The Impact of Jitter on the Signal-to-Noise Ratio in Uniform Bandpass Sampling Receivers

Transcription

1 The Impact of Jitter on the Signal-to-Noise Ratio in Uniform Bandpass Sampling Receivers Boern Almeroth and Gerhard Fettweis Vodafone Chair Mobile Communications Systems Technische Universität Dresden, 0106 Dresden, Germany {boern.almeroth, Abstract Receiver front-ends, enabling multi-mode multiband operation, are essential for future ef cient mobile communications and require a proper parametrization to achieve certain performance requirements. A key component in the receive chain is the analog-to-digital converter ADC). To determine feasible con gurations of the ADC, an abstract model is investigated in order to evaluate the performance in terms of the signal-to-noise ratio SNR) of bandpass sampling receivers. It models the available types of sampling circuits, the impact of stationary and non-stationary itter processes, as well as limited quantization resolution. The derived ADC model is used to determine the dominating itter effect, either aperture or clock itter, depending on the receiver setup. Furthermore, required root mean square itter values are derived analytically for a prede ned receiver noise gure. A properly designed bandpass sampling receiver, matching the proposed maximum itter requirements, avoids signi cant SNR performance losses and can be employed in mobile communications. I. INTRODUCTION In these days, devices for mobile communications typically support a variety oommunications protocols. This is realized by implementing dedicated receiver front-ends in parallel. The approach is simple, but it lacks in terms of size and costs. Hence, reusing key processing elements in the receiver chain is much smarter. Another trend in receiver architectures, motivated by Mitola s vision of a software radio architecture [1], pushes the digital signal processing closer to the receive antenna to have a more versatile front-end. As a consequence, the ADC is shifted towards the antenna. Thus, the selected sampling rate f s and quantization resolution b are set much higher to achieve a competitive performance while sampling the signal in bandpass BP) domain. Furthermore, the impact of sampling time uncertainty, also called itter, has to be treated more carefully for BP signal reception, especially if the carrier frequency of the receive signals is in the gigahertz range. Basic investigations have been carried out by [], rst. In [3], itter is categorized into three types: 1) input signal itter, ) sampling circuit itter, and 3) sampling clock itter. Here, we focus on the impact of sampling circuit itter, most likely aperture itter, and sampling clock itter, called clock itter, on the overall system performance. A general analysis of this two types of itter has been presented in [4]. In order to design bandpass sampling receiver frontends, a link budget analysis of the ADC has to be performed to explore the performance limiting elements cp. [5]). But Fig. 1: System model of the bandpass sampling ADC. still, there is no detailed analysis in the literature about nding feasible parameter ranges to calibrate the ADC. In this paper an abstract model of an ADC, applicable for bandpass sampling applications, is investigated as shown in Fig. 1. It comprises three main parts. First, the sampling stage modeling the impact of realistic sampling circuits by means of a linear lter function h s t) concatenated with the ideal sampler operating at the sampling rate f s. The second stage models the impact of time itter as an additive noise source n [m] cp. [6]). In the last part, the additive error n q [m] models quantization error. It depends on the limited resolution b in bits, and the properties of the signal as peakto-average-power ratio PAPR) and oversampling ratio OSR). The presented model is used to evaluate the acceptable itter of the ADC, analytically. In addition, the model is utilized to assess the absolute power levels of the desired and the interference signals as well as their SNR to perform a detailed link budget analysis. The reminder of this paper is organized as follows. In Section II, the system model and the metric of the effective SNR γ eff is introduced for general system evaluation. Then Section III investigates the impact of stationary and nonstationary itter processes on the SNR performance of the ADC. Incorporating this result, Section IV discusses the derivation of the required itter performance and studies the link budget of the complete receiver chain. Finally, Section V draws the conclusions. II. SYSTEM MODEL The actual sampling time t m of the m-th sample is modeled as the ideal sampling time mt s and an additive stochastic itter process J with it realizations J m. t m = mt s + J m m Z 1)

