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Speech ProcessingSpeech Coding
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*Veton Kpuska*Speech CodingDefinition:Speech Coding is a process
that leads to the representation of analog waveforms with sequences
of binary digits.Even though availability of high-bandwidth
communication channels has increased, speech coding for bit
reduction has retained its importance.Reduced bit-rates
transmissions required for cellular networksVoice over IPCoded
speech Is less sensitive than analog signals to transmission
noiseEasier to: protect against (bit) errorsEncryptMultiplex,
andPacketizeTypical Scenario depicted in next slide (Figure
12.1)
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*Veton Kpuska*Digital Telephone Communication System
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*Veton Kpuska*Categorization of Speech CodersWaveform
Coders:Used to quantize speech samples directly and operate at
high-bit rates in the range of 16-64 kbps (bps - bits per
second)Hybrid CodersAre partially waveform coders and partly speech
model-based coders and operate in the mid bit rate range of 2.4-16
kbps.VocodersLargely model-based and operate at a low bit rate
range of 1.2-4.8 kbps.Tend to be of lower quality than waveform and
hybrid coders.
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*Veton Kpuska*Quality MeasurementsQuality of coding can is
viewed as the closeness of the processed speech to the original
speech or some other desired speech waveform.NaturalnessDegree of
background artifactsIntelligibilitySpeaker identifiabilityEtc.
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*Veton Kpuska*Quality MeasurementsSubjective
Measurement:Diagnostic Rhyme Test (DRT) measures
intelligibility.Diagnostic Acceptability Measure and Mean Opinion
Score (MOS) test provide a more complete quality judgment.Objective
Measurement:Segmental Signal to Noise Ratio (SNR) average SNR over
a short-time segmentsArticulation Index relies on an average SNR
across frequency bands.
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*Veton Kpuska*Quality MeasurementsA more complete list and
definition of subjective and objective measures can be found
at:J.R. Deller, J.G. Proakis, and J.H.I Hansen, Discrete-Time
Processing of Speech, Macmillan Publishing Co., New York, NY,
1993S.R. Quackenbush, T.P. Barnwell, and M.A. Clements, Objective
Measures of Speech Quality. Prentice Hall, Englewood Cliffs, NJ.
1988
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Statistical Models
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*Veton Kpuska*Statistical ModelsSpeech waveform is viewed as a
random process.Various estimates are important from this
statistical perspective:Probability DensityMean, Variance and
Autocorrelation
One approach to estimate a probability density function (pdf) of
x[n] is through histogram.Count up the number of occurrences of the
value of each speech sample in different ranges: for many speech
samples over a long time duration.Normalize the area of the
resulting curve to unity.
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*Veton Kpuska*Statistical ModelsThe histogram of speech
(Davenport, Paez & Glisson) was shown to approximate a gamma
density: where x is the standard deviation of the pdf.Simpler
approximation is given by the Laplacian pdf of the form:
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*Veton Kpuska*PDF of Speech
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*Veton Kpuska*PDF of Modeled Speech
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*Veton Kpuska*PDF of SpeechKnowing pdf of speech is essential in
order to optimize the quantization process of analog/continous
valued samples.
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Quantization of Speech & Audio SignalsScalar
QuantizationVector Quantization
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*Veton Kpuska*Time Quantization (Sampling) of Analog
SignalsAnalog-to-Digital Conversion. Continuous Signal x(t).
Sampled signal with sampling period T satisfying Nyquist rate as
specified by Sampling Theorem. Digital sequence obtained after
sampling and quantization x[n]
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*Veton Kpuska*ExampleAssume that the input continuous-time
signal is pure periodic signal represented by the following
expression:
where A is amplitude of the signal, 0 is angular frequency in
radians per second (rad/sec), is phase in radians, and f0 is
frequency in cycles per second measured in Hertz (Hz).
Assuming that the continuous-time signal x(t) is sampled every T
seconds or alternatively with the sampling rate of fs=1/T, the
discrete-time signal x[n] representation obtained by t=nT will
be:
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*Veton Kpuska*Example (cont.)Alternative representation of
x[n]:
reveals additional properties of the discrete-time signal.
The F0= f0/fs defines normalized frequency, and 0 digital
frequency is defined in terms of normalized frequency:
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*Veton Kpuska*Reconstruction of Digital SignalsDigital-to-Analog
Conversion. Processed digital signal y[n]. Continuous signal
representation ya(nT). Low-pass filtered continuous signal
y(t).
