Quantization process

quantisation parade quantisation is a process of mapping an infinite present of scalar or transmitter quantities by a finite set of scalar or sender quantities. quantisation has applications in the argonas of signal processing, spoken communication processing and visualise processing. In speech cryptograph, quantisation is indispensable to reduce the list of bits use for representing a consume of speech signal there by the bit-rate, difficultness and retentivity requisite laughingstock be reduced. quantisation guides in the expiry in the superior of a speech signal, which is undesirable. So a agree must(prenominal) be made amongst the reduction in bit-rate and the quality of speech signal. Two types of quantisation proficiencys embody they ar scalar quantization and sender quantization. Scalar quantization deals with the quantization of samples on a sample by sample basis, while transmitter quantization deals with quantizing the samples in groups called transmitters. sender quantization summations the optimality of a quantizer at the monetary value of increased computational complexity and computer keeping requirements. Shannon theory states that quantizing a sender contri ande be more effective than quantizing individual scalar values in terms of ghostly strain. According to Shannon the place of a transmitter elect greatly affects the trans crookion of quantization. transmitters of greatheartedr belongings produce dampen quality when comp argond to senders of little property and in vectors of smaller dimension the transp bency in the quantization is not good at a ill-tempered bit-rate chosen 8. This is because in vectors of smaller dimension the correlation that exists in the midst of the samples go out be lost and the scalar quantization itself destroys the correlation that exists mingled with successive samples so the quality of the approximate speech signal depart be lost. Therefore, quantizing corre lated entropy requires proficiencys that withhold the correlation mingled with the samples, such a proficiency is the vector quantization technique (VQ). sender quantization is the simplification of scalar quantization. transmitters of larger dimension produce transp arncy in quantization at a peculiar(a) bit-rate chosen. In transmitter quantization the data is quantize in the skeleton of contiguous draw a blanks called vectors rather than individual samples. But later with the development of better mark techniques, it is made possible that transp atomic number 18ncy in quantization good deal also be achieved til now for vectors of smaller dimension. In this thesis quantization is performed on vectors of profuse length and on vectors of smaller dimensions for a inclined bit-rate 4, 50. An example of 2-dimensional vector quantizer is shown in flesh 4.1. The 2-dimensional division shown in physical body 4.1 is called as the voronoi region, which in turn contains sev eral bets of small hexangular regions. The hexagonal regions defined by the blue b secerns ar called as the en scratch regions. The green dots represent the vectors to be quantize which fall in disparate hexagonal regions and the red dots represent the commandmentwords (centroids). The vectors (green dots) travel in a particular hexagonal region quite a little be best represented by the codeword (red dot) falling in that hexagonal region 51-54. transmitter quantization technique has become a great tool with the development of non variational design algorithms identical the Linde, Buzo, Gray (LBG) algorithm. On the other hand besides phantasmal optical aberration the vector quantizer is having its own limitations handle the computational complexity and memory requirements needed for the researching and storing of the codebooks. For applications requiring risqueer(prenominal)(prenominal) bit-rates the computational complexity and memory requirements increases expone ntially. The block draw of a vector quantizer is shown in physique 4.2. permit be an N dimensional vector with truly cute samples in the range. The superscript T in the vector form-to doe withs the transpose of the vector. In vector quantization, a real valued N dimensional input vector is matched with the real valued N dimensional codewords of the codebook Ci , the codeword that best matches the input vector with net torturing is taken and the input vector is replaced by it. The codebook consists of a finite set of codewords C=Ci,, where , where C is the codebook, L is the length of the codebook and Ci denote the ith codeword in a codebook. In LPC coding the high bit-rate input vectors be replaced by the low bit-rate codewords of the codebook. The parameters apply for quantization are the line apparitional frequencies (LSF). The parameters apply in the analysis and synthesis of the speech signals are the LPC coefficients. In speech coding the quantization is not perform ed directly on the LPC coefficients, the quantization is performed by transforming the LPC coefficients to other forms which ensure come home perceptual constancy after quantization. another(prenominal) reason for not utilise LPC coefficients is that, LPC coefficients have a wide alive(p) range and so the LPC permeate easily becomes unstable after quantization. So LPC coefficients are not utilise for quantization. The alternative