The Computation of Line Spectral Frequencies Using
Chebyshev Polynomials
PETER KABAL AND RAVI PRAKASH RAMACHANDRAN

Abstract-Line spectral frequencies provide an alternate parameter- LSF was introduced by Itakura [6]. LSF’s encode speechization of the analysis and synthesis filters used in linear predictive
spectral information more efficiently than other transmiscoding (LPC) of speech. In this paper, a new method of converting
sion parameters [2]-[4] , 171. This can be attributed to thebetween the direct form predictor
coefficients and line spectral freintimate
relationship between the LSF’s and the formant
quencies is presented. ,The system polynomial for the analysis filter isfrequencies.
Accordingly, LSF’s can be quantized taking
converted to two even-order symmetric polynomials with interlacingroots on the unit circle. The line spectral frequencies are given by the
into account spectral features known to be important inpositions of the roots of these two auxiliary polynomials. The response perceiving speech signals. In addition, LSF’s lend themof each of these polynomials on the unit circle is expressed as a series
selves to frame-to-frame interpolation with smooth specexpansion in Chebyshev polynomials. The line spectral frequenciesaretralchanges because of their frequency domain interprefound using an iterative root finding algorithm which searches for realtation.
roots of a real function. The algorithm developed is simple in structure
and is designed to constrain the maximum numberof evaluations of
The emphasis of this work is on the efficient computathe series expansions. The method is highly accurate and canbe usedtion of line spectral frequencies. This will involve an itin a form that avoids the storage of trigonometric tables or the comerative root finding algorithm for a series representation
putation of trigonometric functions. The reconversion of line spectral
frequencies to predictor coefficients uses an efficient algorithm derived in Chebyshev polynomials.. The algorithm developed issimple in structure and constrains the maximum number
by expressing the root factors as an expansion in Chebyshev polynomials.
of function evaluations. These considerations are impor-

I. INTRODUCTION

tant if LSF’s are to be used in a real-time environment.
Thereconversion of LSF’stopredictor
coefficients isbased onreconstructing the expansionin Chebyshev polynomials from ‘the root factors.

N many speech coders, the parameters of the all-zero
11. LINE SPECTRALFREQUENCIES
predictor filter or the corresponding all-pole synthesis
filter are coded and sent as
part of the information stream.
The starting point for deriving the LSF’s is the response
Recently, there has been a growing interest in the use of of the prediction error filter withP coefficientsline spectral frequencies (LSF’s) to code the filter paramP
eters for linearpredictive coding (LPC)of speech [ 11-[4].
LSF’s are an alternative to thedirect form predictor coefA(z) = 1 a(k)z-!
(1)
k= 1
ficients or the lattice form reflection coefficients for representing the filter response.
The ( a ( k ) }are the direct form predictor coefficients. The
The direct form coefficient representation of the LPC
corresponding all-pole synthesis filter is l/A(z). A minifilters is not conducive to efficient quantization. Instead,mum phase prediction error filter (i.e., one with all its
nonlinear functions of the reflection coefficients (e.g.,roots within the unit circle) has a corresponding synthesis
log-area ratio or inverse sine of the reflection coefficient)filter which is stable.
are often used as transmission parameters [5]. These paAsymmetricpolynomial Fl(z) andanantisymmetricrameters are preferable because they have a relatively low
polynomial F2(z)related to A(z) are formed by adding andspectral sensitivity.
subtracting the time-reversed system function
Line spectral frequencies are an alternate parameterization of the filter with a one-to-one correspondence with
F,(z) = A(z) z-(p+l)A(z-’),the direct form predictor coefficients. The concept of an

I

c

+

F&)
Manuscript received November 22, 1985; revised February 15, 1986.
P. Kabal is with the Department of Electrical Engineering, McGill University,Montreal, P.Q., Canada H3A 2A7, and INRS-TBlCcommunications, UniversitC du QuBbec, Verdun, P.Q., Canada H3E 1H6.
R. P. Ramachandran is with the Department of Electrical Engineering,
McGill University, Montreal, P:Q., Canada H3A 2A7.
IEEE Log Number 8610167.

