The Weighted Sum of the Line Spectrum Pair for Noisy Speech

Transcription

1 HELSINKI UNIVERSITY OF TECHNOLOGY Deartment of Electrical and Communications Engineering Laboratory of Acoustics and Audio Signal Processing Zhijian Yuan The Weighted Sum of the Line Sectrum Pair for Noisy Seech Master s Thesis submitted in artial fulfilment of the requirements for the degree of Master of Science in Technology. Esoo, 0 May, 003 Suervisor: Professor Paavo Alku

2 Abstract Linear rediction (LP) rovides a robust, reliable and accurate method for estimating the arameters that characterize the linear time-varying system. It has become the redominant technique for estimating the basic seech arameters such as formants. Linear rediction exloits the redundancies of a seech signal by modelling the seech signal as an all-ole filter. The filter coefficients are obtained from standard autocorrelation method or covariance method. In this work, we focus on a new develoed method, the Weighted Sum of the Line Sectrum Pair (WLSP), which is based on the Line Sectrum Pair (LSP) decomosition. In conventional LP, the LSP decomosition is comuted to quantise the LP information. WLSP utilises the LSP decomosition as a comutational tool, and it yields a stable all-ole filter to model the seech sectrum. In contrast to the conventional autocorrelation method of LP, WLSP takes advantage of the autocorrelation of the inut signal also beyond the time index determined by the rediction order in order to obtain a more accurate all-ole model for the seech sectrum. With the hel of the sectral distortion method, the exeriments results show that WLSP can distinguish the most imortant formants more accurately than the conventional LP. Keywords: Linear rediction, Line Sectrum Pair, sectral modelling

3 Acknowledgments I would like to exress my deeest gratitude to my suervisor, Professor Paavo Alku, for his continuous suort and guidance throughout my graduate studies at the Acoustics and Audio Signal Processing Laboratory, Helsinki University of Technology. His knowledge, atience and valuable advice did much to bring this work to a successful conclusion. I am thankful to all the fellows in the Acoustics and Audio Signal Processing Laboratory, ast and resent, for their good research environment. My gratitude also goes to my family for their love, suort and encouragement. Otaniemi, May 0, 003 Yuan Zhijian 3

4 Contents Chater Introduction Seech roerties..4. Seech coding Performance criteria of seech coding.6.. Source Coders.8..3 Hybrid Coders 8..4 Performance Criteria of Seech Coding..9.3 Objectives of this research Chater Linear Prediction.... Linear rediction modelling... Estimation of linear rediction coefficients.3.. Windowing.4.. Autocorrelation method 4..3 Covariance.5..4 Numerical solution of Linear Prediction linear equations 6..5 Stability Comuting the gains.8.3 Line sectral air Line sectral air olynomials.9.3. The root roerties of Line Sectral Pair olynomials Comuting line sectral frequency.. Chater 3 Distortion Measurements 3 3. Time domain measures.3 3. Sectral envelo distortion measures..4 4

5 Chater 4 Weighted Sum of the Line Sectrum Pair.9 4. Weighted Sum of the Line Sectral Pair Polynomials Algorithm of Weighted Sum of the Line Sectrum Pair..34 Chater 5 Exeriments Weighted Sum of the Line Sectrum Pair with clean seech Weighted Sum of the Line Sectrum Pair with noisy seech...40 Chater 6 Conclusions...44 References

6 List of abbreviations AbS AC ADM ADPCM APC APCM AR ARMA ASRC CCITT CELP db DCT DFT DM DPCM IR LAR LP LPC LSP MA MPE PCM RC RPE SNR SNRseg WLSP Analysis-by-Synthesis Autocorrelations Adative Delta Modulation Adative Differential Pulse Code Modulation Adative Predictive Coding Adative Pulse Code Modulation Autoregressive Autoregressive Moving Average Arcsine of Reflection Coefficients International Telegrah and Telehone Consultative Committee Code-Excited Linear Predictive decibel Discrete Cosine Transform Discrete Fourier Transform Delta Modulation Differential Pulse Code Modulation Imulse Resonse Logarithm Area Ratio Linear Prediction Linear Predictive Coding Line Sectrum Pair Moving Average Multi-Pulse Excited Pulse Code Modulation Reflection Coefficient Regular-Pulse Excited Signal-to-Noise Ratio Segmental SNR Weighted Sum of the Line Sectrum Pair 6

7 Chater Introduction Seech coding is an imortant asect of modern telecommunications. Seech coding is the rocess of digitally reresenting a seech signal. The rimary objective of seech coding is to reresent the seech signal with the fewest number of bits, while maintaining a sufficient level of quality of the retrieved or synthesized seech with reasonable comutational comlexity. To achieve high quality seech at low bit rate, coding algorithms aly sohisticated methods to reduce the redundancies, that is, to remove the irrelevant information from the seech signal. In addition, a lower bit rate imlies that a smaller bandwidth is required for transmission. Although in wired communications very large bandwidths are now available as a result of the introduction of otical fiber, in wireless and satellite communications bandwidth is limited. At the same time, multimedia communications and some other seech related alications need to store the digitised voice. Reducing the bit rate imlies that less memory is needed for storage. These two alications of seech comression make seech coding an attractive field of research.. Seech roerties Before elaborating on seech coding, some seech roerties have to be discussed. Seech is highly redundant. For examle, multile eak cliing of the seech signal (i.e. reducing it to binary waveform) eliminates virtually all amlitude information, yet listeners easily understand seech distorted rovided that the samling frequency is large enough. Seech signals are non-stationary and at best can be considered as quasi- stationary over short segments, tyically 5-0 ms. The statistical and sectral roerties of seech are thus defined over short segments. Seech can generally be classified as voiced (e.g., /a/, /i/, etc.), unvoiced (e.g., /s/), or mixed. Time and frequency domain lots for voiced and unvoiced segments are shown in Figure.. Voiced seech is quasi-eriodic in the time domain and harmonically structured in the frequency domain while unvoiced seech is noise-like and broadband. In addition, the energy of voiced segments is generally higher than the energy of unvoiced segments. An imortant feature in the sectrum is the resonant structure (eaks) known as 7