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Scalar Quantization
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*Veton Kpuska*Conceptual Representation of ADC This conceptual
abstraction allows us to assume that the sequence is obtained with
infinite precession. Those values are scalar quantized to a set of
finite precision amplitudes denoted here by . Furthermore,
quantization allows that this finite-precision set of amplitudes to
be represented by corresponding set of (bit) patterns or symbols, .
x(t)
C/DQuantizer Coder
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Without loss of generality, it can be assumed that Input signals
cover finite range of values defined by minimal, xmin and maximal
values xmax respectively. This assumption in turn implies that The
set of symbols representing is finite.
Encoding:The process of representing finite set of values to a
finite set of symbols is know as encoding; performed by the coder,
as in Figure in previous slide. Mapping: Thus one can view
quantization and coding as a mapping of infinite precision value of
to a finite precision representation picked from a finite set of
symbols.
*Veton Kpuska*
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*Veton Kpuska*Scalar QuantizationQuantization, therefore, is a
mapping of a value x[n], xmin x xmax, to. The quantizer operator,
denoted by Q(x), is defined by:
where denotes one of L possible quantization levels where 1 i L
and xi represents one of L +1 decision levels.
The above expression is interpreted as follows; If , then x[n]
is quantized to the quantization level and is considered quantized
sample of x[n]. Clearly from the limited range of input values and
finite number of symbols it follows that quantization is
characterized by its quantization step size i defined by the
difference of two consecutive decision levels:
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*Veton Kpuska*Scalar QuantizationAssume that a sequence x[n] was
obtained from speech waveform that has been lowpass-filtered and
sampled at a suitable rate with infinite amplitude precision.x[n]
samples are quantized to a finite set of amplitudes denoted by
.Associated with the quantizer is a quantization step
size.Quantization allows the amplitudes to be represented by finite
set of bit patterns symbols.Encoding:Mapping of to a finite set of
symbols.This mapping yields a sequence of codewords denoted by c[n]
(Figure 12.3a in the next slide).Decoding Inverse process whereby
transmitted sequence of codewords c[n] is transformed back to a
sequence of quantized samples (Figure 12.3b in the next slide).
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*Veton Kpuska*Scalar Quantization
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*Veton Kpuska*FundamentalsAssume a signal amplitude is quantized
into M levels.Quantizer operator is denoted by Q(x); Thus
Where denotes L possible reconstruction levels quantization
levels, and1i Lxi denotes L +1 possible decision levels with 0i LIf
xi-1< x[n] < xi, then x[n] is quantized to the reconstruction
level is quantized sample of x[n].
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*Veton Kpuska*FundamentalsScalar Quantization Example:Assume
there L=4 reconstruction levels.Amplitude of the input signal x[n]
falls in the range of [0,1]Decision levels and Reconstruction
levels are equally spaced:Decision levels are
[0,1/4,1/2,3/4,1]Reconstruction levels assumed to be
[0,1/8,3/8,5/8,7/8]Figure 12.4 in the next slide.
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*Veton Kpuska*Example of Uniform 2-bit Quantizer
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*Veton Kpuska*ExampleAssume there are L = 24 = 16 reconstruction
levels. Assuming that input values fall within the range [xmin=-1,
xmax=1] and that the each value in this range is equally likely.
Decision levels and reconstruction levels are equally spaced; =i,=
(xmax- xmin)/L i=0, , L-1.,
Decision Levels:
Reconstruction Levels:
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*Veton Kpuska*16-4bit Level Quantization Example
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*Veton Kpuska*Uniform QuantizerA uniform quantizer is one whose
decision and reconstruction levels are uniformly spaced.
Specifically:
is the step size equal to the spacing between two consecutive
decision levels which is the same spacing between two consecutive
reconstruction levels.Each reconstruction level is attached a
symbol the codeword. Binary numbers typically used to represent the
quantized samples (as in Figure 12.4 in previous slide).
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*Veton Kpuska*Uniform QuantizerCodebook: Collection of
codewords.In general with B-bit binary codebook there are 2B
different quantization (or reconstruction) levels.Bit rate is
defined as the number of bits B per sample multiplied by sample
rate fs:I=Bfs
Decoder inverts the coder operation taking the codeword back to
a quantized amplitude value (e.g., 01 ).
Often the goal of speech coding/decoding is to maintain the bit
rate as low as possible while maintaining a required level of
quality.Because sampling rate is fixed for most applications this
goal implies that the bit rate be reduced by decreasing the number
of bits per sample
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*Veton Kpuska*Uniform QuantizerDesigning a uniform scalar
quantizer requires knowledge of the maximum value of the
sequence.Typically the range of the speech signal is expressed in
terms of the standard deviation of the signal.Specifically, it is
often assumed that: -4xx[n]4x where x is signals standard
deviation.Under the assumption that speech samples obey Laplacian
pdf there are approximately 0.35% of speech samples fall outside of
the range: -4xx[n]4x.Assume B-bit binary codebook 2B. Maximum
signal value xmax = 4x.