to LPC coefficients is the use of line unearthly oftenness (LSF) parameters which ensure filter stability after quantization. The filter stability buttocks be checked easily just by observing the pasture of the LSF samples in an LSF vector after quantization. If the LSF samples in a vector are in the ascending or descending differentiate the filter stability can be ensured otherwise the filter stability cannot be ensured 54-58. The angular positions of the roots of and gives us the line ghostlike frequencies and lapses in complex conjugate pai rs. The line spiritual frequencies range from. The line religious frequencies have the following properties All the roots of and must lie on the whole circle which is the required condition for stability. The roots of and are arranged in an alternate means on the unit circle i.e., The roots of equivalence (4.6) can be grasped employ the real root order 31 and is The coefficients of equatings (4.6) and (4.7) are symmetrical and so the order p of equations (4.6) and (4.7) get reduces to p/2. transmitter quantization of speech signals requires the multiplication of codebooks. The codebooks are intentional using an repetitious algorithm called Linde, Buzo and Gray (LBG) algorithm. The input to the LBG algorithm is a cultivation sequence. The planning sequence is the concatenation of a set LSF vectors chanceed from people of incompatible groups and of different ages. The speech signals use to obtain procreation sequence must be free of background noise. The speech signals utilize for this purpose can be recorded in sound proof booths, computing device rooms and establish environments. In this work the speech signals are recorded in computer rooms. In practice speech data bases like TIMIT database, YAHOO data base are available for use in speech coding and speech recognition. The codebook extension using LBG algorithm requires the contemporaries of an sign codebook, which is the centroid or look upon obtained from the training sequence. The centroid, so obtained is then classifyted into two centroids or codewords using the break awayting method. The iterative LBG algorithm splits these two codewords into four, four into eight and the process get out be act till the required material bodys of codewords in the codebook are obtained 59-61. The flow chart of LBG algorithm is shown in Fig 4.3. The LBG algorithm is properly implemented by a recursive procedure wedded below1. Initially the codebook multiplication requires a training sequence o f LSF parameters which will be the input to LBG algorithm. The training sequence is obtained from a set of speech samples recorded from different groups of people in a computer room.2. permit R be the region of the training sequence.3. Obtain an initial codebook from the training sequence, which is the centroid or mean of the training sequence and let the initial codebook be C.4. dismantle the initial codebook C into a set of codewords and where is the minimum mistake to be obtained in the midst of old and radical codewords.5. Compute the difference surrounded by the training sequence and individually of the codewords and and let the difference be D.6. set off the training sequence into two regions R1 and R2 depending on the difference D between the training sequence and the codewords and. The training vectors closer to falls in the region R1 and the training vectors closer to falls in the region R2.7. Let the training vectors falling in the region R1 be TV1 and the training sequence vectors falling in the region R2 be TV2.8. Obtain the new centroid or mean for TV1 and TV2. Let the new centroids be CR1 and CR2.9. Replace the old centroids and by the new centroids CR1 and CR2.10. Compute the difference between the training sequence and the new centroids CR1 and CR2 and Let the difference be .11. repeat steps 5 to 10 until .12. Repeat steps 4 to 11 till the required physique of codewords in the codebook are obtained. Where N=2b represents the number of codewords in the codebook and b represents the number of bits used for codebook generation. represents the difference between the training sequence and the old codewords, represents the difference between the training sequence and the new codewords. The quality of the speech signal is an Copernican parameter in speech coders and is surveyd in terms of spectral twirl measured in decibels (dB). The spectral distortion is measured between the LPC force out spectra of the quantized and unquantized speec h signals. The spectral distortion is measured mannequin wise and the average or mean of the spectral distortion careful over all tacks will be taken as the last-place value of the spectral distortion. For a quantizer to be transparent the mean of the spectral distortion must be less(prenominal) than 1 dB without for individually one(prenominal) audible distortion in the reconstructed speech. But the mean of the spectral distortion is not a sufficient measure to find the doing of a quantizer, this is because the military personnel ear is sensitive to large quantization errors that occur occasionally. So in addition to measuring the mean of the spectral distortion it is also necessary to have another measure of quality which is the share number of frames having a spectral distortion greater than 2dB and less than 4dB and the