the line spectral frequency model of a finite-length sequence

646 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 4, NO. 3, JUNE 2010

The Line Spectral Frequency Model ofa Finite-Length Sequence

Satya Sudhakar Yedlapalli and K. V. S. Hari, Senior Member, IEEE

Abstract—The line spectral frequency (LSF) of a causal finitelength sequence is a frequency at which the spectrum of the se-quence annihilates or the magnitude spectrum has a spectral null. Acausal finite-length sequence with � �� samples having exactly

-LSFs, is referred as an Annihilating (AH) sequence. Using somespectral properties of finite-length sequences, and some model pa-rameters, we develop spectral decomposition structures, which areused to translate any finite-length sequence to an equivalent set ofAH-sequences defined by LSFs and some complex constants. Thisalternate representation format of any finite-length sequence is re-ferred as its LSF-Model. For a finite-length sequence, one can ob-tain multiple LSF-Models by varying the model parameters. TheLSF-Model, in time domain can be used to synthesize any arbitrarycausal finite-length sequence in terms of its characteristic AH-se-quences. In the frequency domain, the LSF-Model can be used toobtain the spectral samples of the sequence as a linear combinationof spectra of its characteristic AH-sequences. We also summarizethe utility of the LSF-Model in practical discrete signal processingsystems.

Index Terms—Annihilating (AH) sequence, antisymmetric,fixed-point approximation, Levinson–Durbin, line spectral fre-quency (LSF), line spectral pair (LSP), linear prediction, linearphase, minimum phase, normalized phase, roots of a polynomial,spectral decomposition, symmetric.

I. INTRODUCTION

I N MANY signal processing systems, one can easily identifytwo basic types of operations: 1) time-domain synthesis of a

sequence; and 2) spectral analysis/estimation of a sequence. Forexample, a speech encoder captures the spectral envelope [1] ofa segment of speech samples using the Linear Predictor [2]. A

th-order Linear-Predictor [3] transforms the auto-cor-relation values of a sequence to a set of -Linear-Prediction Co-efficients (LPCs) and a residual prediction error with the helpof Levinson–Durbin algorithm [3]. The resulting LPC sequencehas always a minimum-phase spectrum [3]. The speech decoder,synthesizes the speech samples from these LPCs and other aux-iliary information [3].

In speech coders, a unique mapping of the real min-imum-phase LPC sequence to a pair of AH-sequences (charac-terized by LSFs), was proposed by Itakura [4]. The LSFs are

Manuscript received March 31, 2009; revised October 27, 2009; acceptedDecember 10, 2009. First published April 15, 2010; current version publishedMay 14, 2010. The associate editor coordinating the review of this manuscriptand approving it for publication was Prof. Hsiao-Chun Wu.

The authors are with the Statistical Signal Processing Lab, Department ofElectrical Communication Engineering, Indian Institute of Science (IISc), Ban-galore 560 012, India (e-mail: [email protected]; [email protected]; [email protected]).

Digital Object Identifier 10.1109/JSTSP.2010.2048233

well known for their robustness to quantization and interpola-tion characteristics and hence play a vital role in many speechanalysis and synthesis algorithms. Many researchers practicallyconfirmed their relative merit over other representation formats[5]. References [6] and [7] present the significance of theseLSFs in the context of speech signals (real) and [8] uses a fastFourier transform (FFT) for efficient real-time computation ofthese LSFs.

In signal processing, apart from the basic discrete Fouriertransform (DFT), alternate tools like cepstrum [9] and ana-lytic sequence were used to capture the spectral informationof a sequence. Reference [10] invokes the duality of timeand frequency domains and interprets, the zero-crossings ofa sequence, as a dual form of the LSFs in the time domain.Reference [11] gives a decomposition scheme which leads tothe representation of signals by their envelope, instantaneousfrequency, and zero-crossings; and [12] constructs an analyticalsequence for extracting modulated components.

In this paper, we define LSFs of any causal finite length com-plex sequence with the help of some model parameters. ThisLSF-Model is independent of the characteristics of the sourceof the complex sequence and can be used as an alternate form ofthe sequence for both time/frequency domain operations.

We now give some basic definitions to describe the problemmore precisely. A finite-length sequence ,with , can be interpreted as the

coefficients of a polynomial with a

complex variable . We denote this finite-length sequence asand is also referred as its -transform [9].

This finite-length sequence can also be denoted as a columnvector with each row element as a com-plex number. For convenience, the sequence issimply denoted as with an implied dimension .

As the characteristics of are due to , we attribute allthe polynomial properties of to the sequence . For a basicfactor , with zero at ,

and the roots are synonymously

referred as the roots of . Here uniquely relateto the coefficients . When , (or )

is monic with and .

As any complex number can be expressed with the pair(Appendix A), the phasor with

and . The NPV is alsoreferred as the normalized (dimensionless) frequency [9].

