1 Quantization Noise Shaping for Information Maximizing ADCs Arthur J. Redfern and Kun Shi

Abstract—ADCs sit at the interface of the analog and dig- be done via changing the constant bits vs. frequency profile ital worlds and fundamentally determine what information is for delta sigma ADCs on a per channel basis [3], projecting available in the digital domain for processing. This paper shows the received signal on a basis optimized for the signal before that a configurable ADC can be designed for signals with non constant information as a function of frequency such that within conversion [4], [9] and by allocating more ADCs to bands a fixed power budget the ADC maximizes the information in the where the signal has a higher variance [10]. Efficient shaping converted signal by frequency shaping the quantization noise. for these cases relies on having a sufficiently large number Quantization noise shaping can be realized via loop filter design of ADCs such that the SNR of the received signal does not for a single channel delta sigma ADC and extended to common change significantly within the band converted by the 1 of the time and frequency interleaved multi channel structures. Results are presented for example wireline and wireless style channels. individual constant bit vs. frequency ADCs which comprise the multi channel ADC structure. Index Terms—ADC, quantization noise, shaping. Compressive sensing ADCs provide an implicit example of bits vs. frequency shaping for cases when the input signal I.INTRODUCTION is sparse in frequency and the sampling rate is proportional to the occupied signal bandwidth rather than the total system T’S common for electronic devices to operate with con- bandwidth [5]. Ignoring noise folding issues which arise when I strained power budgets. Within these devices ADCs sit at unwanted signals are also present, this can be viewed as an the interface of the analog and digital worlds and fundamen- on/off shaping of the quantization noise where 0 bits are tally determine what information is available in the digital assigned to frequencies where there is no signal and a constant domain for processing. Sampling at a high frequency with number of bits are assigned to frequencies where there is a a high number of bits allows a sizable contiguous block of signal. As in the multi channel case, this can be viewed as frequencies to be reproduced with high fidelity, but also has block constant frequency shaping. the drawback of requiring a large amount of ADC power Given the existence of signals with information content P ∝ ∆f2b where ∆f is the bandwidth and b is the number ADC which varies with frequency and ADCs which can be designed of bits [6], [7]. explicitly or implicitly for quantization noise shaping, the Considering the analog signal in more detail, there are question arises as to what is the optimal quantization noise cases where the signal resides within a contiguous band shape for an ADC with a fixed power budget to maximize of frequencies but within those frequencies the information the information content in the converted signal. The key content of the signal is non constant. As an example, consider theoretical result in Section II is the derivation of an equation a multicarrier wireline or wireless communication system with which answers this question and is independent of a specific bit loading where large constellation sizes are used in high ADC architecture. Section III then connects this theory to SNR regions and small constellation sizes are used in low a common ADC design by showing that quantization noise SNR regions (Fig. 1). shaping can be achieved through the design of the loop filter Traditionally, an ADC for this type of system would be arXiv:1305.2801v1 [cs.IT] 13 May 2013 in a delta sigma ADC. Optimal quantization noise shaping designed with a number of bits capable of supporting the is extended to time and frequency interleaved multi channel largest constellation size across the entire band. However, ADC structures in Section IV and conclusions are provided in this is power inefficient in the low SNR regions as many Section V. more bits are resolvable than the information content of the signal. Likewise, the impact of the quantization noise is a SNR SNR nonuniform degradation of the received signal SNR, as an equivalent amount of quantization noise added to a high SNR region results in a larger degradation of SNR than if it was added to a low SNR region. dB dB To address this, various ADCs have been proposed which allow for shaping the ADC quantization noise and thus the bits vs. frequency profile of the ADC. For example, multi f f channel ADCs in the literature have shown how shaping can Hz Hz

A. Redfern and K. Shi are with the Texas Instruments Systems and Fig. 1. Received signal SNR as a function of frequency for an example Applications R&D Center, Dallas, TX 75243 USA (e-mail: {redfern, k- wireline (left) and wireless (right) channel. shi}@ti.com). 2