= A(z) - z-(P+l)A(z-1).

(2)

The roots of these two auxiliary polynomials determine
the line spectral frequencies. The two polynomials also
have the interpretation of‘being the system polynomials
for a P
1 coefficient .predictor derived from a lattice
structure. The first P stages of the lattice have the same+0096-3518/86/1200-1419$01.00 O 1986 IEEE

1420

IEEE TRANSACTIONSON

ACOUSTICS, SPEECH,
AND

6 , DECEMBER 1986

SIGNAL PROCESSING, VOL. ASSP-34,
NO.

response as the original P stage predictor. An additional
stage is added with reflection coefficients equal to 1 or
- 1 to give the response Fl(z) or F2(z),respectively [2].
Soong and Juang [2] have shown that if A(z) is minimum phase, then 1) the roots of Fl(z) and F2(z) are on theunit circle; and 2) the roots are simple and separate eachother. In addition, any procedure which determines an ordered set of LSF’s can be used to construct a minimum
phase prediction error filter.
The polynomials Fl(z)and F2(z)being symmetrical andantisymmetrical, respectively, have roots at z = 1 and/or z = - 1 which can be removed by polynomial division++

0 roots of FZ(z)0 roots of A ( z )These polynomial divisions can be performed by additions and subtractions of the coefficients of Fl(z) and F2(z).
The resulting G,(z) and G2(z)are symmetric polynomialsof even order. Since theroots occur in complex conjugate
pairs, it is o’nly necessary to determine the roots locatedon the upper semicircle. The roots of interest are exp j w i
f o r i = 1, 2,, P. Theline spectral frequencies are
the angular positions of the roots 0 < wi < T .
Fig. 1 shows the arrangementof zeros of Fl(z) and F2(z)for both even and odd P. These plots show the actual root
positions for a voiced segment of speech (8 kHz samplingrate). The polynomials Gl(z)and G2(z)have the same zeros as Fl(z) and F2(z), respectively, except for the zeros
at z = f 1. It can be noted that for any order, the lowest
frequency LSF corresponds to aroot of G,(z).These plotsshow that for roots of A(z) near the unit circle, a pair of
LSF’s tends to bracket the angular position of the root ofA(z). However, it alsoindicatesthatthe
difference between pairs of LSF’s is not necessarily a good indicatorof how close a root of A(z) is to the unit circle. Interpretation of the LSF’s in terms of formant resonances for P
odd is more tenuous due to the influence of the real axis
root of A @ ) .
A stability theorem which uses a form similar to the
LSF formulation has been formulated by Schussler [8].
The auxiliary symmetric and antisymmetric polynomials
defined by Schussler become the same as F l ( z ) and F2(z)if A ( z ) is considered to be a polynomial of degree P + 1
with a(P + 1) = 0. The root locations of the auxiliary
polynomials given by a direct application of Schussler’s
theorem (i.e., without appendingazero
valued coefficient), and those given by an LSF formulation are compared in Appendix A. Thisexamination provides additional insight as to the relationship of the LSF’s to theroots of A(z).
The cases of an odd number and an evennumber of
LSF’s differ in some details. Let the order of the poly-

9
@

“\

1’

-

Fig. 1. Root locations. (a) P even (shown for P = 6 ) . (b) P odd (shownfor P = 7 ) .

nomials Gl(z)and G2(z)be 2M1 and 2M2, respectively,
M1 =

P

P
and M - -,2-2

P even,Then explicitly showing the symmetry of the polynomial
coefficients

G,(z) = 1 + gI(1)z-l +* *+ gl(MJz-M1 +* **+ gl(l)z -(2M1- 1) + y 2 M 1 ,
G ~ ( z=) 1

+ g2(l)z-’ ++ g2(l>z-(2M2-1)* * *+ g2(M2)zPM2+ .

+ y2M2.

