Multimedia Systems Giorgio Leonardi A.A Lecture 4 -> 6 : Quantization
|
|
- Flora Underwood
- 6 years ago
- Views:
Transcription
1 Multimedia Systems Giorgio Leonardi A.A Lecture 4 -> 6 : Quantization
2 Overview Course page (D.I.R.): Consulting: Office hours by appointment: Office #182 (in front of Sala Seminari ) me any time
3 Outline (of the following lectures) Discrete signals: quantization Linear quantization Mid-riser quantization Mid-tread quantization Non-linear quantization µ-law A-law Quantization error, noise and proper quantization settings Vector quantization Storage of discrete digital signals: tradeoff between quality and space Reconstruction of discrete digital signals
4 Architecture of a A/D converter Sample And hold Quantizer Sampling clock An analog/digital converter is a device which samples the input signal at fixed time intervals and produces the corresponding digital version ADC is composed by: A low (or band) -pass filter, to filter out part of the analog signal s noise A clock, regulating the sampling intervals A quantizer, transforming into discrete values the sampled data
5 Architecture of a A/D converter We want to transform an analog signal into a discrete and digital signal Sampling is the step of transforming the analog signal into a discrete one, using a sampling frequency fs, properly chosen Quantization is the step of digitalizing the sampled values into digital codewords, representing discretized amplitude values The final output will be a digital signal, which is discretized both in time and amplitude
6 Quantizer As Sampling, quantization is a lossy operation: since amplitude values (real, continuous values) are quantized into discrete levels, the original values are lost Different types of quantizers exist: Scalar quantizers: Uniform (mid-reiser, mid-tread) Non-uniform (A-law, µ-law) Vector quantizers Adaptive quantizers
7 Scalar quantizers
8 Definition of a Scalar Quantizer Scalar Quantization maps a scalar input value x to a scalar value y q by a function Q: Where: y q = Q(x) q = 0, 1,2,, M 1 is called the quantization index M is the number of quantization levels y q is called the reconstruction (or quantizing) level Q is called the input-output function (or characteristic function)
9 Scalar quantization A scalar quantizer partitions the codomain of a signal into M subsets, called quantization regions Each interval I q, q= 0, 1,, M-1, is represented by: An integer quantization index q A quantization level, also called reconstruction level y q A binary codeword Quantization boundaries Reconstruction levels Quantization regions Binary codewords Quantization indexes
10 Scalar quantization The scalar quantizer processes one sample at time, and: For each sample, substitutes its value with the reconstruction level of the quantization region it falls in Usually, reconstruction levels are in the middle of the intervals Reconstruction levels Quant. indexes Binary codewords V 3V 1.5V -1.5V -3V -4.5V
11 Scalar quantization Reconstruction levels Quant. indexes Binary codewords V 3V 1.5V -1.5V -3V -4.5V Sampling and quanization will generate the following sequences: Reconstruction levels Quantization indexes Binary codewords
12 Scalar quantization (possible) reconstruction from samples: Reconstruction levels Quant. indexes Binary codewords V 3V 1.5V -1.5V -3V -4.5V
13 Scalar quantization Summarizing, given a value x of the input signal, scalar quantization maps it into a new value: y q =Q(x), if and only if x I q = [x q, x q+1 ) I q = [x q, x q+1 ), q = 0,, M 1 are M non-overlapping intervals The M values x q, q= 0,1,,M, are called (boundary) decision levels If the input signal is unbounded, x 0 = and x M =+ The length Δq = x q+1 x q of each I q is called step size
14 Scalar quantization For each input value x, the function Q performs three stages: 1. Classification: finds the value q such that x Iq 2. Reconstruction: given the classification q, maps x into the reconstruction level y q I q 3. Encoding: the reconstruction levels are encoded as binary numbers
15 Scalar quantization Classification: the amplitude range of the input signal is classified into the correct interval I q = [x q, xq +1 ), finding the correct value of q
16 Scalar quantization Reconstruction: each interval I q is represented by a reconstruction level y q I q which implements the mapping y q =Q(x), x I q
17 Scalar Quantization Encoding: each reconstruction level y q is represented by one of the M R-bits binary numbers the number of bits R is called quantization rate In case of fixed-length binary numbers, R = log 2 M
18 Quantization error Quantization causes loss of information: the reconstructed quantized value obtained applying Q(x) is different than the input x This difference is called quantization error e q (x) e q (x) = x Q(x) It is also referred to as quantization distortion or quantization noise
19 Quantization error In audio, it is commonly perceived as background noise
20 Wanna hear it?
21 Quantization error In images causes a contouring effect Original image Quantization at 2 levels Quantization at 4 levels Quantization at 8 levels
22 Quantization error Quantization error for a sample x generates two types of noise: Granular quantization noise: caused when x lies inside an interval I q Overload quantization noise: the value x lies over the designed quantization boundaries Reconstruction levels Quant. indexes Binary codewords Granular quant. noise Overload quant. noise V 3V 1.5V -1.5V -3V -4.5V
23 Overload quantization noise It is not always bad: it can be due to a bad design of the quantizer, but it could be a precise design choice: Bad design: the signal s main shape lies beyond the quantizing region Not necessarily a bad choice: the very few spots over the quantization region can be considered outliers
24 Example/Exercise For the following sequence: {1.2,-0.2,-0.5,0.4,0.89,1.3, 0.7} Quantize it using a quantizer dividing the range of (-1.5,1.5) with 4 equal levels, and write: the quantized sequence; the binary codewords; and the quantization error for each sample
25 Example/Exercise Binary codewords: Sequence to be quantized: {1.2,-0.2,-0.5,0.4,0.89,1.3, 0.7} Suggestion: 1.2 fall between 0.75 and 1.5, and hence is quantized to 1.125, with codeword 11 and quantization error: e1= x Q(x) = = 0.075
26 Types of scalar quantizers Scalar quantizers can be classified in: Uniform quantizers: all the intervals I q have the same size Non-uniform quantizers: The size of the intervals I q can be adapted to match the signal shape
27 Uniform quantizers
