least squares theory and design of optimal noise shaping filters

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