The spectrum is denoted by thepair so that and

1932-4553/$26.00 © 2010 IEEE

YEDLAPALLI AND HARI: LINE SPECTRAL FREQUENCY MODEL OF A FINITE LENGTH SEQUENCE 647

. The functions are dis-criminated as the -transform and the discrete time Fouriertransform (DTFT) [9] of based on the argument of (com-plex- or real-NPV- ).

As is a product of -basic factors defined by

, is indeed a spectral de-

composition of in terms of , the spectrumof . This decomposition in time domain im-plies, can be synthesized as the cumulative convolution

, with, . As the magnitude and phase spectra

are effectively controlled by the spectralfunctions , we focus on the spectral properties ofsequences with some specific root patterns.

A sequence with roots is re-ferred as a Minimum-Phase (MP)-Sequence [9] when

. A sequence is referred as a Linear Phase(LP)-sequence when (linearw.r.t. ) where are two constants.

Consider an AH-sequence , with rootsdefined by the NPV’s . The LSFs

which define the spectral nulls ( for )can be sorted as:

. Hence, an AH-sequence withcan be defined/stored either as a set of

complex coefficients or real signed fractions(LSFs) .

From the spectral properties of basic sequences (Sections IIIand V) with characteristic spectra like MP/LP/AH-sequences,we notice that an AH-sequence has interesting properties whichcan be exploited for time domain synthesis as well as for spec-tral sampling. As these properties cannot be exploited by anyarbitrary sequence, this motivates us to look for an alternate rep-resentation format of in terms of AH-sequences defined byLSFs.

1) Problem Addressed: The LSF-Model of is defined bytwo types of model parameters and consists of i) a setwhich comprises of sets of AH-sequences(given by LSFs) andsome complex constants, ii) a Spectral Decomposition Structure(SDS), which is a set of vector operations (Appendix B). In timedomain, the LSF-Model gives a hierarchical structure(/method)for synthesizing an arbitrary sequence as a linear combinationof smaller length AH-sequences. In the frequency domain, thespectral functions of the AH-sequences in the LSF-Model canbe linearly combined to obtain . The LSF-Model, servesas an alternate representation format of and allows any signalprocessing application to efficiently perform the time domainsynthesis and spectral sampling of any arbitrary sequence interms of the characteristic AH-sequences.

As the LSF-Model of requires a spectral decomposition,we consider some spectral properties of sequences to developfour LSF-Models viz LSF-A, LSF-B, LSF-C, and LSF-D whichare based on spectral decomposition structures SDS-A, SDS-B,SDS-C, and SDS-D.

Using some properties of MP-sequences, we also develop theLSF-Format of , which maps each sample independentlyto a pair of LSFs. Unlike the LSF-Model, the LSF-Format nei-

ther uses any model parameters nor relies on the root structure.The LSF-Format is more appropriate for storage of in prac-tical discrete systems.

2) Outline: We first factorize a monic aswith as the Non-Annihilating (NA)

factor with coefficients , , and. The spectral properties of an LP-sequence

are used to identify a basic operator for decomposinginto a pair of spectral functions . Usingthese properties, we prove that has a linear-phasespectrum with some spectral nulls but is not always anAH-sequence. This principle of spectral decomposition of

is used in LSF-C/LSF-D and it requires only the LSFs(i.e., the AH-factor ). The LSF-A/LSF-B is

also based on this principle but requires all the roots .The operator for constructing spectral functions

is based on an arbitrary linear-phase spectrum of the formwith as a constant phase shift and as the slope

of the linear-phase (delay/prefix length Appendix B in timedomain). For each choice of these model parameters , thespectral functions are unique.

The properties of sequences, which are easy to derive aresummarized in Appendix D. These properties are presented inorder of relevance and also summarize alternate forms of previ-ously presented properties. Some of these results which requiredetailed proofs are given as theorems with their proofs deferredto Appendixes E and F. These properties are used to develop theLSF-Models and the LSF-Format in Sections VI and VIII. Therelative merit of the LSFs over the coefficients is justi-fied in Section VII.

II. OPERATORS FOR ISOLATING THE MAGNITUDE

AND PHASE SPECTRA

From the properties of DTFT [9], a causal sequence whichcaptures is a sequence given as follows.

Definition 1: Let denote a polyno-mial with coefficients, . Here is referredas the Conjugate Reflected Polynomial (CRP) of and isreferred as the Conjugate Reflected Sequence (CRS) of .

The non-causal auto-correlation [3] sequencedefined for , can be made causal

with a delay of -samples as follows.Definition 2: For ,

defines the causal auto-correlation sequence

; and defines an

all-pass function of [13].When ; , , and

. This trivial case effectively reveals the following op-erators for a complex number.

Definition 3: For a complex number , withand ; 1) , 2)

, and 3) with ,or .

From Definition 2 and Definition 3, we observe thatand . The pair uniquely map to

the pair , as isolates the magnitude and retains


TABLE ICLASSIFICATION OF BASIC FACTORS

the phase . Property 1 shows that iso-late .

For a polynomial (Property 2), the functionsbeing a product and ratio of

also manifest similar relationships (on ) as the complexnumbers . The functions isolate themagnitude and phase spectra and hencejointly characterize . From Property 2, the phase spectrumand magnitude spectrum are trivial or donot capture any significant characteristics of .

