least squares theory and design of optimal noise shaping filters

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Verhelst and De Koning Least Squares Noise Shaping proximated by the inverse of an equi-loudness curve. The so-called E-weighting and F-weighting models for the 15-phon audibility function have been used as the perceptual weighting function in [2] and [3], respectively. As in SBM-1, this results in quantization noise that is by itself minimally audible (i.e., minimally audible in silence) and makes the noise shaping filter independent of the input signal, thus avoiding the need for on-line optimization of (5). Interestingly, [2] and [3] use a dithered quantizer, while [5] uses non-dithered quantization. 2. A LEAST SQUARES THEORY From equations (5), (4), and (9) the optimal noise shaping filter can be found by minimizing � � Å� �ÆË � � � Ò � � Ò� ��Ò � È� � �� Ï � �� (10) Observe that, while this problem may be difficult to solve in general, in the case at hand Ï � is a perceptual weighting function. Therefore, Ï � is real and positive, such that this specific problem is actually a least-squares problem that can easily Ô be solved. To demonstrate this, let us define Î � � Ï � È� �� , 1 and denote its inverse Fourier transform by Ú Ò . Parseval’s theorem can now be applied to transform the problem to time domain: �ÆË equals the energy contained in the sequence that would be obtained by filtering Ú Ò with the noise shaping filter, i.e., �ÆË � � Ò� � Å� �� Ú Ò � � (11) This quantity can be straightforwardly minimized by requiring that �ÆË � � � � � � ��Å� (12) which leads to the so-called normal equations: with Ê � � � � Ê� � Ö (13) Ö Ö �� Ö Å Ö Ö Ö Å . . .. . Ö Å Ö Å �� Ö � � � � . � Å Ö � � � Ö � . Ö Å � � � � � � 1 This can be simplified to Î � � Ô Ï � if the quantizer’s error spectrum is white, as in the case of properly dithered quantizers or when the input level is high compared to the quantizer stepsize. and Ö � � � Ò� Ú Ò Ú Ò � (14) is the autocorrelation function of Ú Ò . Equations (13) and (14) are the so-called autocorrelation formulation of the LPC analysis of signal Ú Ò . The properties of these equations have been studied for several digital signal processing applications in the past, e.g., for speech processing in [8]. For example, it is well known that, without computation errors, the solution is indeed a minimum phase filter [9]. Thus, our theory provides an alternative and straightforward way to prove Gerzon and Craven’s proposition. As another example, it is also known that the weighted and shaped error spectrum � À � �� È� � �� Ï � will be maximally flat, such that the noise shaping filter spectrum will approximate the inverse of the weighted error spectrum Ï � È� � �� . Further, this least squares theory also shows how truly optimal noise shaping filters can be designed. Indeed, linear matrix equation (13) can be easily solved with any of a number of well-known methods and produces the required optimal noise shaping filter. Furthermore, Ê being a symmetric Toeplitz matrix, there is a choice of efficient and robust solution algorithms for solving (13) (such as [10], for example). Traditionally, the quantizer has been assumed to produce a white error signal � Ò . In that case, the autocorrelation function of Ú Ò is by definition equal to the inverse Fourier transform of the perceptual weighting function Ï � . In practice, Ö � �� Å can therefore be approximated by sampling Ï � and computing the inverse FFT: Ö � � Æ Æ � �� Ï � Æ � �� Æ �� Å (15) By using enough frequency samples Æ �� Å, the total approximation error can be made arbitrarily small. Given a desired filter order Å, the solution of matrix equation (13) produces the filter that does minimize (5). While numerical optimization of (5) is too slow for online applications and may fail to converge in practice, our method is computationally efficient and robust. Furthermore, this method can also be applied in case of nonwhite quantization error, such as in the non dithered case. Indeed, in that case it suffices to perform the same operations on Ï � È� � �� (instead of Ï � ). In SBM the inverse noise shaping filter is approximated by applying an LPC modellisation to the inverse of the desired noise shaping spectrum. If we let Ï � be equal to the inverse of the desired noise shaping spectrum, then the above theory proves that the SBM filter is the opti- AES 22 Ò� International Conference on Virtual, Synthetic and Entertainment Audio 4