II.INFORMATION MAXIMIZATION where λ is a Lagrange multiplier. As both the information loss The purpose of this section is to determine the ADC (3) and the power constraint formed from (6) are convex, their quantization noise shape that maximizes the information in sum (7) is also convex [2]. the signal after the ADC. Before the ADC, when the signal Taking first order partial derivatives with respect to Sq(k) and noise are uncorrelated and the noise is additive colored and λ, setting the results to 0 and using the assumption that Gaussian, the maximum information in a signal occupying Sq(f) is small relative Sv(f) creates the system of equations frequencies f to f is 1 A B ∂J − 2 2(fB −fA) log2(e) Sq(k) =0 ⇒ Sq (k)≈ , (8) Z fB   ∂Sq(k) λ Sv(k) Sx(f) C = log 1 + df (1) K √ b 2 1 f Sv(f) ∂J 1 X − 2 12P A =0 ⇒ Sq (k) = . (9) ∂λ K fB − fA where f is frequency, Sx(f) is the signal PSD and Sv(f) is k=1 the noise PSD [1]. Substituting (8) into (9), solving for λ, then substituting the Modeling the effect of the ADC as adding shaped quan- result into (8) and solving for Sq(k) results in tization noise with PSD Sq(f) to the signal, the maximum 2   3 fB −fA PK Sq (k) information in the signal after the ADC is 2 3 K k=1 Sv (k) Sq(k) = Sv (k)  √  . (10) Z fB   Sx(f) 12P Ca ≈ log2 1 + df (2) fA Sv(f) + Sq(f) While (10) relates Sq(k) to Sv(k), it’s somewhat cumber- where the approximation is due to the quantization noise some to use as Sq(k) occurs on both sides of the equation. To having a uniform PDF and signal correlation. get rid of the summation term with Sq(k) on the right hand The loss of information due to the ADC is found by side form an equivalent summation term on the left hand side subtracting (2) from (1) and solve for the summation term, then substitute back into (10) to get Z fB  S (f)  C =C −C ≈ log 1 + q df (3) ∆ b a 2 − 1 2 S (f) " fB −fA PK 3 # fA v 2 S (k) 3 K k=1 v Sq(k) = Sv (k) √ . (11) and assuming that the noise PSDs Sq(f) and Sv(f) are small 12P relative to the signal PSD S (f). x Letting K → ∞ in (11) yields While small, the quantization noise is not arbitrarily small  1 2 or 0 because the ADC is limited in power. The quantization R fB − 3 2 Sv (f)df 3 fA noise PSD and number of bits are related by Sq(f) = Sv (f)  √  (12) 12P −2b(f) Sq(f) = 2 /12 (4) which explicitly relates the optimal quantization noise shape and the ADC power and number of bits are related by to the signal noise shape. Z fB Considering (12) in more detail, the squared term in brackets 1 b(f) PADC = 2 df, (5) on the right hand side is a constant which is made smaller c fA by increasing the power of the ADC. Thus, the optimal 2 3 where c is a proportionality constant that for convenience we quantization noise shape is proportional to Sv (f). Without can absorb in the definition of P ≡ cPADC. Using (4) and (5) 2 the 3 power, the optimal quantization noise PSD would be a the quantization noise PSD and the ADC power are related as fixed offset from the noise PSD regardless of the level of the 2 f Z B 1 √ noise PSD. The 3 power effectively shrinks the gap between − 2 Sq (f) = 12P. (6) the optimal quantization noise PSD and the noise PSD in low fA noise regions. As such, while additional power in the ADC The smaller the quantization noise, the larger the power of the is allocated to low noise frequencies relative to high noise ADC. frequencies, the amount of additional power is constrained. To determine the optimal quantization noise PSD shape Figs. 2 and 3 show examples of the optimal quantization which minimizes the information loss of the signal after the noise PSD for maximizing information after the ADC. Equa- ADC (3) given the power constraint (6), integrals are converted tion (12) was used to generate the analytical quantization noise into Riemann sums by dividing the band from fA to fB curves. The numerical quantization noise curves were gener- into K subchannels of bandwidth (fB − fA)/K indexed by ated by a stochastic search algorithm designed to minimize k = 1,...,K and forming the Lagrangian (3) given the power constraint (6) and serve as a check on the theoretical result. K   fB − fA X Sq(k) J[λ, Sq(k)] = log2 1 + K Sv(k) k=1 III.SINGLECHANNEL ADC QUANTIZATION NOISE K √ ! SHAPING 1 1 X − 2 12P + λ Sq (k) − , (7) Delta sigma ADCs achieve noise shaping through oversam- K fB − fA k=1 pling and a feedback loop with an embedded quantizer (see 3

