 5GHz center requency with a 2MHz channel spacing A direct implementation o BPS would lead to an undersampling o 25 times In addition, since the sampling device, eective bandwidth B e = 1/(4RC) = π 3dB /2, has to be much aster than the highest input signal requency the noise aliasing is even worse Jitter is also always present in sampling systems, and the eects can be serious especially when the input signal requency is high Focusing on these implementation problems in BPS systems, the generalized bandpass sampling combined with FIR iltering was proposed and analyzed in a series o papers [4] to [7] In this paper, complex FIR iltering [8] and IIR iltering is added and implementations using Switched-pacitor (SC) circuit techniques are presented These results are based on the recent PhD thesis work by [9] Both analysis and simulations show that the designed complex SC ilter ulils the expected improvement o generalized bandpass sampling as compared to LPS This paper is organized as ollows In section 2, the concept o generalized bandpass sampling is briely reviewed In section 3, active and passive sampling schemes are discussed and compared In section 4, the implementation o generalized bandpass sampling combined with complex FIR/IIR iltering is shown and discussed Finally, conclusions are drawn in section 5 2 GENERALIZED BANDPASS SAMPLING Second order generalized bandpass sampling with irst order FIR iltering can be described as shown in Fig 1 assuming x(t) is a bandpass signal with a carrier requency o c, a band-pass sampling rate o s = 1/T s, and the second sampling branch lags behind the irst by T D Two special x(t) m δ(t mt s ) m δ(t mt s T D ) β β 1 x s (t) Figure 1: Second order sampling with irst order FIR iltering cases o generalized bandpass sampling are o speciic interest: Generalized Quadrature BandPass Sampling (GQBPS) with T D = 1/(4 c ) and Generalized Uniorm BandPass Sampling (GUBPS) with T D = 1/(N 1) s ) [9] Generalizing Fig 1 to N1 branches the sampled-data signal x s (t) is given by x s (t) = x(t) N n= m= β n δ(t mt s nt D ) (1) The corresponding Fourier transorm o x s (t) is given by X s ( ) = 1 T s H( ) = H(k s )X( k s ), (2) k= N n= β n e j2π nt D (3) 27 EURASIP 1975

2 15th European Signal Processing Conerence (EUSIPCO 27), Poznan, Poland, September 3-7, 27, copyright by EURASIP Note that the iltering unction is perormed at discrete requencies = k s o the ilter transer unction This property is used to completely eliminate signal and noise aliasing as well as reduce jitter eects This result is in sharp contrast to the charge sampling techniques [1], [11] the iltering is perormed at continuous requencies X s ( ) = 1 T s H ( ) = H ( k s )X( k s ), (4) k= N n= β nsinc( t)e j2π n t (5) t is the integration time o the charge sampler and sinc(x) = sin(πx)/(πx) Furthermore, as seen in the previous equations the proposed techniques does not have the lowpass character limitation as charge sampling has In most cases GQBPS results in nonuniorm sampling since normally T D = 1/(4 c ) is not rationally related to T s Circuit implementation o uniorm sampling is easier to realize compared to nonuniorm sampling Thereore, in the ollowing discussion only GUBPS will be considered For GUBPS with complex FIR iltering, the coeicients β n are given by [8] β n = exp ( j2π s ) n, (6) (N 1) ρ denotes the largest integer less than or equal to ρ, and s is chosen according to s = 2 c /(2m1), m =, 1, 2, to avoid harmul signal spectral olding To illustrate the improvements o GUBPS Figure 2 shows the theoretical and simuluated SNR or noise and jitter aliasing as a unction o the ilter order or a complex FIR ilter implementation or an undersampling o 9 times Note that N = 1 corresponds to traditional BPS For noise aliasing, the let igure, when the ilter order N = 63 the eective sampling rate s,e = (N 1)F s 2B e and no urther improvement is obtained For the simulation parameters chosen in this example the inal improvemnt is obtained already at N = 49 SNR (in db) Simulation Theoretical Order o FIR iltering SNR (in db) Simulation Theoretical Order o FIR iltering Figure 2: SNR improvement or GUBPS with complex FIR iltering, s = 2 c /9, B e = 1 c Jitter N(,σ τ ) with σ τ = 3/ s Let: Noise Aliasing, Right: Jitter aliasing 3 PASSIVE AND ACTIVE SAMPLING Passive sampling circuits consisting o only capacitors and switches permit very ast sampling and can handle a large input bandwidth [12] The circuit bandwidth is determined by the ON-resistance o the switch and the sampling capacitance It has been shown that passive sampling can successully sample a 9MHz