28 Uniform quantization For amplitude bounded signals with amplitude in the interval S= [X mix, X max ], S is split into M uniform non-overlapping intervals Iq = [x q, x q+1 ), q = 0,, M 1 Size of I q = Δ = Xmax Xmin M Q(x q+1 ) Q(x q ) = x q+1 x q, q = 0,, M 1
29 Quantization function Quantization function is shown as follows: Output level Q(x) Q(x) Reconstruction levels y q Input level x Xmin Xmax Tread Riser Decision boundaries: Iq= [ Δ 2, 3Δ 2 )
30 Uniform quantizers On the basis of the quantization function, different uniform quantizers exist. For signals ranging from [-Xmax, Xmax]: Mid-Riser Mid-Tread Special case for signals ranging from [0, Xmax]
31 Mid-Tread quantizer To be used when M is odd The output scale is centered on the value Q(0) = 0. It can code the value 0 in output
32 Mid-Tread quantizer Δ = Xmax Xmin M Q x = sgn x x Index q: q = sgn x x Reconstruction level y q : y q = q Binary code: binary code of q (in 2-complement) (ex. q= 2 010)(ex. q = )
33 Exercise: Mid-tread Quantization Given a signal g whose amplitude varies in the range: [Xmin, Xmax] = [ 1.5, 1.5] and the following signal values: {1.2, 0.2, 0.5,0.4,0.89,1.3} quantize it with 5 levels (3 bits) and write down: The index values The reconstruction levels The binary codes of the index values The quantization errors
34 Exercise: Mid-tread Quantization M = 5 Δ = Xmax Xmin M = 1.5 ( 1.5) 5 x= 1.2 Q x = y q = sgn x x = = = q = sgn x x = = Binary code: q=2 BC = 010 Quantization error eq= x Q(x) = = 0 = 0. 6 Index q Binary code Reconstruction level Boundary levels [ 1.5, 0.9) [ 0.9, 0.3) [-0.3, 0.3) [0.3, 0.9) [0.9, 1.5) =
35 Mid-Riser quantizer To be used when M is even NO decision level = 0 in output
36 Mid-Riser quantizer Δ = Q x = Xmax Xmin M x Index q: q = x Reconstruction level y q : y q = q Binary code: binary code of q (in 2-complement) (ex. q= 2 010)(ex. q = )
37 Exercise: Mid-riser Quantization Given a signal g whose amplitude varies in the range: [Xmin, Xmax] = [ 1.5, 1.5] and the following signal values: {1.2, 0.2, 0.5,0.4,0.89,1.3} quantize it with 6 levels (how many bits?) and write: The index values The reconstruction levels The binary codes of the index values The quantization errors
38 Exercise: Mid-riser Quantization M = 6 Δ = Xmax Xmin M = 1.5 ( 1.5) 6 = 0. 5 Index q Binary code Reconstruction level Boundary levels [ 1.5, 1) [ 1, 0. 5) [ 0.5, 0) [0, 0.5) [0.5, 1) [1, 1.5) x x= 1.2 Q x = y q = + 1 = = = = q = x = = 2 Binary code: q=2 BC = 010 Quantization error eq= x Q(x) = = 0.05
39 Special case for positive signals Q x Δ = Xmax M = x Index q: q = x Reconstruction level y q :y q = q Binary code: binary code of q (ex. q= 2 10)
40 How to choose the correct value of M?
41 Performance criteria Our goal is to minimize the overall difference between the quantized samples and the original (continuous) values. We must find a measure which can predict a correct value of M on the basis of the signal properties Common performance criteria: Mean-square error (MSE) Signal-to-noise ratio (SNR)
42 Mean-square error Quantifies the difference between the values implied by the M-levels quantizer Q(x), and the real values x, in the interval [x i, x i+1 ]: MSEq = M 1 x i+1 i=0 e q x 2 f q x dx = M 1 i=0 x i x i+1 = x Q(x) 2 f q x dx x i F q (x) is the pdf (probability density function) of the error distribution Sometimes, MSE is referred to as average distortion D, with (x Q(x)) 2 being called distortion d(x)
43 Signal-to-noise ratio Measures the strength of the signal wrt the background noise Where: P signal and P noise are the RMS (Root Mean Square) power of the signal and the noise, respectively A signal and A noise are the RMS amplitude of the signal and the noise, respectively Usually expressed in decibels
44 Decibel The Decibel (db) is a logarithmic unit that indicates the ratio of a physical quantity P (usually power or intensity) relative to a given reference level P 0 Formally, a decibel is one tenth of a bel (B) 1 B = 10 db
45 Signal-to-noise ratio Signal-to-Noise Ratio (SNR or S/N) in decibels:
46 Signal-to-Quantization-Noise Ratio (SQNR): measure the strength of the sampled signal wrt the quantization error Where: δ G 2 is the variance of the input signal probability distribution with pdf f G and mean μ G MSE q is the mean-square quantization error
47 Signal-to-Quantization-Noise Ratio Signal-to-Quantization-Noise Ratio (SQNR) in decibels:
48 Choice of a good quantizer Given the properties introduced, a quantizer can be defined «good», if: its MSEq approaches to zero, or its SQNR is very high
49 Measures for uniform quantizers Uniformly distributed input We are going to analyze MSEq and SQNR of the noise generated by uniformly distributed input in uniform quantizers This noise has a «sawtooth» shape, with amplitude in the interval : e q = [- 2, + 2 ] Noise shape proper of uniform quantizers
50 MSEq for uniformly distributed input MSEq = M 1 i=0 x i+1 x i x Q(x) 2 f q x dx In case of uniformly distributed input, we know from literature that pdf is the following: Substituting, we have: We solve the integral using the formula:
51 MSEq for uniformly distributed input MSEq = 2 12
52 SQNR for uniformly distributed input In case of uniformly distributed signal, the pdf f g (x) has variance: Substituting, we have: M SQNR db = 10 log R = log 2 M M = 2 R = 20 log 10 M = 20 log 10 2 R = 20R log 10 2 SQNR db = 6. 02R
53 Choice of a good quiantizer
54 Sampling parameters for CD quality For what concerns sampling frequency: Add that the spectrum of signals a human can hear is approximately the range [20Hz, 20KHz] What frequency would you choose to sample the music, given these information?
55 Sampling parameters for CD quality For what concerns quantization bit depth, here is an excerpt from an audiophile book: Signal To Noise Ratio (SN or SNR) - measured in db, a measure of the ratio between the wanted signal - the music being recorded and played back - against unwanted noise introduced by the reproduction system - tape hiss, vinyl noise, turntable rumble, etc. For comfortable listening, say a good amplifier, this should be at least 100dB. Audio Cassette (AC) was typically around 40-50dB, perhaps 60dB using Dolby. By contrast, CD systems should be as good as an amplifier, around 100dB, although scratches or other damage to CDs and wear in the player can both cause jumps in the sound that are similar in effect to scratches on vinyl. What bit depth R would you choose to quantize music for a CD, given these information? And if you would quantize for audio cassette quality?