Signal, Noise and Quantization PSD and NTF(z) are the signal and noise transfer functions given 10 by 0 H(z) 1 STF(z) = and NTF(z) = . (14) −10 1 + H(z) 1 + H(z) Note that using the structure in Fig. 4 shapes the quantiza- −20 tion noise PSD as 2 −30 ∆   2 j2πf/fs Sq(f) = NTF z = e , (15) dB 12fs −40 where ∆ is the quantization step size and fs is the sampling −50 frequency. Noise shaping can thus be achieved through the design of the loop filter H(z). Since it is possible to control −60 the filter coefficients through adjusting feedback currents in Signal −70 Noise the analog IC, noise shaping can be decided in the digital Quant (Numerical) Quant (Analytical) domain according to (12) and then realized in the ADC though −80 0 1 2 3 4 5 6 7 8 9 10 controlling feedback currents. 5 Hz x 10 As an example, a 4th order delta sigma ADC with an Fig. 2. A wireline style channel example optimal quantization noise PSD oversampling ratio of 12 was simulated with a loop filter computed numerically (blue) and analytically (magenta) from (12) for max- H(z) optimized to achieve quantization noise shaping as in imizing information after the ADC with signal PSD (red) and noise PSD (12) for the example where the channel has a shaped noise (green). spectrum. The resulting signal, noise, ideal quantization and actual quantization PSDs are shown in Fig. 5. Signal, Noise and Quantization PSD 10

fs 0 H z)( Decimation −10 Loop Filter Quantizer −20

−30 Fig. 4. A delta sigma ADC. dB −40

−50 Signal, Noise and Quantization PSD

0 −60 −20 Signal −70 Noise Quant (Numerical) −40 Quant (Analytical) −80 0 1 2 3 4 5 6 7 8 9 10 −60 5 Hz x 10 −80 Fig. 3. A wireless style channel example optimal quantization noise PSD dB computed numerically (blue) and analytically (magenta) from (12) for max- −100 imizing information after the ADC with signal PSD (red) and noise PSD (green). −120 Signal −140 Noise Quant (Simulated) Fig. 4). Using a low pass delta sigma ADC as an example, −160 Quant (Analytical) the loop filter is designed such that the gain is large inside 1 2 3 4 5 6 7 8 9 10 5 the signal band and small outside the signal band to allow the Hz x 10 input signal and the analog feedback of the modulator output to match closely within the signal band. Consequently, most of Fig. 5. A wireless style channel example optimal quantization noise PSD the signal difference at the summation node will be at higher from the simulated delta sigma ADC (blue) and analytically (magenta) from (12) for maximizing information after the ADC with signal PSD (red) and frequencies and generate a shaped quantization error with it’s noise PSD (green). power pushed outside the signal band. A delta sigma ADC can be represented in the z domain by [8] IV. MULTI CHANNEL ADC QUANTIZATION NOISE SHAPING Y (z) = STF(z)X(z) + NTF(z)Q(z), (13) Single channel ADCs, each able to optimally shape their where X(z), Y (z) and Q(z) are the z transforms of the ADC quantization noise, can be combined to create a multi channel input, output and quantization error, respectively, and STF(z) ADC using any of the traditional multi channel structures. 4

