DIRECT LINEAR CONVERSION OF LSP PARAMETERS
FOR PERCEPTUAL CONTROLIN SPEECH AND AUDIO CODING
R. Sugiura1 , Y. Kamamoto2 , N. Harada2 , H. Kameoka2 , T. Moriya2
1

Graduate School of Information Science and Technology, the University of Tokyo
2
NTT Communication Science Labs., Nippon Telegraph and Telephone Corp.

ABSTRACT
We have devised a direct and simple scheme for linear conversion of line spectrum pairs (LSP) with low computational
complexity aiming at weighting or inverse weighting spectralenvelopes for noise control in speech and audio coders. Using
optimally prepared coefficients, we can perform the conversion directly in the LSP domain, which ensures low computational costs and also simplifies the check or the modificationof unstable parameters. We show that this method performs
the same as the weighting in the linear prediction coding domain but with lower complexity in a low-bit-rate situation.
The devised method is therefore expected to be useful for lowbit-rate speech and audio coders for mobile communications.
Index Terms— audio coding, signal processing, LSP, linear approximation, TCX
1. INTRODUCTION
The comfortability of voice communication tools, such as
mobile phones greatly depends on the speech coder used inthem. Although the technology in speech coders has been
progressing for years, there is still a demand for higher quality not only in speech but also in the other audio signals suchas music.
3GPP Extended Adaptive Multi-Rate WideBand (AMRWB+) and MPEG-D Unified Speech and Audio Coding(USAC) [1, 2] are known as the state-of-the-art speech andaudio coders. Both coders have at least two different modes
and use the one that suits the input signals. Voice signals arecoded in the time domain by Algebraic Code Exited Linear
Prediction (ACELP), and the other audio signals are coded in
the frequency domain by Transform Coded eXitation (TCX).
The goal of this work is to design a high-quality low-bit-rate
speech and audio coder based on their ideas with lower delay
aiming at its use in mobile phones. In the context of mobile
communications, the coder is also required to have a low
computational cost since there is a limit in the cost for realtime processing. Lowering the computational complexity ofthe process in the coder allows the coder to meet the limitof the cost when the coder is too complex or makes a room
for additional processes when the cost is low enough, whichleads to higher quality. This paper discusses saving computational costs by focusing on the following conventional
process.
Both AMR-WB+ and USAC, mentioned above, quantize
and code linear prediction coding (LPC) parameters, which

are used for constructing whitening filters or for representing
spectral envelopes. Weighted parameters are also used in order to shape quantization noise perceptually, and this perceptual weighting is often performed in the LPC domain since it
requires only a single parameter for tuning the weights. However, in the coders, these parameters are quantized and codedin the form of line spectrum pairs (LSP) [4], which is robustfor quantization or interpolation and also makes it much easier to guarantee the stability of the parameters. Since the conversion between LPC and LSP parameters costs complexity,
it is reasonable to perform the weighting directly in the LSP
domain. Furthermore, by weighting in the LSP domain, theuseful properties of LSP are available in the process. Here, we
present a low-complexity LSP conversion scheme based on
simple linear operations with trained coefficients that approximate either the perceptual weighting or the inverse weighting.
This method of conversion enables several types of coders to
reduce the computational costs without degrading quality.
In section 2, we explain the perceptual weighting used in
the coder. In section 3, the properties of LSP are described.
Then, in section 4, the method of approximating the weighting in the LSP domain is introduced. Finally, in section 5, theapplications of this conversion are shown and the conversion
is evaluated in those contexts.
2. PERCEPTUAL WEIGHTING IN THE CODER
First of all, we explain perceptual weighting using the TCX
coder [3] as an example. The TCX coder quantizes and codes
two kinds of information: the LPC parameters explained
above and residual spectra, the input frequency spectra divided by their envelopes.
The weighted LPC parameters are used to calculate theenvelopes divided into the spectra. Normally, envelopes canbe computed from the frequency response of an all-pole filter
with the LPC parameters:p∑H(z) = 1/(1 +an z −n )(1)
n=1where an is the LPC parameter of nth order. To control quantization noise in the spectra perceptually, the weighted parameters are applied to the filter asp∑H̃(z) = 1/(1 +γ n an z −n ), (0 < γ < 1)(2)
n=1