56 Non-uniform quantization
57 Uniform/Non-uniform Quantization
58 Non-uniform quantization Problems with uniform quantization Only optimal for uniformly distributed signal Real audio signals (speech and music) are more concentrated near zeros Human ear is more sensitive to quantization errors at small values Solution Using non-uniform quantization quantization interval is smaller near zero
59 Non-uniform quantizers Quantizing intervals are not uniform In general, Δ i Δ j for i j Suitable when the input signal is not uniformly distributed E.g., Gaussian distribution,...
60 Non-uniform quantizers A non-uniform quantizer can be designed as: A pre-defined function Q(x) written to map input and output using different quantization intervals, or: A Companding quantizer
61 Companding Quantization A compander consists of three stages: Compression Uniform quantization Expansion
62 Companding quantization Compression The input signal is compressed with a nonlinear characteristic C (e.g., C(x)=log(x)) Uniform Quantization The compressed signal is quantized with the uniform characteristc Q Expansion The uniformly quantized signal is expanded inversely with the expansion characteristic E=C 1
63 Companding quantization Compression Uniform quantization Expansion Resulting quantizer
64 Companding quantization The result of this process is a non-uniform quantizer: x Q n (x) = E(Q(C(x)))
65 Companding vs non-linear Q(x) Non-linear Q(x): Faster then companding: the function Q(x) is immediately applicable to the input x Design is not straightforward we need algorithms for adaptive quantization Companding quantizer: Not faster than non-linear Q(x) logarithmic functions must be applied to transform x But look-up tables can be built to speed-up calculations Design and implementation are simpler modular architecture, we just need to configure the function C(x), the rest is a classic uniform quantizer
66 Companding quantizer: the µ-law µ-law is a configurable compression function used in North America and Japan for telecommunications It generates different uniform quantizations by changing the value of µ (N.A. and Japan use µ= 255)
67 µ-law quantizer Compression: compress the sample value x with the formula: x Xmax log 1 + μ log 1 + μ y = C(x) = sgn(x) Uniform Quantization: quantize y with a uniform R-bits quantizer: y = Q(y) Expansion: transform back the value of y using the inverse formula: x = C 1 (y ) = sgn(y ) Xmax μ log 1+μ 10 Xmax y 1
68 Exercise For the following sequence {1.2,-0.2,-0.5,0.4,0.89,1.3 }, Quantize it using a µ -law quantizer (with µ = 9), in the range of [-1.5,1.5] with 4 levels and, and write: the quantized sequence the sequence of binary codes the quantization error for each sample Solution (indirect method): apply the inverse formula to the partition and reconstruction levels found for the uniform quantizer at 4 levels. Because the mu-law mapping is symmetric, we only need to find the inverse values for y = 0.375, 0.75, μ=9, Xmax=1.5, >0.1297, 0.75->0.3604, > Then quantize each sample using the above partition and reconstruction levels.
69 Exercise x = C 1 y = sgn(y ) Xmax 10 μ log 1+μ Xmax y For example: C -1 (-1.125)= sgn log = = = = 0.767
70 µ-law quantizer Uniform, M = 6 µ-law, M = 6; µ= 16
71 A-Law Companding quantizer A-Law is the compression function used in Europe for telecommunications When connecting international systems, A- Law becomes the standard over the µ-law It generates different uniform quantizations by changing the value of compression parameter A (in Europe, A= 87.7, or A= 87.6) Same principles as µ-law, using the following compression and decompression functions:
72 Exercise (for home!) For the following sequence {1.2,-0.2,-0.5,0.4,0.89,1.3 }, Quantize it using a A -law quantizer (with A = 87.7), in the range of [-1.5,1.5] with 4 levels and, and write: the quantized sequence the sequence of binary codes the quantization error for each sample Solution (indirect method): apply the inverse formula to the partition and reconstruction levels found for the uniform quantizer at 4 levels. Because the A-law mapping is symmetric, we only need to find the inverse values for y = 0.375, 0.75, Then quantize each sample using the above partition and reconstruction levels. A=87.7, Xmax= 1.5, >?, >?, >?
73 Adaptive Quantization
74 Adaptive quantization Adaptive quantization allows non-uniform quantizers to decrease the average distortion by assigning more levels to more probable regions. For given M and input pdf, we need to choose {x i } and {y i } to minimize the distortion
75 Lloyd-Max Scalar Quantizer Also known as pdfoptimized quantizer For given M, to reduce MSE (σ q 2 ), we want narrow regions when f(x) is high and wider regions when f(x) is low x 0 x 1 x 2 x 3 x 4 x 5 pdf Given M, the optimal bi and yi that minimize MSE satisfy the Lagrangian condition: σ q 2 regions σ q 2
76 Lloyd-Max Scalar Quantizer Solving the differential equations, we obtain the Lloyd-Max conditions: y i = x i x i 1 x i x i 1 xf x dx f x dx x i = y i + y i+1 2 x i-1 x i X i+1 Given {x i }, it is easy to calculate {y i } Given {y i }, it is easy to calculate {x i } Problem: How can we calculate {x i } and {y i } simultaneously? {x i } depends on {y i }; at the same time {y i } depends on {x i } Solution: iterative method
77 An iterative algorithm known as the Lloyd algorithm solves the problem by iteratively optimizing the encoder and decoder until both conditions are met with sufficient accuracy. The Lloyd Algorithm
78 Lloyd-Max Scalar Quantizer Input: threshold ε Output: values of {y i } and {x i } 1. Initialize all y i ; let j = 1; d 0 = + (distortion). 2. Update all decision levels: 3. Update all y i : y i = 4. Compute MSE (d j ): x i x i 1 x i x i 1 x i = y i + y i+1 2 xf x dx f x dx 5. If d j 1 d j < ε, stop. d j 1 otherwise, set j = j + 1; go to step 2
79 Adaptive quantization Adaptive quantization can be used to reduce the number of colors of an image to a finite number (e.g. 256 for GIF). 7 colors
80 Vector quantization
81 Vector quantization Quantization can be extended to more than one dimension: Scalar quantizer quantizes one sample at a time Vector quantizer quantizes a vector of N samples into a vector of N quantized values Useful when: Input signals show a strong correlation to the patterns stored into the quantizer vectors The sampled signal has codomain with cardinality > 1, e.g. R 3 Mostly used for: Color quantization: Quantize all colors appearing in an image to L (few) colors Image quantization: Quantize every NxN block into one of the L typical patterns (obtained through training).
