This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Author(s): Fabián Esqueda, Stefan Bilbao and Vesa Välimäki Title: Aliasing reduction in clipped signals Year: 2016 Version: Author accepted / Post print version Please cite the original version: Fabián Esqueda, Stefan Bilbao and Vesa Välimäki. Aliasing reduction in clipped signals. IEEE Transactions on Signal Processing, Vol. 64, No. 20, pp. 5255-5267, October 2016. DOI: 10.1109/TSP.2016.2585091 Rights: © 2016 IEEE. Reprinted with permission. In reference to IEEE copyrighted material which is used with permission in this thesis, the IEEE does not endorse any of Aalto University's products or services. Internal or personal use of this material is permitted. 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Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Powered by TCPDF (www.tcpdf.org) This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication. The final version of record is available at http://dx.doi.org/10.1109/TSP.2016.2585091 REVISED MANUSCRIPT, JUNE 2016. 1 Aliasing Reduction in Clipped Signals Fabian´ Esqueda, Stefan Bilbao, Senior Member, IEEE and Vesa Valim¨ aki,¨ Fellow, IEEE Abstract—An aliasing reduction method for hard-clipped sam- the aliased components are sufficiently attenuated, their effects pled signals is proposed. Clipping in the digital domain causes a become inaudible and can therefore be neglected [1], [2]. large amount of harmonic distortion, which is not bandlimited, A large share of earlier research on clipped signals has so spectral components generated above the Nyquist limit are reflected to the baseband and mixed with the signal. A model focused on declipping or the reconstruction of the underlying for an ideal bandlimited ramp function is derived, which leads original unclipped signal. Abel and Smith [3] introduced to a post-processing method to reduce aliasing. A number of optimization methods to reconstruct the clipped samples based samples in the neighborhood of a clipping point in the waveform on constraints. Recent work on declipping has considered are modified to simulate the Gibbs phenomenon. This novel methods based on matching pursuits [4], compressed sensing method requires estimation of the fractional delay of the clipping point between samples and the first derivative of the original [5], social sparsity [6], sparse and co-sparse regularization signal at that point. Two polynomial approximations of the [7], and non-negative matrix factorization [8]. Declipping can bandlimited ramp function are suggested for practical implemen- improve the audio quality of nonlinearly-compressed sound tation. Validation tests using sinusoidal, triangular, and harmonic files [9] and help speaker recognition [10], for example. signals show that the proposed method achieves high accuracy in The purpose of this work is not signal reconstruction but aliasing reduction. The proposed 2-point and 4-point polynomial correction methods can improve the signal-to-noise ratio by 12 dB enhancement, by allowing clipping to occur and by sup- and 20 dB in average, respectively, and are more computationally pressing the aliasing introduced. Practical applications for the efficient and cause less latency than oversampling, which is the proposed approach are found in systems in which clipping is standard approach to aliasing reduction. An additional advantage implemented digitally. One application in radio broadcasting of the polynomial correction methods over oversampling is that and music production is the limiting of the maximum values they do not introduce overshoot beyond the clipping level in the waveform. The proposed techniques are useful in audio and of the signal, such as in dynamic range compressors and other fields of signal processing where digital signal values must limiters, which are known to introduce distortion and aliasing be clipped but aliasing cannot be tolerated. [11]–[13]. In such applications, declipping is out of question, Index Terms—Antialiasing, interpolation, nonlinear distortion, because it would cancel the limiting effect, which is necessary signal denoising, signal sampling. for maximizing the signal level. However, the antialiasing method proposed in this work can be useful, since it cleans the clipped signal by suppressing the aliasing, and restores the I. INTRODUCTION limited distribution of sample values, as required. LIPPING is a form of distortion that limits the values Other practical applications involving digital clipping are C of a signal that lie above or below certain threshold. simulations of analog and physical systems in which signal In practice, signal clipping may be necessary due to system values are naturally limited. In digital simulation of analog fil- limitations, e.g. to avoid overmodulating an audio transmitter. ters, hard clipping is used as a simple model for the saturation In discrete systems, it can be caused unintentionally due to of large signals inside analog filters [14], [15]. In the digital data resolution constraints, such as when a sample exceeds modeling of vacuum-tube amplifiers and guitar effects, the the maximum value that can be represented, or intentionally saturating characteristics must also be implemented digitally as when simulating a process in which signal values are [16]–[18]. Often hard clipping is used in connection with soft constrained. Clipping is a nonlinear operation and introduces clipping so that the latter works at small signal values while the frequency components not present in the original signal. In the former saturates (clips) the large signal values to a maximum digital domain, when the frequencies of these new components or minimum value. Antialiasing for the combination of a soft exceed the Nyquist limit, the components are reflected back and a hard clipper in this context was the first application into the baseband, causing aliasing. This paper proposes a for the method discussed in this paper [19]. Similar saturating novel method to reduce aliasing in clipped signals. modules appear in physical models of musical wind and string Aliasing can severely affect the quality of a digital signal instruments in which a saturating waveshaper simulates the by corrupting the data it represents. For example, in audio nonlinear interaction between the excitation and state of an applications, aliasing can cause severe audible effects such as acoustic resonator, such as a tube or a string [20], [21]. beating, inharmonicity and heterodyning [1]. Nevertheless, if Full-wave and half-wave rectification are special cases of hard clipping. Thus, the proposed method can enhance such This work was supported by the CIMO Center for International Mobility digitally rectified signals, which have applications in various and the Aalto ELEC Doctoral School. F. Esqueda and V. Valim¨ aki¨ are with the Department of Signal Pro- fields. For example in music processing, full-wave rectification cessing and Acoustics, Aalto University School of Electrical Engineer- offers an easy implementation of an octaver, a device which ing, FI-00076 AALTO, Espoo, Finland (e-mail: fabian.esqueda@aalto.fi; doubles the fundamental frequency of a signal [13]. vesa.valimaki@aalto.fi). S. Bilbao is with the Acoustics and Audio Group, University of Edinburgh, The only previous general method for reducing aliasing in Edinburgh EH9 3JZ, UK (e-mail: [email protected]). digitally-clipped signals is oversampling [11], [16], [18], [22]– Copyright (c) 2016 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected]. This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication. The final version of record is available at http://dx.doi.org/10.1109/TSP.2016.2585091 REVISED MANUSCRIPT, JUNE 2016. 2 [24]. Its use is limited to cases where the input signal of a ideal bandlimited case. To do so, we begin by considering the nonlinear device can be accessed. In oversampling, the input simple case of a continuous-time sinewave signal of the nonlinear device is upsampled (e.g. by factor of eight) so that most of the spectral components generated by s(t) = cos (2πf0t) ; (1) the clipper can be suppressed using a digital lowpass filter. where t is time and f0 is fundamental frequency in Hz. Next, Then, the output signal is downsampled back to the original we define c(t), a time-domain expression for this signal after rate. The level of aliasing reduction will then depend on the bipolar (positive and negative) clipping: oversampling factor and order of the antialiasing lowpass filter, which will together also determine the computational cost of c(t) = sgn [s(t)] min (js(t)j;L) : (2) the process as a whole. Here, 0 < L < 1 is the clipping threshold relative to a The concept of aliasing suppression has been previously normalized peak amplitude of 1 and sgn(·) is the sign function. studied in the design of digital oscillators such as those Now, in order to generate a digital bandlimited clipped used in music synthesis [25].