III. SPECTRAL DECOMPOSITION OF A SEQUENCE

As the spectral properties of a sequence are dictated by itsroot pattern (Property 2), we classify the roots of a sequenceas in Tables I and II. When is real, the redundancy in rootsmanifests as complex-conjugate-pairs. When is complex, theonly form of redundancy that can be easily identified is when itscoefficients are symmetric/anti-symmetric. For example, given

, one cannot recognizewithout the complete set . We now focus on thespectral properties of each type of basic factor of whicheventually lead to their spectral decomposition structures fortheir LSF-Models in Section VI.

From Property 3, the all-pass functionsdepend on , but their

product is independent of . The all-functionssimplify to phasors defined by

. From Property 1, a linear-phase spectrum ofrequires to be phasor (independent of ).

For an LP-sequence (Property 4), the phasespectrum varying linearly with is equivalent to theall-pass function ,being independent of . This linear-phase characteristic interms of all-pass functions reveals a relationship between

and . Property 4 also shows that linear-phasecharacteristic explicitly manifests as Symmetry/Anti-Symmetryof the coefficients when are all real numbers.

TABLE IIFACTORS OF �� WITH COEFFICIENTS �� AND ��

From Property 4, a Non-Linear-Phase (NLP) sequencecan be recognized with ,

i.e., cannot be independent of . We now splitin terms of and obtain alternate expressionsfor as in Property 5.

1) AH-Sequence: From Property 4, the NPV associ-ated with the LSFs , can also be identified as in Property6. From Table II, the sequence with ,has complex coefficients or real coefficients. Exceptfor the two trivial cases , we see that thereal coefficients can be alternately represented with halfof these values which are not only real but also signed frac-tions. Similarly, for , the real coeffi-cients can be conveyed with only which are onlypositive fractions. Hence, the sequence with

and real coefficients can be conveyedby only real fractions.

As is a linear-phase spectrum, we see that isan LP-sequence and further it is also an extreme case of MP-se-quence as the roots are on . The Hilbert-relationshipsbetween the magnitude and phase spectra of causal sequences,are well-known in the cepstral domain [9]. References [15] and[16] exploit some form of these redundancies for the reconstruc-tion of sequences from its phase or magnitude spectra.

For an AH-sequence , the intimacy between the magni-tude and phase spectra is governed by a set of simple linear-


phase functions as in Property 7. In this case,

we see that are not required. From Prop-erty 7, the magnitude and phase spectra of can becomputed independently with an annihilating spectral function

which intersects the -axis at . Thisfunction differs from the magnitude spectrum onlyin the sign. This sign change reflects in the linear-phase spec-trum as a term for .

Hence, the LSFs are the only parameters required forthe time domain synthesis of and its spectral sampling

as in Property 8 and Property 9. These remarkableproperties of AH-sequences defined by LSFs, motivate us toprobe further into the spectral properties of the NA-factors (MP,NP, SM) which help in evolving the spectral decompositionstructures for the LSF-Models in Section VI. The LSF-Modelsindirectly extend the ease of time domain synthesis and fre-quency sampling of AH-sequences to any arbitrary sequence,in particular a NA-sequence.

2) NP-Sequence: From Property 3, we see that the CRPis an MP-factor and hence the NP-sequence can

be derived from an MP-sequence as in Property 10.3) SM-Sequence: The redundancy in the coefficients of the

SM-sequence can be expressed in terms of an MP-sequenceas follows.

Theorem 1: A Symmetric sequencecan be characterized with a monic Minimum-Phase sequence

and as follows:

• ;

• for , ;• .

In Theorem 1 (proof in Appendix E) the polynomials,have a common term and

is identified directly from the coefficients . Property 11shows similar relationships for the SM-sequences and .

IV. SPECTRAL FUNCTIONS OF AN NA-SEQUENCE

From Property 3, for an NPV- and ,. The term can be recognized as a

trivial polynomial which can be usedto generate a zero-prefixed sequence as in Property 12.

We now define a pair of LP-sequences [17] which capture thespectra of as follows.

Definition 4: For an NA-sequence and twomodel parameters :

• defines the general-ized CRP of ;

• defines a spectralpolynomial with coefficients as ,

.Remark 1: Consider for two NPV’s and

:

: ;

: .

As , we can discriminate for only. For a real sequence ,

will always result in real sequences .

Fig. 1. Vector Synthesis of �� , �� , and �� . (a) Synthesis of �� .(b) Synthesis of �� .

From Property 4 of , the characteristics of the aresummarized in Property 13. In Property 13, is referredto as the Cosine-Spectral function and as the Sine-Spec-tral function as they have cosine and sine multiplicative fac-tors. The Spectral polynomials are discriminated as theSum-Spectral polynomial and the Difference-Spectral polyno-mial of .