82 Vector quantization The encoder is loaded with a Codebook: list of predetermined quantizer vectors, each one with an associated index The same Codebook is loaded in the decoder
83 Vector quantization Encoding: given an input vector of N consecutive signal samples, the encoder searches for the most similar vector in the codebook The corresponding index is stored as the output
84 Vector quantization Decoding: the decoder receives the encoded index, and retrieves the corresponding vector in the codebook as reconstruction levels
85 Vector quantization Performance of VQ is asymmetric: Encoding is a heavy process: not straightforward to find the closest vector in the codebook Decoding is very easy: just the retrieval of an indexed vector
86 Encoding performance Encoding: comparing an input vector I v, to all the vectors in the Codebook (V 1, M ), to find the closest one (1-NN) We must calculate the distance between I v and all the M Codebook vectors Distance measure: usually euclidean distance Given two vectors: N V1= <x 1, x 2,, x n > 2 V2= <y 1, y 2,, y n > d V 1, V 2 = x i y i i=1 This is also the Quantization Error!
87 Encoding performance Complexity - if the vectors contain N elements each, and the Codebook contains M vectors: The complexity of calculating the Euclidean Distance on 2 vectors is linear in N Θ(N) Calculating Euclidean Distance between Iv and all the M Codebook vectors is Θ(N*M)
88 Encoding performance But: Quantizing at R bits, means that we have M = 2 R Therefore, the encoder must perform the search of the most closest vector with time: Θ(N* 2 R ) The encoder performance is esponential in the number of bits R!
89 Decoder performance Decoder complexity: the decoder receives the Codebook index, therefore it can immediately retrieve the quantized vector: Θ(1) Many encoders/decoders for video, images and audio have asymmetric performance!!!
90 Quantization Codebook The simplest Codebook can be generated dividing the N-space in M equally shaped regions
91 Quantization Codebook Adaptive techniques can be applied to find the best subdivision of the N-dimensional space, to minimize the quantization error: Adaptive clustering of image samples/colors Generalized Lloyd algorithm
92 Exercise Given the following Codebook, with M = 4 (2 bits): INDEX X1 X2 X And given the following input vectors: <1, 4, 3> <2, 2, 2> <1, 2, 3> <5, 4, 2> Write: The sequence of quantization indexes The quantization error for each vector
93 Reconstruction: main techniques
94 Reconstruction Signal Decoder: Sometimes, it is necessary to convert a digital signal into an analog one: A PC sound card must do it in order to let sounds to be heard through speakers Signal decoders apply different techniques to reconstruct the original signal, using only the sampled and quantized data. The main techniques include: Zero-order hold Linear interpolation Ideal interpolation
95 Zero-order hold the value of the each sample y(n) is held constant for duration T: x(t) = y(n) for the time interval [nt, (n+1)t]
96 Linear interpolation Intuitively, this converter connects the samples with straight lines in the time interval [nt, (n+1)t] x(t)= segment connecting y(n) to y(n+1),
97 Ideal interpolation This converter calculates a smooth curve that passes through the samples The curve can be calculated with methods from numerical analysis. For example, Fourier polynomials or piecewise polynomials (spline)
Multimedia Communications. Scalar Quantization
Multimedia Communications Scalar Quantization Scalar Quantization In many lossy compression applications we want to represent source outputs using a small number of code words. Process of representing
More informationGaussian source Assumptions d = (x-y) 2, given D, find lower bound of I(X;Y)
Gaussian source Assumptions d = (x-y) 2, given D, find lower bound of I(X;Y) E{(X-Y) 2 } D
More informationThe information loss in quantization
The information loss in quantization The rough meaning of quantization in the frame of coding is representing numerical quantities with a finite set of symbols. The mapping between numbers, which are normally
More informationScalar and Vector Quantization. National Chiao Tung University Chun-Jen Tsai 11/06/2014
Scalar and Vector Quantization National Chiao Tung University Chun-Jen Tsai 11/06/014 Basic Concept of Quantization Quantization is the process of representing a large, possibly infinite, set of values
More informationPulse-Code Modulation (PCM) :
PCM & DPCM & DM 1 Pulse-Code Modulation (PCM) : In PCM each sample of the signal is quantized to one of the amplitude levels, where B is the number of bits used to represent each sample. The rate from
More informationPrinciples of Communications
Principles of Communications Weiyao Lin, PhD Shanghai Jiao Tong University Chapter 4: Analog-to-Digital Conversion Textbook: 7.1 7.4 2010/2011 Meixia Tao @ SJTU 1 Outline Analog signal Sampling Quantization
More informationCS578- Speech Signal Processing
CS578- Speech Signal Processing Lecture 7: Speech Coding Yannis Stylianou University of Crete, Computer Science Dept., Multimedia Informatics Lab yannis@csd.uoc.gr Univ. of Crete Outline 1 Introduction
More informationThe Secrets of Quantization. Nimrod Peleg Update: Sept. 2009
The Secrets of Quantization Nimrod Peleg Update: Sept. 2009 What is Quantization Representation of a large set of elements with a much smaller set is called quantization. The number of elements in the
More informationC.M. Liu Perceptual Signal Processing Lab College of Computer Science National Chiao-Tung University
Quantization C.M. Liu Perceptual Signal Processing Lab College of Computer Science National Chiao-Tung University http://www.csie.nctu.edu.tw/~cmliu/courses/compression/ Office: EC538 (03)5731877 cmliu@cs.nctu.edu.tw
More information7.1 Sampling and Reconstruction
Haberlesme Sistemlerine Giris (ELE 361) 6 Agustos 2017 TOBB Ekonomi ve Teknoloji Universitesi, Guz 2017-18 Dr. A. Melda Yuksel Turgut & Tolga Girici Lecture Notes Chapter 7 Analog to Digital Conversion
More informationCMPT 889: Lecture 3 Fundamentals of Digital Audio, Discrete-Time Signals
CMPT 889: Lecture 3 Fundamentals of Digital Audio, Discrete-Time Signals Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University October 6, 2005 1 Sound Sound waves are longitudinal
More informationVector Quantization. Institut Mines-Telecom. Marco Cagnazzo, MN910 Advanced Compression
Institut Mines-Telecom Vector Quantization Marco Cagnazzo, cagnazzo@telecom-paristech.fr MN910 Advanced Compression 2/66 19.01.18 Institut Mines-Telecom Vector Quantization Outline Gain-shape VQ 3/66 19.01.18
More informationExample: for source
Nonuniform scalar quantizer References: Sayood Chap. 9, Gersho and Gray, Chap.'s 5 and 6. The basic idea: For a nonuniform source density, put smaller cells and levels where the density is larger, thereby
More informationFinite Word Length Effects and Quantisation Noise. Professors A G Constantinides & L R Arnaut