For the case of time interleaving, a set of N individual REFERENCES ADCs with appropriate time offsets and matching can be [1] C. Shannon, “Communication in the presence of noise,” Proceedings of combined to form an ADC with an overall quantization noise the IRE, pp. 10-21, 1949. PSD shape that resembles a Nx bandwidth expanded version [2] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge Uni- versity Press, 2004. of the quantization noise PSD of an individual ADC. [3] R. Cormier, T. Sculley and R. Bamberger, “Combining subband decom- For the case of frequency interleaving, the total bandwidth position and sigma delta modulation for wideband A/D conversion,” can be divided into N contiguous bands such that the power IEEE ISCAS, pp. 357-360, 1994. [4] S. Hoyos and B. Sadler, “Ultra-wideband analog-to-digital conversion given in (6) is equal for each ADC. Note that this may result via signal expansion,” IEEE Transactions on Vehicular Technology, pp. in an unequal distribution in frequency of the total bandwidth. 1609-1622, 2005. Using the wireline style system as an example, Fig. 6 shows [5] S. Kirolos et. al., “Analog-to-information conversion via random demod- ulation,” IEEE CAS Workshop on DAIS, pp. 71-74, 2006. how the total bandwidth is split between N = 4 ADCs such [6] B. Murmann, “A/D converter trends: power dissipation, scaling and that the power of each ADC is equal and the information after digitally assisted architectures,” IEEE CICC, pp. 105-112, 2008. the multi channel ADC structure is maximized. [7] B. Murmann, “ADC Performance Survey 1997-2012,” [Online]. Avail- able: http://www.stanford.edu/ ∼murmann/adcsurvey.html. To simplify ADC design and combining, an additional [8] J. M. Rosa, “Sigma-Delta Modulators: Tutorial Overview, Design constraint of equal or integer scale factors of bandwidth could Guide, and State-of-the-Art Survey,” IEEE Transactions on Circuits and included. Systems-I: Regular Papers, pp. 1-21, 2011. [9] F. Wang and T. Zhi, “Wideband receiver design in the presence of strong narrowband interference,” IEEE Communications Letters, pp. 484-486, Signal, Noise and Quantization PSD − Vertical Lines Indicate ADC Bands 10 2008. [10] F. Yang, X. Jiang, J. Hu and S. Li, “Design of frequency domain A/D conversion based UWB receiver in the presence of quantization noise,” 0 ICCCAS, pp. 314-318, 2008.

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−20 Arthur Redfern Arthur J. Redfern received a B.S. in 1995 from the University of Virginia and M.S. −30 and Ph.D. in 1996 and 1999, respectively, from

dB the Georgia Institute of Technology, all in elec- −40 trical engineering. While at Georgia Tech he was supported by the Robert G. Shackelford fellowship −50 from the Georgia Tech Research Institute and a Graduate Student Research Program fellowship from

−60 the National Aeronautics and Space Administration. Following his thesis work on data aided and blind Signal equalization of nonlinear communication channels −70 Noise Quant (Numerical) modeled by the Volterra series, Arthur joined the Systems and Applications Quant (Analytical) R&D Center at Texas Instruments where he currently manages the Signal −80 0 1 2 3 4 5 6 7 8 9 10 Processing for Analog Systems branch. His activities at TI have spanned the 5 Hz x 10 areas of ADC architectures and compensation, antenna tuning, PA compensa- tion, speaker protection, touch screen controllers, wireless systems (DTV and Fig. 6. A wireline style channel example. Vertical black lines indicate the BAN) and wireline systems (DSL and SerDes). He has been granted over 20 partitioning of the total bandwidth to the individual ADCs such that the power US patents. of each ADC is equal and the information after the multichannel ADC is Arthur’s hobbies include cars, poker and quantitative trading. maximized.

Kun Shi Kun Shi received a B.S. in 2002 from V. CONCLUSIONS the Beijing University of Posts and Telecommuni- cations, M.S. in 2005 from Tsinghua University and This paper derived the optimal quantization noise PSD Ph.D. in 2008 from Georgia Institute of Technology, shape to maximize the information content in a signal after an all in electrical engineering. ADC with a power constraint. It was shown that quantization Since January 2009, Kun has been with the Sys- tems and Applications R&D Center at Texas Instru- noise shaping can be realized via loop filter design for a single ments. His research interests are in the general areas channel delta sigma ADC and extended to common time and of statistical signal processing, nonlinear systems frequency interleaved multi channel structures. and adaptive methods.