2 The impact of the timing error J m on the amplitude distortion n [m] =smt s + J m ) smt s ) is determined by the rstorder Taylor series expansion [6]: dst) s mt s + J m ) smt s )+J m dt. ) t=mts This holds for any sampled signal st) that has its highest frequency components at f max 1 J. Here, 1 J denotes the inverse standard deviation of the itter process J. The evaluation of the system performance of the bandpass ADC is carried out by using the effective in-band SNR γ eff at its output. It is described as the ratio between the inband signal power x = E[x[m] ] to the contributions of the ltered thermal noise power n w, and the power by the ittered sampling n and the quantization stage n q : x γ eff = n w + n + n. 3) q Here, the sampled signal is de ned as the sum of the desired signal and the noise: s[m] =x[m]+n w [m]. We will now rewrite 3) as a function of the SNRs of each stage to ease the computation of the effective in-band SNR γ eff. Moreover, this also allows us to introduce an input-output relation of the in-band SNR, which is used to nd feasible con gurations for a prede ned noise gure α of the ADC. In what follows, we will model the SNRs after the deterministic sampling γ s, after the itter stage γ, and after the quantization γ q in greater detail. The signal-to-sampling-noise ratio SSNR) γ s, after itterfree sampling, is de ned as a function of the selected sampling rate f s and the type of sampling circuit with transfer function H s f,d) = F{h s t, d)} with variable duty cycle d. Here, the duty cycle denotes the ratio while the sampling switch is closed to the sampling period T s. Here, we use the track-andhold circuit with the characteristic of resistor-capacitor RC) circuit. Furthermore, we assume that the carrier frequency and the bandwidth B of the receive signal xt) are known such that we de ne the SSNR as follows: γ s f s,d)= x n w = γ in f H c,b sf,d) df i f H c,b sf if s,d) df f s,d) α nw f s,d). 4) Here, the power spectral density PSD) of the input signal is assumed to be rectangular shaped with center and width B = f g. The term f g is the cutoff frequency of the signal xt). This allows to detach the impact of the sampling circuit H s f,d) from the input signal characteristics. First, the in-band SNR of the input signal γ in is determined by the ratio of the signal power x and the noise power n w. Then the sampling circuit characteristics are modeled for the desired signal and the noise, separately. The term αf s,d) = f H c,b sf,d) df describes the gain of the desired signal due to sampling stage, and α nw f s,d) models the gain of the noise. The term inherently includes the receive signal properties as carrier frequency and bandwidth B, the impact of the sampling circuit H s f,d), and the impact of the sampling rate f s. As a result, the SSNR is modeled as the the input SNR times the degradation factor /α nw. It should be noted that sampling rates are most likely considered to be in the range of B <f s <, since we focus on applications of bandpass sampling ADCs. The impact of time itter on the sampled signal is generally modeled as the signal-to-itter-noise ratio SJNR). It can be de ned as the ratio of the total input power, here the contribution from desired signal and noise, to the itter error power n : γ = x + n w α nw n. 5) This term for evaluating the SJNR will be replaced by a itter speci c expression in Section III. At this stage it used to derive the general relation between effective in-band SNR at the ADC output to its input SNR and the ADC noise gure. Finally, the impact due to the quantization of the sampled signal is formulated by the signal-to-quantization-noise ratio SQNR) as γ q b, β, η) = x + n w α nw + n n q 3 4 b β η. 6) The additive quantization error n q [m], and the quantization noise power n q can be assumed to be generated by a white noise process for suf cient large resolution b cp. [7]). Eq. 6) can now be approximated by a general expression to evaluate the SQNR as a function of the resolution of b bits, an oversampling ratio OSR) β, and a peak-to-average-power ratio PAPR) η. Using 4), 5), and 6) to rewrite 3) as a function of the individual SNRs leads to the following equation of the effective in-band SNR at the ADC output: γ eff = γ q γ γ s γ q +1)γ + γ s +1)+γ γ s. 7) This approach eases the understanding of the in uences of the single stages of the ADC on the overall SNR. Furthermore, it allows to trade-off individual ADC parameters in each stage. To obtain a direct input-output relation of the SNR, we have to subsitute 4) into 7) yielding γ q γ γ in γ eff = ) = γ in γ q +1) γ +1) αnw + γ in + γ γ α. 8) in Now, we can de ne the denominator, α, as the SNR loss or ADC noise gure between the input SNR and the SNR at the output of the bandpass ADC. Further, it is used as a performance indicator of the receiver front-end. It can be assumed that α 1 for realistic ADC models. In the following sections, the parametrization of the ADC bases on the assumption of a maximum SNR loss α.

