Undersampling RF-To-Digital CT Σ∆ Modulator with Tunable Notch Frequency and Simpliﬁed Raised-Cosine FIR Feedback DAC

Undersampling RF-to-Digital CT Σ∆ Modulator with Tunable Notch Frequency and Simpliﬁed Raised-Cosine FIR Feedback DAC

Sohail Asghar, Roc´ıo del R´ıo and Jose´ M. de la Rosa Instituto de Microelectronica´ de Sevilla, IMSE-CNM (CSIC/Universidad de Sevilla) C/Americo´ Vespucio, 41092 Sevilla, SPAIN E-mail: [asghar,rocio,jrosa]@imse-cnm.csic.es

Abstract—This paper presents a continuous-time fourth-order [5] and undersampling or subsampling [2], [3], [7], [9], [10]. Σ∆ band-pass Modulator for digitizing radio-frequency signals The latter allows to digitize RF signals placed at fRF > fs/2 in software-defined-radio mobile systems. The modulator archi- (with f being the sampling frequency), while still keeping tecture is made up of two resonators and a 16-level quantizer s in the feedforward path and a raised-cosine finite-impulsive- high values of the OverSampling Ratio (OSR), since the signal response feedback DAC. The latter is implemented with a reduced bandwidth is typically Bw (fRF, fs) [9], [10]. number of filter coefficients as compared to previous approaches, The performance of undersampling Σ∆Ms is mainly de- which allows to increase the notch frequency programmability graded by two major limiting factors. One is the attenuation from 0.0375fs to 0.25fs, while keeping stability and robustness of the RF alias signal in the Nyquist band and the other is to circuit-element tolerances. These features are combined with undersampling techniques to achieve an efficient and robust the reduction of the quality factor of the noise transfer func- digitization of 0.455-to-5GHz signals with scalable 8-to-15bit tion. These problems can be partially mitigated by combin- effective resolution within 0.2-to-30MHz signal bandwidth, with ing Finite-Impulsive-Response (FIR) filters and raised-cosine a reconfigurable 1-to-4GHz sampling frequency1. waveform for the Digital-to-Analog Converter (DAC) in the feedback path of the modulator [7], [11], [12]. However, this I.INTRODUCTION approach increases in practice the number of Σ∆M loop-filter Future generations of Software-Defined-Radio (SDR) coefficients, with the subsequent penalty in terms of sensitivity transceivers will perform most of the signal processing in the to technology process variations and limited range of notch digital domain, thus allowing to increase the programmability frequency programmability while keeping stable behavior. and adaptability to a large number of standards and operation This problem, which is aggravated as the loop-filter order modes of the hand-held devices based on this technology. The increases, is addressed in this paper by simplifying the imple- most critical key building block enabling such a technology mentation of undersampling CT BP-Σ∆Ms with raised-cosine is the Analog-to-Digital Converter (ADC). This circuit should FIR DACs. The proposed architecture reduces the required be ideally placed at the antenna so that Radio Frequency (RF) number of feedback paths and gain coefficients, while provid- signals could be directly digitized, thus being processed in a ing excess loop delay compensation and guaranteeing stability flexible way by running software on a Digital Signal Processor and the equivalence with the corresponding Discrete-Time (DSP) [1]. (DT) system in a widely programmable range of 0.0375-to- Unfortunately, the efficient implementation of SDR mobile 0.25fs notch frequencies. These features are put together and terminals is still far from a consumer product deployment, applied to the system-level design of a fourth-order topology mostly limited by the unfeasible power-hungry specifica- with embedded 4-bit quantization intended for RF-to-Digital tions required for the ADC. However, recent advances in conversion in SDR applications. Continuous-Time (CT) Sigma-Delta Modulation (Σ∆M) techniques – fueled by the continuous downscaling of CMOS pro- II.DESIGN CONSIDERATIONS OF RF Σ∆MS cesses – are pushing RF digitization forward, taking significant Fig. 1 shows the conceptual block diagram of a SDR steps towards making SDR-based phones a reality [2]–[7]. receiver, where after being filtered and preamplified by the The majority of reported RF digitizers are Band-Pass (BP) LNA, RF signals are digitized by a CT BP-Σ∆M and most of CT-Σ∆Ms which include diverse strategies to reduce the re- the signal processing is carried out by a DSP. The majority of quired sampling frequency and mitigate the impact of interfer- reported BP-Σ∆Ms use a fixed notch frequency, fn, and hence ence signals. Among others, some of the proposed techniques a widely programmable frequency synthesizer is required to are frequency-translation [8], out-of-band embedded filtering place the in-coming signal within the passband of the BP- Σ∆M. Alternatively, the design of the synthesizer can be 1This work was supported by the Spanish Ministry of Science and Inno- vation (with support from the European Regional Development Fund) under simplified if a reconfigurable BP-Σ∆Ms with tunable fn is contract TEC2010-14825/MIC. used instead [6]. Digital Control w s x(t) wNs N Y(n) 2 2 2 2 wN +s wN +s RF Digital Decimator Signal Signal phi1 + LNA ADC Raised 1-cos!DAC" Cosine f f DSP m21 m20 m11 m10 c1 c0 RF n DAC