and this weighting smooths the envelopes. The residual spec-

tra, which are divided by the weighted envelopes, are scalarquantized, followed by entropy coding. Here, the weighted
envelopes shape the quantization noise into approximatelyH̃(z)/H(z), eventually causing smaller distortion in the
spectral peaks, which is more important than the spectral valleys for human perception. It is experimentally known thatthe perceptual weighting works well when 0.92 is chosen for
the value of the parameter γ.
This method of weighting is widely used since it needs
only one parameter γ and p operations of multiplication.
3. PROPERTIES OF LSP
LSP parameters have roughly two intuitive properties, whichmakes them worth using in audio coders. One is about the stability of the parameters [5]. The all pole filter eq. (1) is stable
if and only if it can be written with real value LSP parameters,
ω1 , · · · , ωp , when p = 2m − 1 as
H(z)−1 = (1 − z −2 )m−1∏

(1 − 2 cos(ω2i )z −1 + z −2 )

i=1

+

m
∏

(1 − 2 cos(ω2i−1 )z −1 + z −2 )

(3)

i=1

and when p = 2m as
H(z)−1 = (1 − z −1 )m−1∏

(1 − 2 cos(ω2i )z −1 + z −2 )i=1

+ (1 + z −1 )m∏

(1 − 2 cos(ω2i−1 )z −1 + z −2 )(4)i=1

with 0 < ω1 < ω2 < ... < ωp < π satisfied. This property
makes it easy to not only detect the stability of the parameters
but also to modify the parameters to stabilize them if they
become unstable by quantization or other transformations.
The other property of LSP is the relation between the parameters and the envelopes represented by them. The arrangement of the parameters influences the steepness and smoothness of the envelopes. Fig. 1 shows an envelope and its LSP
parameters. The envelope gets steeper when the parameters
are closely neighboring each other, and vice versa. For thisreason, an intuitive process can be taken, such as the simple
interpolation in an ACELP coder, which interpolates linearly
the LSP parameters in subframes from those in current andpast frames.
4. PERCEPTUAL WEIGHTING IN LSP DOMAIN
4.1. Linear conversion model
To approximate the perceptual weighting in the LSP domain,
several assumptions should be made by considering the following facts.
First, the weights decrease the steepness of the envelopes,
thus the weighted LSP parameters get farther away from eachother, ending up in even intervals where γ = 0, in which theenvelopes become completely flat. Fig. 2 shows an example of the change in LSP parameters when the corresponding
LPC parameters are weighted by γ. It can be seen that inFig. 1. LSP parameters and the envelope represented by themin normalized frequency axis. The blue solid line indicatesthe spectral envelope. The red and black vertical dashed lines
indicate the LSP parameters for the respective order.
γ = 1, which indicates the unweighted parameters, the LSP
parameters are arranged unevenly, causing peaks and valleys
in the envelope. This uneven arrangement of LSP parameters
gradually becomes even as γ gets smaller.
Second, the change in the parameters due to γ is large
when the parameters are far from their corresponding frequencies in γ = 0. Fig. 3 displays the gradients of the respective LSP parameters by their displacements. The gradients are
given byωi (0.92) − ωi (1)(5)
0.92 − 1
where ωi (γ) indicates the i-th LSP parameter weighted by γ,
and the displacements are defined asπi.
(6)ωi (1) − ωi (0) = ωi (1) −p+1This figure shows that the gradients and the displacements
have a positive correlation, proving the possibility of linear
approximation of the weighting.
Third the shapes of the envelopes are mainly influenced
by combinations of only a few LSP parameters around therespective frequency.
With the considerations above, we make the following assumptions for the direct weighting in LSP domain:
1. It can be performed by linear operations.
2. Changes in LSP parameters depend only on their values
and the values of the neighboring parameters.
3. Evenly arranged LSP parameters do not change no matter
what γ is.
These assumptions lead to the conversion model:

π 




ω1 (γ1 ) − p+1ω̂1 (γ2 )ω1 (γ1 )..
.
 ..  = K  (γ2 −γ1 )+ ..  (7).
.
πpω̂p (γ2 )ωp (γ1 ) − p+1
ωp (γ1 )
where



x1 y1
0
 z2 x2 y2
K= ... ... ...  .
0
zp xp

(8)

where

 (n)
d1



L(n) = 


(n)

d2

0
(n)

d1

(n)

d2

(n)

d3

(n) (n) (n)d2 d3 d4

..

0
(n)
di =

weighting parameter γ. The leftmost group is the group of
unweighted LSP and the green vertical dashed line indicates
γ = 0.92, which is actually used for the weighting.






.

(
)
πi
(n)
ωi (γ1 ) −(γ2 − γ1 )p+1

x = (x1 , y1 , z2 , x2 , y2 , ..., zp , xp )T .

Fig. 2. The transition of 16th-order LSP parameters due to the



(12)

Therefore, the objective function of eq. (9) can be transformedas}∑{xT L(n)T L(n) x − 2∆(n)T L(n) x + ∆(n)T ∆(n) ≡ G(x)n∆(n) ≡ (ω (n) (γ2 ) − ω (n) (γ1 )).

(13)

Since this function G(x) is convex, it can be minimized at thestationary point of x found as:}∑{dG(x) =
2L(n)T L(n) x − 2L(n)T ∆(n) = 0
dx
n
(
)−1 (
)
∑
∑
⇐⇒ x =
L(n)T L(n)L(n)T ∆(n) (. 14)nn

By changing the training data, this model can approximate
not only the perceptual weighting but also the inverse weighting or a more complex weighting mentioned in the following
section.
Fig. 3. The gradients in γ ∈ [0.92 1] at each LSP parameter

position (total 15044 frames, 16th-order LSP).
When γ1 = 1 and γ2 = 0.92, this model approximates theweighting, and when γ1 = 0.92 and γ2 = 1, this model approximates the inverse. This conversion needs only 3p operations of multiplication, which is a minor complexity cost forthe coders.
4.2. Optimizing the coefficients of the model
The coefficients for the model proposed above are decided
in advance by means of an optimization for training data.
This optimization can be represented in the following minimization problem, with LSP parameters ωi (γ1 ) and ωi (γ2 )(i = 1, ..., p) given from the data by weighting in the LPC
domain:∑|| ω (n) (γ2 ) − ω̂ (n) (γ2 )||22
(9)
min
K

n

where K is the one in eq. (8), and(n)ω (n) (γ) = (ω1 (γ), ..., ωp(n) (γ))T(n)ω̂ (n) (γ) = (ω̂1 (γ), ..., ω̂p(n) (γ))T .

(10)