*(5)

Gl(z) contributes M1 pairs of conjugate zeros and G2(z)contributes M2 pairs of conjugate zeros ( M I + M2 = P ) .
On the unit circle, the linear phase term can be removed
to give two zero phase series expansions in cosines
Gl(ejo,)= e-jwMl G!
7

RAMACHANDRAN:
KABAL AND

1421

COMPUTATIONFREQUENCIES
SPECTRALOF LINE

mapping x = cos w maps the upper semicircle in the zplane to the real interval [ - 1, 11. Therefore, all theGi(w) = 2 COS M ~ u 2g1(1) COS (MI - l ) ~ * *roots xi lie between - 1 and 1, with the root corresponding to the lowest frequency LSF being the one nearest+ 2g1(M1 - 1) cos w + g d M , ) ,+l.
G ~ ( w=) 2 COS M ~ w 2g2(1) COS (M2 - 1) w.* *The Chebyshev polynomial series lends itself to an efficient and accurate evaluation which bypasses an expan2g2(M, - 1) cos w
gz(M2).
(7)
sion in powers of x. Let the series to be evaluated be repVarious methods to locate the roots of Gi(w)and G$(o) resented ashave been suggested. The procedure developed by SoongN- 1
and Juang [2] evaluates (7) on a fine grid by using a discretecosinetransform.
Sign changes atadjacentgrid
points isolate intervals containing roots and further bisec- Consider the backward recurrence relationshiption of these intervals gives an approximation to the rootbk(X) = &bk+l(x) - b k + 2 ( x ) + ck?
(13)
positions.
Kang and Fransen [7] have proposed two other methods with initial conditions bN(x) = bN+l(x) = 0. This recurfor finding the LSF's. In one method, the autocorrelation sion is used to calculate bo@)and b2(x). Then Y(x) can be) used expressed in terms of bo(x)and b2(x)functions of the coefficients of Gi(w) and G ~ ( wareto calculate power spectra. The locationsof the local minN- 1
ima of thepowerspectragivethe
LSF's. The second
y(x) =[bk(x) - 2xbk + 1(x) + bk + 2(x)1 &(x)method uses an all-pass ratio filter

where

+

+

+

+

+

+

+

+

k=O

L

The phase spectrum of the ratio filter is evaluated andwhenever the phase response takes on a value which is amultiple of ?r,the corresponding frequency is an LSF. Allof the above methods require the evaluation of trigonometric functions and may not be appropriate for a realtime environment.
The method proposed in this paper requires no priorstorage or calculation of trigonometricfunctions. The
method is introduced in two steps. In the next section, theuse of a Chebyshev polynomial expansion is discussed.
Subsequently, an efficient numerical algorithm to find theroots with this formulation is established.

111. CHEBYSHEVSERIES.FORMULATION
Consider the frequency mapping x = cos w . Thencos mw = T,(x),(9)

where T,(x) is an mth-order Chebyshev polynomial in x .
The Chebyshev polynomials satisfy the order recursion- Tk-2(x),(10)
with initial conditions, To(x)= 1 and Tl(x) = x. The series expansions in cosines (7) can now be expressed inTk(x)= 2xTk-l(x)terms of Chebyshev polynomials

G;(x) = 2Grr,(x) + 2g1(1)T,,-

1(x)

,+ -

*

+ 2glWl - 1)Tl(X) + gl(Ml),
G$(x)= 2TM2(x) + 2g2(1) TM2- 1 ( ~ ) +*+ 2g2W2 - 1) Tl(4 + 82"(11)Once the roots {xi] of Gi(x) and G$(x)are determined,
the corresponding LSF's are given by wi = arccos xi.The