8 formants (see the sectrum of the voiced seech of Fig... These formants are known to be ercetually imortant in the recognition of seech sounds, articularly for vowels. Figure. An unvoiced to voiced seech transition, the underlying excitation signal and short-time sectra.. Seech coding Over the ast few decades, a variety of seech coding techniques has been roosed, analyzed, and develoed. Here, we briefly discuss techniques, which are used today, and those which may be used in the future [45]. Traditionally, seech coders are divided into two classes waveform coders and source coders (also known as arametric coders or vocoders). Tyically waveform coders oerate at high bit-rates, and give very good seech quality. Source coders are used at very low bit-rates, but tend to roduce synthetic quality seech. Recently, a new class of coders, called hybrid coders, was introduced which uses techniques from both source and waveform coding, and gives good quality seech at intermediate bitrates. Figure. shows the tyical behavior of the seech quality versus bit-rate curve for the two main classes of seech coders... Waveform Coders 8

9 Waveform coders attemt to reroduce the inut signal waveform. They are generally designed to be signal indeendent so they can be used to code a wide variety of signals. Generally they are low comlexity coders roducing high quality seech at rates above about 6 kbs. Waveform coding can be carried out either in the time domain or in the frequency domain. Figure. Seech quality versus bit-rate for common classes of coders Time Domain Coders Time domain coders erform the coding rocess on the time samles of the signal data. The well known coding methods in the time domain are [3, 45]: Pulse Code Modu-lation (PCM), Adative Pulse Code Modulation (APCM), Differential Pulse Code Modulation (DPCM), Adative Differential Pulse Code Modulation (ADPCM), Delta Modulation (DM), Adative Delta Modulation (ADM), and Adative Predictive Coding (APC). In the following, we briefly describe some imortant coding schemes in the time domain. PCM Coders Pulse code modulation is the simlest tye of waveform coding. It is essentially just a samle-by-samle quantization rocess. Any form of scalar quantization can be used with this scheme, but the most common form of quantization used is logarithmic quantization. The International Telegrah and Telehone Consultative Committee s (CCITT) Recommendation G.7 determines 8 bit A-law and µ-law PCM as the standard method of coding telehone seech. DPCM and ADPCM Coders 9

10 PCM makes no assumtions about the nature of the waveform to be coded in most of the cases, hence it works well for non-seech signals. However, when coding seech there is a very high correlation between adjacent samles. This correlation could be used to reduce the resulting bit-rate. One simle method to do this is to transmit only the differences between each samle. This difference signal will have a much lower dynamic range than the original seech, so it can be effectively quantized using a quantizer with fewer reconstruction levels. In the above method, the revious samle is being used to redict the value of the resent samle. The rediction would be imroved if a larger block of the seech is used to make the rediction. This technique is known as differential ulse code modulation (DPCM). Its structure is shown in Fig..3. Figure.3 General differential PCM system: coder on left, decoder on right. The inverse quantizer simly converts transmitted codes back into a signal u ˆ( n) value. An enhanced version of DPCM is Adative DPCM in which the redictor and quantizer are adated to local characteristics of the inut signal. There are a number of ITU recommendations based on ADPCM algorithms for narrowband (8 khz samling rate) seech and audio coding e.g., G.76 oerating at 40, 3, 4 and 6 kbs. The comlexity of ADPCM coders is fairly low. Frequency Domain Coders Frequency domain waveform coders slit the signal into a number of searate frequency comonents and encode these searately. The number of bits used to code each frequency comonent can be varied dynamically. Frequency domain coders are divided into two grous: subband coders and transform coders. Subband Coders Subband coders emloy a few bandass filters (i.e., a filterbank) to slit the inut signal into a number of bandass signals (subband signals) which are coded searately. At the receiver the subband signals are decoded and summed u to reconstruct the outut signal. The main advantage of subband coding is that the quantization noise roduced in one band is connected to that band. The ITU has a 0

11 standard on subband coding (i.e., G.7 audio coder [30]) which encodes wideband audio signals (7 khz bandwidth samled at 6 khz) for transmission at 48, 56, or 64 kbs. Transform Coders This technique involves a block transformation of a windowed segment of the inut signal into the frequency, or some other similar, domain. Adative Coding is then accomlished by assigning more bits to the more imortant transform coefficients. At the receiver the decoder carries out the inverse transform to obtain the reconstructed signal. Several transforms like the Discrete Fourier transform (DFT) or the Discrete Cosine Transform (DCT) can be used... Source Coders Source coders oerate using a model of how the source was generated, and attemt to extract, from the signal being coded, the arameters of the model. It is these model arameters which are transmitted to the decoder. Source coders for seech are called vocoders, and use the source-filter model of seech roduction as shown in Fig..4. This model assumes that seech is roduced by exciting a linear time-varying filter (the vocal tract) by a train of ulses for voiced seech, or white noise for unvoiced seech segments. Vocoders oerate at around kbs or below and yield synthetic quality. Deending uon the methods of extracting the model arameters, several different tyes of vocoders have been develoed, viz, channel vocoder, homomorhic vocoder, formant vocoder, linear rediction vocoder [6, 45]...3 Hybrid Coders Hybrid coders attemt to fill the ga between waveform and arametric coders. Waveform coders are caable of roviding good quality seech at bit-rates around 6 kbs; on the other hand, vocoders oerate at very low bit-rates (.4 kbs and below) but cannot rovide natural quality. Although other forms of hybrid coders exist, the most successful and commonly used are time domain Analysis-by-Synthesis (AbS) coders. Such coders use the same linear rediction filter model of the vocal tract as found in LPC vocoders. However, instead of alying a simle two-state voiced/unvoiced model to find the necessary inut to this filter, the excitation signal is chosen by matching to match the reconstructed seech waveform as closely as ossible to the original seech waveform. A general model for AbS coders is shown in Fig..5. AbS coders were first introduced in 98 by Atal and Remde with what was to become known as the Multi-Pulse Excited (MPE) coder. Later the Regular- Pulse Excited (RPE), and the Code-Excited Linear Predictive (CELP) coders were introduced. Many variations of CELP coders have been standardized, including [7, 3] G.73. oerating at 6.3/5.3 kbs, G.79 oerating at 8 kbs, G.78 a low delay coder oerating at 6 kbs, and all the digital mobile telehony encoding standards including [7, 8, ] GSM, IS-54, IS-95, and IS-36. The waveform interolation coder that will be discussed in the subsequent chaters is also a hybrid coder.