Verhelst and De Koning Least Squares Noise Shaping Magnitude (dB) 30 20 10 0 -10 -20 -30 0 5 10 Frequency (kHz) 15 20 Figure 4: A � Ø� order FIR noise shaping design (dashed line) compared to the F-weighting equi-loudness curve. The equi-loudness curve was downward shifted 16 dB for easy comparison: £ ÐÓ� Ï � � was plotted. Because the noise shaping filter is minimum phase, its spectrum on a log scale averages to zero [7]. mum one. However, this is only valid under the assumption that È� � �� is constant. Because SBM uses a non-dithered quantizer, it is doubtful that this would be satisfied for the more critical low level signals. This suggests that SBM could be improved by applying the LPC modellisation to the inverse of the desired noise shaping spectrum multiplied by the quantizer’s error spectrum È� � �� or an estimate thereof. 3. EXPERIMENTS 3.1. Duplication of the F-weighted design To illustrate its optimality, we applied the proposed design procedure to the problem of designing a minimally audible dither signal using the F-weighting curve as the perceptual weighting function Ï � , as was discussed in section 1.3. Fig. 4 illustrates the result for a sampling frequency of �� ÀÞ. A� Ø� order filter approximation was used. The filter coefficients we obtained differed less than 0.1% from those obtained by direct optimization of (5) that were published in [3]. This small deviation is probably due to the approximation of the autocorrelation function by inverse FFT, and could be further reduced by using more frequency samples of Ï � . 3.2. Noise shaping experiment 3.2.1. Experimental setup A software version of the psychoacoustical noise shaping requantizer (Fig. 3) was implemented. 44.1 kHz sampling frequency and � Ø� order FIR designs for À Þ were used. The filter coefficients were obtained by solving (13) as described above. Ö � was approximated by a 512 point inverse FFT of the sampled weighting function Ï �� Æ �� Æ � Æ �� Ï �� was updated every 256 input samples and corresponded to the inverse masking curve of a simplified psychoacoustic model: Ï �� ¬ÈÜÜ �� ¬ ÈÌÉ �� (16) ¬ � � Æ � (17) Here ÈÜÜ �� represents the energy spectrum of a 512 point hanning windowed input segment. Its spectral resolution is comparable to the critical bandwidth of hearing at about �ÀÞ. ÈÌÉ �� is the hearing threshold in quiet, and � is the number of bits of the input signal representation. In our experiments � � � and È ÌÉ �� was approximated by the energy spectrum of the � Ø� order F-weighting filter (dashed curve in Fig. 4). As with other perception models, ¬ depends on the expected sound pressure level of the input signal. The value in (17) compensates for the scale factor incurred with our choice of ÈÌÉ �� and is appropriate when the loudest signal portions are played at 84 dB SPL. Sixteen bit test data from good-quality CDs was used in the experiments. Six different fragments were used, containing different types of instruments, vocals, and musical styles. They had a total duration of 1 minute 33 seconds. The fragments were first requantized using straightforward requantization (i.e., rounding) to a precision where the requantization errors became clearly audible. The fragments were then requantized to the same precision (between 5 and 7 bits, depending on the fragment) with three additional methods, resulting in four different versions: Version 1. Straightforward requantization Version 2. Requantization with standard dither Version 3. Non-dithered requantization with fixed noise shaping (F-weighting) Version 4. Non-dithered requantization with adaptive psychoacoustic noise shaping 3.2.2. Informal diagnostic evaluation Two adults with normal hearing participated in this informal evaluation. They were allowed to listen to the different quantized and original sound fragments in any desired order and as often as they desired. The experiment was performed in a quiet office and the sound files were AES 22 Ò� International Conference on Virtual, Synthetic and Entertainment Audio 5
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