The benefit of this formulation is that errors in the evaluation of bo(x)and bz(x)tend to cancel [9]. This results
in a numerically stable evaluation of the Chebyshev polynomial series. Neglecting,the factor of 2 which does notaffect the root locations, each evaluation can be computedwith about N multiplies and 2N additions.
IV. NUMERICAL
SOLUTION
FOR THE LINE SPECTRALFREQUENCIESIn this section, a numerical algorithm to find the roots
corresponding to thelinespectralfrequencies
is developed. The basic task is to isolate the roots of Gi(x) bysearching incrementallyforintervalsin
which thesignchanges. The search proceeds backwards from x = 1 since
Gi(x) has the root nearest x = 1. The location of the rootin an interval containing a sign change is refined by successive bisection of the root interval. The function values
are determined using the backward recursion given in theprevious section to compute the Chebyshev polynomials
at a given argument value. In this way, only two function
values at a time need bestored. Given theinterlacing
property of the roots, the search for a root of G$(x)starts
from the position of the 'root of Gi(x)just found. The algorithm continues as before, but interchanges the roles ofthe functions as each root is found.
Two different precisions must be specified for the numerical algorithm. The initial evaluation interval 6 mustbe sufficiently small so that two ormore roots of the samefunction do not occur in the same interval. Let the roots
be denoted by {xi} for i = 1, 2,, P , andletthem
be ordered such thatxi > xi- 1 . The roots of G[(x)(xiwith
i odd) interlace with the roots of G$(x)(xiwith i even).

--

1422

IEEE
TRANSACTIONS

VOL. ASSP-34, NO. 6, DECEMBER 1986

ON ACOUSTICS,
SPEECH,AND
SIGNALPROCESSING,

.

6ot n
bin width 0.01
A.

I

c

0.2

0.4

0.6

1.2

1.0

0.8

I
-1

0

-0.5

0.5

1

2-

Fig. 2. Plots of G;(x) and G;(x) ( P = 10).

Then the initial evaluation interval must satisfy

6 < min (xi- x i - 2 ) .

(15)

I

A second increment E specifies the acceptable uncertainty in root position. This value must be small enoughso that in switching the search for roots from one function
to the other, a root is not missed or 1-oots are not interchanged in order. TO guarantee this, E must be smaller
than the minimum spacing between pairs of roots, one
taken from each function,E< min (xi - xii

(16)

0.005

I

0.01

expanded first bin

bin width 0.01
0.2

0.4

0.6

0.8

1.0

1.2

(c)
Fig. 3. Histograms of root differences. (a) Differences for roots of G;(x).
(b) Differences for roots of G;(x). (c) Differences between roots of G;(x)and G;(x).

Experiments with speech data were conducted to determine reasonable values for 6 and E . Five utterances comTABLE I
prising 10 s of speech sampled at 8 kHz were used. Three
ROOTDIFFERENCES
utterances were spoken by males and two by females. In
Minimum Difference
Maximum Difference
all cases, a 20 ms Hamming window was used to perform
a 10th-order autocorrelation analysis. The ensures that theG ; ( x )only
0.0232
1.121
predictor is minimum phase [lo]. The root locations cor- G;(n)1.195only
0.0564
responding to LSF's were determined to a high precision. G;(x)and G W
0.0015
0.946
A plot of G;(x)and G$(x)for a voiced segment of speech
is shown in Fig. 2. It shows a tendency of roots to paircoarse quantization of LSF's below 300 Hz does not afwhich requires E to be significantly smaller than 6.
Histograms of root differences are shown in Fig. 3 . The fect speech quality.
As a last step, the root position is estimated by linearlyextreme values of the root differences are summarized inTable 1. These results indicatethat 6 = 0.02 is suffi- interpolating between the function values determined inciently small to avoid missing sign changes. Each interval thelast bisection step.This results inanaverageerrorof length 6 will be bisected to further resolve the root lo- which is significantly smaller than the worst case value
cation. From the table, E should be chosen to be less than given by E . The given root finding algorithm has been for0.0015, implying that 4 bisections will be sufficient (for mulated to limit the number of function evaluations. The
6 = 0,02). While the worst case uncertainty in the x-do- initial search for intervals containing roots uses approximain is constant, the uncertainty in the w-domain varies mately 2/6 + P evaluations. Bisection then uses an ad) P evaluations. For the values of 6with w due to the nonlinear relationship between x and w . ditional ( [log2 (6l~)I
For the parameters given above andassuming 8 kHz sam- and E given above, the number of function evaluations forpling, the worst case uncertainty in the LSF's varies be- a 10th-order LPC system is less than 150. The actual
tween 64 Hz at low and high frequencies down to l .6 Hz number is somewhat smaller than this value,sincethe
at the middle frequencies. However, the uncertainty re- search for roots can be terminated when all roots are
mains less than 10 Hz for the frequencies between 200 found.
The algorithm to find the LSF's is extremely simple inand 3800 Hz. Xang and Fransen1'71 suggest a 10 H z resolution in evaluating LSF's and, furthermore, find that structure and keeps the number of function evaluations