12 Figure.4 The source-filter model of seech roduction used by vocoders...4 Performance criteria of seech coding There are different dimensions of erformance of seech coders. To judge a articular seech coder certain erformance criteria should be considered. Some of the major erformance asects of seech coders are discussed below: One of the major criteria is seech quality. Seech coders tend to roduce the least audible distortion at a given bit rate. Naturalness and intelligibility of the roduced sounds are imortant and desired criteria. The seech quality can be determined by listening tests, which comute the mean oinion of the listeners. The quality of seech can also be determined in some cases in terms of the objective measures such as SNR, rediction gain, log sectral distortion. Seech coders strive to make the decoded or synthesized seech signal as close as ossible to the original signal. Another imortant issue is bit rate. The bit rate of the encoder is the number of bits er second the encoder needs to transmit. The objective of the coding algorithm is to reduce the bit rate but maintain the high quality of seech. Seech coding algorithms are tyically imlemented on DSP chis. These chis have limited memory (RAM) and seed (MIPS-million instructions er second). Consequently, seech coding algorithms should not be so comlex that their requirements exceed the caacity of modern DSP chis.

13 Figure.5 Analysis-by-Synthesis (Abs) coder structure. (a) encoder and (b) decoder. Often, seech coding algorithms rocess a grou of samles together. If the number of samles is too large, it introduces an additional delay between the original and the coded seech. This is undesirable in the case of real time transmission, but it is tolerable to a larger extent in the case of voice storage and layback. Bandwidth of the seech signal that needs to be encoded is also an issue which should be taken into account. Tyical telehony requires Hz bandwidth. Wideband seech coding techniques (useful for audio transmission, tele-conferencing and tele-teaching) require 7 0 khz bandwidth. The seech coding algorithms must be robust against channel errors. Channel errors are caused by channel noise, inter-symbol interference, signal fading. While seech signals are transmitted in real alications, they are distorted by different tyes of background acoustic noises such as street noise, car noise, and office noise. Seech coding algorithms should be caable of maintaining a good quality even in the resence of such background noises..3 Objective of this research Linear rediction analysis of seech is historically one of the most imortant seech analysis techniques. The traditional linear rediction exloits the redundancies of a seech signal by modelling the seech signal as a linear filter. The basis is the sourcefilter model where the filter is constrained to be an all-ole linear filter. The filter coefficients are derived in such a way that the energy at the outut of the filter is 3

14 minimized. However, the conventional linear rediction has some limitations. As we will see late, it cannot detect the formants very recisely. In this work, we will focus on a new method develoed by T. Bäckström and P. Alku [,, 4], the Weighted Sum of the Line Sectrum Pair (WLSP), which is based on the LSP decomosition. WLSP has good roerties and behaviours comaring the traditional linear rediction with the same order. We erform WLSP in both the clean and noisy seech. 4

15 Chater Linear Prediction During the ast two decades, LPC has become one of the most revalent techniques for seech analysis. In fact, this technique is the basis of all the sohisticated algorithms that are used for estimating seech arameters, e.g., itch, formants, sectra, vocal tract and low bit reresentations of seech. The basic rincile of linear rediction states that seech can be modelled as the outut of a linear, time-varying system excited by either eriodic ulses or random noise.. Linear rediction modelling The most general redictor form in linear rediction is the autoregressive moving average (ARMA) model where a seech samle s(n) is redicted from ast redicted seech samles s ( n ),, s( n ) with the addition of an excitation signal u(n) according to the following s ( n) = ak s( n i) + G blu( n l), b = 0, (.) k = q l= 0 where G is the gain factor for the inut seech and ak and bl are filter coefficients. Equivalently, in the frequency domain, the related transfer function H (z) is H (z) is also referred to a ole-zero model in which the olynomial roots of the denominator and the numerator are the oles and zeroes of the system, resectively. When a = 0 for k, H (z) becomes an all-zero or moving average (MA) k q l + bl z S( z) l= H ( z) = = G. (.) U ( z) k a z k = k 5

16 model since the outut is a weighted average of the q rior inuts. Conversely, when b l = 0 for l q, H (z) reduces to an all-ole or autoregressive (AR) model in which case the transfer function is simly as H ( z) = i= a k z k =. A( z) (.3) Figure. shows the two reresentations of all-ole model. u(n) H ( z) = G k = a k z k s(n) G u(n) k = a s( n k) k + - Σ s(n) Linear Predictor of Order Figure.. Two reresentations of discrete all-ole model. Estimation of linear rediction coefficients There are two widely used methods for estimating the LP coefficients: Autocorrelation. Covariance. Both methods choose the LP coefficients { a k } in such a way that the residual energy is minimized. The classical least square technique is used for that urose. In the following sections, we will go through the major issues related to comutation of LP. 6

