Scalar and Vector Quantization

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Optimal ∆ that minimizes MSQE Given pdf f X (x) of the source, let’s design an M-level mid-rise uniform quantizer that minimizes MSQE: σ 2 q = 2 −1 M 2 ∑∫ i= 1 i∆ ( i−1) ∆ ⎛ ⎜ ⎝ x − 2i −1 ⎞ ∆⎟ 2 ⎠ 2 f X ( x) dx → Granular error + 2 ∫ ∞ ⎛ M ⎞ ⎜ −1⎟∆ ⎝ 2 ⎠ ⎛ ⎜ ⎝ x − M −1 ⎞ ∆⎟ 2 ⎠ 2 f X ( x) dx. → Overload error x – Q(x) Overload error Granular error 18/55
Solving for Optimum Step Sizes Given an f X (x) <strong>and</strong> M, we can solve for ∆ numerically: dσ 2 q d∆ = − −1 M 2 ∑ i= 1 − ( M (2i −1) −1) ∫ ∞ ∫ i∆ ( i−1) ∆ ⎛ M ⎞ ⎜ −1⎟∆ ⎝ 2 ⎠ ⎛ ⎜ ⎝ ⎛ ⎜ ⎝ 2i −1 ⎞ x − ∆⎟ f 2 ⎠ M −1 ⎞ x − ∆⎟ f 2 ⎠ ( x) dx ( x) dx = Optimal uniform quantizer ∆ for different sources: X X 0. 19/55
Page 1 and 2: Scalar and Vector Quantization Nati
Page 3 and 4: The Quantization Problem Encoder m
Page 5 and 6: Quantization Function A quantizer
Page 7 and 8: Quantization Error If the quantiza
Page 9 and 10: Quantized Bitrate with VLC For VLC
Page 11 and 12: Uniform Quantizer All intervals ar
Page 13 and 14: Uniform Quantization of Uniform Sou
Page 15 and 16: The SNR of Quantization For n-bit
Page 17: Quantization of Non-uniform Sources
Page 21 and 22: Variance Mismatch Effects (1/2) Ef
Page 23 and 24: Distribution Mismatch Effects Give
Page 25 and 26: Forward Adaptive Quantization (FAQ)
Page 27 and 28: Speech Quantization Example (2/2)
Page 29 and 30: Backward Adaptive Quantization (BAQ
Page 31 and 32: Output Levels of 3-bit Jayant Quant
Page 33 and 34: Picking Jayant Multipliers We must
Page 35 and 36: Jayant Quantizer Performance To av
Page 37 and 38: pdf-optimized Quantization Given f
Page 39 and 40: Lloyd-Max Algorithm (2/3) Begin wi
Page 41 and 42: Example: pdf-Optimized Quantizers
Page 43 and 44: Companded Quantization (CQ) In com
Page 45 and 46: Example: CQ (2/2) The expandar fun
Page 47 and 48: Entropy Coding of Quantizer Outputs
Page 49 and 50: Why Vector Quantization (1/2)? Cor
Page 51 and 52: Vector Quantization Rule Vector qu
Page 53 and 54: The Linde-Buzo-Gray Algorithm 1. St
Page 55: Impact of Training Set The trainin

Scalar and Vector Quantization

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