The superscript n indicates the frame number of the data. The
conversion model eq. (7) can also be written asω̂ (n) (γ2 ) = L(n) x + ω (n) (γ1 )(11)5. APPLICATIONS OF THE DIRECT LSPCONVERSION
5.1. Inverse weighting in low-bit-rate TCX coder
The low-bit-rate TCX coder used in the hybrid TCX andACELP coding is one possible application of the conversion
scheme proposed in this paper.
The TCX coder, as mentioned above, quantizes and codes
LSP parameters and spectra divided by their weighted envelopes. Therefore, the TCX coder itself needs only theweighted parameters. However, in the hybrid coder, the unweighted LSP parameters are also used to interpolate between
the next frame when the mode is switched to ACELP.
Especially in low-bit-rate situations where sufficient bits
cannot be allocated for the parameters, the accuracy of theweighted parameters is more important for the TCX coder.
As explained above, LSP parameters become farther awayfrom each other when weighted, which decreases the dynamic
range of the vectors of the parameters. For these reasons, it
can be more efficient to quantize the weighted LSP parameters instead of unweighted ones. When the weighted parameters are quantized, the unweighted ones must also be calculated by inverse weighting. Here, the conversion scheme
makes a difference in the computational complexity. The conversion of LPC to LSP costs much more than the inverse conversion: LPC to LSP costs about 0.4 weighted Million Operations Per Second (wMOPS [6]), while LSP to LPC andthe devised conversion in the LSP domain cost about 0.06
wMOPS and 2.4 × 10−3 wMOPS respectively, in the case of
16th order at 20 ms per frame at a 16-kHz sampling rate. By
inverse weighting the parameters in the LSP domain directly,
we can reduce the additional conversion of LPC to LSP and
thereby lower the computational costs. Moreover, it is mucheasier in the LSP domain to detect or modify the parametersthat are destabilized by the inverse weighting.
We conducted an experiment to simulate the situation
mentioned above. LSP parameters are often quantized byvector quantization [7–9]. Therefore, we used a two-stage
vector quantization with first order moving average interframe prediction as shown in Fig. 4. This quantizer has
several kinds of codebooks: one in the first stage stores vectors for all the orders of LSP parameters and the others inthe second stage store vectors for partial orders of them.
(t) (t)(t)
The outputs in frame t of the quantizer ξ1 , ξ2 , ..., ξp arecalculated by adding together for the respective orders thevectors from the codebooks in each stage. Then, the outputs
are added with biases and the outputs of the previous frame
as:
(t)

ω̂i

(t)

(t−1)

= ξi + aξi+ bi , (i = 1, ..., p)

Fig. 4. Outline of an example of two-step vector quantizer for
LSP parameters with first order moving average interframe
prediction. ω̂ indicates the quantized parameters of the respective order.
Table 1. Spectral distortion (SD) by the LSP quantization
calculated in each criteria. 15044 vectors of 16th-order LSP
were tested at 16-kHz sampling rate.
(a) SD in weighted envelope given by eq. (2).
Unweighted LSP quantized
Weighted LSP quantized
Av.SD
Outliers (in %)Av.SD
Outliers (in %)(in dB)
2-4 dB> 4 dB(in dB)
2-4 dB> 4 dB
1.76
22.7
2.30
1.63
18.3
1.81
1.58
15.8
1.33
1.51
14.4
0.96
1.41
9.87
0.72
1.40
10.1
0.53
1.22
5.58
0.17
1.18
5.93
0.09
1.04
3.12
0.05
1.01
3.01
0.01
0.87
1.67
0.01
0.83
1.12
0.00

(15)
Bits
used
10
15
20
25
30
35

where bi is a bias of the ith order. The target parameters arequantized to the closest ω̂ in Euclidean distance. We used 0.3
for a.
Table 1(a) shows distortion in the weighted spectral envelopes by either quantizing the unweighted LSP parametersor the weighted ones. The spectral distortion (SD) was calculated from the mean square errors of the log spectral envelopes [10]. It can be seen that especially at lower bit rates,
quantizing weighted parameters has more advantages compared with quantizing unweighted ones in representing theweighted envelopes.
Table 1(b) compares the inverse weighting after quantizing the weighted LSP parameters. The weighted LSP parameters were inverse weighted in two ways after quantized. In
the first case, the parameters were transformed into LPC parameters and then the LPC parameter of respective order ai
was divided by the ith power of the weighting parameter γ.
In the second case, the parameters were inverse weighted directly in the LSP domain by the proposed conversion with
optimal coefficients. The direct inverse weighting in the LSP
domain performed the same as the inverse weighting in the
LPC domain. Fig. 5 shows an example of the results for the
inverse weighting. The red solid line is the envelope given by
inverse weighting the blue dotted line in the LSP domain. The
spectrum and its envelopes are presented in a linear amplitude
domain since the spectrum are quantized linearly in the basecoder. The inverse weighted envelope became similar to theblack dashed line of the original unweighted envelope.