Rx Filter Reconfigurable f s − 5.0 − 5.0 −1 SET CT Band-Pass Digital z z z Q S Sigma-Delta Control Q CLR R phi1 PLL-based Q SET Synthesizer S

Q CLR R phi1

Fig. 1. Conceptual diagram of a SDR receiver based on a CT BP-Σ∆M. Fig. 2. Block diagram of the proposed undersampling CT BP-Σ∆M. A. Modulator Specifications alternative values of fs and fn for different wireless standards In order to find out the specifications of a BP-Σ∆M in under consideration. Therefore, taking into account the three a SDR receiver like that shown in Fig. 1, a number of mentioned conditions (a)-(c) and the whole range of fRF to be wireless standards were considered, namely: GSM, EDGE, covered by the undersampling BP-Σ∆M in Fig. 1, fs should GPRS, CMDA-2000, WCDMA, Bluetooth, WLAN (IEEE be programmable to the following values: 1, 2, 2.5 and 4GHz. 802.11a/b/n), WiMAX and LTE. Handling all these operating In addition, a widely programmable value of fn is required, modes in the SDR of the system in Fig. 1 requires digitiz- ranging from 0.0375fs to 0.237fs. ing signals with carrier frequencies ranging from 0.455GHz (CDMA-2000) to 5.093GHz (WiMAX) with channel band- III.PROPOSED RF-TO-DIGITAL Σ∆MARCHITECTURE widths (Bw) varying from 0.2MHz (GSM) to 30MHz (LTE). Fig. 2 shows the block diagram of the proposed CT BP- The calculation of the required Dynamic Range (DR) in- Σ∆M. It consists of a fourth-order single-loop topology made volves considering the different ranges of input signals for up of two resonators. In order to reduce the impact of alias each standard, ranging from the minimum level (determined by signal attenuation a raised cosine DAC has been used and the standard sensitivity) to the strongest out-of-band blockers, combined with half-delayed FIR based feedback loop in order while satisfying the required Noise Figure (NF) and linearity. to increase the degrees of freedom in the synthesis process As a case study, and considering a LNA with 15-dB gain, 3- when applying a DT-to-CT transformation method [11]. dB NF and an off/on chip blocker attenuation of 15dB, the Two additional feedback paths, with gains c0 and c1, are computation of the BP-Σ∆M effective resolution – beyond included to compensate for the excess loop delay error. The the scope of this paper – yields to an scalable DR of 90dB to latch driving the first compensation path, i.e. the path with 50dB from Bw = 200kHz to Bw = 30MHz, respectively. gain c0, has the opposite phase to the clock signals driv- B. Benefits of Undersampling ing the quantizer and the other feedback paths. The second compensation path is introduced with a full sampling-period The modulator requirements detailed above imposes very delay. This way, the modulator exhibits a full digital delay aggressive specifications, particularly in terms of the sampling between the quantizer output and main DAC inputs. Also a frequencies required by the ADC. These requirements can be minimum of one full delay does exist between the modulator relaxed by using