Finite Word Length Effects and Quantisation Noise 1 Finite Word Length Effects Finite register lengths and A/D converters cause errors at different levels: (i) input: Input quantisation (ii) system: Coefficient
More informationEE368B Image and Video Compression
EE368B Image and Video Compression Homework Set #2 due Friday, October 20, 2000, 9 a.m. Introduction The Lloyd-Max quantizer is a scalar quantizer which can be seen as a special case of a vector quantizer
More informationQuantization 2.1 QUANTIZATION AND THE SOURCE ENCODER
2 Quantization After the introduction to image and video compression presented in Chapter 1, we now address several fundamental aspects of image and video compression in the remaining chapters of Section
More informationReview of Quantization. Quantization. Bring in Probability Distribution. L-level Quantization. Uniform partition
Review of Quantization UMCP ENEE631 Slides (created by M.Wu 004) Quantization UMCP ENEE631 Slides (created by M.Wu 001/004) L-level Quantization Minimize errors for this lossy process What L values to
More informationCommunication Engineering Prof. Surendra Prasad Department of Electrical Engineering Indian Institute of Technology, Delhi
Communication Engineering Prof. Surendra Prasad Department of Electrical Engineering Indian Institute of Technology, Delhi Lecture - 41 Pulse Code Modulation (PCM) So, if you remember we have been talking
More information4. Quantization and Data Compression. ECE 302 Spring 2012 Purdue University, School of ECE Prof. Ilya Pollak
4. Quantization and Data Compression ECE 32 Spring 22 Purdue University, School of ECE Prof. What is data compression? Reducing the file size without compromising the quality of the data stored in the
More informationDesign of Optimal Quantizers for Distributed Source Coding
Design of Optimal Quantizers for Distributed Source Coding David Rebollo-Monedero, Rui Zhang and Bernd Girod Information Systems Laboratory, Electrical Eng. Dept. Stanford University, Stanford, CA 94305
More informationat Some sort of quantization is necessary to represent continuous signals in digital form
Quantization at Some sort of quantization is necessary to represent continuous signals in digital form x(n 1,n ) x(t 1,tt ) D Sampler Quantizer x q (n 1,nn ) Digitizer (A/D) Quantization is also used for
More informationDigital Signal Processing 2/ Advanced Digital Signal Processing Lecture 3, SNR, non-linear Quantisation Gerald Schuller, TU Ilmenau
Digital Signal Processing 2/ Advanced Digital Signal Processing Lecture 3, SNR, non-linear Quantisation Gerald Schuller, TU Ilmenau What is our SNR if we have a sinusoidal signal? What is its pdf? Basically
More informationLecture 20: Quantization and Rate-Distortion
Lecture 20: Quantization and Rate-Distortion Quantization Introduction to rate-distortion theorem Dr. Yao Xie, ECE587, Information Theory, Duke University Approimating continuous signals... Dr. Yao Xie,
More informationETSF15 Analog/Digital. Stefan Höst
ETSF15 Analog/Digital Stefan Höst Physical layer Analog vs digital Sampling, quantisation, reconstruction Modulation Represent digital data in a continuous world Disturbances Noise and distortion Synchronization
More informationClass of waveform coders can be represented in this manner
Digital Speech Processing Lecture 15 Speech Coding Methods Based on Speech Waveform Representations ti and Speech Models Uniform and Non- Uniform Coding Methods 1 Analog-to-Digital Conversion (Sampling
More informationVector Quantization and Subband Coding
Vector Quantization and Subband Coding 18-796 ultimedia Communications: Coding, Systems, and Networking Prof. Tsuhan Chen tsuhan@ece.cmu.edu Vector Quantization 1 Vector Quantization (VQ) Each image block
More informationProyecto final de carrera
UPC-ETSETB Proyecto final de carrera A comparison of scalar and vector quantization of wavelet decomposed images Author : Albane Delos Adviser: Luis Torres 2 P a g e Table of contents Table of figures...
More informationVector Quantization Encoder Decoder Original Form image Minimize distortion Table Channel Image Vectors Look-up (X, X i ) X may be a block of l
Vector Quantization Encoder Decoder Original Image Form image Vectors X Minimize distortion k k Table X^ k Channel d(x, X^ Look-up i ) X may be a block of l m image or X=( r, g, b ), or a block of DCT
More informationCompression methods: the 1 st generation
Compression methods: the 1 st generation 1998-2017 Josef Pelikán CGG MFF UK Praha pepca@cgg.mff.cuni.cz http://cgg.mff.cuni.cz/~pepca/ Still1g 2017 Josef Pelikán, http://cgg.mff.cuni.cz/~pepca 1 / 32 Basic
More informationAudio /Video Signal Processing. Lecture 2, Quantization, SNR Gerald Schuller, TU Ilmenau
Audio /Video Signal Processing Lecture 2, Quantization, SNR Gerald Schuller, TU Ilmenau Quantization Signal to Noise Ratio (SNR). Assume we have a A/D converter with a quantizer with a certain number of
More informationDigital Signal Processing
COMP ENG 4TL4: Digital Signal Processing Notes for Lecture #3 Wednesday, September 10, 2003 1.4 Quantization Digital systems can only represent sample amplitudes with a finite set of prescribed values,
More informationCHAPTER 3. Transformed Vector Quantization with Orthogonal Polynomials Introduction Vector quantization
3.1. Introduction CHAPTER 3 Transformed Vector Quantization with Orthogonal Polynomials In the previous chapter, a new integer image coding technique based on orthogonal polynomials for monochrome images
More information1. Probability density function for speech samples. Gamma. Laplacian. 2. Coding paradigms. =(2X max /2 B ) for a B-bit quantizer Δ Δ Δ Δ Δ
Digital Speech Processing Lecture 16 Speech Coding Methods Based on Speech Waveform Representations and Speech Models Adaptive and Differential Coding 1 Speech Waveform Coding-Summary of Part 1 1. Probability
More informationSignals, Instruments, and Systems W5. Introduction to Signal Processing Sampling, Reconstruction, and Filters
Signals, Instruments, and Systems W5 Introduction to Signal Processing Sampling, Reconstruction, and Filters Acknowledgments Recapitulation of Key Concepts from the Last Lecture Dirac delta function (
More informationLab 4: Quantization, Oversampling, and Noise Shaping
Lab 4: Quantization, Oversampling, and Noise Shaping Due Friday 04/21/17 Overview: This assignment should be completed with your assigned lab partner(s). Each group must turn in a report composed using
More informationModule 3. Quantization and Coding. Version 2, ECE IIT, Kharagpur
Module Quantization and Coding ersion, ECE IIT, Kharagpur Lesson Logarithmic Pulse Code Modulation (Log PCM) and Companding ersion, ECE IIT, Kharagpur After reading this lesson, you will learn about: Reason
More informationE303: Communication Systems
E303: Communication Systems Professor A. Manikas Chair of Communications and Array Processing Imperial College London Principles of PCM Prof. A. Manikas (Imperial College) E303: Principles of PCM v.17
More informationSignal types. Signal characteristics: RMS, power, db Probability Density Function (PDF). Analogue-to-Digital Conversion (ADC).