17 .. Windowing Seech is a time varying signal. In signal analysis, one tyically assumes that the roerties of a signal usually change relatively slowly with time. This allows for short-term analysis of a signal. The signal is divided into successive segments, analysis is done on these segments and some dynamic arameters are extracted. The signal s(n) is multilied by a fixed length analysis window w(n) to extract a articular segment at a time. This is called windowing. The simlest analysis window is a rectangular window of length N:, 0 n N, w ( n) = (.4) 0, otherwise. A rectangular window has an abrut discontinuity at the edge in the time domain. As a result there are large side lobes and undesirable ringing elects in the frequency domain. To avoid this roblem, the Hamming window was used in this research. It is actually a raised cosine function: πn cos( ), w( n) = N 0, 0 n N, otherwise. (.5) There are also other tyes of taered windows, such as the Hanning, Blackman, Kaiser and the Bartlett window [6]... Autocorrelation method By using Hamming window with length N, we get a windowed seech segment s N (n), where s N ( n) = s( n) w( n). Then the energy in the residual signal is minimized. The residual energy E is defined as: E = n= e ( n) = n= ( s N ( n) a k s N ( n k)). (.6) The values { a } that minimize E are found by assigning the artial derivatives of E k with resect to { a } to zeroes. We get the following variables { a }: k k equations with unknown k = a s k N n= ( n i) s N ( n k) = s N n= ( n i) s N ( n), i. (.7) By introducing the autocorrelation function, noticing that the windowed seech signal s N ( n) = 0 outside the window w(n), N n= i R ( i) = s ( n) s ( n i), 0 i, (.8) N N 7

18 The equation (.7) becomes k = R( i k ) a = R( i), i. (.9) k The set of linear equations can be exressed in the following matrix form: R(0) R() M R( ) R() R(0) M R( ) L L O L R( ) a R( ) a M M R(0) a R() () R =. M R( ) (.0) Or Ra = r. (.) The resulting matrix R is a Toelitz matrix where all elements along a given diagonal are equal. This allows the linear equations to be solved by the Levinson-Durbin algorithm or the Schur algorithm [4]. Among the different variations of LP, the autocorrelation method of linear rediction is the most oular. In this method, a redictor (an FIR of order m) is determined by minimizing the square of the rediction error, the residual, over an infinite time interval [9]. Poularity of the conventional autocorrelation method of LP is exlained by its ability to comute with a reasonable comutational load a stable allole model for the seech sectrum, which is accurate enough for most alications when resented by a few arameters. The erformance of LP in modelling of the seech sectrum can be exlained by the autocorrelation function of the all-ole filter, which matches exactly the autocorrelation of the inut signal between 0 and m when the rediction order equals m []...3 Covariance method The covariance method is very similar to the autocorrelation method. The basic difference is the length of the analysis window. The covariance method windows the error signal instead of the original signal. The energy E of the windowed error signal is E = en n= ( n) = n= e ( n) w( n). (.) If we assign the artial derivative equations: E a k to zero, for k, we have the following k = φ ( i, k) a = φ( i,0), i, (.3) k 8

19 Where φ ( i, k) is the covariance function of s(n) which is defined as: (, ) φ i k = w( n) s( n i) s( n k). (.4) n= The equation above can be reresented in the following matrix form: φ(,) φ(,) M φ (,) φ(,) φ(,) M φ(,) L L O L φ(, ) a φ(, ) a M M φ(, ) a ϕ() () ϕ =. M ϕ ( ) (.5) Where ϕ ( i) = φ( i,0) for i =,, L,. Or φ a = ϕ (.6) φ is not a Toelitz matrix, but it is symmetric and ositive definite. The Levinson- Durbin algorithm cannot be used to solve these equations. These equations can be solved by using the decomosition method...4 Numerical solution of LP linear equations The following two sections discuss the two numerical methods to solve the linear equations in autocorrelation and covariance methods to get the LP coefficients. Levinson-Durbin rocedure: The correlation method In 947, N.Levinson [8] resented a recursive algorithm for solving a general set of Ra = r linear symmetric Toelitz equations. Later, in 96, Durbin [] imroved the Levinson recursive algorithm for the secial case in which the right-hand side of the Toelitz equations is a unit vector. Let a k (m) be the kth coefficient for a articular frame in the mth iteration. The Levinson recursive algorithm solves the following set of ordered equations recursively for m =,, K, : k( m) = R( m) m k = a m ( m) = k( m) a ( m ) R( m k), k a ( m) a ( m ) k( m) a ( m ), k m, k = k m k < ( k( m) ) E( ), E( m) = m (.7) (.8) (.9) (.0) 9

20 where initially E ( 0) = R(0) and a ( 0) = 0. At each iteration, the mth coefficient a m (m) for k =,, K, m describes the otimal mth order linear redictor; and the minimum error E(m) is reduced by a factor ( k( m) ). Since E(m) (squared error) is never negative, k( m) <. The details of the Levinson-Durbin (LD) algorithm are shown in the following Table. Table Levison-Durbin algorithm for solving a Toelitz system Inut: Predictor order, Autocorrelation coefficient R (0),K, R( ). ( ) ( ) Outut: LP coefficients a = a, K, a = a. E0 R(0) for i := to do k i R( i) ( i) i i k = E a i a ki for j := to i do a a + k a ( i ) k ( i) ( i ) ( i ) j j i i j end for E i k end for ( i ) Ei R( i k) Decomosition Method: the covariance method The decomosition method is generally used for solving the covariance equations. Due to the symmetric and ositive definite nature of the covariance matrix φ, it can be decomosed as (the Cholesky decomosition) T φ = CC, (.) Where C is a lower triangular matrix, the diagonal elements of which are all ositive. This equation can be written as φ( i, j) = j k = C( i, k) C( j, k). (.) Then we get C( i, j) = φ ( i, j) C( i, k) C( j, k), i > j, (.3) C( j, j) = j k = φ( j, j) j k = C 0 ( j, k). (.4)