constituent number of frames having a spectral distortion greater than 4dB. The frames having spectral distortion between 2 to 4dB and greater than 4 dB are called as outlier frames 54. In order to measure objectively the distortion between the quantized and unquantized outputs, a method called the spectral distortion is often used in narrowband speech coding. For an ith frame the spectral distortion (in dB), is condition by equation (4.19). (4.19) Where and are the LPC powerfulness spectra of the unquantized and quantized ith frame respectively. The frequency f is in Hz and the frequency range is tending(p) by f1 and f2. The frequency range used in practice for narrowband speech coding is 0-4000 Hz 12, 33. The conditions for transparent speech coding are The average or mean of the spectral distortion (SD) must be less than or equal to 1dB. There must be no outlier frames having a spectral distortion greater than 4dB. The number of outlier frames between 2 to 4dB must be less than 2%.These three conditions are required to evaluate the murder of a quantizer. At a apt(p) bit-rate, an optimization process has to be carried out so as to obtain better cognitive operation i.e., accepting a large average spectral distortion for a few outliers. In the design of a vector quantizer instead of using the mean square offd error (MSE) distance measure the heavy LSF distance measurement is used. This is done to place emphasis on the low frequency LSFs and on the LSFs with higher power spectrum. The weights used can be of two types they are static or dynamic 54. unbending or Static weights These are used to place emphasis on the low frequency LSFs in order to account for the sensitivity of human ear for low and high frequency LSFs. Varying or energetic weights These are used to place emphasis on the LSFs with high power spectrum. There exist a number of vector quantization techniques distributively one is having its own advantages and wrongs. Each technique is developed to decrease the parameters like spectral distortion, computational complexity and memory requirements. The vector quantization techniques tha t exist are the change integrity vector quantization (SVQ) technique, Multi represent transmitter quantisation (MSVQ) technique, crack-Multi map vector quantisation (S-MSVQ) technique and Switched rupture sender Quantization (SSVQ) technique. As marketability and cost of a reaping depends on the complexity and memory requirements, the performance of the vector quantization techniques is measured in terms of spectral distortion in decibels, computational complexity in kilo flops per frame and memory requirements in floats. The performance of a vector quantization technique importantly depends on how efficiently the codebook is generated. The codebook can be generated efficiently using a large training set and using more number of bits for codebook generation. The goal abstruse in the design of for each one vector quantization technique is to benefit the technique to use more number of training vectors and less number of bits for codebook generation there by the spectral distortion, computational complexity and memory requirements can be reduced. It has been observed that as the number of bits used for codebook generation decreases the computational complexity and memory requirements decreases but the spectral distortion increases, this increase in spectral distortion can be reduced by increasing the number of training vectors used for codebook generation 62-71. The block draw of an unconstrained vector Quantizer (UVQ) is shown in Fig 4.4. unconstrained vector Quantization technique is the most awful vector quantization technique used for achieving lowest distortion at a given bit-rate and dimension. In LPC-10 the order of the filter chosen is 10 and so the length of each LSF vector will be 10. In Unconstrained transmitter Quantization technique the quantization is done on vectors of full length i.e., using 10 samples of an LSF vector. From Fig 4.4 S1, S2, S3Sn are the input LSF vectors to be quantized using the Unconstrained transmitter Quanti zer. The main advantage of this vector quantization technique is that it is expect to give lowest quantization distortion at a given bit-rate as the correlation that exists between the samples of a vector is preserved. But the disadvantage with this quantization technique is that as vectors of full length are used, at higher bit-rates the computational complexity and memory requirements increases in an exponential manner making it impractical for applications requiring higher bit-rates. Another problem with this quantization technique is that at higher bit-rates the size of the codebook will be large and the generation of the codebook for this type of quantization technique will be difficult on general purpose computers as the memory available with them is limited. So the number of training vectors used for codebook generation must be limited in number or the length of each vector must be reduced. In practice on general purpose computers the codebook cannot be generated even with t raining vectors less than the number of codewords in a codebook at higher bit-rates. But the number of training vectors required to generate the codebook must be large than the number of codewords in a codebook otherwise there will be too