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*Veton Kpuska*Uniform QuantizerFor the uniform quantization step
size we get:
Quantization step size relates directly to the notion of
quantization noise.
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*Veton Kpuska*Quantization NoiseTwo classes of quantization
noise:Granular DistortionOverload DistortionGranular Distortion
x[n] un-quantized signal and e[n] is the quantization noise.For
given step size the magnitude of the quantization noise e[n] can be
no greater than /2, that is:
Figure 12.5 depicts this property were:
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*Veton Kpuska*Quantization Noise
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*Veton Kpuska*ExampleFor the periodic sine-wave signal use 3-bit
and 8-bit quantizer values. The input periodic signal is given with
the following expression:
MATLAB fix function is used to simulate quantization. The
following figure depicts the result of the analysis.
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*Veton Kpuska*L=23=8 & 28= 256 Levels Quantization Plot a)
represents sequence x[n] with infinite precision, b) represents
quantized version L=8, c) represents quantization error e[n] for
B=3 bits (L=8 quantization levels), and d) is quantization error
for B=8 bits (L=256 quantization levels).
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*Veton Kpuska*Quantization NoiseOverload DistortionMaximum-value
constant:xmax = 4x (4xx[n]4x)For Laplacian pdf, 0.35% of the speech
samples fall outside the range of the quantizer.Clipped samples
incur a quantization error in excess of /2.Due to the small number
of clipped samples it is common to neglect the infrequent large
errors in theoretical calculations.
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*Veton Kpuska*Quantization NoiseStatistical Model of
Quantization NoiseDesired approach in analyzing the quantization
error in numerous applications.Quantization error is considered an
ergodic white-noise random process.The autocorrelation function of
such a process is expressed as:
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*Veton Kpuska*Quantization ErrorPrevious expression states that
the process is uncorrelated.Furthermore, it is also assumed that
the quantization noise and the input signal are uncorrelated,
i.e.,E(x[n]e[n+m])=0, m.Final assumption is that the pdf of the
quantization noise is uniform over the quantization interval:
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*Veton Kpuska*Quantization ErrorStated assumptions are not
always valid.Consider a slowly varying linearly varying signal then
e[n] is also changing linearly and is signal dependent (see Figure
in the next slide).Correlated quantization noise can be
annoying.When quantization step is small then assumptions for the
noise being uncorrelated with itself and the signal are roughly
valid when the signal fluctuates rapidly among all quantization
levels.Quantization error approaches a white-noise process with an
impulsive autocorrelation and flat spectrum.
One can force e[n] to be white-noise and uncorrelated with x[n]
by adding white-noise to x[n] prior to quantization.
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*Veton Kpuska*Example of Quantization Error due to
CorrelationExample of slowly varying signal that causes
quantization error to be correlated. Plot represents sequence x[n]
with infinite precision, represents quantized version , represents
quantization error e[n] for B=3 bits (L=9 quantization levels), and
is quantization error for B=8 bits (L=256 quantization levels).
Note reduction in correlation level with increase of number of
quantization levels which implies degrease of step size .
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*Veton Kpuska*Quantization ErrorProcess of adding white noise is
known as Dithering.This decorrelation technique was shown to be
useful not only in improving the perceptual quality of the
quantization noise but also with image signals.Signal-to-Noise
RatioA measure to quantify severity of the quantization
noise.Relates the strength of the signal to the strength of the
quantization noise.
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*Veton Kpuska*Quantization ErrorSNR is defined as:
Given assumptions forQuantizer range: 2xmax, andQuantization
interval: = 2xmax/2B, for a B-bit quantizerUniform pdf, it can be
shown that:
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*Veton Kpuska*Quantization ErrorThus SNR can be expressed
as:
Or in decibels (dB) as:
Because xmax = 4x, thenSNR(dB)6B-7.2
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*Veton Kpuska*Quantization ErrorPresented quantization scheme is
called pulse code modulation (PCM).B-bits per sample are
transmitted as a codeword.Advantages of this scheme: It is
instantaneous (no coding delay)Independent of the signal content
(voice, music, etc.)Disadvantages:It requires minimum of 11 bits
per sample to achieve toll quality (equivalent to a typical
telephone quality) For 10,000 Hz sampling rate, the required bit
rate is: B=(11 bits/sample)x(10000 samples/sec)=110,000 bps=110
kbpsFor CD quality signal with sample rate of 20,000 Hz and
16-bits/sample, SNR(dB) =96-7.2=88.8 dB and bit rate of 320
kbps.
