least squares theory and design of optimal noise shaping filters

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Verhelst and De Koning Least Squares Noise Shaping It can be noted that digitizing a � ÀÞ tone at �� ÀÞ does indeed result in a sequence with a period of 441 samples explaining the harmonics of ÀÞ seen in Fig. 1a. Most energy is in the fourth harmonic (� ÀÞ). When the quantization step-size increases, the amount of energy outside this main harmonic increases (Fig. 1b). The use of dither noise before quantization works to destroy the quantized signals harmonicity (Fig. 1c). With or without dither, the requantization error can be made minimally audible by proper noise shaping. It suffices to change the shape of the error spectrum such that it becomes minimally audible in the presence of the audio signal. The specifications for the optimal noise shaping filter can be determined from the input signal using a psychoacoustic model. However, these design specifications are time-varying since they depend on the global masking properties of the audio input. Thus, because the noise shaping filter coefficients need to be updated on-line, it is crucial that an efficient filter design technique be used that is reliable (for example, that is guaranteed to produce stable filters). In Super Bit Mapping (SBM) [5] for example, the optimal filter coefficients are obtained by ¯ approximating the inverse of the desired transfer function with an LPC synthesis filter (stable allpole filter) ¯ inverting the LPC synthesis filter, which results in a minimum phase FIR filter. More simple noise shaping with a fixed noise shaping filter has also been proposed [2], [3], [5]. There, it is considered that quantization errors are likely to be most audible and disturbing when the input signal level is low. Therefore, a fixed noise shaping can be used that shapes the error spectrum according to an equi-loudness curve (the hearing threshold in SBM-1 [5] and the 15 phon curve in F-weighting [3]). While the resulting fixed noise shaping filters lead to a much simpler system, it is clear that the time varying strategy is preferable from the point of view of the overall sound quality: that strategy minimizes the audibility of the error in the presence of the actual audio signal, while the fixed noise shaping minimizes the audibility of the error itself (when played in quiet). In this paper, we present a new theory for optimal noise shaping of audio signals. This theory is based on a Least Squares interpretation of the problem. It provides a shorter and more straightforward proof of known properties of dithered and non-dithered noise shaping. Moreover, and in contrast with the standard theory, this new approach does show how noise shaping filters that attain the theoretical optimum can be designed in practice. It turns out that the method used in SBM is an optimum method. Specifically, our theory proves that given an allowed filter order, the filter design method of SBM x(n) x’(n) + - H(z) dither + - e(n) + Q y(n)={ x’(n)+e(n) x(n)+e’(n) Figure 2: Dithered requantization with error feedback filter À Þ and requantization error � Ò will produce the FIR filter that does minimize the perceptually weighted error spectrum. In fact, we independently arrived at the very same design method in [6]. Finally, we also present results from an experimental noise shaping system for minimally audible signal requantization that is based on our filter design method and a simple masking model. In listening experiments, this system was unanimously prefered over the alternatives which included straightforward requantization, dithered requantization and optimized fixed noise shaping [2], [3]. The rest of this paper is organized as follows. The standard theory of noise shaping [7] and minimally audible dither signals [2]-[4] is reviewed in section 1. The newly proposed theory and the practical design method that it entails are described in section 2. The experiments described in section 3 show unanimous preference amongst listeners for our experimental noise shaping system. Finally, section 4 concludes the paper. 1. REQUANTIZATION AND NOISE SHAPING 1.1. General concept While a white noise dither signal can already improve the quality of low level requantized signals, noise shaping can additionally be applied in order to make the requantization error minimally audible [2]. Fig. 2 illustrates the general scheme for signal requantization with noise shaping. In this scheme, Q represents the quantizer and À Þ is the error feedback filter. Due to the requantization error � Ò , the output Ý Ò differs from Ü Ò and from Ü Ò . The error feedback filter has to be controlled such that the difference between Ý Ò and Ü Ò becomes minimally audible. With signals defined as shown in Fig. 2, and using ztransforms, we have � Þ � � Þ À Þ � Þ (1) � Þ � � Þ � Þ � Þ (2) where � Þ represents quantizer Q’s error signal and � Þ AES 22 Ò� International Conference on Virtual, Synthetic and Entertainment Audio 2
