Lecture 4: Quantization Eytan Modiano AA Dept Eytan Modiano
Lecture 4: Quantization Eytan Modiano AA Dept. Eytan Modiano Slide 1
Sampling Sampling provides a discrete time representation of a continuous waveform Sample points are real-valued numbers In order to transmit over a digital system we must first convert into discrete valued numbers Quantization levels Sample points What are the quantization regions What are the quantization levels Eytan Modiano
Sampling • Sampling provides a discrete-time representation of a continuous waveform – Sample points are real-valued numbers – In order to transmit over a digital system we must first convert into discrete valued numbers Quantization levels Q3 Q2 Q1 � � � � � � � � � � Sample points What are the quantization regions What are the quantization levels Eytan Modiano Slide 2
Uniform Quantizer 3△-2△-△ △ 3△ All quantization regions are of equal size(4 Except first and last regions if samples are not finite valued With n quantization regions, use log2(N)bits to represent each quantized value Eytan Modiano
∆ Uniform Quantizer ∆ ∆ 3∆ −3∆ −2∆ −∆ ∆ 2 • All quantization regions are of equal size (∆) – Except first and last regions if samples are not finite valued • With N quantization regions, use log2(N) bits to represent each quantized value Eytan Modiano Slide 3
Quantization Error e(x=Q(x)-x Squared error: D= E[e(x ]=E[(Q(X)-x 1 SQNR:EⅨ2]/E(Q(x)-×)1] Eytan Modiano
Quantization Error e(x) = Q(x) - x Squared error: D = E[e(x) 2] = E[(Q(x)-x) 2] SQNR: E[X 2]/E[(Q(x)-x) 2] Eytan Modiano Slide 4
Example x is uniformly distributed between-A and a f(x=1/2A,A<EX<=A and o otherwise · Uniform quantizer with n levels=>Δ=2AN Q(x]=quantization level midpoint of quantization region in which x lies D= E[e(x) 2] is the same for quantization regions D=Ee(x)2|x∈ R=2x2(x)s4 12 EIX=I xdx A2/3A2/3 SON N2,(△=2A/N) △2/12(2N2/12 Eytan Modiano
∆ E X Example • X is uniformly distributed between -A and A – f(x) = 1/2A, -A<=x<=A and 0 otherwise • Uniform quantizer with N levels => ∆ = 2A/N – Q(x) = quantization level = midpoint of quantization region in which x lies • D = E[e(x)2] is the same for quantization regions D E e x 2 ∈ ∆ / 2 1 ∆ / 2 ∆2 = [( ) | x Ri] = ∫−∆ / 2 x2 f(x)dx = ∆ ∫−∆ / 2 x2 dx = 12 1 A 2 A2 [] = 2 A ∫− Ax dx = 3 A2 / 3 A2 / 3 SQNR= ∆2 /12 = (2 A N)2 /12 = N2 , (∆ = 2 A/ N) / Eytan Modiano Slide 5