21 These can be used to found the elements of the lower triangular matrix C. Solution for a can then be found by using forward elimination and backward substitution...5 Stability A causal all-ole filter is stable if all its oles lie inside the unit circle. The oles of H (z) are simly the roots of A(z), where H (z) and A(z) are defined in equation (.3). Using the autocorrelation method, if the coefficients R ( i) are ositive definite, the solution of the autocorrelation equation (.0) gives redictor arameters which guarantee that all the roots of A(z) lie inside the unit circle [3, 9]. In other words, the filter / A( z) is stable. However, using covariance method, the redictor arameters from the solution of the equation (.5) cannot in general be guaranteed to form a stable all-ole filter. The comuted filter tends to more stable as the number of signal samles N is increased. Comaring autocorrelation method and covariance method, the covariance method is quite general and can be used with no restrictions. The only roblem is that of stability of the resulting filter, which is not a severe roblem generally. In the autocorrelation method, on the other hand, the filter is guaranteed to be stable, but the roblems of arameter accuracy can arise because of the necessity of windowing the time signal. This is usually a roblem if the signal is a ortion of an imulse resonse...6 Comuting the gains Levinson-Durbin's algorithm gives the gain as G = calculated as: E (0) / E ( ). This may also be G = a k r k (.5) r 0 k =.3 Line sectral air The Line Sectrum Pair (LSP) decomosition was first introduced by Itakura in 975 []. It is mainly used as a convenient reresentation of Linear Prediction Coding (LPC). There are also some other reresentations of LP arameters, such as reflection coefficients (RC), autocorrelations (AC), log area ratios (LAR), arcsine of reflection coefficients (ASRC), imulse resonse of LP synthesis filter (IR). The Line Sectrum Pair (LSP) decomosition has advantageous roerties over others [4]. In this technique, the minimum hase redictor olynomial comuted by the autocorrelation method of linear rediction is slit into a symmetric and an antisymmetric olynomial. It has been roved that the roots of these two olynomials, the LSPs, are located interlaced on the unit circle, if the original LP redictor is minimum hase

22 [9]. Furthermore, it has been shown that LSPs behave well when interolated [5]. Due to these roerties, the LSP decomosition has become the major technique in quantisation of LP information and it is used in various seech coding algorithms..3. Line sectral air olynomials Consider the conventional LP olynomial with order, i A ( z) = a z, (.6) i= 0 i Where a 0 =. Define two LSP olynomials P + ( z) and Q +( z) of order + as following: Figure. Demonstration of the unit circle roerty and the intra-model interlacing roerty of LSP. White circles deict the zeroes of the original LPC olynomial A(z), squares and filled circles are the zeroes of LSP olynomials P(z) and Q(z), resectively. The arrows show how the zeroes are transformed in LSP decomosition. Model order is m=8 [3]. P Q + + ( z) = A ( z) = A ( z) + z ( z) z A ( z A ( z ) ). (.7)

23 It is easy to see that LSP olynomials P + ( ) and Q ( ) are symmetric and antisymmetric, resectively; and trivially, z + z A = [ P + ( z) + Q ( z) ]. + (.8).3. The root roerties of LSP olynomials The LSP olynomials have the following roerties. Proerty (Unit circle roerty) Zeroes of the LSP olynomials P + ( ) and Q ( ) are on the unit circle. z + z Figure.3 Tyical root locus corresonding to a fourth-order minimum-hase A(z). The intersection oints ( k = ±) are the roots of LSP olynomials. That is, if and z, are zeroes of olynomials P + ( ) and Q + ( z), then z P i zq,, = i jβ Q, i z = e Q, i z P, i Q i i =. Therefore, the zeroes can be exressed as z = P and. We call theses angles α P,i and β Q, i Line Sectrum Frequencies (LSF). z jα, P, i e 3

24 Proerty (Intramodel interlacing roerty) Zeroes of the LSP olynomials P + ( z) and Q ( ) are searated and interlaced if the zeroes of olynomial (z) are inside + z the unit circle. Figure. demonstrates the first two roerties. Proerty 3 (Stability of reconstructed olynomial) If the zeroes of the LSP olynomials P + ( ) and Q ( ) are searated and interlaced on the unit circle, then z + z the zeroes of the reconstructed olynomials (z) A A are inside unit circle Proerty 4 (Intermodel interlacing roerty) The zeroes of the LSP olynomial P + ( ) are interlaced with the zeroes of the olynomial (z), and the zeroes of the z LSP olynomial Q ( ) are interlaced with the zeroes of the olynomial Q (z. + z ) Proerty 5 (Roots loci roerty) Consider P (z) and (z) as members of the family of olynomials Q P + ) F( z, k) = A ( z) + kz A ( z ( ) (.9) Where k is a real arameter. Then P (z) and Q (z) are F (z, and F( z, ), ) resectively. The roots loci is shown in Figure.3 and it was studied in []..3.3 Comuting line sectral frequencies It is easy to see that the LSP olynomials P + ( ) and Q + ( z) with even order have trivial zeroes at z z = and z =, resectively. While the order is odd, Q has trivial zeroes at z = ±, but P + ( z) has no trivial zeroes. Removing the roots at ± gives two new olynomials: ( + z ) G P + ( z), = P +, + z odd even G Q + ( z), = z Q +, z odd. even (.30) It is obvious that both G ( z ) and G ( z) have even order ( M and M ). So either jω olynomial can be rewritten as after substituting z = e : M M jiω jmω jm G( z) = gie = e ck cos( kω) e i= 0 k = 0 ω Y ( ω), (.3) 4

25 Where c 0 = and c g, k =, L, M. g M k = M k Soong and Juang [4] roosed a numeric method by converting Y (ω ) to a olynomial in x = cosω using multi-angle relations to find the roots of Y (ω ), while Kabal and Ramachandran used Chebyshev olynomials [3]. 5

26 Chater 3 Distortion measurements A distortion measure is an assignment of a nonnegative number to an inut/outut air of a system. The distortion between an inut or original and an outut or reroduction reresents the cost or distortion resulting when that inut is reroduced by that outut. To be useful, a distortion measure must ossess to a certain degree the following roerties [5, 6]: It must be subjectively meaningful in the sense that small and large distortion corresond to good and bad subjective quality, resectively; It must be tractable in the sense that it is amenable to mathematical analysis and leads to ractical design techniques; It must be comutable in the sense that the actual distortions resulting in a real system can be efficiently comuted. Distortion measures have a wide variety of alications in the design and comarison of systems. It lays an imortant role in seech coding. One use of the distortion measures is to evaluate the erformance of seech coding system. In this chater, we will summarize the most often used distortion measures. 3. Time domain measures The signal-to-noise ratio (SNR) and the segmental SNR (SNRseg) are the most common time-domain measures of the difference between original and coded seech signals. 6

