PAPERS J. A. S. Angus, “Modern Sampling: A Tutorial” J. Audio Eng. Soc., vol. 67, no. 5, pp. 300–309, (2019 May.). DOI: https://doi.org/10.17743/jaes.2019.0006 Modern Sampling: A Tutorial JAMIE A. S. ANGUS, AES Fellow (j.a.s.angus@gmail.com) University of Salford, Salford, Greater Manchester, M5 4WT, UK High-resolution audio has started to use modern sampling principles to further enhance the quality of digital audio reproduction. This tutorial seeks to explain how these new methods are an improvement over traditional Shannon sampling. The tutorial first reviews the relevant parts of traditional sampling and then goes on to introduce the modern method based on splines, which are one of many possible approaches. It shows that it is possible to provide better signal reconstruction with practical filters compared to traditional sampling. Where possible a non-mathematical approach is used. 1 INTRODUCTION Sampling is a critical process in digital audio. Sampling is the process of converting a continuous time signal that exists for all time values, into one that exists only at discrete time values. Continuous time signals must be converted to discrete time ones in order to provide a list or sequence of signal values that can be processed or stored. In audio this is usually achieved by taking snapshots, or samples, of the signal at regular time intervals T s and therefore at a constant sampling frequency F s . The process can be described as the product of a series of sampling impulses and the original signal, as shown in Fig. 1. The most obvious application of sampling is at the analog to digital interface, where the analog signal must be sampled so that it can be quantized into a digital word. Likewise at the digital to analog output the sampled signal must be con- verted back, or reconstructed, into a continuous time one. However, sampling, or re-sampling, also occurs implicitly in any process that changes the sample rate of the incoming digital signal. In these sample rate change processes the sampled signal must be converted to a continuous, or very high sample rate, signal and then re-sampled at the new output sample rate. Thus sampling processes may occur both within an au- dio processing system, as well as at its analog inputs and outputs. It is also important to note that sampling, the dis- cretization of the signal in time, is independent of the pro- cess of quantization, which is the discretization of the input signal in amplitude. Although in many analog to digital converters the two operations are entwined, they remain distinct processes theoretically. Sampling can be, in principle, “lossless” in that, under the right conditions, a finite bandwidth continuous signal can be sampled and reconstructed without error, as shown by Shannon, Nyquist, Whitaker, and Kotel’nikov [1–4]. On the other hand, quantization always causes some error to the signal, as either distortion or noise and, thus, is never “lossless.” This tutorial concerns itself with only the sampling pro- cess. The reason for this is that modern sampling ap- proaches are being used in some modern high-resolution audio formats such as MQA [5]. There has also been re- search [6] that seems to show that wider bandwidth, and thus higher sampling rate, material may be preferred over 44.1 kHz or 48 kHz sampled material. High-resolution au- dio is concerned with obtaining the highest possible audio quality and modern sampling approaches allow better re- construction of the audio waveform with finite length filters compared to the traditional approach. This tutorial will look at modern theories of sampling, and explain them, in as much of a non-mathematical way as is possible. Where there are equations, they are there to add some detail and rigor to the explanations. However, they can be skipped over without compromising the explanation and be revisited, if required, at the reader’s leisure. We will first examine the traditional sampling process and the classical theory behind it. Then the modern tech- niques will be discussed. It will show that sampled audio, when properly reconstructed, preserves more of the spec- trum of the original signal. 2 TRADITIONAL SHANNON-NYQUIST SAMPLING: THE FREQUENCY DOMAIN EXPLANATION Multiplying a time domain signal by a series of regu- larly spaced impulses results in copies or aliases of the 300 J. Audio Eng. Soc., Vol. 67, No. 5, 2019 May