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: fredfern, k- wireline (left) and wireless (right) channel. [email protected]). 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 p 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 2 3 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) 4 p 5 : (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) p : (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 ! 1 in (11) yields While small, the quantization noise is not arbitrarily small 2 1 32 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) 4 p 5 (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 p 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.