27 Signal-to-Noise Ratio The signal-to-noise ratio (SNR) can be defined as the ratio between the inut signal ower and the noise ower, and is given in decibels (db) as: SNR = 0log 0 E E s e = 0log 0 n= n= s ( n) ) (in db). (3.) ( s ( n) s ( n)) Where s ) (n) is the coded version of the original seech samle s(n). The rincial benefit of the SNR quality measure is its mathematical simlicity. The measure reresents an average error over time and frequency for a rocessed signal. However, SNR is a oor estimator for a broad range of seech distortions. The fact that SNR is not articularly well related to any subjective attribute of seech quality and that it weights all time domain errors in the seech waveform equality, makes it a oor measure. Segmental Signal-to-Noise Ratio An imroved quality measure can be obtained if SNR is measured over short frames and the results are averaged. The frame-based measure is called the segmental SNR (SNRseg) and is formulated as: M m j s ( n) = = + SNRseg = 0log0 ) (in db). (3.) M j 0 n m N ( s( n) s( n)) j Where m0, m, L, m M are the end-times for the m frames, each of which is length N. The segmentation of the SNR ermits the objective measure to assign equal weights to loud and soft ortions of the seech. In some cases, roblems might arise with the segmental SNR measure if frames of silence are included, since large negative SNR values bias the overall measure of SNRseg. A threshold can be used to exclude any frames that contain unusually high or low SNR values. An extension to the segmental SNR is the frequency weighted segmental SNR measure and it can be used to match the listener s ercetion of quality. It has the following form: SNRfw = M M j= 0 s ( n) W ( j, i)0 log0 ) ( s( n) s( n)) W ( j, i) i i (in db). (3.3) Where the j is the segment index, i is the frequency band index and W ( j, i) is the frequency weight. 7

28 3. Sectral envelo distortion measures A sectral distortion measure is a function of two sectral densities, f and f ) for ) examle, which assigns a nonnegative number d( f, f ) to reresent the distortion in using f ) to reresent f. The most common such measures are difference distortion ) measures where one uses an L norm on the difference f f. These are metrics or ) ) distance in the sense that they satisfy a symmetry requirement d ( f, f ) = d( f, f ) and a triangular inequality: d ( f, g) d( f, h) + d( h, g). (3.4) Usually, the sectral distortion should measure the discreancies between the original and coded sectral enveloes that will lead to sounds being distinguished as honetically different. The disarities between the original and coded sectral enveloes include the following: Significantly different center frequencies for the resonaces or formants of the original and coded sectral enveloes. Alteration of the formant bandwidths caused by the coded sectral enveloes. The Log Sectral distortion measure The L norm-based log sectral distance measure is d π π SD = 0log0 S( ω) 0log0 S( ω) π ) dω (3.5) where the frequency magnitude sectrum S (ω ) is ( G G S ω ) = =. (3.6) jω A( e ) jnω a e G is the LP filter gain factor, and are { n= a n n } the LP coefficients. When =, we have the L norm or root mean square (rms) log sectral distortion measure. The rms log sectral distortion measure is defined in db as: d SD = w u w l wu wl 0 log 0 S( ) ) (in db). dω (3.7) S( ω) ω where w and w define the lower and uer frequency limits of integration. Ideally, wl l is equal to zero, and u wu corresonds to half the samling frequency. 8

29 The Itakura-Saito distortion measure The Itakura-Saito measure generally corresonds better to the ercetual quality of seech. Also known as likelihood ratio distance measure, it measures the energy ratio between the residual signal that results when using the quantized LP filter and the one that results when using the unquantized LP filter. It is defined as follows: d V ( ω [ e V ( ω) ] π ) IS = π π dω (3.8) where the log sectral difference V (ω ) between the two sectra is defined as ) V ( ω) = log S( ω) log S( ω). (3.9) Evaluating the integrals, this measure can be exressed as the olynomial d ) T ) G a Ra G = ) log (3.0) T G a Ra G IS ) ) ) ) K ) Where a = [, a, a,, ] T, and a [, a, a,, ] T a ) =, and R is the autocorrelation K matrix. When the gains are assumed to be equal, the Itakura-Saito measure is d IS ) a = a T T ) Ra. Ra a (3.) However, the Itakura-Saito measure is not symmetric. For symmetry, a modified Itakura measure can be used: d ) a = a T ) Ra a ) Ra a IS T T ) T ) Ra ). Ra (3.) A weighting term can be introduced to the Itakura-Saito measure to take advantage of the ercetual discrimination roerties of the human ear and is formulated as: d IS π ω V ( ω ) = ( ) ( ω) ω. W e e V d π π j [ ] jω Some weighting schemes W ( e ) are roosed in [9]. The Log-Area Ratio measure (3.3) The log-area ratio measure is based on the set of reflection coefficients and defined as: 9

30 ) k i ki d LAR = log log ) n= + ki + ki (3.4) with ki being the set of reflection coefficients and k ) i their quantized counterart. The Cestral distance The L cestral distance is defined as: d CD = n= ( c n ) c ) (3.5) n And is directly related to the rms log sectral distance: d CD = n= ( c n ) π ) (3.6) cn ) = log S( ω) log S( ω) dω. π π Using Parseval s equality and the fact that c n = c and c 0. n 0 = The log sectral distortion measure suffers from the drawback that Fourier transform and logarithm comutations are required for each oint in the summation. The cestral distance can be comuted efficiently by truncating the summation to a finite number terms N, usually three times the order of the LP analysis filter: d CD N ) = 0log0 e ( cn cn ) db. n= The introduction of a weighting term in the cestral distance has been investigated by several researchers: d CD N ) = 0log0 e wn ( cn cn ) db, (3.7) n= where the weighting term can be: w ( n) = n, called quefrency weighted cestral distance [9]. w ( n) = / v( n), v (n) being the variance of the cestral coefficients [9]. The Weighted Euclidean LSF distance measure The Euclidean distance measure between two vectors is simly as: ) ) T ) ) d( x, x) = ( x x) ( x x) = x x. (3.8) In general, we minimize this mean squared error between the unquantized and quantized vector to select the best codeword vector. Since LSF s have a direct 30