3 of the mean SJNR. Substituting 10) into 9) leads to: Fig. : Effect of aperture itter ap =0.5 ps) on the SJNR γ of three different input signals: 1) BP signal with rectangular PSD at =10GHz and variable bandwidth B =f g, ) bandlimited BL) white Gaussian noise with B =f g baseband), and 3) sine wave signal at frequency f g. III. JITTER IN BANDPASS ADCS This section investigates the impact two common types of itter processes, stationary and non-stationary, on the SJNR of a bandpass sampling ADC. The following investigations are based on the analysis of the mean SJNR values, as shown in [4]. To simplify the notation, the lowest signal frequency is de ned as = f g and the highest frequency as f h = + f g. In general, the SJNR γ is de ned as γ = M S ss f)df M 1 m=0 S ss f)1 E{e πfjm }) df. 9) Here, S ss f) refers to the rectangular shaped PSD of the sampled signal s[m] and M is the number of samples per block. Furthermore, E{e πfjm } describes the characteristic function of the underlying itter process J. To incorporate this equation into the overall SNR of the BP ADC, we will investigate a simpli ed rule for computation and to determine the dominating itter effect. A. Stationary Processes: Aperture Jitter The time uncertainty in the state transition of the sampling circuit switches is called aperture itter. It is mainly caused by thermal noise in electrical circuits and it is described as an i.i.d. Gaussian distributed process J ap) with zero mean and variance ap. Applying this properties, the characteristic function for the aperture itter is de ned as E{e πfjap) m } = e π f ap 1 π f ap. 10) The resulting term is independent from the actual sampling time instance m. Hence, it is stationary. Furthermore, a linear approximation, under the assumption that highest signal frequency f max ap 1, is applied to simplify the computation γ ap) S ss f)df ap ap 4π f S ss f)df. 11) ap is evaluated for In Fig., the results for the SJNR γ ap) three common signal classes. The standard deviation of the itter is assumed to be ap =0.5 ps. The dotted line shows the SJNR of an sinusoidal input signal located at frequency f g and the dashed gives the SJNR performance for baseband white Gaussian noise signal with cutoff frequency f g. By applying a bandpass signal with a xed carrier frequency of = 10 GHz and a variable bandwidth of B = f g,we can now observe the following. Given that we use narrowband signaling f g / 1 then the SJNR of the BP signal is independent from the signal bandwidth B and can be approximated by using the sine wave equation at carrier frequency. Hence, we can make use of the existing sine wave approximation to calculate the SJNR for narrow-band BP signals as γ ap) 4π fc ap ) 1 if f g 1 f h ap 1. 1) B. Non-Stationary Processes: Clock Jitter In contrast to the aperture itter, the clock itter process is non-stationary in its nature. Clock itter describes the variation of the sampling period duration caused due to the oscillator phase noise. For the free-running oscillator, it is modeled as accumulated itter with i.i.d. Gaussian increments i N0,cT s ). The derived clock itter process J clk) N 0,clk mt ) s)=cmt s depends on the cycle-tocycle itter ct s, has a linearly increasing variance by the time index m, and is hence non-stationary. Applying a similar assumptions as in 10) leads to the following equation for the characteristic function: E{e πfjclk) m } = e π f clk mts) 1 π f clkmt s ). 13) The only difference to 11) is that the SJNR, assuming clock itter, is now dependent on the number of samples per block M. Thus, the mean SJNR can be evaluated by deriving a mean clock itter variance clk : γ clk) clk M 1 1 M clkmt s ) m=0 }{{} clk S ss f)df 4π f S ss f)df 14) For suf cient block length, the mean clock itter variance can be computed by clk = ct sm 1)/ ct s M/ for M 1. To simplify 14) further, the PSD S ss f) is assumed to be rectangular shaped in the desired band as used earlier. γ clk) clk ) 3 clk π B +1fc ). 15)