of the envelopes is written as

5.2. Weighting resolution-warped envelopes

Hk =Another possible application of the conversion scheme is forenvelopes with warped frequency resolution [11]. The envelopes are calculated from a model that uses frequency warping based on spares non-negative matrices, which make it possible to warp the frequency resolution arbitrarily. The model(b) SD in unweighted envelope given byeq. (1)
Unweighted LSP quantized
Bits
Av.SD
Outliers (in %)used (in dB)
2-4 dB> 4 dB
10
3.26
67.7
20.8
15
3.02
66.6
16.5
20
2.77
64.7
12.3
25
2.42
49.4
8.20
30
2.08
31.9
5.00
35
1.76
20.3
2.75

Bits
used
10
15
20
25
30
35

Weighted LSP quantized
LSP→LPC→inv. weight→LSP
Inv. weighted directly in LSP(conventional inverse weighting)(proposed inverse weighting)
Av.SD
Outliers (in %)Av.SD
Outliers (in %)(in dB)
2-4 dB> 4 dB(in dB)
2-4 dB> 4 dB
3.30
66.0
22.1
3.33
66.4
22.6
3.21
66.1
20.3
3.21
66.8
20.3
3.17
69.3
18.7
3.13
70.0
18.1
2.97
62.0
15.1
2.85
62.6
14.3
2.66
55.1
10.8
2.56
55.0
9.82
2.38
40.2
8.08
2.22
40.3
6.39

∑
l

Ukl 1 +p∑−1

an e−j N n
2πl, (k = 0, · · · , N − 1)n=1where N and U indicate the length of the envelopes and theinverse warping matrix, respectively. To approximate the perceptual weighting in the warped domain, the weighting parameter γ has to be modified as

Fig. 5. 16th-order envelope inverse weighted in the LSP do-

main.
H̃k =

∑

Ukl 1 +

p
∑

−1
f (l)

γ l n an e−j N n(16)

Ukn (n − 1).

(17)

2πl

n=1

l

where
f (k) =

N∑
n=1

This means the extent of the weighting changes over the frequencies so that fast Fourier transform cannot be used for calculating the envelopes, which makes this method complex incomputation. However, since the shapes of envelopes are influenced by the arrangement of the neighboring LSP parameters, this weighting can be performed in the LSP domain.
For example, if the envelopes are warped logarithmically,
the weighting must be greater for the lower frequencies, andthis can be achieved by using higher values for the upper leftcoefficients of K in the model eq. (7).
In this context, the training data for weighted LSP parameters ω(γ2 ) to optimize the coefficients of the model cannot be strictly given. Therefore, we approximate the training data using the LSP parameters calculated from the autocorrelation function given by Fourier transforming the powerof the weighted envelopes before they are inverse warped:
1+

p
∑

−2
f (k)

γ k n an e−j N n
2πk

.

(18)

n=1

Fig. 6 shows an example of applying the optimized conversion in the LSP domain. The red solid line is the envelope weighted in the LSP domain, while the black dashed lineis the envelope strictly weighted by eq. (16). The weighting model succeeded in approximating the weighting in the
resolution-warped envelope. This method of weighting enables us to calculate the weighted envelopes by fast Fouriertransform as in the case of normal envelopes and costs only
additional 3p operations of multiplication. When p = 16, thismethod reduces the complexity if the length of envelopes isless than 256 points.
6. SUMMARY
We devised a method of direct conversion in the LSP domain
for perceptual weighting with low computational complexity.
By considering the useful properties of LSP, the weighting

Fig. 6.

16th-order frequency-resolution-warped envelope
weighted in the LSP domain. We used the Mel-frequency
scale for warping.
or the inverse weighting can be simply approximated in a
linear operation form. This method can reduce the computational costs either in the case of a low-bit-rate TCX
coder or frequency-resolution-warped envelopes. Furthermore, weighting or inverse weighting in the LSP domain also
allows us to easily check the stability of the parameters.
LSP domain perceptual weighting enables us to weight
the envelopes adaptively among the frequencies with a small
amount of computation. A future issue is whether there are
any weights that make the perceptual control of the quantization noise more efficient.
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