V. SPECTRAL POLYNOMIALS

From Property 13, the spectral polynomials of can alsobe characterized as in Property 14. The linear-phase autocor-relation polynomial (Property 2), is a single polynomialobtained as the product of the pair with

. This product annihilates the phase spectrum .The two spectral polynomials constructed as a linear-combination of the pair (Property 14) retain

(i.e., complete ) in an alternate format.Property 15 summarizes a spectral decomposition principle

of , as in Fig. 1.From Property 14, as occur as common

factors for , the synthesis of can be done in any one ofthe following three methods.

1) For ; .2) For ; .3) For ; .The first and second methods require the isolation of

from but the third method does not.The lengths of the spectral polynomials in the three methodsare dependent on , where .As the complexity of convolution is length dependent, we focusonly on the first two methods which require .

As is also a NA-sequence , from Property15, can be further characterized as follows.

Theorem 2: For an MP-sequence andmodel parameters , isan AH-sequence.


Fig. 2. Alternate LSF-Models of the four basic sequences.

Fig. 3. LSF-Models �� , �� and �� of AH-sequences.

As are three forms of MP-sequences, fromTheorem 2 (proof in Appendix F), their spectral polynomialscan be summarized with independent model parameters as inProperty 16. From Property 16, we infer that an MP-polyno-mial and its CRP (NP-polynomial) can be jointlysynthesized as the sum and difference of two AH-polynomials

. The LSFs encountered in most speech coding applica-tions [5] correspond to the case of with (LPC order10), , . The principle of spectral decompositionof can be obtained by replacing the CRP in The-orem 1, with its generalized CRP as in Property 17.

VI. LSF MODELS OF SEQUENCES

From Table II, we construct LSF-Type-1 and LSF-Type-2 foreach of the four basic sequences (AH/MP/NP/SM) as in Fig. 2.

As an AH-sequence is the only sequence which can be di-rectly defined by its LSFs, its LSF-Model (/vector synthesisstructure) does not require any model parameters and hence itsLSF-Model can be simply denoted as in Fig. 3.

From the spectral decomposition principles in Property 10,Property 11, Property 16, and Property 17, we obtain the LSF-Models of (Fig. 4), (Fig. 5), and (Fig. 6).

From the spectral decomposition of in Table II and theLSF-Models of basic sequences, the LSF-Models LSF-A andLSF-B are obtained in Fig. 7. The spectral decomposition prin-ciples in Property 15 to Property 17 are used to develop theLSF-Models LSF-C and LSF-D in Fig. 8.

As the LSF-Model of a NA-sequence involves manyLSF-Models of characteristic MP-sequences, the complex con-stants and LSF-sets in are discriminated as in Table III. For

Fig. 4. �� , the LSF-Model of an MP-sequence �� .

Fig. 5. �� , the LSF-Model of an NP-sequence �� .

simplicity, only LSF-Type-1 LSF-Models are assumed withinLSF-A/B/C/D in Table III.


Fig. 6. �� , the LSF-Model of an SM-sequence �� .

Fig. 7. �� , the LSF-Model LSF-A/LSF-B of a sequence �� and ��. (a) LSF-A.(b) LSF-B.

The spectral decomposition in Fig. 8, is based on the SM-se-quences , with obtained as a linear combi-nation of (Fig. 1). With three independent model pa-rameters , the LSF-Modelscan only synthesize zero-prefixed SM-sequences.

Fig. 8. �� , the LSF-Model LSF-C/LSF-D of a sequence �� and ��.

Let denote a zero-prefixed pair, with aszero-prefixed sequence of , such that .The zero-prefixed sequence pairs in Fig. 8 can now be summa-rized as: , , ,

, , . As the linearcombination of requires their lengths to be equal,we match their lengths with two independent zero-prefixlengths as in Fig. 8. Despite the independent choice


TABLE IIILSF SET �� OF A SEQUENCE �� WITH �� . (a) THE

LSF-SET �� IN LSF-A/LSF-B. (b) THE LSF-SET �� IN LSF-C/LSF-D

of and the disparity (if any) , thedelay matching always yields the pairs: ,

with output as summarized in Fig. 8.Example 1 illustrates the four LSF-Models of a sequenceobtained by the following Procedure 1.

Procedure 1: The LSF-Set of the LSF-Model of can becomputed as follows.

1.1) Identify the AH-sequence defined by the LSFs .1.2) LSF-A/LSF-B: isolate the roots , , and

as defined in Table II. From Property 10, Property 11,Property 16 and Property 17, compute the LSF-Models(Type-1/Type-2) of , , and and as inFigs. 4–6.

1.3) LSF-C/LSF-D: From Property 15 to Property 17,obtain the desired LSF-Model as in Fig. 8 based on

. The LSF-D requires of .From the LSF-Models of , we infer that the coefficients

can be conveyed with a set of characteristic AH-se-quences and some complex constants, based on an impliedspectral decomposition structure, which dictates a hierarchicalmethod of combining these AH-sequences with vector oper-ations in Appendix B. The number of LSFs required is morethan the coefficients as the LSF-Models are defined by some

model parameters which stretch the dimensions(/lengths) ofthe sequences. The simplest LSF-Model of a sequence refers toan AH-sequence as its roots directly map to LSFs.