much over fitting of the training set 54. The computational complexity and memory requirements of a b bit, n dimensional vector quantizer are calculated as follows 54 To calculate the mean square error (MSE) between two vectors of n dimension, n subtractions, n multiplications and n-1 additions are required. So a total of 3n-1 flops are required. To search a codebook of 2b code vectors, (3n-1)2b flops are required in addition to the minimum distortion search requiring 2b-1 flops. So the number of computations made by a b bit, n dimensional vector quantizer is Total complexity = (3n-1)2b + 2b-1 = 3n2b-1 flops per vector. (4.24)In the computing the complexity each addition, multiplication and comparison is considered as one floating acid operation . So a b bit n dimensional vector quantizer requires a codebook of 2b code vectors, it needs to store n2b floating point values, it computes 3n2b 1 flops per vector. Instead of the mean square error distance measure if weighted distance measure is used in the design of a vector quantizer the complexity increases from 3n2b 1 to 4n2b 1 flops per vector. The computational complexity of an Unconstrained vector Quantizer is given by equation (4.25)Where n is the dimension of the vectorb is the number of bits allocated to the vector quantizer. The fund requirements of an Unconstrained Vector Quantizer is given by equation (4.26) Exhaustive search vector quantizers achieve lowest distortion at the expense of complexity and memory requirements at higher bit- rates. So to make the vector quantizers more practical for vectors of larger dimension and higher bit-rates structural constraints are imposed on the design of a vector quantizer or codebook. One way of achieving this is to decompo se the codebook into a Cartesian product of smaller codebooks i.e., C = C1 * C2 * C3 . ..*Cm. The advantage with smaller codebooks is that the computational complexity and memory requirements can be reduced to a very great extent. This is because the number of bits used for codebook generation will be split up among the sets of the decomposed codebook 12, 18. Examples of product code vector quantization techniques are Split Vector Quantization (SVQ), Multi full point Vector Quantization (MSVQ), Split-Multi typify Vector Quantization (S-MSVQ), Switched Split Vector Quantization (SSVQ). In this thesis two product code vector quantization techniques are proposed they are Switched Multi point in time Vector Quantization (SWMSVQ) and Multi Switched Split Vector Quantization (MSSVQ) techniques 54, 72. The main disadvantage of Unconstrained Vector Quantizer is that the complexity, memory requirements are very high and the generation of codebook is a very difficult task as vectors of full length are used for quantization without any structural constraint. As a result more number of training vectors and bits cannot be used for codebook generation. With these constraints the quantizer cannot produce better quality quantized outputs. So to improve the performance of Unconstrained Vector Quantization technique a well know technique called Split Vector Quantization has been developed. The concept behind Split Vector Quantization is that, in it vectors of larger dimensions are splitted into vectors of smaller dimensions and the bits allocated to the quantizer are divided up among the splits (separate). Due to splitting the dimension of a vector gets fall hence more number of training vectors and bits are used for codebook generation. As a result the performance of quantization is increased, the complexity and memory requirements are reduced. But the main disadvantage with this technique is that, due to splitting the additive and non linear dependencies that exist betwee n the samples of a vector will be lost and the shape of the quantizer cells will be affected. As a result the spectral distortion increases slightly. This increase in spectral distortion can be compensated by increasing the number of training vectors and using more number of bits for codebook generation. The number of splits in this type of quantizer must be limited in number otherwise the vector quantizer will act as a scalar quantizer. In Split Vector Quantization the training sequence used for codebook generation will also be splitted into vectors of smaller dimension and each split of the training sequence is used to generate separate sub codebooks, there by independent vector quantizers exist and the bits must be allocated to each of them. As a result less number of bits will be available at each quantizer, the computational complexity and memory requirements gets reduced as they depend on the number of bits allocated to the quantizer and on the dimension of the vector to be q uantized. The block draw of a three part Split vector quantizer is shown in Fig 4.5. From Fig 4.5 it can be observed that a vector S1 of dimension n is quantized by splitting it into sub-vectors S11, S12, S13 of smaller dimensions. Each of these sub-vectors are quantized using their respective codebooks. In this work the order of the filter is taken as 10 and so the LSF vector contain 10 samples and these 10 samples are splitted into three parts of 3, 3, 4 samples 54, 73-75. From results of Split Vector Quantization technique it is proved that the computational complexity and memory requirements