signal [13] It might be a good candidate to GUBPS implementations Passive sampling is extensively used in high-speed timeinterleaved A/D converters To increase the eective sampling rate, several sampling branches can operate in parallel which is similar to the sampling architecture o generalized bandpass sampling However, the total number o converters in time-interleaved A/D converters is equal to the total number o parallel sampling branches while only two A/D converters are needed in the proposed sampling architecture, one or the real- and one or the imaginary data path The conventional passive sampling architecture is very sensitive to clock skew between parallel sampling channels An improved passive sampling technique that is insensitive to skew was proposed by Gustavsson and Tan [14] by introducing a global sampling clock in the sampling system Passive sampling can encounter oset or distortion problems in RC sampling network [15] Active sampling circuits consisting o switches, sampling capacitors and operational ampliiers (opamp) can realize oset cancellation or autozero by using a unit gain eedback [16] However, active sampling takes more time than passive sampling due to the settling o the opamp limiting the sampling rate Additionally, the circuit bandwidth is limited by both the RC sampling network and the opamp bandwidth 4 IMPLEMENTATIONS OF COMPLEX FIR AND IIR FILTERING For GUBPS with complex FIR iltering, a complex signal multiplication is needed The output ater each multiplication consists o real and imaginary parts which are the inputs to the ollowing digital processing As shown in Figure 3, the interesting inormation band o sampled-data signal by GUBPS is centered at s /2B/2, s is the sampling rate or each sampling branch The sampled data is modulated at this low IF A bandpass ilter located at the interesting band is needed prior to the second requency downconversion (or decimation) and quantization process Such a bandpass ilter is normally realized as a complex ilter since either the positive or negative requency components are needed This ilter could be either an FIR, IIR ilter or a combination o them In the ollowing, GUBPS with complex FIR iltering will be described using SC circuit techniques Due to the sampled-data nature o SC circuits, FIR iltering can be implemented in a parallel structure, see Figure 4 To simpliy it is assumed that the input-oset voltage o the opamp is neglected and no oset compensation is considered in the circuit The corresponding clock scheme is also shown in the igure The input voltage is sampled by N 1 parallel sampling channels, the sampling clock on each channel lags behind the one on the previous channel by T D On each sampling channel, the voltage is sampled by two dierent capacitors in parallel C bi, C bi1 (i = 2n,n =,1,2,N) The sampled voltage levels must be held by using an opamp with a capacitor and a reset switch in the negative eedback loop Using charge conservation analysis, one can easily get the transer unction o the sampling circuit (see Appendix C 27 EURASIP 1976

3 15th European Signal Processing Conerence (EUSIPCO 27), Poznan, Poland, September 3-7, 27, copyright by EURASIP X( ) a k = k = 1 k = 2 k = 3 Al( ) k = 4 k = 5 b) S S S1 S2 S3 S4 S6 S7 S8 S9 k = 9 k = 8 k = 7 k = 6 Ar( ) k = 5 k = 4 k = 3 1 S4 c) S9 S8 S7 S6 S5 S3 S2 S1 S X( ) k = 6 k = 7 k = 2 k = 8 k = 1 k = 9 k = a) Ts/2 d) Cb Cb1 2 Cb2 Cb a X( ) 2N Cb2N 2N 2N e) 2N1 Cb2N1 2N 2N1 Figure 3: Signal spectral olding o 2nd order GUBPS, s = 2 c /9 TD 1 in [9]): H FIR (z) = z 1/2 N [ Cb2i j C ] b 2i1 z i, (7) i= C h C h 2 3 2N 2N1 a Ts = NTD the values o complex impulse response coeicients C b2i /C h, C b2i1 /C h are determined by eq (6), and the unittime delay element z 1 is given by z 1 = e j2πk st D It is obvious that this sampling circuit perorms Nth order complex FIR iltering The extra delay element z 1/2 in the numerator only causes a phase shit The sampled voltage levels determined by the capacitor ratios represent the real and imaginary voltages, respectively The equivalent sampling rate o the circuit is s,e = (N 1) s An extra switch across the opamp is operating at a lower rate s realizes a decimation operation at the output In order to improve selectivity, we extend the FIR iltering unction by adding a irst order complex IIR ilter It is easy to see that the circuit shown in Figure 5 is a irst order real IIR lowpass ilter, and are a pair o complementary nonoverlapping clocks The corresponding transer unction is given by H IIRLP (z) = C b z 1 (8) (C a C h ) C h z1 A irst order complex bandpass ilter can