VII. LSFS IN PRACTICAL DISCRETE SYSTEMS

Any practical discrete system can only store/process a realsequence by representing each sample with only finitenumber of bits [18]. In any fixed point representation (FPR) ofa sequence (Appendix C), one has to identify an appropriateformat [19] such that and thenonzero minimum value . As the word length

is fixed in practical systems, the tradeoff betweenis inevitable. This tradeoff is relaxed only for two extreme caseswhen the numbers are 1) pure integers and2) pure fractions . Hence, in any FPR of , one canonly approximate with within an acceptable nonzero errorlimit by balancing the extremum values and within thefinite word length limit .

The FPR of inevitably requires a pre-scaling based on somecriteria [9]. The scaling effectively defines ,where is a scale factor usually chosen as one of the norms

[18], [20] which are dependent onall the samples of . After scaling, the extremas of are withinthe feasible limits of the chosen FPR . As this processeffectively approximates to (maps to ), using a pre-scaling,one cannot minimize the error in representing the largest andsmallest nonzero values of simultaneously. For example, when

, , the precision of representingdictates the error in fixed point approximation.

Unlike the case of any arbitrary sequence , which requires(for pre-scaling), an AH-sequence can be easily

represented/conveyed by its LSFs which are signed realfractions. This fractional representation in the LSF-Modelevolves from the spectral decomposition structure of andnot from the time domain value which depends directly onthe coefficients .

VIII. LSF-FORMAT OF A SEQUENCE

In speech coding, as LSFs are derived from the LPCs, theirproperties are interpreted w.r.t linear-prediction in [6], [7], [21],and [22]. For an LSF , is widely referred as a LineSpectral Pair (LSP) [22] and [23].

From Theorem 2, we interpret the LSFs of a basic MP-factorin Property 18 and pictorially in Fig. 9. Property 18 is used todevelop the LSF-Format of a complex number as in Property19. As the sequence , is a set of complex numbers,each can be represented independently in its LSF-Format(angles) as in Property 20. From Property 19, all the complexconstants in Table III can also be expressed in their LSF-Format.With this, the set is a complete set of fractions (angles) andintegers (exponents).

The LSF-Format of is illustrated in Example 2. Unlike theLSF-Models (LSF-A/B/C/D), this LSF-Format does not exploitthe root structure of , but only uses the complex number struc-ture of each sample . Hence, if sequence denotes a per-mutation of the samples within , the LSF-Format can bederived by following the same permutation of the vectors within (Property 20). As roots of differ from the roots of


Fig. 9. Geometrical interpretation of the LSFs of the basic factor � �� with model parameters � � �, � � �.

, the LSF-Model cannot be derived from or must beindependently computed.

IX. CONCLUSION

The LSF-Model (LSF-A/B/C/D) of a causal finite-length se-quence is a fundamental decomposition independent of theoriginating source of the sequence. It relies on 1) the root struc-ture of the sequence, 2) the principle of spectral decomposition(SDS-A/B/C/D), and 3) the model parameters .By varying the model parameters, one can obtain multiple LSF-Models of the same sequence .

In the time-domain, the LSF-Model LSF-A/B/C/D can beused to synthesize by following the vector synthesis struc-ture in SDS-A/B/C/D. In the frequency domain, it can be usedto sample the DTFT at discrete values of , by followingthe spectral decomposition structure SDS-A/B/C/D.

As an Annihilating (AH) sequence has good properties intime and frequency domains, the same can be exploited by anyarbitrary causal finite length sequence (even Non-Annihilating),when represented in its LSF-Model with the choice of model pa-rameters . The LSF-Model reveals that an AH-se-quence is a basic sequence that can be used to characterize anyarbitrary causal finite-length sequence.

The LSF-Format of a causal finite-length sequence is yet an-other representation format, which also uses only fractions andintegers but does not use any information pertaining to the rootsor model parameters as in the LSF-Model. The LSF-Format ismore appropriate for storing/processing of a sequence in em-bedded systems.

When the complex constants in the LSF-Model are alsomapped to their LSF-Formats, the LSF-Model contains only

fractions (LSFs) and integers (exponents). With this mapping,any causal finite length sequence can be easily represented inembedded systems, which inevitably approximate real numberswith finite number of bits.

Hence, the LSF-Model extends all the remarkable propertiesof an AH-sequence to any . As fractions and integers are moreamenable to fixed point representation relative to the coefficients

, these two alternate representation formats (LSF-Modeland LSF-Format) can be exploited by any signal processingsystem working with causal finite length sequences both in timeand frequency domains.

APPENDIX ANORMALIZED PHASE

Any phasor with an absolute phase of can al-ways be expressed as: , where

. The phase is referred as the normalized phase of .Consider a complex variable with . As

, is referred as a Normalized Phase Variable(NPV) of . For example

.The NPV of is obtained with as

.(1)

The (modulo) function ensures that andis a modulo operator which yields the normalized

phase. For two phasors , the phase valuescan be compared only with their normalized phase values givenas: , . The two real values

of are denoted as and is alternatelywritten as (example ). For

; and .