1423

KABAL AND RAMACHANDRAN: COMPUTATION OF LINE SPECTRAL FREQUENCIES

in the representation of YN+l(x). This equation defines one
relatively constant.The
numerical analysisliterature
abounds with more sophisticated root finding algorithms. step in the recursion to determine the coefficients of the
These will tend to find the roots with fewer function eval- Chebyshev representation from the root factors.
The coefficients for Gl(z) and G2(z) can be determineduations 6nthe average.However, most of these algorithms are inappropriate in a real-time environment since directly from Gi(x) and Gi(x). This involves applying a
the maximum number of function evaluations is indeter- factor of $ to all but one of the coefficients. In fact, multiplication by this factor can be avoided if the recursion isminate.
A slightly more complex root finding algorithm which modified to directly produce the coefficients of Gl(z) andcombinesbisection with inverseparabolic interpolation Gz(z).
As the penultimate step, Gl(z) and G2(z) must be mul[111 merits consideration for some applications. It can beused to refine the root position when an interval contain- tiplied by the polynomial terms with roots at f l to giveon half of the total
ing a root has been identified. The worst case number of Fl(z) and F2(z). This can be carried out
function evaluationsforthis
algorithm is two or three number of coefficients in theseauxiliary polynomials
times that for simple bisection, but the average number is (using symmetry) and involves only additions and subtractions. Finally, the coefficients of the prediction error
smaller.
filter are determined from
V . CONVERSION
OF LSF’s TO PREDICTORCOEFFICIENTSThe conversionof LSF’s to predictor coefficients is less
The reconstruction procedure described requires about
computationally intensive than deriving LSF’s from preP2
/ 4 - PI2 multiplications and P 2 / 2
2P - 4 addidictor coefficients. Each LSF oi gives rise to a secondtions.
This
is
less
than
of
the
numberofmultiplications
order polynomial factor of the form 1 - 2 cos q z - l
zP2. These can be multiplied together to form the auxil- and additions cited for the reconstruction procedure sugiary polynomials directly. In this section, an alternate re- gested b y Kang and Fransen [7].
A minimum phase prediction error filter results. This
construction process using the Chebyshev series represenfollows
directly from the fact that the reconstruction protation will beformulated.Thisleads
to an efficientcedureisthe step-by-step inverse of the procedure to findreconstruction process whichtakes symmetries in the auxthe
LSF’s.
As long as the LSF’s are distinctand Fl(z) andiliary polynomials into account.
F2(z)
are
formedfrom alternatingroots,theminimum
The polynomials Gf(x) and Gi(x) are reconstructed fromphase
propertyofthereconstructedprediction
error
filtertheir roots by successive polynomial multiplication of theisguaranteed.
appropriate first-order LSF polynomials
An alternative to converting the LSF’s
to predictor coefMI
ficients is the use of filter structures that use the LSF’s
Gi(X) = II 2(X - X 2 k - 1 ) ,directly as parameters. This kind of structure implements
k= 1
Fl(z)
and F2(z) directly as cascaded second-order sections
M2
and
can
be used as the basis of both the analysis (predicGi(X) = I
I 2(X - x 2 k ) .
( 17)
k= 1
tion error) filter and the correspondingsynthesis filter [7].
However, it is not the coefficients of the powers of x that However, this form of filter requires more arithmetic opare desired, but the coefficients of the Chebyshev poly- erations per sample than a directform filter using the prenomial terms.Consider
an Nth-order polynomial ex- dictor coefficients. The tradeoff is then between this extra
computation which occurs for each sample of data propressed as a Chebyshev series
cessed and the computation required to convert LSF’s toNpredictor coefficients. Kang and Fransen [7] show that for&(X) =
Ck T k ( X ) .
(18) reasonable frame sizes in an LPC coder, conversion tok=O
predictor coefficients and the use of a direct form filter
Adding one more root factor toform an N
1st-order structure results in a lower operations count than the use
Chebyshev representation,of an LSF based filter structure.Thisconclusionisstrengthened
by
the
more
efficientprocedure
toconverttoY N + l(x) = 2(X yN(x)direct form coefficients described here.
N+ 1
The reconstruction procedure given in this section starts=[ c k - 1 - h r c k + c k +T1k1( x ) .
(19)
with
the roots in the x-domain and thus ‘complements thek = -1algorithm to derive them from the predictor coefficients.
This expression has been put in this form by applying the In any transmission system, the LSF’s must be quantized)defining c k to and coded. In order to avoid conversion to the w-domain,
relation 2xTk(x) = Tk- 1 ( ~ ) Tk+ 1 ( ~andbe zero fork < 0 and k > N. In addition, note that T- 1 ( ~ )‘These counts apply for P even. For P odd, the number of operations is= Tl(x),which means that the term for k = - 1 shouldslightly smaller than given by these formulas. Note also that the counts
be combined with the term for k = 1. The bracketed term do not include P‘multiplications by the factor and P multiplications byin (19) is the coefficient of the kth Chebyshev polynomial the factor 2.