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*Veton Kpuska*Nonuniform QuantizationUniform quantization may
not be optimal (SNR can not be as small as possible for certain
number of decision and reconstruction levels)Consider for example
speech signal for which x[n] is much more likely to be in one
particular region than in other (low values occurring much more
often than the high values).This implies that decision and
reconstruction levels are not being utilized effectively with
uniform intervals over xmax.A Nonuniform quantization that is
optimal (in a least-squared error sense) for a particular pdf is
referred to as the Max quantizer.Example of a nonuniform quantizer
is given in the figure in the next slide.
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*Veton Kpuska*Nonuniform Quantization
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*Veton Kpuska*Nonuniform QuantizationMax QuantizerProblem
Definition: For a random variable x with a known pdf, find the set
of M quantizer levels that minimizes the quantization
error.Therefore, finding the decision and boundary levels xi and
xi, respectively, that minimizes the mean-squared error (MSE)
distortion measure:D=E[(x-x)2]E-denotes expected value and x is the
quantized version of x.
It turns out that optimal decision level xk is given by: ^^^
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*Veton Kpuska*Nonuniform QuantizationMax Quantizer (cont.)The
optimal reconstruction level xk is the centroid of px(x) over the
interval xk-1 x xk:
It is interpreted as the mean value of x over interval xk-1 x xk
for the normalized pdf p(x). Solving last two equations for xk and
xk is a nonlinear problem in these two variables.Iterative solution
which requires obtaining pdf (can be difficult).^~^
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*Veton Kpuska*Nonuniform Quantization
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CompandingA fixed non-uniform quantizer
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*Veton Kpuska*CompandingAlternative to the nonuniform quantizer
is companding.It is based on the fact that uniform quantizer is
optimal for a uniform pdf. Thus if a nonlinearity is applied to the
waveform x[n] to form a new sequence g[n] whose pdf is uniform
thenUniform quantizer can be applied to g[n] to obtain g[n], as
depicted in the Figure 12.10 in the next slide. ^
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*Veton Kpuska*Companding
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*Veton Kpuska*CompandingA number of other nonlinear
approximations nonlinear transformation that achieves uniform
density are used in practice which do not require pdf measurement.
Specifically and A-law and law companding.-law coding is give
by:
CCITT international standard coder at 64 kbps is an example
application of -law coding.-law transformation followed by 7-bit
uniform quantization giving toll quality speech. Equivalent quality
of straight uniform quantization achieved by 11 bits.
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Ulaw Input-Output Function*Veton Kpuska*
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Adaptive Coding
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*Veton Kpuska*Adaptive CodingNonuniform quantizers are optimal
for a long term pdf of speech signal.However, considering that
speech is a highly-time-varying signal, one has to question if a
single pdf derived from a long-time speech waveform is a reasonable
assumption.Changes in the speech waveform:Temporal and spectral
variations due to transitions from unvoiced to voiced speech,Rapid
volume changes.Approach: Estimate a short-time pdf derived over
20-40 msec intervals.Short-time pdf estimates are more accurately
described by a Gaussian pdf regardless of the speech class.
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*Veton Kpuska*Adaptive CodingA pdf (e.g., Gaussian) derived from
a short-time speech segment more accurately represents the speech
nonstationarity.One approach is to assume a pdf of a specific shape
in particular a Gaussian with unknown variance 2.Measure the local
variance then adapt a nonuniform quantizer to the resulting local
pdf.This approach is referred to as adaptive quantization.For a
Gaussian we have:
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*Veton Kpuska*Adaptive CodingMeasure the variance x2 of a
sequence x[n] and use resulting pdf to design optimal max
quantizer.Note that a change in the variance simply scales the time
signal:If E(x2[n]) = x2 then E[(x [n])2] = 2 x2 Need to design only
one nonuniform quantizer with unity variance and scale decision and
reconstruction levels according to a particular variance.Fix the
quantizer and apply a time-varying gain to the signal according to
the estimated variance (scale the signal to match the
quantizer).
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*Veton Kpuska*Adaptive Coding
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*Veton Kpuska*Adaptive CodingThere are two possible approaches
for estimation of a time-varying variance 2[n]:Feed-forward method
(shown in Figure 12.11) where the variance (or gain) estimate is
obtained from the inputFeedback method where the estimate is
obtained from a quantizer output.Advantage no need to transmit
extra side information (quantized variance)Disadvantage additional
sensitivity to transmission errors in codewords.Adaptive quantizers
can achieve higher SNR than the use of law companding.law
companding is generally preferred for high-rate waveform coding
because of its lower background noise when transmission channel is
idle. Why?Adaptive quantization is useful in variety of other
coding schemes.
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Differential and Residual Quantization
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*Veton Kpuska*Differential and Residual QuantizationPresented
methods are examples of instantaneous quantization.