Signal types. Signal characteristics:, power, db Probability Density Function (PDF). Analogue-to-Digital Conversion (ADC). Signal types Stationary (average properties don t vary with time) Deterministic
More informationChapter 10 Applications in Communications
Chapter 10 Applications in Communications School of Information Science and Engineering, SDU. 1/ 47 Introduction Some methods for digitizing analog waveforms: Pulse-code modulation (PCM) Differential PCM
More informationOptical Storage Technology. Error Correction
Optical Storage Technology Error Correction Introduction With analog audio, there is no opportunity for error correction. With digital audio, the nature of binary data lends itself to recovery in the event
More informationEE 230 Lecture 40. Data Converters. Amplitude Quantization. Quantization Noise
EE 230 Lecture 40 Data Converters Amplitude Quantization Quantization Noise Review from Last Time: Time Quantization Typical ADC Environment Review from Last Time: Time Quantization Analog Signal Reconstruction
More informationAudio Coding. Fundamentals Quantization Waveform Coding Subband Coding P NCTU/CSIE DSPLAB C.M..LIU
Audio Coding P.1 Fundamentals Quantization Waveform Coding Subband Coding 1. Fundamentals P.2 Introduction Data Redundancy Coding Redundancy Spatial/Temporal Redundancy Perceptual Redundancy Compression
More informationELEN E4810: Digital Signal Processing Topic 11: Continuous Signals. 1. Sampling and Reconstruction 2. Quantization
ELEN E4810: Digital Signal Processing Topic 11: Continuous Signals 1. Sampling and Reconstruction 2. Quantization 1 1. Sampling & Reconstruction DSP must interact with an analog world: A to D D to A x(t)
More informationQuantisation. Uniform Quantisation. Tcom 370: Principles of Data Communications University of Pennsylvania. Handout 5 c Santosh S.
Tcom 370: Principles of Data Communications Quantisation Handout 5 Quantisation involves a map of the real line into a discrete set of quantisation levels. Given a set of M quantisation levels {S 0, S
More informationAnalysis of Finite Wordlength Effects
Analysis of Finite Wordlength Effects Ideally, the system parameters along with the signal variables have infinite precision taing any value between and In practice, they can tae only discrete values within
More informationLecture 7 Predictive Coding & Quantization
Shujun LI (李树钧): INF-10845-20091 Multimedia Coding Lecture 7 Predictive Coding & Quantization June 3, 2009 Outline Predictive Coding Motion Estimation and Compensation Context-Based Coding Quantization
More informationQuantization. Introduction. Roadmap. Optimal Quantizer Uniform Quantizer Non Uniform Quantizer Rate Distorsion Theory. Source coding.
Roadmap Quantization Optimal Quantizer Uniform Quantizer Non Uniform Quantizer Rate Distorsion Theory Source coding 2 Introduction 4 1 Lossy coding Original source is discrete Lossless coding: bit rate
More informationIntroduction p. 1 Compression Techniques p. 3 Lossless Compression p. 4 Lossy Compression p. 5 Measures of Performance p. 5 Modeling and Coding p.
Preface p. xvii Introduction p. 1 Compression Techniques p. 3 Lossless Compression p. 4 Lossy Compression p. 5 Measures of Performance p. 5 Modeling and Coding p. 6 Summary p. 10 Projects and Problems
More informationChapter 2: Problem Solutions
Chapter 2: Problem Solutions Discrete Time Processing of Continuous Time Signals Sampling à Problem 2.1. Problem: Consider a sinusoidal signal and let us sample it at a frequency F s 2kHz. xt 3cos1000t
More informationECE521 week 3: 23/26 January 2017
ECE521 week 3: 23/26 January 2017 Outline Probabilistic interpretation of linear regression - Maximum likelihood estimation (MLE) - Maximum a posteriori (MAP) estimation Bias-variance trade-off Linear
More informationProc. of NCC 2010, Chennai, India
Proc. of NCC 2010, Chennai, India Trajectory and surface modeling of LSF for low rate speech coding M. Deepak and Preeti Rao Department of Electrical Engineering Indian Institute of Technology, Bombay
More informationEE-597 Notes Quantization
EE-597 Notes Quantization Phil Schniter June, 4 Quantization Given a continuous-time and continuous-amplitude signal (t, processing and storage by modern digital hardware requires discretization in both
More informationCODING SAMPLE DIFFERENCES ATTEMPT 1: NAIVE DIFFERENTIAL CODING
5 0 DPCM (Differential Pulse Code Modulation) Making scalar quantization work for a correlated source -- a sequential approach. Consider quantizing a slowly varying source (AR, Gauss, ρ =.95, σ 2 = 3.2).
More informationLecture 5 Channel Coding over Continuous Channels
Lecture 5 Channel Coding over Continuous Channels I-Hsiang Wang Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw November 14, 2014 1 / 34 I-Hsiang Wang NIT Lecture 5 From
More informationEE67I Multimedia Communication Systems
EE67I Multimedia Communication Systems Lecture 5: LOSSY COMPRESSION In these schemes, we tradeoff error for bitrate leading to distortion. Lossy compression represents a close approximation of an original
More informationDigital Image Processing Lectures 25 & 26
Lectures 25 & 26, Professor Department of Electrical and Computer Engineering Colorado State University Spring 2015 Area 4: Image Encoding and Compression Goal: To exploit the redundancies in the image
More informationNumber Representation and Waveform Quantization
1 Number Representation and Waveform Quantization 1 Introduction This lab presents two important concepts for working with digital signals. The first section discusses how numbers are stored in memory.