undersampling techniques. To this end, let output to the input of the quantizer through both compensation us consider that RF signals with bandwidth, Bw, are located paths, which gives a full sampling-period delay margin for the at a given frequency, fRF, and they are processed in Fig. 1 by quantizer+DAC operation. a BP-Σ∆M working at fs. In terms of the carrier signal at Compared to previous approaches of CT BP-Σ∆Ms based fRF, the signal is undersampled, since fs < 2fRF. However, on raised-cosine FIR feedback DACs [7], [11], the proposed in terms of Bw, the modulating signal, which contains the architecture uses a two-tap FIR structure instead of three-tap information, is oversampled since Bw fs. Therefore, the output spectrum of the BP-Σ∆M in Fig. 1 has a replica of the input (RF) signal with noise shaped at a TABLE I ALTERNATIVES VALUESOF fn AND fs FOR DIFFERENT STANDARDS lower frequency, fn, given by [13]: ( Standard fRF (GHz) fs (GHz) fn (GHz) fn/fs rem(f , f ) b 2fRF c is even 0.25 0.045 0.18 RF s fs fn = 2f (1) CDMA-450 0.455 0.50 0.045 0.09 f − rem(f , f ) b RF c is odd 2 0.455 0.28 s RF s fs 0.5 0.119 0.24 GSM (850-MHz Band) 0.882 where rem(x, y) stands for the reminder of x/y. Note from 4 0.882 0.22 1 0.14 0.14 the above expression that in order to downconvert a RF signal CDMA (2.14-GHz Band) 2.14 2 0.14 0.07 without overlapping, three conditions need to be fullfiled: (a) 1.25 0.094 0.08 fs > 2Bw (Nyquist theorem); (b) fn − Bw/2 > 0 and (c) WiMax (5-GHz Band) 5.09 2.5 0.094 0.04 5 0.094 0.02 fn < fs − Bw/2. As an illustration, Table I shows several based implementation. For instance, the modulator in [11] expressions of TFmuv (z) in (3), the modulator loop-filter co- uses 16 feedback coefficients instead of only 6 coefficients efficients, m1k, m2k and ck, with k = 0, 1, can be obtained for used in the proposed architecture. This simplification of the all required values of fn. As an illustration, Table II shows the loop-filter implementation results in an improvement of the values of and obtained from replacing ( ) in (3) ak bk TFmuv z √ sensitivity of the modulator with respect to circuit-element for fn = 0.2fs (p1 = p2 = −0.309 ± 0.95i, where i ≡ −1). tolerances and mismatches. The latter is particularly critical in In this case, the corresponding TFs for the compensation paths −1 −2 SDR applications, since a high degree of tunability is required are TFc0 = 2z and TFc1 = 2z , and the resulting loop- to have a widely programmable notch frequency as stated in filter coefficients are m10 = 0.3, m11 = 0.53, m20 = 0.048, previous section. m21 = 0.037, c0 = 0 and c1 = 0.25. The modulator is synthesized in two steps. First, a DT BP- Following the same procedure, loop-filter coefficients can be Σ∆M is obtained from an equivalent Low-Pass (LP) system calculated for different values of fn as illustrated in Table III. which fulfills the required specifications. Second, a DT-to-CT transformation is applied to obtain the desired CT system. IV. SIMULATION RESULTS