Those approaches do not take advantage of the fact that speech
is highly correlated signal:Short-time (10-15 samples), as well
asLong-time (over a pitch period)
In this section methods that exploit short-time correlation will
be investigated.
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*Veton Kpuska*Differential and Residual QuantizationShort-time
Correlation:Neighboring samples are self-similar, that is, not
changing too rapidly from one another.Difference of adjacent
samples should have a lower variance than the variance of the
signal itself.This difference, thus, would make a more effective
use of quantization levels:Higher SNR for fixed number of
quantization levels.Predicting the next sample from previous ones
(finding the best prediction coefficients to yield a minimum
mean-squared prediction error same methodology as in Liner
Prediction Coefficients - LPC). Two approaches:Have a fixed
prediction filter to reflect the average local correlation of the
signal.Allow predictor to short-time adapt to the signals local
correlation.Requires transmission of quantized prediction
coefficients as well as the prediction error.
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*Veton Kpuska*Differential and Residual QuantizationIllustration
of a particular error encoding scheme presented in the Figure 12.12
of the next slide.In this scheme the following sequences are
required:x[n] prediction of the input sample x[n]; This is the
output of the predictor P(z) whose input is a quantized version of
the input signal x[n], i.e., x[n]r[n] prediction error signal;
residualr[n] quantized prediction error signal.
This approach is sometimes referred to as residual
coding.~^^
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*Veton Kpuska*Differential and Residual Quantization
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*Veton Kpuska*Differential and Residual QuantizationQuantizer in
the previous scheme can be of any
type:FixedAdaptiveUniformNonuniformWhatever the case is, the
parameter of the quantizer are determined so that to match variance
of r[n].Differential quantization can also be applied to:Speech
signalParameters that represent speech:LPC linear prediction
coefficientsCepstral coefficients obtained from Homomorphic
filtering.Sinewave parameters, etc.
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*Veton Kpuska*Differential and Residual QuantizationConsider
quantization error of the quantized residual:
From Figure 12.12 we express the quantized input x[n] as:^
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*Veton Kpuska*Differential and Residual QuantizationQuantized
signal samples differ form the input only by the quantization error
er[n].Since the er[n] is the quantization error of the residual: if
the prediction of the signal is accurate then the variance of r[n]
will be smaller than the variance of x[n] A quantizer with a given
number of levels can be adjusted to give a smaller quantization
error than would be possible when quantizing the signal
directly.
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*Veton Kpuska*Differential and Residual QuantizationThe
differential coder of Figure 12.12 is referred to:Differential PCM
(DPCM) when used with a fixed predictor and fixed
quantization.Adaptive Differential PCM (ADPCM) when used
withAdaptive prediction (i.e., adapting the predictor to local
correlation)Adaptive quantization (i.e., adapting the quantizer to
the local variance of r[n])ADPCM yields greatest gains in SNR for a
fixed bit rate.The international coding standard CCITT, G.721 with
toll quality speech at 32 kbps (8000 samples/sec x 4 bits/sample)
has been designed based on ADPCM techniques.To achieve higher
quality with lower rates it is required to:Rely on speech
model-based techniques and The exploiting of long-time prediction,
as well as Short-time prediction
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*Veton Kpuska*Differential and Residual QuantizationImportant
variation of the differential quantization scheme of Figure
12.12.Prediction has assumed an all-pole model (autoregressive
model).In this model signal value is predicted from its past
samples:Any error in a codeword due to for example bit errors over
a degraded channel propagate over considerable time during
decoding.Such error propagation is severe when the signal values
represent speech model parameters computed frame-by frame (as
opposed to sample-by-sample).Alternative approach is to use a
finite-order moving-average predictor derived from the residual.
One common approach of the use of the moving-average predictor is
illustrated in Figure 12.13 in the next slide.
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*Veton Kpuska*Differential and Residual Quantization
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*Veton Kpuska*Differential and Residual QuantizationCoder Stage
of the system in Figure 12.13:Residual as the difference of the
true value and the value predicted from the moving average of K
quantized residuals:
p[k] coefficients of P(z)Decoder Stage:Predicted value is given
by:
Error propagation is thus limited to only K samples (or K
analysis frames for the case of model parameters)
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Vector Quantization
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*Veton Kpuska*Vector Quantization (VQ)Investigation of scalar
quantization techniques was the topic of previous sections.A
generalization of scalar quantization referred to as vector
quantization is investigated in this section.In vector quantization
a block of scalars are coded as a vector rather than
individually.An optimal quantization strategy can be derived based
on a mean-squared error distortion metric as with scalar
quantization.