31 relationshi to the shae of the sectral enveloe, we associate them with this measure. The Euclidean distance measure allocates equal weights to individual comonents of the LSF vector. Sectral sensitivities can be taken into account with a weighted Euclidean distance, which is defined as: ) ) T ) d( x, x) = ( x x) W ( x x), (3.9) where W is an m m symmetric and ositive definite weighting matrix which may be deendent on x. If W is a diagonal matrix with elements w ii > 0, the distance can also be exressed as: m ) d( x, x) = i= w ( x ii i ) x ) i. (3.0) Some weighting schemes have been given by Paliwal and Atal [36], and Laroia et al [3]. 3

32 Chater 4 Weighted Sum of the Line Sectrum Pair Since the criterion of otimisation in the conventional LP analysis is the minimization of the residual energy, all the frequencies of the inut signal are treated equally. In other words, the all-ole model comuted by the conventional LP favors high-energy regions of signal sectrum regardless of at which frequencies these occur. This equal treating of the frequencies of the inut signal is inconsistent with, for examle, the roerties of human hearing, which is known to be frequency deendent (i.e, the sectral resolution decreases towards higher frequencies) []. Therefore, linear redictive methods have been develoed that utilize frequency selectivity of human hearing [40, 4]. The equal treating of the inut frequencies embedded in the conventional LP analysis is also inconsistent from the oint of view of seech roduction, because the most imortant formants, the first and second formant, are tyically located at frequencies below khz. Hence, both from the seech roduction s and ercetion s oint of view it would be desirable to obtain all-ole models of seech with imroved resolution on the frequency range where the lowest two lowest formants are located rather than modelling high-energy regions over the entire frequency range of the inut signal (see Figure 4.). In this chater, we focus on the algorithm of the Weighted-sum of the Line Sectrum Pair (WLSP), which is a newly develoed linear redictive algorithm by T. Bäckström and P. Alku [,, 4]. WLSP yields an all-ole filter of order m to model the seech sectrum. In contrast to the conventional autocorrelation method of LP, WLSP takes advantage of the autocorrelation of the inut signal also beyond time index m in order to obtain a more accurate all-ole model for the seech sectrum. WLSP utilises the LSP decomosition in a manner different from that tyically used in seech coding: the LSP decomosition is not comuted in order to quantise the LP information but rather as a comutational tool, using which stable all-ole filters with 3

33 the roosed autocorrelation matching roerty are defined. WLSP method could distinguish these ercetually most imortant formants more accurately than conventional LP. Figure 4. Illustration of the failure of the conventional LP in modelling of the second formant (marked F). The FFT sectrum of the inut signal (vowel /a/) is deicted by thin line and the all-ole sectrum of the conventional LP (m = 6) is deicted by thick line. 4. Weighted Sum of the LSP Polynomials The conventional LP redictor of order m is given by m i m ( z) = + ai z. i= coefficient vector a = <ai> i=0 m, where a 0 =, can be solved from the normal equations Ra = σ [,0, 0] T, where the autocorrelation matrix R is defined as the exected value of correlation E{xx T }, and the residual energy is σ = a T Ra. ) (where a is the LP coefficient vector), the symmetric and antisymmetric LSP olynomials P(z) and Q(z) can be exressed equally in the matrix form as ) ) ) ) = a + q = a, (4.) By defining the zero extended LP coefficient vector a = [ a ] T, 0 T a R a R A The Where subscrit R denotes reversal of rows. 33

34 The WLSP method is based on the redictor olynomial D ( z,, which is defined as [35] D( z, = λp( z) + ( Q( z), (4.) that is, D ( z, is a weighted sum, with weighting arameter λ, of the LSP olynomials P(z) and Q(z) in the olynomial sace. It is worth noticing that in the oen interval λ (0,) olynomial D ( z, is minimum-hase. Further, setting λ = yields the original LP olynomial A m (z). Choosing the value of λ (0,) which minimises the residual energy yields the LP olynomial A ( z m+ ). In addition, in the oen interval λ (0,) the autocorrelation function of the inverse of the redictor D ( z, fits exactly the first m- values of the autocorrelation of the inut [,, 3, 4]. Residual of D ( z, T For the zero extended LP coefficient vector = [ a, 0] T Ra ) = [ σ,0,,0, γ ] K T where m γ = a R( m + ) = yields thus i i i 0 a ), we have. The symmetric LSP olynomial σ + γ 0 ) ) R ( a + a R ) = M, (4.3) 0 σ + γ and the antisymmetric LSP olynomial yields similarly ) ) R( a a R ) = T [ σ γ 0 L 0 γ σ ]. Polynomial D ( z, can be written in vector form as and we have d( = λ + ( q, (4.4) σ + (λ ) γ 0 Rd ( = M. 0 (λ ) σ + γ (4.5) From this equation, we can write the following exression for the residual energy T T [ ( n) ] E[ d ( xx d( ] E e T = = d( Rd( = σ + (λ ) γ + (λ ) σ. 34 (4.6)

