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7/30/2019 Max Lloyd Quantization
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MaxMax--LloydLloyd
QuantizerQuantizer
Quantization ConceptQuantization Concept
z L-level Quantization
Minimize errors for this lossy process
What L values to use?
Map what range of continuous values to each of L values?
tmin tmax
z
Uniform partition Maximum errors = ( tmax - tmin ) / 2L = 2V / 2L
over a dynamic range of 2V
Best solution?
X Consider minimizing maximum absolute error (min-max) vs. MSE
X what if the value between [a, b] is more likely than other intervals?
tmin tmax
tk tk+1
(tmaxtmax)/2L
quantization
errorUMCPENEE631Slides(createdbyM.Wu2001/2004)
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Consider the Probability DistributionConsider the Probability Distribution
z Minimize error in a probability sense
MMSE (minimum mean square error)X assign high penalty to large error and to likely occurring values
X squared error gives convenience in math.: differential, etc.
z An optimization problem
What {tk} and {rk} to use?
Necessary conditions: by setting partial differentials to zero
t1 tL+1
p.d.f pu(x)
r1 rL
z Allocate more reconstruct. values in more probable ranges
UMCPENEE631
Slides(createdbyM.Wu2001)
Derivation of MMSE (MaxDerivation of MMSE (Max--Lloyd)Lloyd) QuantizerQuantizer
z MSE of L-level quantizer
z Optimal (MMSE) quantizer necessary conditions
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MMSEMMSE QuantizerQuantizer(Lloyd(Lloyd--Max)Max)z Reconstruction and decision levels need to satisfy
z Solve iteratively
Choose initial values of {tk}(0)
, compute {rk}(0)
Compute new values {tk}(1), and {rk}
(1)
z For large number of quantization levels
Approx. constant pdf within t[tk, tk+1), i.e. p(t) = p(tk) for tk=(tk+tk+1)/2
Reference: S.P. Lloyd: Least Squares Quantization in PCM, IEEE Trans. Info.
Theory, vol.IT-28, March 1982, pp.129-137
UMCPENEE631Slides(createdbyM.Wu2001/2004)
Observations on the MMSEObservations on the MMSE QuanitizerQuanitizer
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Example: Uniform DensityExample: Uniform Density
Uniform quantizer is the optimal mean square quantizer for uniform density.
MMSEMMSE QuantizerQuantizerfor Uniform Distributionfor Uniform Distribution
z Uniform quantizer
Is optimal for uniformly distributed r.v. in MMSE sense
MSE = 2 / 12 with = 2V / L
z SNR of uniform quantizer
SNR = 6n dB with L = 2n
X 1 bit is worth 6 dB.t1 tL+1
2V
1/(2V)
p.d.f. of uniformdistribution
t1 tL+1
UMCPENEE631S
lides(createdbyM.Wu2001/2004)
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z Note: Quantization is a Lossy Stepin SourceCoding
z Quantization is a lossy way to data compression
Lloyd-Max quantizer minimizes MSE distortion with a
number of output bits n (i.e. a give L=2n)
UMCPENEE739MSlides(createdbyM.Wu2002)