4 Fig. 3: Comparing the impact of aperture itter and clock itter on the SJNR γ for two different sampling rates f s,1,f s, and a xed sampling block duration of M T s =1ms. Under the narrow-band assumption as in 1), the SJNR due to clock itter can be computed by the following approximation: γ clk) 1 4π fc clk if f 1 g 1 f h clk. 16) C. Find the Prevailing Type of Jitter After characterizing both types of itter by their SJNRs separately, the prevailing type of itter for a given application scenario shall be derived. The results of this section have a direct relation to determine the itter speci cations for circuit design. In Fig. 3, the SJNRs for a given aperture itter according to 1) and clock itter from 16) are shown. The dominating type of itter can now be found under the assumption of a xed block length M T s, e.g., a block length of 1 ms for one transmission time interval TTI) in LTE, and a sampling rate f s. Furthermore, the intersection between the two types, aperture itter solid line) and clock itter dashed or dotted line), can be determined analytically by using 1) and 15) in order to nd the prevailing type of itter: M =1+ ap ct s 1 3 fg ) ) 1 +1 ap. 17) ct s As a result, the number of samples per block M for narrowband signals is obtained by the ratio of the aperture itter variance ap to half of the cycle-to-cycle itter variance ct s. For the given scenario above, the number of samples per block for f s,1 is M = 4787 and for f s, is M = We can compare the results from the former analysis 17) with the minimum required number of samples per block M = T f s in a practical scenario, where T is the reference duration of a block, to nd the dominating itter effect: { n ap if M M, 18) clk if M M. Fig. 3 examines this analysis for two sampling rates f s,1 and f s, with a reference block length of T =1ms. For both rates, the number of samples in one block is much bigger than the calculated M M 1 M. Thus, the dominating itter is due to the used clock signal and limits the SJNR to γ 9.8 db. In this case, the effect of aperture itter can be neglected. In addition, we can not observe a direct in uence of the chosen sampling rate f s on the SJNR γ. Thus, the carrier frequency of the bandpass signal has a maor impact on the SJNR cp. Fig. 3). In conclusion, the limiting type of itter for the application scenario is determined by the sampling block duration and the sampling rate as well as the properties of the sampling circuit and sampling clock. IV. REQUIRED JITTER AND LINK BUDGET ANALYSIS Traditionally, the required values for the itter are determined by assuming that the itter noise power n has to be smaller than the quantization noise power n q cp. [8]). This approach is valid for the conversion of baseband or low-if signals with a Nyquist-ADC, but it does not take into account the characteristics of the input signal and the sampling frontend, including sampling circuit h s t) and sampling rate f s, in case direct RF sampling applications. Hence, the proposed approach incorporates the complete bandpass sampling ADC based on the evaluation of the effective in-band SNR γ eff at the ADC output. The basic relation between the output SNR γ eff as a function of the input SNR γ in and a prede ned SNR loss α has been introduced in 8). The link budget, in terms of power levels and SNR at the output of each element, for a direct RF sampling receiver and an exemplary signal reception is shown in Fig. 4. The individual parts of the receiver are connected in serial fashion and are explained in Tab. I. In this example, we concentrate on the comparison of itter-free sampling with ittered sampling. The in-band SNR at the input of the ADC is set to γ in =41dB. This results from a signal power of 60 dbm and a in-band noise power of 101 dbm. After sampling, the in-band SNR of the signal s[m] is reduced to 40 db due to the effect of aliasing. Observing the individual power levels after the sampling circuit, it can be seen that the absolute power of signal and noise is heavily attenuated by 13 db due to its lowpass characteristics. Now, for the case of ittered sampling, the SJNR is further limited to a calculated value of 9.8 db, which can also be observed in Fig. 3. This is 10 db less compared to the itterfree case. The results would change for an smaller input SNR of γ in < 30 db. In this case, the SNR degradation due to itter will be very minor since the itter error n [m] is smaller then the noise contributions of the preceding elements. A similar behavior can be observed for the impact of the quantization noise power on the SNR performance of the