35 Vector d( corresonds to the LP redictor of order m + if λ is chosen such that the last row in Eq. 4.6 becomes equal to zero, that is, = γ λ σ + This is, in fact, one iteration ste of the slit Levinson-Durbin recursion [, 8]. (4.7) Incrementing the redictor order in LP decreases the energy of the residual. Thus, the residual energy for the model order m should be greater than or equal to the residual energy for the model order m + (Eq. 7), that is, σ σ + (λ ) γ + (λ ) σ (4.8) Substituting γ extracted from Eq. 4.8 yields 4 ( λ, which holds true when λ is chosen such that + λ,. That is, if λ is chosen within these limits, the residual energy given by D ( z, (of order m + ) is smaller than the residual energy given by the LP redictor of order m. Autocorrelation of D ( z, Given an autocorrelation R( i) (0 i m) of an inut signal, each ossible value of R( m +) (ossible values are such that matrix R is invertible) has a corresonding value of λ. That is, if λ is chosen such that it obeys Eq. 4.8 and γ is calculated with a given value of R ( m +), then D ( z, is the LP redictor of order m +and all autocorrelation values will be exactly fitted u to m +. If λ is chosen otherwise, the model then matches some other autocorrelation R ) ) ( i), 0 i m + where R(i) = R(i) for 0 i m. In other words, D ( z, always matches autocorrelation values on range 0 i m. Root tracks of D ( z, The LSP olynomials P (z) and Q (z) are symmetric and antisymmetric, resectively, and thus form an orthogonal basis in the olynomial sace. Polynomial D( z, is a linear combination, and thus a continuous transformation in the sace sanned by P (z) and Q (z). We can readily rove that also the roots of olynomial D ( z, follow continuous tracks in the Z-lane as functions of λ. The olynomial root sace of D ( z, for λ R is shown in Figure

36 Figure 4. Root tracks of D ( z, for λ R, an examle with m = 0. Small circles inside the unit circle corresond to the roots of the LP olynomial A m (z), and small circles outside the unit circle corresond to their mirror image artners (i.e. roots of A m ( z ) ). The LSP olynomials roots are at the intersections of the unit circle and the D ( z, root tracks. Figure 4.3 Model erformance (m = 0) in the autocorrelation domain. Dotted line is the original autocorrelation (male vowel /a/). Autocorrelation functions given by LP and WLSP, are denoted by dashed and solid line, resectively. The arrow indicates the interval used in the otimisation of λ. 36

37 Aart from being continuous, the root tracks of D ( z, can be roved to form closed aths, that is, their ending oints coincide. In fact, when λ goes to infinity (either ositive or negative), then roots of D( z, will become roots of A m ( z ) (i.e., the mirror image artners of the roots of A m (z) ). That is, D( z, λp( z) + ( Q( z) lim = lim λ ± + λz λ ± + λz m Am ( z) + (λ ) z Am ( z ) = lim = z λ ± + λz m A ( z m ). (4.9) Concluding, olynomial D ( z, is well-behaved and in the root sace a continuous function of λ. The all-ole filter D ( z, is stable in the oen interval λ (0,) and otherwise unstable. Further, setting λ = yields the original LP olynomial A m (z) of γ order m- and λ = + yields the LP olynomial A m (z) of order m. σ 4. Algorithm of WLSP Given an inut signal x(n), defining a stable WLSP all-ole filter of order m comrises the following stages. (The time index of the autocorrelation function is denoted by i).. Calculate LP olynomial A ( z m ) for signal x(n) using conventional LP with the autocorrelation criterion.. Construct the LSP olynomials P(z) and Q(z) from A ( z m ). 3. Identify the first eak of the autocorrelation function of the inut signal beyond m. Define I as the osition of this eak. 4. Define an all-ole filter D ( z, of order m with additional arameter λ. Otimize the all-ole filter by searching for the value of λ that minimizes the absolute error between the autocorrelations of x(n) and D ( z, for m i I. (Notice that the autocorrelations of x(n) and D ( z, λ ) are equal for 0 i<m, when the order of D ( z, equals m.) See Figure 4.3 for illustration. 37

38 Chater 5 Exeriments In this chater, we aly WLSP to clean and noisy seech and comared it with the linear rediction. As a distortion measurement, rms log sectral distortion [5, 6] was used to analysis the behaviours of WLSP and conventional linear rediction. 5. WLSP with clear seech Conventional LP might result in oor modelling of those formants of wide-band seech that are close to each other at low frequencies, see Figure 4.. Therefore, we aimed at analysing whether the WLSP method could distinguish these ercetually most imortant formants more accurately. The Finish vowels /a/, /e/, /i/, /o/, /u/ and /y/ with five male and female seakers were studied by using linear rediction and WLSP. The two linear redictive analyses were comuted using a rediction order m = 0, a 0 ms Hamming window and a samling frequency of 8kHz. The sectra of vowels /a/, /e/, /i/, /o/, /u/ and /y/ are shown in Figure 5. to 5.6. Comaring the sectra of LP and WLSP for the Finnish male vowel /a/, as seen in Figure 5., we found that the WLSP detects the first four formants, but, the conventional linear rediction cannot find the formants as clear as WLSP, esecially for the second formant. The same henomenon haened when analysing Finnish male vowel /e/, /y/ and /i/ (see Figure ). Figure 5. shows the sectrum of the Finnish male vowel /o/, the conventional linear rediction cannot find the second formant while WLSP detects it. 38

39 Figure 5. The FFT sectrum of a male vowel /a/ together with all-ole sectra of LP and WLSP. The order is 0 and Hamming window is 0 ms. Figure 5. The FFT sectrum of a male vowel /o/ together with all-ole sectra of LP and WLSP. The order is 0 and Hamming window is 0 ms. 39

40 Figure 5.3 The FFT sectrum of a male vowel /e/ together with all-ole sectra of LP and WLSP. The order is 0 and Hamming window is 0 ms. Figure 5.4 The FFT sectrum of a male vowel /y/ together with all-ole sectra of LP and WLSP. The order is 0 and Hamming window is 0 ms. 40

41 Figure 5.5 The FFT sectrum of a male vowel /i/ together with all-ole sectra of LP and WLSP. The order is 0 and Hamming window is 0 ms. Figure 5.6 The FFT sectrum of a male vowel /u/ together with all-ole sectra of LP and WLSP. The order is 0 and Hamming window is 0 ms. 4