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*Veton Kpuska*Vector Quantization (VQ)MotivationAssume the vocal
tract transfer function is characterized by only two resonance's
thus requiring four reflection coefficients.Furthermore, suppose
that the vocal tract can take on only one of possible four
shapes.This implies that there exist only four possible sets of the
four reflection coefficients as illustrated in Figure 12.14 in the
next slide.Scalar Quantization considers each of the reflection
coefficient individually:Each coefficient can take on 4 different
values 2 bits required to encode each coefficient.For 4 reflection
coefficients it is required 4x2=8 bits per analysis frame to code
the vocal tract transfer function.Vector Quantization since there
are only four possible vocal tract positions of the vocal tract
corresponding to only four possible vectors of reflection
coefficients.Scalar values of each vector are highly
correlated.Thus 2 bits are required to encode the 4 reflection
coefficients. Note: if scalars were independent of each other
treating them together as a vector would have no advantage over
treating them individually.
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*Veton Kpuska*Vector Quantization (VQ)
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*Veton Kpuska*Vector Quantization (VQ)Consider a vector of N
continuous scalars:
With VQ, the vector x is mapped into another N-dimensional
vector x:
Vector x is chosen from M possible reconstruction (quantization)
levels:^^
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*Veton Kpuska*Vector Quantization (VQ)TT
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- *Veton Kpuska*Vector Quantization (VQ)VQ-vector quantization
operatorri-M possible reconstruction levels for 1i
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*Veton Kpuska*Vector Quantization (VQ)Properties of VQ:P1:In
vector quantization a cell can have an arbitrary size and shape. In
scalar quantization a cell (region between two decision levels) can
have an arbitrary size, but its shape is fixed.P2:Similarly to
scalar quantization, distortion measure D(x,x), is a measure of
dissimilarity or error between x and x. ^^
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*Veton Kpuska*VQ Distortion MeasureVector quantization noise is
represented by the vector e:
The distortion is the average of the sum of squares of scalar
components:
For the multi-dimensional pdf px(x):
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*Veton Kpuska*VQ Distortion MeasureGoal to minimize:
Two conditions formulated by Lim:C1:A vector x must be quantized
to a reconstruction level ri that gives the smallest distortion
between x and ri. C2:Each reconstruction level ri must be the
centroid of the corresponding decision region (cell Ci)
Condition C1 implies that given the reconstruction levels we can
quantize without explicit need for the cell boundaries.To quantize
a given vector the reconstruction level is found which minimizes
its distortion.This process requires a large search active area of
research.Condition C2 specifies how to obtain a reconstruction
level from the selected cell.
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*Veton Kpuska*VQ Distortion MeasureStated 2 conditions provide
the basis for iterative solution of how to obtain VQ codebook.Start
with initial estimate of ri.Apply condition 1 by which all the
vectors from a set that get quantized by ri can be determined.Apply
second condition to obtain a new estimate of the reconstruction
levels (i.e., centroid of each cell)Problem with this approach is
that it requires estimation of joint pdf of all x in order to
compute the distortion measure and the multi-dimensional
centroid.Solution: k-means algorithm (Lloyd for 1-D and Forgy for
multi-D).
Veton Kpuska
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*Veton Kpuska*k-Means AlgorithmCompute the ensemble average D
as: xk are the training vectors and xk are the quantized
vectors.Pick an initial guess at the reconstruction levels {ri}For
each xk select closest ri. Set of all xk nearest to ri forms a
cluster (see Figure 12.16) clustering algorithm. Compute the mean
of xk in each cluster which gives a new ris. Calculate D.Stop when
the change in D over two consecutive interactions is
insignificant.
This algorithm converges to a local minimum of D. ^
Veton Kpuska
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*Veton Kpuska*k-Means Algorithm
Veton Kpuska
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*Veton Kpuska*Neural Networks Based Clustering
AlgorithmsKohonens SOFMTopological Ordering of the SOFMOffers
potential for further reduction in bit rate.
Veton Kpuska
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*Veton Kpuska*Use of VQ in Speech TransmissionObtain the VQ
codebook from the training vectors - all transmitters and receivers
must have identical copies of VQ codebook.Analysis procedure
generates a vector xi. Transmitter sends the index of the centroid
ri of the closest cluster for the given vector xi. This step
involves search.Receiving end decodes the information by accessing
the codeword of the received index and performing synthesis
operation.
Veton Kpuska
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Model-Based CodingLiner Prediction Method (LPC)
Veton Kpuska
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*Veton Kpuska*Model-Based CodingThe purpose of model-based
speech coding is to increase the bit efficiency to achieve
either:Higher quality for the same bit rate orLower bit rate for
the same quality.Chronological perspective of model-based coding
starting with:All-pole speech representation used for coding:Scalar
QuantizationVector QuantizationMixed Excitation Linear Prediction
(MELP) coder:Remove deficiencies in binary source
representation.Code-excited Linear Prediction (CELP) coder:Does nor
require explicit multi-band decision and source characterization as
MELP.