More informationTime-domain representations
Time-domain representations Speech Processing Tom Bäckström Aalto University Fall 2016 Basics of Signal Processing in the Time-domain Time-domain signals Before we can describe speech signals or modelling
More informationVID3: Sampling and Quantization
Video Transmission VID3: Sampling and Quantization By Prof. Gregory D. Durgin copyright 2009 all rights reserved Claude E. Shannon (1916-2001) Mathematician and Electrical Engineer Worked for Bell Labs
More informationImage Compression. Fundamentals: Coding redundancy. The gray level histogram of an image can reveal a great deal of information about the image
Fundamentals: Coding redundancy The gray level histogram of an image can reveal a great deal of information about the image That probability (frequency) of occurrence of gray level r k is p(r k ), p n
More informationL. Yaroslavsky. Fundamentals of Digital Image Processing. Course
L. Yaroslavsky. Fundamentals of Digital Image Processing. Course 0555.330 Lec. 6. Principles of image coding The term image coding or image compression refers to processing image digital data aimed at
More informationOn Optimal Coding of Hidden Markov Sources
2014 Data Compression Conference On Optimal Coding of Hidden Markov Sources Mehdi Salehifar, Emrah Akyol, Kumar Viswanatha, and Kenneth Rose Department of Electrical and Computer Engineering University
More informationencoding without prediction) (Server) Quantization: Initial Data 0, 1, 2, Quantized Data 0, 1, 2, 3, 4, 8, 16, 32, 64, 128, 256
General Models for Compression / Decompression -they apply to symbols data, text, and to image but not video 1. Simplest model (Lossless ( encoding without prediction) (server) Signal Encode Transmit (client)
More informationRoundoff Noise in Digital Feedback Control Systems
Chapter 7 Roundoff Noise in Digital Feedback Control Systems Digital control systems are generally feedback systems. Within their feedback loops are parts that are analog and parts that are digital. At
More informationECE 521. Lecture 11 (not on midterm material) 13 February K-means clustering, Dimensionality reduction
ECE 521 Lecture 11 (not on midterm material) 13 February 2017 K-means clustering, Dimensionality reduction With thanks to Ruslan Salakhutdinov for an earlier version of the slides Overview K-means clustering
More informationSignal Modeling Techniques in Speech Recognition. Hassan A. Kingravi
Signal Modeling Techniques in Speech Recognition Hassan A. Kingravi Outline Introduction Spectral Shaping Spectral Analysis Parameter Transforms Statistical Modeling Discussion Conclusions 1: Introduction
More informationLecture 12. Block Diagram
Lecture 12 Goals Be able to encode using a linear block code Be able to decode a linear block code received over a binary symmetric channel or an additive white Gaussian channel XII-1 Block Diagram Data
More informationChapter 3. Quantization. 3.1 Scalar Quantizers
Chapter 3 Quantization As mentioned in the introduction, two operations are necessary to transform an analog waveform into a digital signal. The first action, sampling, consists of converting a continuous-time
More informationUNIT I INFORMATION THEORY. I k log 2
UNIT I INFORMATION THEORY Claude Shannon 1916-2001 Creator of Information Theory, lays the foundation for implementing logic in digital circuits as part of his Masters Thesis! (1939) and published a paper
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Lesson 7 Delta Modulation and DPCM Instructional Objectives At the end of this lesson, the students should be able to: 1. Describe a lossy predictive coding scheme.
More informationL11: Pattern recognition principles
L11: Pattern recognition principles Bayesian decision theory Statistical classifiers Dimensionality reduction Clustering This lecture is partly based on [Huang, Acero and Hon, 2001, ch. 4] Introduction
More informationMultimedia Networking ECE 599
Multimedia Networking ECE 599 Prof. Thinh Nguyen School of Electrical Engineering and Computer Science Based on lectures from B. Lee, B. Girod, and A. Mukherjee 1 Outline Digital Signal Representation
More informationHARMONIC VECTOR QUANTIZATION
HARMONIC VECTOR QUANTIZATION Volodya Grancharov, Sigurdur Sverrisson, Erik Norvell, Tomas Toftgård, Jonas Svedberg, and Harald Pobloth SMN, Ericsson Research, Ericsson AB 64 8, Stockholm, Sweden ABSTRACT
More informationEE 5345 Biomedical Instrumentation Lecture 12: slides
EE 5345 Biomedical Instrumentation Lecture 1: slides 4-6 Carlos E. Davila, Electrical Engineering Dept. Southern Methodist University slides can be viewed at: http:// www.seas.smu.edu/~cd/ee5345.html EE
More informationOptimization of Quantizer s Segment Treshold Using Spline Approximations for Optimal Compressor Function
Applied Mathematics, 1, 3, 143-1434 doi:1436/am1331 Published Online October 1 (http://wwwscirporg/ournal/am) Optimization of Quantizer s Segment Treshold Using Spline Approimations for Optimal Compressor
More informationFACULTY OF ENGINEERING MULTIMEDIA UNIVERSITY LAB SHEET
FACULTY OF ENGINEERING MULTIMEDIA UNIVERSITY LAB SHEET ETM 3136 Digital Communications Trimester 1 (2010/2011) DTL1: Pulse Code Modulation (PCM) Important Notes: Students MUST read this lab sheet before
More informationVarious signal sampling and reconstruction methods
Various signal sampling and reconstruction methods Rolands Shavelis, Modris Greitans 14 Dzerbenes str., Riga LV-1006, Latvia Contents Classical uniform sampling and reconstruction Advanced sampling and
More informationWhat does such a voltage signal look like? Signals are commonly observed and graphed as functions of time:
Objectives Upon completion of this module, you should be able to: understand uniform quantizers, including dynamic range and sources of error, represent numbers in two s complement binary form, assign
More informationLogarithmic quantisation of wavelet coefficients for improved texture classification performance
Logarithmic quantisation of wavelet coefficients for improved texture classification performance Author Busch, Andrew, W. Boles, Wageeh, Sridharan, Sridha Published 2004 Conference Title 2004 IEEE International
More informationCMPT 365 Multimedia Systems. Final Review - 1
CMPT 365 Multimedia Systems Final Review - 1 Spring 2017 CMPT365 Multimedia Systems 1 Outline Entropy Lossless Compression Shannon-Fano Coding Huffman Coding LZW Coding Arithmetic Coding Lossy Compression
More informationAnalysis of methods for speech signals quantization
INFOTEH-JAHORINA Vol. 14, March 2015. Analysis of methods for speech signals quantization Stefan Stojkov Mihajlo Pupin Institute, University of Belgrade Belgrade, Serbia e-mail: stefan.stojkov@pupin.rs
More information1. Quantization Signal to Noise Ratio (SNR).