be obtained by requency shiting or modulating a lowpass ilter [17]: z 1 (z z) 1, (9) z 1 = e j2π T s = cos(2π T s ) j sin(2π T s ) = α jβ, (1) is the center requency o the bandpass ilter, and s is the operating requency o the IIR ilter The transer unction o the IIR bandpass ilter transormed rom H IIRLP (z) is obtained as H IIRBP (z) = C b (α jβ)z 1 (11) (C a C h ) C h (α jβ)z1 t = (n 1)Ts Figure 4: SC FIR ilter using passive sampling and the corresponding clocking scheme Cb Vout(nTs) Figure 5: An example o irst order lowpass ilter The corresponding circuit implementation or a real input signal is shown in Figure 6 The modulation actor z introduces two cross-coupled capacitors between the inputs and outputs o the opamps in the real and imaginary datapaths, and also cross terms rom the signal input Assuming = and = 1 at t = nt s and it toggles to = 1 and = at t = (n1)t s The input-output equation can be written as V in (n)c b α Vout(n)C Im h β = Vout(n1)(C Re h C a ) Vout(n)C Re h α V in (n)c b β Vout(n)C Re h β (12) = Vout(n1)(C Im h C a ) Vout(n)C Im h α One can easily veriy that the transer unction o the circuit is exactly the same as eq (11) This circuit can also be extended to a complex input signal V in (t) = Vin Re (t) jv Im in (t) by adding two more cross terms or the imaginary part The circuit architectures discussed above are single ended output implementations It is known that ully dierential opamps provide a larger output voltage swing than their single ended counterparts This is important when the 27 EURASIP 1977

4 15th European Signal Processing Conerence (EUSIPCO 27), Poznan, Poland, September 3-7, 27, copyright by EURASIP (1 α) (1 α) Cbα Cbα Cbα V Re out(nts) V Re out(nts) (1 α) (1 α) Cbβ Figure 6: First order single-ended complex IIR BP-ilter Cbβ Cbβ (1 α) V Im out(nts) V Im out(nts) (1 α) input supply voltage is small Two outputs with complementary signs are obtained at the same time in the dierential output Even-order distortion can also be rejected by a ully dierential circuit [18] In addition, it is seen rom eq (12) that two cross-coupled capacitors C h β cause opposite signs in charge conservation because the upper branch is using inverting bottom plate sampling and the lower one use toggle switching So do the cross terms C b α and C b β at the signal input Toggle switching capacitors are parasitics sensitive and normally undesired in practice [19] They can be avoided by using a ully dierential technique A ully dierential implementation o the irst order complex IIR bandpass ilter with a perect balance circuit is shown in Figure 7 which replaces all the toggle switches by inverting bottom plate sampling Note that this IIR complex bandpass iltering implementation is only valid or α,β >, but it is easy to be adjusted or both positive and negative α and β by changing the components in the eedback loops [17] The numerator o H IIRBP (z) in eq (11) is determined by the passive sampling array in the irst order complex IIR bandpass ilter It can be easily replaced by the sampling array with intrinsic FIR iltering to achieve a composite FIR/IIR iltering unction A single ended implementation o GUBPS with composite FIRIIR iltering and the corresponding clock scheme is shown in Fig 8 As discussed in [7], the wanted band is centered at s /2 B/2 ater GUBPS The IIR ilter is combined with the extra decimation switch so that the operating rate o IIR iltering is s According to eq (1), α,β < or GUBPS implementations The irst order complex IIR bandpass ilter shown in Figure 6 is used ater minor changes The transer unction o the circuit is obtained using charge conservation analysis, see Appendix C in [9] H FIRIIR = z1 N i= (C b 2i jc b2i1 )z i (C h C a ) C h (α jβ)z (N1) (13) The numerator is determined by the passive sampling array with intrinsic FIR iltering in a high operating rate and the denominator determines the pole o the irst order complex IIR ilter in a 1/N times lower operating rate as compared Figure 7: First order dierential complex IIR BP-ilter 2 2N 1 2N N 3 2N1 b b TD Cb Cb1 Cb2 Cb3 Cb2N Cb2N1 2 2N 2 2N Ts = NTD N1 2N1 b b b b b b b b b β b (1 α) b b b β t = (n1)ts b b b (1 α) b b Figure 8: GUBPS with composite complex FIRIIR iltering with the corresponding clocks to FIR iltering Two more toggle switches are introduced to compensate the negative α As mentioned previously they can be completely avoided in ully dierential implementations A sharper roll-o in the gain response o IIR ilter can 27 EURASIP 1978