Veton Kpuska
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*Veton Kpuska*Basic Linear Prediction Coder (LPC)Recall the
basic speech production model of the form: where the predictor
polynomial is given as:
Suppose:Linear Prediction analysis performed at 100 frames/s13
parameters are used:10 all-pole spectrum parameters, PitchVoicing
decisionGainResulting in 1300 parameters/s.Compared to telephone
quality signal:4000 Hz bandwidth 8000 samples/s (8 bit per
sample).1300 parameters/s < 8000 samples/s
Veton Kpuska
- *Veton Kpuska*Basic Linear Prediction Coder (LPC)Instead of
prediction coefficients ai use:Corresponding poles biPartial
Correlation Coefficients ki (PARCOR)Reflection Coefficients ri,
orOther equivalent representation.Behavior of prediction
coefficients is difficult to characterize:Large dynamic range (
large variance)Quantization errors can lead to unstable system
function at synthesis (poles may move outside the unit
circle).Alternative equivalent representations:Have a limited
dynamic rangeCan be easily enforced to give stability because
|bi|
-
*Veton Kpuska*Basic Linear Prediction Coder (LPC)Many ways to
code linear prediction parameters:Ideally optimal quantization uses
the Max quantizer based on known or estimated pdfs of each
parameter.Example of 7200 bps coding:Voice/Unvoiced Decision: 1 bit
(on or off)Pitch (if voiced): 6 bits (uniform)Gain: 5 bits
(nonuniform)Each Pole bi: 10 bits (nonuniform)5 bits for bandwidth
5 bits for center frequency Total of 6 poles
100 frames/s 1+6+5+6x10=72 bitsQuality limited by simple
impulse/noise excitation model.
Veton Kpuska
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*Veton Kpuska*Basic Linear Prediction Coder (LPC)Improvements
possible based on replacement of poles with PARCOR.Higher order
PARCOR have pdfs closer to Gaussian centered around zero nonuniform
quantization.Companding is effective with PARCOR:Transformed pdfs
close to uniform.Original PARCOR coefficients do not have a good
spectral sensitivity (change in spectrum with a change in spectral
parameters that is desired to minimize). Empirical finding that a
more desirable transformation in this sense is to use logarithm of
the vocal tract area function ratio:
Veton Kpuska
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*Veton Kpuska*Basic Linear Prediction Coder (LPC)Parameters
gi:Have a pdf close to uniformSmaller spectral sensitivity than
PARCOR:The all pole spectrum changes less with a change in gi than
with a change in kiNote that spectrum changes less with the change
in ki than with the change in pole positions.Typically these
parameters can be coded at 5-6 bits each (significant improvement
over 10 bits):100 frames/sOrder 6 of the predictor (6
poles)(1+6+5+6x6)x100 bps = 4800 bpsSame quality as 7200 bps by
coding pole positions for telephone bandwidth speech.
Veton Kpuska
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*Veton Kpuska*Basic Linear Prediction Coder (LPC)Government
standard for secure communications using 2.4 kbps for about a
decade used this basic LPC scheme at 50 frames per second.Demand
for higher quality standards opened up research on two primary
problems with speech codes base on all-pole linear prediction
analysis:Inadequacy of the basic source/filter speech production
modelRestrictions of one-dimensional scalar quantization techniques
to account for possible parameter correlation.
Veton Kpuska
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*Veton Kpuska*A VQ LPC CoderVQ based LPC PARCOR coder.K-means
algorithm
Veton Kpuska
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*Veton Kpuska*A VQ LPC CoderUse VQ LPC Coder to achieve same
quality of speech with lower bit-rate:10bit code book (1024
codewords)800 bps 2400 bps of scalar quantization44.4 frames/s440
bits to code PARCOR coefficients per second.8 bits per frame for:
PitchGainVoicing1 bit for frame synchronization per second.
Veton Kpuska
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*Veton Kpuska*A VQ LPC CoderMaintain 2400 bps bit rate with a
higher quality of speech coding (early 1980):22-bit codebook 222 =
4200000 codewords.Problems: Intractable solution due to
computational requirements (large VQ search)Memory (large Codebook
size)VQ based spectrum characterized by a wobble due to LPC-based
spectrum being quantized:Spectral representation near cell boundary
wobble to and from neighboring cells insufficient number of
codebooks.Emphasis changed from improved VQ of the spectrum and
better excitation models ultimately to a return to VQ on the
excitation.