+

+

+

a

+

1424

IEEE
TRANSACTIONSON ACOUSTICS, SPEECH,
AND
SIGNALthe quantization procedure must be modified to work directly on the xi values. For scalar quantization in the wdomain, an equivalent scheme in the x-domaincan be formulated by simply applying the cosine nonlinearity to thequantizer boundaries and output values. Kang and Fransen [7] describe a scheme to code the center frequency
and difference frequency of a pair of LSF's. This procedure can be viewed in two dimensions as a quantizer with(rotated) rectangularboundaries. In thex-domain,theequivalent quantizer boundaries are curved, but locally,
can be well approximated by straight lines. With straight
lineboundaries,the x-domain quantizer can be implemented with only a small increasein complexity and storage over the o-domain quantizer. Cawing out the quantization in the x-domain eliminates the need to evaluate
transcendental functions in the conversion to orfrom predictor coefficients.

PROCESSING, VOL. ASSP-34,
NO.

6, DECEMBER 1986

when these are close to the unit circle. Some insight into
the clustering phenomenon can be obtained by examining
another formulation related to that for the LSF's.
Schussler [8] has given a stability theorem for a polynomial &). The stability condition is expressed in termsof two auxiliary polynomials

E,(z) = A(z) + z-p A(z-'),&)= A(z) - z-p A(z-').