5 Fig. 4: 1) Power levels and ) SNR values at the output of each element. Parameters are chosen according to Tab. I. Receiver Stage BPF LNA AGC SC Sampling Jitter Quantization DSP BP Filter TABLE I: Con guration Parameters Parameter and Description Passband:.3.4) GHz, 0 db attenuation Stopband: 30 db attenuation Gain =15dB; NF =db; no lter characteristics Gain =5dB; NF =1dB; no lter characteristics Track-and-Hold with f 3dB = 545 MHz f s = 383 MHz Aperture Jitter: ap =0.5 ps RMS Clock Jitter with c =10 0 s, M = : clk =.4 ps see Fig. 3) Resolution of b =8bit and PAPR =1dB MC) Out-of-Band attenuation of 30 db receiver. In case of itter-free sampling, a SNR performance degradation is observable in Fig. 4, while ittered sampling shows approximately no SNR loss. In order to nd the required itter values, which will then give us a similar performance as in the itter-free case, we use 8) to derive the required SJNR as αnw + γ in 1 + γ q ) γ γ in,γ q,α) αγ q γ in αnw 1 + γ q ). 19) Moreover, the relation is used to get the required root-mean squared RMS) itter value by applying 1) or 16): 1 3 4b β η α αnw b β η γ in. 0) π αnw + γ in b β η As a consequence, the obtained RMS itter values can be used to select feasible oscillators for the appropriate application based on their oscillator constant c, e.g., the required oscillator constant to achieve the same performance as in the itter-free case for the given example is c req = s. Furthermore, the required quality of the sampling circuit switches based on their RMS aperture itter can be de ned in advance. V. SUMMARY In this paper we have modeled and investigated the effective in-band SNR performance of the bandpass sampling receiver architecture with ittered sampling. We have shown how to determine the prevailing type of itter for a given system architecture and the application based on sampling block length M, signal characteristics,b), and the speci c itter values for aperture ap and mean clock itter clk. Furthermore, we gave guidelines to determine the required itter values for the design of a feasible receiver. Finally, an exemplary link budget analysis is performed for the complete direct RF sampling receiver. Here, the input signal is assumed to be bandpass. It has been shown that the SNR performance is quite sensitive to itter errors in the SNR range of γ in > 30 db. Thus, itter errors have to be considered carefully in bandpass sampling architectures. REFERENCES [1] J. Mitola, The software radio architecture, IEEE Communications Magazine, vol. 33, no. 5, pp. 6 38, May [] A. Balakrishnan, On the problem of time itter in sampling, IRE Transactions on Information Theory, vol. 44, pp , 196. [3] M. Shinagawa, Y. Akazawa, and T. Wakimoto, Jitter analysis of highspeed sampling systems, Solid-State Circuits, IEEE Trans. on, vol. 5, no. 1, pp. 0 4, [4] M. Löhning and G. Fettweis, The effects of aperture itter and clock itter in wideband ADCs, in in Proc. of the International Workshop on ADC Modelling and Testing IWADC), Perugia, Italy, 003, pp [5] S. Rodriguez-Parera, A. Bourdoux, F. Horlin, J. Carrabina, and L. Van der Perre, Front-End ADC Requirements for Uniform Bandpass Sampling in SDR, in in Proceedings of the 65th IEEE Vehicular Technology Conference VTC-Spring), Dublin, Ireland, 007, pp [6] N. Da Dalt, M. Harteneck, C. Sandner, and A. Wiesbauer, On the itter requirements of the sampling clock for analog-to-digital converters, IEEE Trans. on Circuits and Systems I: Fundamental Theory and Applications, vol. 49, no. 9, pp , Sep. 00. [7] B. Widrow and I. Kollar, Statistical theory of quantization, IEEE Trans. on Instrumentation and Measurement, vol. 45, no., pp , [8] V. Arkestein, A. Klumperink, and B. Nauta, Jitter requirements of the sampling clock in software radio receivers, IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 53, no., pp , 006.