APPENDIX BBASIC OPERATORS FOR FINITE-LENGTH SEQUENCES

Fig. 10 summarizes the basic vector operations which areused to denote basic linear operations in the spectral decompo-sition of causal finite length sequences. Each node (circle) de-notes a causal finite length sequence which is a one-dimensionalvector. The sample index is internally used in each vector oper-ation. For the Zero-Prefix operation in Fig. 10(e), the sequence

is the delayed version of . For the Zero-Pad operation inFig. 10(f), the last samples of are zeros. Hence thisZero-Pad operation is never unique even thoughor for any .

APPENDIX CFIXED POINT APPROXIMATION OF REAL NUMBERS

For and bit , consider a [bit]binary code word with asthe sign-bit and binary point between . In the ab-sence of the sign bit , the code represents an unsignedvalue.

As the position of the binary point in is fixed, each bitcontributes a value to its numerical value . This de-fines two fixed point representation (FPR) [18] formats


Fig. 10. Basic vector processing blocks for causal finite length sequenceswith �� and � �� , � � � � . (a) Convolution:� �� . (b) Addition/Subtraction: � �� .(c) Conjugate-Reflection: � �� . (d) Scale: � �� . (e)Zero-Prefix: � �� . (f) Zero-Pad:� �� or � �� independent of .

(signed) and (unsigned) with the numerical value dis-criminated as and

with . For , ,

and , these two formats can be characterized withcodes and values as follows.

• The FPR :• ; , ;• denote codes;• , ;• for and the code

• The FPR :• ; , ;• denote codes;• , ;• For and the code

Here defines the extremum value of and denotes thesmallest nonzero value or the precision. For a fixed word length

(or ), it is easy to see that as increases, decreasesand consequently one cannot simultaneously increase anddecrease . Hence, with can be stored/conveyedas a pure integer with an implied scale factor oras a pure fraction with a factor of . Similarly, thevalue can also be conveyed with implied scale factorsas .

APPENDIX DPROPERTIES

Property 1: For the basic factor :• the CRP of is ; ;• ; ;• ; ;• for ; ;• ; ;• ;• when , and

varies linearly with .

Property 2: For a monic sequence :

• the CRP of is ; ;

• ; ;

• ; ;

• ; ;• ; ;• ; ;• for ; ;• in the LP-sequence , the samples ;

can be mapped to a monic MP-sequence(predictor polynomial) by solving the

Yule–Walker equation [3].

Property 3: For the basic factors in Table I:• ;• ;• ;• ; ;

• ;• ;• ;• ;• ;

• for , ;

• the -LSFs of are .

Property 4: For a monic LP-sequence :• ; ;• ;•

;• , ;• ; ;• ; ;• ;• ;• ;

• ; ;• for , ;• ; is Symmetric;• for , and

;


• when (Even), ,;

• when (Odd), i.e., ;• when is a real sequence, with

i.e., ;• when is real with ,

• when is real with ,

Property 5: For a monic sequence :• ; ;• ; ;• ;• with ;• ;• is a common factor of and ;

• ;• ;• ;

• ; ;

• ; ;

• ; ;

• ; .

Property 6: For a monic AH-sequence :

• ;

• ; ;

• the sum is independent of the LSFs ;

• ;

• .

Property 7: For :• ;• ;•

•

Property 8: An AH-sequence in Table II withcan be synthesized with cumulative convolution as follows:

• , with ,for ;

• , with ,; for ;

• can be obtained as ;• this synthesis requires only samples of

for and .

Property 9: A spectral sample ,with can be obtained using Property 7 asfollows:

• ;

• ;

• ;

• ;

• ;

•

• this sampling requires only the samples of for

and .

Property 10: For a monic NP-sequence :• ; ;• is an MP-polynomial (Table II) given as follows:

Property 11: The SM-sequences ,in Table II can be characterized with monic MP-se-

quences , as:• ; ;• for ; ;

;• ;

;• for ; ;

;• ;

.

Property 12: For an NPV and an integer, a zero-prefixed sequence Fig. 10(e) of

with has;• ; ;• .

Property 13: For two model parameters ,, :

• ;


• , ,;

• ;

• ;• ;

• ;• ;

• for , ;

• for , .

• the LSFs of are distinct, i.e., and arecontrolled by the phase spectrum .

Property 14: For and a monic sequence :

• ; ;• ;• ;• for

• for ,

Property 15: For a set , the spectral decomposition ofmonic can be summarized with ,

, as follows:

• for , ; for , ;• ; ;• ; :

• in general , ; but ;• have LSFs ;

• ; in SDS-C/SDS-D;

• ;• ;• the trivial LSFs of are the LSFs which are inde-

pendent of ;• the non-trivial LSFs of are which are controlled

by .

Property 16: For a monic MP-sequence andmodel parameters , the spectral polynomials withcoefficients are as follows:

• ; ;• ; :

• the LSFs of the AH-polynomials aregiven by , and :

• for , ; for , .