Verhelst and De Koning Least Squares Noise Shaping is the additive quantization distortion at the output of the noise shaping requantizer. Subtracting equation (2) from (1), one finds that � Þ � À Þ � Þ (3) The requantization error � Ò therefore has power spectrum È� � �� À � �� È� � �� (4) As we can see, the quantizer’s error spectrum gets shaped by a noise shaping filter that depends on the error-feedback filter À Þ . Note that this result was obtained without reference to the quantizer Q. Therefore, it applies for any type of quantizer, whether it is dithered or not, linear or nonlinear, whether it uses rounding or truncating,. . . In order to achieve minimal audibility of the requantization error, À �� can be designed to minimize the total amount of perceptually weighted noise power Æ Û: ÆÛ � � � � È� � �� Ï � �� (5) Ï � is a perceptual weighting function that approximates the relative audibility of noise power at the different frequencies. If a properly dithered quantizer Q is used, the power spectrum of the quantization error � Ò will be constant such that eq. (4) then reduces to È� � �� É (6) È� � �� À � �� É (7) 1.2. The Gerzon-Craven theory [7] Let us assume that the desired shape È� �� of the error spectrum is given. Thus, from eq. (4), the noise shaping filter À Þ has to be determined such that � À � �� È� � �� «È� � �� (8) with minimal «. In principle, there are several filters À Þ that satisfy eq. (8); the different solutions correspond to noise shaping filters À Þ with a same power spectral shape and different phase characteristics. Based on information theoretic considerations, it was proven by Gerzon and Craven that the noise shaping filter À Þ that satisfies (8) with the smallest possible output error power is the filter that leaves the information capacity of the channel unaltered (maximum), and that this is attained when the filter is minimum phase. Therefore, it was suggested that a practical method to design optimal noise shaping filters would be to use any filter design program to approximate the desired spectral shape and mirroring any zeroes that lie outside the unit circle (for reasons of stability and causality, all poles would already have to be inside the unit circle). x(n) x’(n) + - H(z) + - e(n) Psycho-Acoustic Model dither + Q y(n) = { x’(n)+e(n) x(n)+e’(n) Figure 3: Psychoacoustic requantization with timevarying noise shaping filter À Þ derived from a perceptual masking model 1.3. Some noise shaping implementations Super Bit Mapping (SBM) [5] follows this suggested strategy and introduces a clever trick to design a minimum phase FIR noise shaping filter with a given power spectral shape. Note that in order to avoid delayless loops in Fig. 2, itis required that the FIR noise shaping filter can be written as À Þ � Å� Ò� � Ò Þ Ò � where � � � (9) It was observed in [5] that an Å Ø� -order inverse LPC filter is minimum phase (guaranteed if obtained from the autocorrelation formulation [8]) and satisfies (9). Thus, the required minimum phase FIR noise shaping filter can be obtained by approximating the inverse of the desired noise shaping spectrum with an LPC synthesis filter and inverting the result. In SBM-1, the desired noise shaping spectrum is taken to be the hearing threshold in quiet. Although SBM-1 can be successfully applied to make the quantization error minimally audible in quiet, it could be preferable to use a spectral shaping that minimizes the audibility of the requantization error in the presence of the actual audio. As illustrated in Fig. 3, in SBM-2 the noise shaping filter is a time-varying filter that is designed to approximate the instantaneous masking threshold of the signal. In order to avoid artifacts from abruptly changing error feedback filters, smoothing of the time-varying filter characteristics is applied in the autocorrelation domain. The design of À � �� by optimization of (5) with numeric optimization techniques has also been considered [2],[3]. As this is a rather difficult problem, only fixed noise shaping filters have been designed in this way. Because the noise is obviously most audible in quiet fragments, the perceptual weighting function Ï � was ap- AES 22 Ò� International Conference on Virtual, Synthetic and Entertainment Audio 3
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