eee508_QuantizationInfoTheory_Basics

eee508_QuantizationInfoTheory_Basics - Qua t at o...

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Quantization Some sort of quantization is necessary to represent continuous signals in digital form Quantize 2D Sample x x n n x ( n 1 , n 2 ) Quantizer 2D Sampler ( t 1 , t 2 ) q ( n 1 , n 2 ) Digitizer (A/D) Quantization is also used for data reduction in virtually all lossy coding schemes EEE 508 - Lecture 6
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Quantization – Basic Concepts In quantization, range of input values x is divided into countable non-overlapping subsets, called “quantization levels” ^ ` M j j V 1 Each “quantization level” (subset of input range) k=1,…,M is assigned an index and a representative value , called “reconstruction value” or “reconstruction level” , k V AquantizerQ()withquamtizationlevels and ^ ` M j j k r r 1 ± M V A quantizer Q(.) with quamtization levels and reconstruction levels is a mapping: Q ( x ) = r k ; where x ± V k ^ ` M j j r 1 ^ ` j j 1 EEE 508 - Lecture 6
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Quantization – Basic Concepts Two main types of quantizers: ¾ If the domain of the input signal is R k or C k , i.e. the input x is a k- dimensional vector ± quantizer is a vector quantizer ¾ In the special case where the dimension k =1, i.e. input is scalar ± quantizer is a scalar quantizer ¾ A scalar quantizer is a special case of a vector quantizer Exp. 1 : scalar quantizer (M=5) Exp. 2 : vector quantizer (k=2, M=6) x 01 . 5 3 5 x r 1 =-1 r 2 =1 r 3 =2.25 r 4 =4 r 5 =6 5 V 4 V 3 V 6 V 5 u u u u 2 x =(x 1 , x 2 ) V 1 V 2 V 3 V 4 V 5 Usually, r i = central point of V i ; i =1,2,…6 5 V 1 V 2 u u 0 x 1 EEE 508 - Lecture 6
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Quantization – Basic Concepts In the context of a communication system, quantizer = composition of two mappings: ¾ Encoder Mapping E: X I ¾ Decoder Mapping D: I { r i } ¾ Q = D q E When a variable-length encoding is allowed, an entropy coder L can be included as part of the quantizer for convenience (or for joint optimization) ¾ Q = D q L -1 q L q E EEE 508 - Lecture 6
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Quantization – Basic Concepts Quantizer Design ¾ Objective: optimize the performance of the quantizer given some constraints on its structure. ¾ Reason: 9 Quantizer introduces noise and distortion 9 It is a lossy and irreversible operation need to optimize to minimize distortion EEE 508 - Lecture 6
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Quantization – Basic Concepts Quantizer Design ¾ Performance evaluation: 9 Performance usually evaluated in terms of the reconstruction
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eee508_QuantizationInfoTheory_Basics - Qua t at o...

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