(A. 1)The polynomial A(z) has all its roots within the unit circle
if and only i f I) the roots of E,(z) and P2(z) are on theunit circle; 2) the roots are simple and separate each other;and 3) la(P)I < 1.2 These auxiliary polynomials differ
from those used for LSF's by being of order P instead oforder P
1. As a result, the roots of Schussler's polynomials cannot be used to uniquely reconstruct A ( z ) . An
additional quantity must be specified. This could be thecoefficient a ( P ) , which in addition is known to have magVI.SUMMARYAND CONCLUSIONS
nitudeless than unity. As will be seen, the formulation
This paper has reported a method for converting prederived
fromSchussler's theorem has drawbacks asa
dictor coefficients to aset of linespectralfrequencies
pseudo-LSF
representation.
which can be used for both even- and odd-order LPC sysConsider
rewritingE,(z) and P2(z) as
tems. The proposed method with the given interval parameters is highly accurate. The accuracy can be further
increased by performing more bisections within the root
interval, of course at the cost of more function evaluations. The use of an expansion in Chebyshev polynomials
obviates the calculationof trigonometric functions and/or where the ratio filter R"(z) is defined as
thestorage of trigonometrictables.Theevaluationofthese expansions makes use of an efficient and numerically stable algorithm. The root finding algorithm which
determines the LSF's has been structured tolimit the Note that the only difference between this ratio filter andmaximum number of function evaluations for a given ac- that for the LSF formulation [see(8)] is an extra z-'delay
curacy constraint.
term. Theauxiliary polynomials haveroots at those points
The reconversion of the LSF's to predictor coefficients on the unit circle at which the phase of the all-pass ratio
is formulated in terms of a recursive calculation of the filter passes through multiples of a.
coefficients of the Chebyshev expansion. This gives a
A simple example will point out some of the ramificacomputarionally efficient algorithm which takes into ac- tions of the extra delay term associated with the LSF forcount inherent symmetries in the auxiliary polynomials. mulation. Consider an A ( z ) which has conjugate pairs of
If the LSF's are expressed, in the cosine domain,
trigo- roots near the unit circle. The LSF's coalesce as theroots
nometric computations can be avoided altogether.
of A(z) approach the unit circle. By contrast, the formuAs a test of the overall procedure, LSF's were found lation that arises from Schussler's stability theorem gives
using the procedure described in Section IV for thespeech single roots at the angular position of the roots of A(z)data used previously. The analysis conditions are the same but, in addition, roots appear midway between these poas specified earlier. The 10 LSF's for each frame of speech sitions, Fig. 4 shows a phase plot of both R(e'") (LSF
were reconverted to predictor coefficients by the proce- formulation) and R"(eJ") (Schussler's formulation) for thedure described in Section V. The maximum difference be- case of an A(z) which has 3 pairs of conjugate roots, each
tween a reevaluated predictor coefficient and the original with magnitude 0.99. The angular positions of the roots
predictor coefficient was 3.8 X lop5.
correspond to o equal to aI4, n f 2 , and 3sl4. Symbols areusedto mark the places where the phase angle crosses a
APPENDIXA
multipleof K . In the vicinity of the roots of A ( z ) , the phase
RELATIONSHIPBETWEEN
LSF's AND THE PREDICTOR
undergoesan excursion through nearly 2a rad for both
ROOTS
formulations. However, the phase offset due to the linear
The relationship between the LSF's and theroots of the phase component in theLSFformulation is enough toprediction error filter A(z) is explored in this appendix. In substantially change the positions of the roots of the LSF
the main text, examples for real speech data have shownthat there is indeed a tendency for the LSF's to dusts'The last condition was added by Gnanasekaran 1121. Conditions (1) andaround the angular positions corresponding to roots of A(z) (2) by themselves also hold if A@.) has all its roots outside the unit circle.

+

RAMACHANDRAN:
KABAL
COMPUTATION
AND

- ..

0

OF LINE SPECTRAL FREQUENCIES

x14

TI2

3x14

1425

x

w-

Fig. 4. Phase responses of R(ej“) and 8(eJw)-A(z)has roots at 0.99eti’”’4,
0.99esTn, and 0.99es3”’4.

Fig. 5. Phase response ofR(ej”)-A(z) has roots at r,etiia’4,0.99etiUa, and0.99efi3“‘4.