Property 17: An SM-sequence can becharacterized with and two AH-polynomials ,

defined with , asfollows:

• ; ;

• ;

• ;

• ;

• ;

• for ; is

the zero-prefixed sequence [Fig. 10(e)] of .

Property 18: For with , ,, the spectral polynomials with

model parameters are as follows:• ;• ;• ;• ; and depend

only on . An MP-factor withwill also have the same LSFs ;

• ; anddepend only on . An MP-factor with

will also have the same LSFs ;• is defined by the

two LSFs as in Fig. 9;


• the two LSFs are redundant;• when , ,

Property 19: The LSF-Format of a Complex Number: Anycomplex number can be expressed as ,

. with i) complex sign ,ii) integer exponent , and iii) the LSFscorresponding to as in Fig. 9. When , andcan be defined by only one LSF .

Property 20: The LSF-Format of a Sequence : For a complexsequence :

• , where is the complexsign, , and is an integer exponent;

• each can be mapped to two LSFs ;• the LSF-Format of denoted by the set

can be used to reconstruct . Here; and ;

• each complex sign can be denoted by the FPRso that the codes map to four possiblevalues of complex sign ;

• each LSF can be denoted by the FPR , with(among nonzero values of LSFs);

• each exponent can be denoted by the FPR ,with .

APPENDIX EPROOF OF THEOREM 1

Here and, . As

, .The polynomials and

can be expanded as

(2)

For

(3)

The factor in (3) can be isolated as; where is a polynomial expressed as the

difference of a polynomial of order and the polynomialas

(4)

As is analytic [24] for ; is alsoanalytic within and on the unit-circle of the -plane. As

has no poles; the zeros of cannot lie within the unit-circle. This implies or is an NP-polynomial, andhence is an MP-polynomial.

APPENDIX FPROOF OF THEOREM 2

, implies.

As the all-pass function of is stable [9],from the Maximum Modulus Theorem [25]

.(5)

For , and

.

As only on ,implies .

APPENDIX GEXAMPLES

Example 1: Consider a sequence:, ,

, ,, ,, ,, ,, ,

, , ;with roots; , , ,

, ,, , and

.The LSF-Models (Table III) of with model parameter vec-

tors , are as follows:1) The LSF-A/LSF-B Model :

• ;;

• , ;• ;• ;• ;• ;• ;• .

2) The LSF-C Model ( as in LSF-A):• ; ;

;• ,

;• ,

;•

;• ;


•;

•;

3) The LSF-D Model ( as in LSF-A):• , and are as in LSF-A.• ; ;• , ;• ;• ;• ;• ;• ;• .Example 2: The LSF-Format of in Example 1 is as fol-

lows.• , ;•

;•

;• .

As and(among nonzero LSFs), from Property 20, and .As and

, the real and imaginary parts , requirea FPR with [bits].

REFERENCES

[1] B. S. Atal, “The history of linear prediction,” IEEE Signal Process.Mag., vol. 24, no. 3, pp. 8–10, May 2007.

[2] J. I. Makhoul, “Linear prediction: A tutorial review,” Proc. IEEE, vol.63, no. 4, pp. 561–581, Apr. 1975.

[3] M. H. Hayes, Statistical Digital Signal Processing. New York: Wiley,2003.

[4] A. M. Kondoz, Digital Speech Coding for Low Bit Rate Communica-tion Systems. New York: Wiley, 1990.

[5] F. Itakura, “Line spectrum representation of linear predictor coeffi-cients of speech signal’s,” J. Acoust. Soc. Amer., vol. 57, no. S1, pp.S35–S35, 1975.

[6] T. Backstrom, P. Alku, T. Paatero, and B. Kleijn, “A time-domain in-terpretation for the LSP decomposition,” IEEE Trans. Speech AudioProcess., vol. 12, no. 6, pp. 554–560, Nov. 2004.

[7] W. B. Kleijn, T. Backstrom, and P. Alku, “On line spectral frequen-cies,” IEEE Signal Process. Lett., vol. 3, no. 3, pp. 75–77, May 2003.

[8] S. S. Yedlapalli, “Transforming real linear prediction coefficients toline spectral representations with a real FFT,” IEEE Trans. SpeechAudio Process., vol. 13, no. 5, pp. 733–740, Sep. 2005.

[9] A. V. Oppenheim, The Discrete Time Signal Processing. New Delhi,India: PHI, 1987.

[10] R. Kumaresan and Y. Wang, “On the duality between line-spectral fre-quencies and zero-crossings of signals,” IEEE Trans. Speech AudioProcess., vol. 9, no. 4, pp. 458–461, May 2001.

[11] R. Kumaresan and A. Rao, “On minimum/maximum/all-pass decom-positions in time and frequency domains,” IEEE Trans. Signal Process.,vol. 48, no. 10, pp. 2973–2976, Oct. 2000.

[12] A. Rao and R. Kumaresan, “On decomposing speech into modulatedcomponents,” IEEE Trans. Speech Audio Process., vol. 8, no. 3, pp.240–254, May 2000.