Digital Signal Processing 2/ Advanced Digital Signal Processing Lecture 2, Quantization, SNR Gerald Schuller, TU Ilmenau 1. Quantization Signal to Noise Ratio (SNR). Assume we have a A/D converter with
More informationHigher-Order Σ Modulators and the Σ Toolbox
ECE37 Advanced Analog Circuits Higher-Order Σ Modulators and the Σ Toolbox Richard Schreier richard.schreier@analog.com NLCOTD: Dynamic Flip-Flop Standard CMOS version D CK Q Q Can the circuit be simplified?
More informationPCM Reference Chapter 12.1, Communication Systems, Carlson. PCM.1
PCM Reference Chapter 1.1, Communication Systems, Carlson. PCM.1 Pulse-code modulation (PCM) Pulse modulations use discrete time samples of analog signals the transmission is composed of analog information
More informationOptimal Multiple Description and Multiresolution Scalar Quantizer Design
Optimal ultiple Description and ultiresolution Scalar Quantizer Design ichelle Effros California Institute of Technology Abstract I present new algorithms for fixed-rate multiple description and multiresolution
More informationSeminar: D. Jeon, Energy-efficient Digital Signal Processing Hardware Design Mon Sept 22, 9:30-11:30am in 3316 EECS
EECS 452 Lecture 6 Today: Announcements: Rounding and quantization Analog to digital conversion Lab 3 starts next week Hw3 due on tuesday Project teaming meeting: today 7-9PM, Dow 3150 My new office hours:
More informationBeing edited by Prof. Sumana Gupta 1. only symmetric quantizers ie the input and output levels in the 3rd quadrant are negative
Being edited by Prof. Sumana Gupta 1 Quantization This involves representation the sampled data by a finite number of levels based on some criteria such as minimizing of the quantifier distortion. Quantizer
More informationMaximum Likelihood and Maximum A Posteriori Adaptation for Distributed Speaker Recognition Systems
Maximum Likelihood and Maximum A Posteriori Adaptation for Distributed Speaker Recognition Systems Chin-Hung Sit 1, Man-Wai Mak 1, and Sun-Yuan Kung 2 1 Center for Multimedia Signal Processing Dept. of
More informationEE123 Digital Signal Processing
EE123 Digital Signal Processing Lecture 19 Practical ADC/DAC Ideal Anti-Aliasing ADC A/D x c (t) Analog Anti-Aliasing Filter HLP(jΩ) sampler t = nt x[n] =x c (nt ) Quantizer 1 X c (j ) and s < 2 1 T X
More informationAnalog Digital Sampling & Discrete Time Discrete Values & Noise Digital-to-Analog Conversion Analog-to-Digital Conversion
Analog Digital Sampling & Discrete Time Discrete Values & Noise Digital-to-Analog Conversion Analog-to-Digital Conversion 6.082 Fall 2006 Analog Digital, Slide Plan: Mixed Signal Architecture volts bits
More informationOn Compression Encrypted Data part 2. Prof. Ja-Ling Wu The Graduate Institute of Networking and Multimedia National Taiwan University
On Compression Encrypted Data part 2 Prof. Ja-Ling Wu The Graduate Institute of Networking and Multimedia National Taiwan University 1 Brief Summary of Information-theoretic Prescription At a functional
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Processing Pro. Mark Fowler Note Set #14 Practical A-to-D Converters and D-to-A Converters Reading Assignment: Sect. 6.3 o Proakis & Manolakis 1/19 The irst step was to see that
More informationClassification: The rest of the story
U NIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN CS598 Machine Learning for Signal Processing Classification: The rest of the story 3 October 2017 Today s lecture Important things we haven t covered yet Fisher
More information6.003: Signals and Systems. Sampling and Quantization
6.003: Signals and Systems Sampling and Quantization December 1, 2009 Last Time: Sampling and Reconstruction Uniform sampling (sampling interval T ): x[n] = x(nt ) t n Impulse reconstruction: x p (t) =
More informationThis examination consists of 11 pages. Please check that you have a complete copy. Time: 2.5 hrs INSTRUCTIONS
THE UNIVERSITY OF BRITISH COLUMBIA Department of Electrical and Computer Engineering EECE 564 Detection and Estimation of Signals in Noise Final Examination 6 December 2006 This examination consists of
More informationESE 250: Digital Audio Basics. Week 4 February 5, The Frequency Domain. ESE Spring'13 DeHon, Kod, Kadric, Wilson-Shah
ESE 250: Digital Audio Basics Week 4 February 5, 2013 The Frequency Domain 1 Course Map 2 Musical Representation With this compact notation Could communicate a sound to pianist Much more compact than 44KHz
More informationAn Effective Method for Initialization of Lloyd Max s Algorithm of Optimal Scalar Quantization for Laplacian Source
INFORMATICA, 007, Vol. 18, No., 79 88 79 007 Institute of Mathematics and Informatics, Vilnius An Effective Method for Initialization of Lloyd Max s Algorithm of Optimal Scalar Quantization for Laplacian
More informationON SCALABLE CODING OF HIDDEN MARKOV SOURCES. Mehdi Salehifar, Tejaswi Nanjundaswamy, and Kenneth Rose
ON SCALABLE CODING OF HIDDEN MARKOV SOURCES Mehdi Salehifar, Tejaswi Nanjundaswamy, and Kenneth Rose Department of Electrical and Computer Engineering University of California, Santa Barbara, CA, 93106
More informationSPEECH ANALYSIS AND SYNTHESIS
16 Chapter 2 SPEECH ANALYSIS AND SYNTHESIS 2.1 INTRODUCTION: Speech signal analysis is used to characterize the spectral information of an input speech signal. Speech signal analysis [52-53] techniques
More informationMARKOV CHAINS A finite state Markov chain is a sequence of discrete cv s from a finite alphabet where is a pmf on and for
MARKOV CHAINS A finite state Markov chain is a sequence S 0,S 1,... of discrete cv s from a finite alphabet S where q 0 (s) is a pmf on S 0 and for n 1, Q(s s ) = Pr(S n =s S n 1 =s ) = Pr(S n =s S n 1
More information