auxiliary polynomials from those for Schussler’s formu- Fransen shows that in spite of hurdles in the interpretationof the LSF’s in terms of the spectral features or roots oflation.
Kang and Fransen [4] point out that the group delay of A ( z ) , LSF’s do efficiently represent the perceptually imR(z) increases in the neighborhood of a root of A(z). This portant features of speech spectra.
clearly also applies to @). However, the previous exREFERENCES
ample has shown that peaking of the group delay or theequivalent rapid change in phase angle is not sufficient in [ l ] H. Wakita, “Linear prediction voice synthesizers: Line spectrum pairs(LSP) is the newest of several techniques,” Speech Technol., Fallitself for a pair of LSF’s to occur close together.
1981.
The previous example can be explored further. For this 121 F. K. Soong and B.-W. Juang, “Line spectrum pair (LSP)and speech
data compression,” in Proc. Int. Con5 Acoust., Speech, Signal Prodiscussion, only the LSF formulation will be considered.
cessing, San Diego, CA, Mar. 1984, pp. 1.10.1-1.10.4.
Consider changing the radial position of the first pair of[3] J . R. Crosmer and T. P. Barnwell, 111, “A low bit rate segment voroots of A(z) (at angular position w = a/4) while keeping
coderbase on linespectrumpairs,”
in Proc.Int.Conf.Acousr.,
the otherroot positions constant. As the radius of this pair
Speech, Signal Processing, Tampa, FL, Apr. 1985, pp. 7.2.1-7.2.4.
[4] G. S . Kang and L. J . Fransen, “Application of line spectrum pairs toof roots is reduced to 0.8, the phase change occurs more
low bit rate speech encoders,” in Proc. Int. Con$ Acoust., Speech,
gently. The lowest frequency LSF’s are now spread apart
Signal Processing, Tampa, FL, Apr. 1985, pp, 7.3.1-7.3.4.
(see Fig. 5 ) . When the radius is reduced to 0.5, the phase [5] J. D. Markel and A. H.Gray, Jr., Linear Prediction ofspeech. New
York: Springer-Verlag, 1976.
change is even more gradual and effects the overall phase[6] F. Itakura, “Line spectrum representation of linear predictor coeffiin the vicinity of w = n/2. There are now three closely
cients ofspeechsignal’s,” J. Acoust. SOC. h e r . , vol. 57, S35(A),spaced LSF’s near w = n/2. As the radius is further re1975.
[7] G. S . Kang and L. J. Fransen, “Low bit rate speech encoders based
duced to 0.2, the LSF’s take on a different Configuration.
on line spectrum frequencies (LSFs),” Naval Res. Lab., Rep. 8857,
The lowest LSF has moved to just above w = a/4, theNov. 1984.
second and third LSF’s occur close together
near w =[8] H. W. Schussler, “A stability theorem for discrete systems,” IEEE
Trans.Acoust.,Speech, Signal Processing, vol.ASSP-24, pp. 877r/2, and the fourth LSF liesmidway between these LSF’s
89, Feb. 1976.
and the two paired LSF’s near w = 3n/4.This root con[9] L. Fox and I. B. Parker, Chebyshev Polynomials in Numerical Analysis. London,England:Oxford University Press,1968.
figuration shows that the sharp resonance at w = n/2 issignaled by a close spacing between the second and third [lo] L. R. Rabiner and R . W. Schafer, Digital Processing ofspeech Signals. EnglewoodCliffs, NJ:Prentice-Hall,1978.
LSF’s. In thecoding scheme proposed by Kang and Fran- [ I 11 R. P. Brent, “An algorithm with guaranteed convergence for finding
sen [ 7 ] , LSF’s are coded in pairs specified by a centera zero of a function,” The Comput. J . , vol. 14, pp. 422-425, 1971.
frequency and an offset frequency. In this case, thecoding [12] R . Gnanasekaran, “A note on the new 1-D and 2-D stability theorems
for discrete systems,” IEEE Trans. Acoust.., Speech,Signal Processwould be applied to two pairs of LSF’s, each of whiching, vol. ASSP-29, pp. 1211-1212, Dec. 1981.
has a relatively large offset. After quantization of the offset, this coding scheme may not adequately represent thePeter Kabal received the B.A.Sc., M.A.Sc., andfact that the upper LSF from one pair and the lower LSF
Ph.D. degrees in electrical engineering from thefrom another pair are closely spaced.
University of Toronto, Toronto, Ont., Canada.
He is an Associate Professor in the Department
This appendix has shown examples that point out thatofElectricalEngineering
at McGill University,
the relationship between the positions of the roots of A(z)Montreal, P.Q., Canada, and a Visiting Professor
and the LSF configuration is more fragile than the literaat INRS-Ttltcommunications (a research instituteaffiliated with the Universitt du Qutbec), Verdun,
ture would lead one to believe. Nonetheless, this does not
P.Q., Canada. His currentresearch interests focus
invalidate the use of LSF’s in speech coding. Clearly, theon digital signal processing as applied to speechextensiveperceptual testing carried out by Kang andcoding, adaptive filtering, and data transmission.

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IEEE TRANSACTIONS ON ACOUSTICS, SPEECH,
AND
SIGNALPROCESSING, VOL. ASSP-34,
NO.

6, DECEMBER 1986