[13] P. A. Regalia, S. K. Mitra, and P. P. Vaidyanathan, “The digital all-passfilter: A versatile signal processing building block,” Proc. IEEE, vol.76, no. 1, pp. 19–73, Jan. 1988.

[14] A. H. Sayed and T. Kailath, “A survey of spectral factorizationmethods,” Numerical Linear Algebra with Applications, vol. 8, no.6–7, pp. 467–496, 2001.

[15] M. Hayes, J. Lim, and A. V. Oppenheim, “Signal reconstruction fromphase or magnitude,” IEEE Trans. Acoust., Speech, Signal Process.,vol. SP-28, no. 12, pp. 672–680, Dec. 1980.

[16] B. Yegnanarayana, D. Saikia, and T. Krishnan, “Significance of groupdelay functions in signal reconstruction from spectral magnitude orphase,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-32,no. 3, pp. 610–623, Jun. 1984.

[17] S. S. Yedlapalli, “The line spectral frequency model of a finite lengthsequence,” Ph.D. dissertation, Dept. of Elect. Commun. Eng., IndianInstitute of Science (IISc), Bangalore, 2010.

[18] M. D. Ercegovac and T. Lang, ScienceDirect (Online service. DigitalArithmetic). San Francisco, CA: Morgan Kaufmann, 2004.

[19] I. Koren, Computer Arithmetic Algorithms. Natick, MA: AK Peters,Ltd., 2002.

[20] C. D. Meyer, Matrix Analysis and Applied Linear Algebra. Philadel-phia, PA: SIAM, 2004.

[21] G. A. Mian and G. Riccardi, “A localization property of line spectrumfrequencies,” IEEE Trans. Speech Audio Process., vol. 2, no. 4, pp.536–539, Oct. 1994.

[22] T. Bäckström and C. Magi, “Properties of line spectrum pair polyno-mials-a review,” Signal Process., vol. 86, no. 11, pp. 3286–3298, 2006.

[23] C. H. Wu and J. H. Chen, “A novel two-level method for the compu-tation of the LSP frequencies using a decimation in degree algorithm,”IEEE Trans. Speech Audio Process., vol. 5, no. 2, pp. 106–115, Mar.1997.

[24] M. R. Spiegel, Complex Variables, Schuam’s Outline Series. NewYork: McGraw-Hill, 1981.

[25] P. P. Vaidyanathan, Multirate Systems and Filter Banks, . UpperSaddle River, NJ: Pearson, 1993.

Satya Sudhakar Yedlapalli received the B.E. de-gree in electrical communication engineering fromAndhra University, Visakhapatnam, India, in 1991,the M.Tech. degree in electrical engineering fromthe Indian Institute of Technology (IIT), Madras, in1993, and the Ph.D. degree from the Indian Instituteof Science (IISc), Bangalore, in 2010.

He was a Research Engineer at the Centre forDevelopment of Telematics (C-DOT), Bangalore,(1993–1996), and later was a Lead Engineer atMotorola India Electronics, Ltd. (M.I.E.L.), Banga-

lore, (1996–2001). From 2001 to 2006, he was a Lead Engineer in the BroadBand Communications Group, Texas Instruments (Pvt.), Ltd., Bangalore. Hisresearch interests lie broadly in devising signal processing algorithms forcost-effective realization of embedded systems (communication and biomed-ical).

Dr. Satya is a member of the Institute of Electronics and TelecommunicationEngineers (IETE), India.

K. V. S. Hari (M’92–SM’97) received the B.E.degree in electrical communication engineeringfrom Osmania University, Hyderabad, India, in1983, the M.Tech. degree from the India Instituteof Technology (IIT), Delhi, in 1985, and the Ph.D.degree from the University of California at SanDiego, La Jolla, in 1990.

He was with DLRL, Hyderabad, from 1985 to1987, and at Osmania University as a Scientist from1991 to 1992. Since 1992, he has been a FacultyMember at the Department of Electrical Commu-

nication Engineering, Indian Institute of Science (IISc), Bangalore, where heis currently a Professor and coordinates the activities of the Statistical SignalProcessing Lab in the department. He has been a visiting faculty member atStanford University, Stanford, CA, the Royal Institute of Technology (KTH),Stockholm, Sweden, and the Helsinki University of Technology, Espoo,Finland. His research interests are in digital signal processing with an emphasison developing statistical signal processing algorithms for spectrum estimation,direction-of-arrival estimation, acoustic signal separation, and MIMO wirelesscommunication systems. During his work at Stanford University, he workedon wireless channel modeling and is the coauthor of the WiMAX standard onwireless channel models for fixed-broadband wireless communication systemswhich proposed the Stanford University Interim (SUI) channel models. He iscurrently an Editor of the EURASIP’s Journal on Signal Processing publishedby Elsevier. He is also an academic entrepreneur and is a cofounder of thecompany ESQUBE Communication Solutions, Bangalore.

the line spectral frequency model of a finite-length sequence

Documents