gets lessen when compared to Unconstrained Vector Quantization technique. So Split Vector Quantization technique is superior to Unconstrained Vector Quantization technique in terms of the computational complexity and memory requirements. In a Split Vector Quantizer of n dimension, SP splits, operating at b bits per vector. The vector dummy Rn will be splitted into SP subspaces or splits or parts of bring down dimension then the dimension of each subspace will be and . The number of independent quantizers will be equal to the number of splits and the bits used for quantization are divided among the splits and is . Where is the number of bits allocated to each vector quantizer. The computational complexity of a Split Vector Quantizer is given by equation (4.27)Where ni is the dimension of a sub-vector in ith splitbi is the number of bits allocated to the ith split of a quantizersp is the number of splits. The Memory requirements of a Split Vector Quantizer is given by equation (4.28) Multi map Vector Quantization is a modification of Unconstrained Vector Quantization technique. It is also called as Multistep, Residual or Cascaded Vector Quantization. Multi ramification Vector Quantization (MSVQ) technique preserves all the features of Unconstrained Vector Quantization technique and decreases the computational complexity, memory requirements and spectral distortion w hen compared to it. When compared to Split Vector Quantization technique, Multi point in time Vector Quantization technique shows significant avail in the quality of the speech signal, by decreasing the spectral distortion, but at the expense of increased computational complexity and memory requirements. This is because Split Vector Quantization technique deals with vectors of lower dimensions while Unconstrained and Multi correspond Vector Quantization techniques deal with vectors of larger dimensions. So the complexity and memory requirements are less for Split Vector Quantization technique. Multi gift Vector Quantizer is a cascaded partnership of several vector quantizers, where the output of one stage is given as an input to the next stage and the bits used for quantization are divided among the stages connected in cascade 12, 14. As a result the computational complexity and memory requirements get reduced when compared to Unconstrained Vector Quantizer. The generation of code books at different stages of a three stage MSVQ is shown in Fig 4.6. From Fig 4.6 it can be observed that the codebook at the first stage is generated by taking the training sequence as an input. At the secant stage the codebook is generated using the quantization errors of the first stage, likewise the codebook at the troika stage is generated using the quantization errors of the second stage. This process is continued for the required number of stages 76-80. The block diagram of a three stage Multistage Vector Quantizer is shown in Fig 4.7. Its implementation requires the design of vector quantizers at each stage. In Multistage Vector Quantization the input vector s to be quantized is passed through the first stage of the vector quantizer so as to obtain the quantized version of the input vector i.e., . The quantization error or residual error at the first stage will be computed which is the difference of the input vector and the quantized version of the input vector. The qua ntization error at the first stage is given as an input to the vector quantizer of the second stage so as to obtain the quantized version of the error vector at the first stage i.e., . Likewise the quantization error at the second stage will be given as an input to the vector quantizer of the third stage so as to obtain the quantized version of the error vector at the second stage i.e., and this process can be continued for the required number of stages. Finally the decoder takes the indices Ii from each quantizer stage and adds the corresponding codewords to obtain the quantized version of the input vector i.e., 54. In a Multistage Vector Quantizer each stage acts as an independent vector quantizer and the total bits available for vector quantization will be divided among the stages. Then the complexity of a particular stage becomes, where bj is the number of bits allocated to the jth stage. This is less than the complexity of Unconstrained Vector Quantizer. Likewise the memory re quirements at each stage are , which is less than the memory requirements of Unconstrained Vector Quantizer. The computational complexity of a Multistage Vector Quantizer is given by equation (4.29)Where n is the dimension of the vectorbj is the number of bits allocated to the jth stage P is the number of stages The Memory requirements of a Multistage Vector Quantizer is given by equation (4.30) In order to improve the performance of Multistage Vector Quantization and Split Vector Quantization techniques a hybrid product code vector quantization technique called Split-Multistage Vector Quantization technique has been developed. Split-Multistage Vector Quantization technique is a hybrid of Multistage Vector Quantization and Split Vector Quantization techniques. At reasonable improvement in the quality of the output speech signal, Split-Multistage Vector Quantization technique provides the lowest spectral distortion, computational complexity and