5 15th European Signal Processing Conerence (EUSIPCO 27), Poznan, Poland, September 3-7, 27, copyright by EURASIP be obtained by adding more stages to the irst order IIR ilter, either by cascade coupling or using ladder structures [19] The magnitude responses o 15th order FIR iltering, irst order complex bandpass IIR iltering, and the combination o FIRIIR iltering are shown in Figure 9 The coeicients o Magnitude (in db) Magnitude (in db) > Complex FIR Complex IIR > Composite FIR/IIR Figure 9: Top solid line: 15th order complex FIR ilter, Top dashed line: First order complex IIR ilter, Bottom solid line: Combined 15th order complex FIR and irst order complex IIR ilter, c = 7, B = 5, s = 2 c /9, T D = T s /16, T s = 1/ s The FIR and IIR coeicients determined by eq (6) and eqs (8)(1), respectively with = s /2B/2,C h /C a = 9 FIR iltering are deined by eq (6) or GUBPS implementations It is observed that FIR iltering realizes the maximum gain at the interesting requency band and IIR iltering unctions as a bandpass ilter to select the interesting band Note that the repeated bands rom the IIR ilter are at the notches o the FIR ilter, except or the wanted band The magnitude response o the composite FIRIIR iltering shows good selectivity As shown in Figure 2 and discussed in [8], jitter perormance is also improved to a certain degree by the moving average operation o FIR iltering in the proposed generalized bandpass sampling 5 CONCLUSION Two promising techniques to combat the inherent problems with spectrum aliasing, noise aliasing and jitter sensitivity or bandpass sampling (or subsampling) was presented: GQBPS and GUBPS with complex FIR and IIR iltering As reported in various publications both techniques have the capability to reduce and even eliminate these problems Additionally, two Switched-pacitor realizations o GUBPS with complex FIR ilters and combined with a complex IIR ilter was presented and analyzed The circuits presented operate completely in voltage mode, in contrast to most publications on subsampling, which operate in charge mode Acknowledgement This work was partially supported by the Socware program in Sweden REFERENCES [1] J Mitola, The Sotware Radio Architecture, IEEE Communications Magazine, pp 26-38, May 1995 [2] Y-R Sun and S Signell, Jitter Perormance o Reconstruction algorithms or Nonuniorm Bandpass Sampling, Proc o ECCTD, pp , Krakow, Poland, September 23 [3] B Razavi, Design o Analog CMOS Integrated Circuits, McGrawHill, 2 [4] Y-R Sun and S Signell, A Generalized Quadrature Bandpass Sampling in Radio Receivers, Proc o ASP- DAC, pp , Shanghai, ina, January 25 [5] Y-R Sun and S Signell, Generalized Quadrature Bandpass Sampling with FIR Filtering, Proc o ISCAS, pp , Kobe, Japan, May 25 [6] Y-R Sun and S Signell, Analysis and Implementation o Uniorm Quadrature Bandpass Sampling, Proc o SiPS, Athens, Greece, November 25 [7] Y-R Sun and S Signell, Filtering Transormation in Generalized Quadrature Bandpass Sampling, Proc ICECS, Tunis, Tunisia, December 25 [8] Y-R Sun and S Signell, Generalized Bandpass Sampling with Complex FIR Filtering, Proc APCCAS, Singapore, December 26 [9] Yi-Ran Sun, Generalized Bandpass Sampling Receivers or Sotware Deined Radio, PhD thesis, Royal Institute o Technology (KTH), Sweden, May 26 (Avaiable at [1] K Muhammad and R B Staszewski, Direct RF sampling mixer with recur- sive iltering in charge domain, Proc o IEEE International Symposium on Circuits and System (ISCAS), vol I, pp 57758, 24 [11] S Karvonen et al, A quadrature charge-domain sampler with embedded FIR and IIR iltering unctions, Solid-State Circuits, IEEE Journal o Volume 41, Issue 2, Feb 26 Page(s): [12] M Gustavsson et al, CMOS Data Converters or Communications, Norwell, MA: Kluwer, 2 [13] P Y an et al, A Highly Linear 1-GHz CMOS Downconversion Mixers, ESSCIRC, 1993 [14] M Gustavsson and N N Tan, A Global Passive Sampling Technique or High-Speed Switched-pacitor Time-Interleaved ADCs, IEEE Trans on Circuits and Systems - II: Analog and Digital Signal Processing, vol 47, pp , September 2 [15] K Y Kim, A 1-bit 1 Msample-per-Second Analog-to-Digital Converter in 1µm CMOS Technology, PhD thesis, University o liornia, Los Angeles, 1996 [16] P Allen and D R Holberg, CMOS Analog Circuit Design (2nd Edition), NY Oxord University Press, 22 [17] T Shier and S Signell, Complex Switched pacitor Filter and Designing Method or Such a Filter, United States Patent, no 6,266,689 B1, July 2 [18] P R Gray et al, Analysis and Design o Analog Integrated Circuits (4th Edition), John Wiley & Sons, Inc, 2 [19] S Signell, Studies o Structures or Switched- pacitor Filters with Applications, PhD thesis, Royal Institute o Technology (KTH), Sweden, EURASIP 1979