A. LP-to-BP Transformation Fig. 3 shows the output spectra for different cases of notch frequencies corresponding to RF signals located at 0.44 GHz The desired fourth-order DT BP-Σ∆M is obtained by (Fig. 3(a)), 2.14GHz (Fig. 3(b)) and 5.09GHz (Fig. 3(c)). applying the following transformation [14]: Combining undersampling techniques with reconfigurable fn, the modulator is able to digitize RF signals lying within cos (2πf )z−1 − z−2 −1 → n (2) the ranges 0.7875f -to-f , 1.7875f -to-2f , 2.7875f -to-3f , z −1 s s s s s s cos (2πfn)z − 1 3.7875fs-to-4fs, etc. The modulator can operate in a normal to a LP-Σ∆M which satisfies the required specifications in operation, i.e. without using undersampling, with a variable terms of DR and Bw, for the different standards stated in 0.0375fs-to-0.25fs value of fn. This way, as illustrated in Section II. This way, applying the transformation in (2) to the Table I, a number of different signal frequencies and band- Noise Transfer Function (NTF) of a second-order LP-Σ∆M, widths in the range of Bw = 0.2-to-30MHz, can be handled the desired NTF (z) is obtained for a fourth-order BP-Σ∆M with an scalable 90-to-50dB SNR. As an illustration, Fig. 4 with center frequency at fn. Once the desired BP NTF (z) is shows several SNR-versus-input level curves considering dif- obtained, the loop filter transfer function, can be easily derived ferent cases of input signal bandwidths and notch frequencies as H(z) = 1 − 1/NT F (z). depicted in Fig. 3. As stated in previous sections, one of the benefits of the B. DT-to-CT Transformation presented architecture is the reduced number of coefficients, The loop-filter transfer function, H(s), of the CT BP-Σ∆M which is directly translated in a lower sensitivity to mismatch is derived from the well-known DT-to-CT equivalence: and circuit-element tolerances. As an illustration of the robustness of the proposed Σ∆M, Fig. 5 shows the histogram −1 H(z) ≡ Z{L [H(s) · HCOS-FIR-DAC(s)]} (3) of SNR of a 150-run Monte Carlo simulation for CDMA-450 (Fig. 5(a)) and WiMAX (Fig. 5(b)) standards. In both cases where Z(·) and L(·) denotes the Z-transform and L-transform an standard deviation of 10% was considered in all loop-filter symbols and HCOS-FIR-DAC(s) is the transfer function of the coefficients. It can be shown how the SNR is degraded in raised-cosine FIR-based DAC. As this DAC transfer function 3-4dB in the worst cases. uses half-delay, modified Z-transform is a more suitable technique to compute (3) with delays which are not integral TABLE II multiples of the sampling time [11]. VALUESOF ai, bi AND pi IN (4) FOR fn = 0.2fs In order to calculate the modified Z-transform of (3), partial fraction forms were considered for the different resonator TFmuv a1 a2 b1 b2 TF 0.49 ± 0.35i 0 −1 0 feedback paths including the unit delay in Fig. 2, expressed m10 TFm 0.18 ± 0.58i 0 −0.43 −0.61 as: 11 TFm20 −0.32 ± 0.12i 0.31 ± 0.22i 0.65 0 TFm21 0.39 ± 0.27i 0.38 0.78 0.11 2 ∗ X ak a b1 b2 ( ) = + k + + (4) TFmuv z k ∗ k 2 (pk + z) (p + z) z z TABLE III k=1 k LOOP-FILTER COEFFICIENTS FOR DIFFERENT VALUES OF fn/fs where subscript mu,v denotes the different gains used in the ∗ feedback paths, and u, v = 1, 2 and (pk, pk) stand for the Coefficient fn = 0.06fs fn = 0.15fs fn = 0.246fs complex conjugate poles of the transfer function for a given m10 −1.42 −0.91 3.13 2 m11 0 0.98 0.63 fn. Two additional terms, b1/z and b2/z , result from the m20 −0.14 −0.17 −0.24 mentioned partial form decomposition. These terms can be m21 −0.13 0.15 3.31 cancelled out by the compensation feedback paths, with gains c0 −0.5 0.18 −0.22 c1 0 0.20 0 c0 and c1. This way, applying superposition and replacing all 0 60

−50 40

−100 # of Samples Magnitude (dB) 0 −150 50 52 54 56 58 60 62 SNR(dB) 0 0.5 1 1.5 2 9 Frequency (Hz) x 10 (a) (a) 30

20 −50

−100 # of Samples

Magnitude (dB) 90 92 94 96 98 −150 SNR(dB)

(b) 0 1 2 3 4 5 Frequency (Hz) 8 x 10 Fig. 5. Monte Carlo simulations for two different standards: (a) CDMA-450 (b) (fRF = 0.44GHz) (b) WiMAX (fRF = 5.09GHz). REFERENCES −50 [1] J. Mitola, “The Software Radio Architecture,” IEEE Communications Magazine, pp. 26–38, May 1995. [2] J. Ryckaert, J. Borremans, B. Verbruggen, L. Bos, C. Armiento, J. Cran- −100 inckx, and G. van der Plas, “A 2.4 GHz Low-Power Sixth-Order RF Bandpass ∆Σ Converter in CMOS,” IEEE J. of Solid-State Circuits,

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