memory requirements when compared to Un constrained Vector Quantization, Multistage Vector Quantization and Split Vector Quantization techniques 73-80. The decrease in spectral distortion is due to summing of the quantized errors at each stage. In Split-Multistage Vector Quantization the dimension of the vectors to be quantized has been reduced by means of splitting. Likewise the bits used for quantization are also divided among the stages and among the splits of each stage. As a result the computational complexity and memory requirements get decreased when compared to Unconstrained Vector Quantization, Multistage Vector Quantization and Split Vector Quantization techniques. This is due to the decrease in the dimension of vectors, number of bits used for quantization at each stage and at each split of the vector quantizer. The generation of the codebooks at each stage of the Split-Multistage Vector Quantizer is similar to the codebooks generation at each stage of the Multistage Vector Quantizer. But the difference is tha t each stage of the Split-Multistage Vector Quantizer involves the generation of several sub codebooks. The number of sub codebooks generated at each stage is equal to the number of splits at that stage. In this work, Split-Multistage Vector Quantizer with three parts (splits) and three stages have been developed. The performance of quantization depends on the number of stages and on the number of splits at each stage. As the number of stages increases the quality of the quantized output can be increased, but there must be a limit on the number of stages and on the number of splits at each stage as the number of bits at each stage is limited. The assignation of the bits at each stage is shown in Table 4.1 and the allocation of bits to each split of a stage is shown in Table 4.2. From Tables 4.1 and 4.2 it can be observed that the minimum number of bits at each stage with three parts must be at least three. So with three parts (splits) and three stages, in Split-Multistage Vector Qu antizer the number of bits at a frame cannot be reduced below 9 bits. The block diagram of a Split-Multistage Vector Quantizer with three parts and three stages is shown in Fig 4.8. The block diagram is similar to three stage Multistage Vector Quantizer except for the splits at each stage. In Split-Multistage Vector Quantizer each split is treated as a separate vector quantizer and the vectors at each split are quantized independently. The quantization mechanism involved in Split-Multistage Vector Quantizer is similar to the quantization mechanism involved in Multistage Vector Quantizer, except that in Split-Multistage Vector Quantizer at each stage the sub-vectors are quantized independently. Split-Multistage Vector Quantizer is a hybrid of Split Vector Quantizer and Multistage Vector Quantizer. The equations for computational complexity and memory requirements are derived from the complexity and memory requirement equations of a Split Vector Quantizer and Multistage Vector Quanti zer. Equations (4.31) and (4.32) below are obtained from the equations (4.29) and (4.30) by including the splits (SP) at each stage with a summation term having limits from 1 to SP. The computational complexity of a Split-Multistage Vector Quantizer is given by equation (4.31)Wherenji is the dimension of a sub-vector in jth stage ith splitbji is the number of bits allocated to the jth stage and ith split of a quantizerP is the number of stagessp is the number of splits. The Memory requirements of a Split-Multistage Vector Quantizer is given by equation (4.32) Switched Split Vector Quantization (SSVQ) is one of the latest vector quantization techniques and is developed to improve the performance of Split Vector Quantization technique. Switched Split Vector Quantization technique is a hybrid of Switch Vector Quantization and Split Vector Quantization techniques and is used to exploit the linear and non linear dependencies that exist between the splits of a Split Vector Quantizer. In S witched Split Vector Quantizer initially the Switch Vector Quantizer partitions the entire vector space into voronoi regions and exploits the dependencies that exist across all dimensions of the vector space. Then a Split Vector Quantizer is designed for each of the voronoi regions. As a Split Vector Quantizer is adapted to the topical anesthetic statistics of the Voronoi region the sub optimalitys of the Split Vector Quantizer will be localized. In a Switched Split Vector Quantizer a number of vector quantizers are connected in parallel and it can be implemented in two ways hard finish scheme and soft decision scheme. In hard decision scheme each vector to be quantized is quantized in only one of the codebooks connected in parallel, the selection of a codebook for quantization depends on the nearest codeword selected in the initial codebook. An initial codebook is one which is designed for the selection of a switch. The initial codebook is generated by the training vectors used for the generation of the codebooks at the vector quantizers connected in parallel. The number of codewords or centroids in the initial codebook is equal to the number of switches chosen or number of codebooks connected in parallel and these centroids are used to form the Switch Vector Quantizer. In soft decision sc