Smoothing splines are well known to provide nice curves which smooth discrete, noisy data. We obtain a practical, effective method for estimating the optimum amount of smoothing from the data. Derivatives can be estimated from the data by differentiating the resulting (nearly) optimally smoothed spline. We consider the model yi=g(ti)+e~, i= 1, 2 ..... n, tie[0 , 1], where geW2 ~') = {f: j; f,, .... f(m-i~ abs. cont., f(m~ ~2 [0, 1 ] }, and the {el} are random errors with E e i=0, E eie~=a z 6~j. The error variance a 2 may be unknown. As an estimate ofg we take the solution g,, a to the problem: Find f~ W2 ("~ to minimize 1 1_ ~ (f(tj) -y~)2 + 2 S (f(")(u)) 2 du. The function g,, a is a smoothing polynomial nj=l 0 spline of degree 2m-1. The,parameter 2 controls the tradeoff between the 1 "roughness" of the solution, as measured by S [f(m)(u)]2 du, and the infidelity to 0 the data as measured by -1 ~ (f(t~)_y~)2 and so governs the average square /~j=l error R(2; g)=R()0 defined by R(2)=~ ~ (g,,a(tj)-g(tj)) 2. j=l We provide an estimate ,~, called the generalized cross-validation estimate, for the minimizer of R(2). The estimate 2 is the minimizer of V(1) defined by V (2)

Recent interest in high-resolution digital audio has been accompanied by a trend to higher and higher sampling rates and bit depths, yet the sound quality improvements show diminishing returns and so fail to reconcile human auditory capability with the information capacity of the channel. We propose an audio capture, archiving, and distribution methodology based on sampling kernels having finite length, unlike the "ideal" sinc kernel that extends indefinitely. We show that with the new kernels, original transient events need not become significantly extended in time when reproduced. This new approach runs contrary to some conventional audio desiderata such as the complete elimination of aliasing. The paper reviews advances in neuroscience and recent evidence on the statistics of real signals, from which we conclude that the conventional criteria may be unhelpful. We show that this proposed approach can result in improved time/frequency balance in a high-performance chain whose errors, from the perspective of the human listener, are equivalent to those introduced when sound travels a short distance through air.

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On the topic of high-performance audio, there remains disagreement over the ways in which sound quality might benefit from higher sample-rates or bit-depths in a digital path. Here we consider the hypothesis that if a digital pathway includes any unintended or undithered quantizations, then several types of errors are imprinted, whose nature will change with increased sampling rate and wordsize. Although dither methods for ameliorating quantization error have been well understood in the literature for some time, these insights are not always applied in practice. We observe that it can be rare for a performance to be captured, produced, and played back with a chain "flawless" in this regard. The paper includes an overview of digital sampling and quantization with additive, subtractive, and noise-shaped dither. The paper also discusses more advanced topics such as cascaded quantizers, fixed and floatingpoint arithmetic, and time-domain aspects of quantization errors. The paper concludes with guidelines and recommendations, including for the design of listening tests. SETTING THE SCENEIn [1] we suggested that "High Resolution" should be considered an attribute of a complete system in the analog domain (from microphone to loudspeaker)-rather than of the distributed signal or a specific technology.If the system includes a digital path, higher sample rates enable wider bandwidth and it has been questioned whether a listening preference for wider-bandwidth systems could result from the reproduction of signal frequencies above 20 kHz, or alternatively, whether it might arise as a side-effect of filtering in the chain, such as may be encountered when constraining bandwidth to meet a Nyquist criterion.In [3] and [4] we introduced a hierarchical method by which high resolution, defined as clear separation of temporal events, can be delivered efficiently. Prior to this there has been a tendency to describe resolution in the digital domain by the proxies of sample rate, bandwidth or data rate. We can't listen to a digital file without first converting it back to analog; this paper continues to consider the entire chain.A third frequency-domain hypothesis suggests that wider-band signals may cause misbehavior in playback systems (shown to be improbable in [5]).A fourth possibility, raised here, is that if a chain has defective quantizations in any part of a digital path, then the resulting errors (manifesting as distortion and/or modulation noise) also change with the inter-related variables of sampling rate and wordsize, with collateral consequences. Outline of This PaperThis paper is both a tutorial and call to action, reminding about some nowadays-overlooked fundamentals.Sec. 1 introduces the topic of modulation noise, a type of system error that can disturb or alter perception of background noise, or spatial or low-frequency elements.Sec. 2 reviews quantization distortion and in Sec. 3 we recap the properties of additive, subtractive, and noise-shaped dither, including maintaining linearity to levels well below the LSB (le...