Veton Kpuska
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*Veton Kpuska*Mixed Excitation LPC (MELP)Multi-band voicing
decision (introduced as a concept in Section 12.5.2 not covered in
slides)Addresses shortcomings of conventional linear prediction
analysis/synthesis:Realistic excitation signalTime varying vocal
tract formant bandwidths Production principles of the anomalous
voice.
Veton Kpuska
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*Veton Kpuska*Mixed Excitation LPC (MELP)Model:Different
mixtures of impulses and noise are generated in different frequency
bands (4-10 bands)The impulse train and noise in the MELP model are
each passed through time-varying spectral shaping filters and are
added together to form a full-band signal.MELP unique components:An
auditory-based approach to multi-band voicing estimation for the
mixed impulse/noise excitation.Aperiodic impulses due to pitch
jitter, the creaky voice, and the diplophonic voice.Time-varying
resonance bandwidth within a pitch period accounting for nonlinear
source/system interaction and introducing the truncation
effects.More accurate shape of the glottal flow velocity
source.
Veton Kpuska
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*Veton Kpuska*Mixed Excitation LPC (MELP)2.4 kbps coder has been
implemented based on the MELP model and has been selected as
government standard for secure telephone communications.Original
version of MELP uses:34 bits for scalar quantization of the LPC
coefficients (Specifically the line spectral frequencies LSFs).8
bits for gain7 bits for pitch and overall voicing Uses
autocorrelation technique on the lowpass filtered LPC
residual.5-bits to multi-band voicing.1-bit for the jittery state
(aperiodic) flag.54 bits per 22.5 ms frame 2.4 bps.In actual 2.4
kbs standard greater efficiency is achieved with vector
quantization of LSF coefficients.
Veton Kpuska
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*Veton Kpuska*Mixed Excitation LPC (MELP)Line Spectral
Frequencies (LSFs)More efficient parameter set for coding the
all-pole model of linear prediction.The LSFs for a pth order
all-pole model are defined as follows:Two polynomials of order p+1
are created from the pth order inverse filter A(z) according
to:
LSFs can be coded efficiently and stability of the resulting
syntheses filter can be guaranteed when they are quantized.Better
quantization and interpolation properties than the corresponding
PARCOR coefficients.Disadvantage is the fact that solving for the
roots of P(z) and Q(z) can be more computationally intensive than
the PARCOR coefficients.Polynomial A(z) is easily recovered from
the LSFs (Exercise 12.18).
Veton Kpuska
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*Veton Kpuska*Code-Excited Linear Prediction (CELP)Concept:Core
ideas of CELP:Utilization of long-term as well as short-term linear
prediction models for speech synthesis Avoiding the strict
voiced/unvoiced classification of LPC coder.Incorporation of an
excitation codebook which is searched during encoding to locate the
best excitation sequence.
Code Excited LP comes from the excitation codebook that contains
the code to excite the synthesis filters.
On each frame a codeword is chosen from a codebook of residuals
such as to minimize the mean-squared error between the synthesized
and original speech waveform.The length of a codeword sequence is
determined by the analysis frame length.For a 10 ms frame interval
split into 2 inner frames of 5 ms each a codeword sequence is 40
samples in duration for an 8000 Hz sampling rate.The residual and
long-term predictor is estimated with twice the time resolution (a
5 ms frame) of the short-term predictor (10 ms frame);Excitation is
more nonstationary than the vocal tract.
Veton Kpuska
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*Veton Kpuska*Code-Excited Linear Prediction (CELP)Two approach
to formation of the codebook:DeterministicStochasticDeterministic
codebook It is formed by applying the k-means clustering algorithm
to a large set of residual training vectors.Channel
mismatchStochastic codebook Histogram of the residual from the
long-term predictor follows roughly a Gaussian probability pdf. A
valid assumption with exception of plosives and voiced/unvoiced
transitions.Cumulative distributions are nearly identical to those
for white Gaussian random variables Alternative codebook is
constructed of white Gaussian random variables with unit
variance.
Veton Kpuska
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*Veton Kpuska*CELP CodersVariety of government and International
standard coders:1990s Government standard for secure communications
at 4.8 kbps at 4000 Hz bandwidth (Fed-Std1016) uses CELP
coder:Three bit rates:9.6 kbps (multi-pulse)4.8 kbps (CELP)2.4 kbps
(LPC)Short-time predictor: 30 ms frame interval coded with 34 bits
per frame.10th order vocal tract spectrum from prediction
coefficients transformed to LSFs coded nonuniform
quantization.Short-term and long-term predictors are estimated in
open-loopResidual codewords are determined in closed-loop
form.Current international standards use CELP based
coding.G.729G.723.1
Veton Kpuska
Digital Systems: Hardware Organization and DesignDigital
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