Question

Question 5) a. Determine and illustrate the partition levels and reconstruction levels of a uniform quantizer in the range of (-1,1) with 8 levels. b. Determine and illustrate the partition levels and reconstruction levels of a u-law quantizer in the range of (-1,1) with 8 levels, u = 15. c. For the sequence {0.5,-0.4,-0.7,0.2, 0.15, -0.25, ...}, determine the quantized sequence using the uniform quantizer of part a and the u -law quantizer of part b. Also determine the mean square quantization error and SNR for both quantizers. For this sequence, which method (uniform vs. Į -law) gives a more accurate result?

Answer 1

a data range is 1-(-1) = 2 nog levels = N=8 a = range - 2 = - = 0.25 Ni 8 TQ=0.25) levels Reconstruction -0.875 0.625-0.375 0.125 0.125 0.375 0.625 0.875 -1 -0.75 -0.5 -0,25 0 0.25 0,5 0.75 1 Partition levels b) given M= 15 levels N=4 Because of symmetry of il-law quantizeg; we need to determine mapped values for 0.125,0.25, 0.375, 0.5, 0625, 0.75 0.875 For mapping to the partition and reconstrucleón We apply inverse. ll law formulae. X=po[y] = Xmax [10102407 u 1! Xmax

Here U= 15, xmax = 1 for: y = 0.125 = X = 0.027 y = 0.25 => X = 0.0667 g=0x375 => x = 101218 ya 0.5 - x = 0.2 Y = 0.625 -X = 0.31 4-0.75 -> X = 0.467 Y=0.895 5X = 0.687 Reconstruition levels 027 partition levels © given sequence {0,5,-0.47-0.7.10.270.157,-0.25.} The quantized sequence is 5.0.375, -0.375,-0.625,0.12,5). 0.125, -0.125

The quantization ellor's (past a) q= $(xm) - 6m)) = = [C0.5 -0.345)*+ (0.4-0.335) + (0.7 -0.625)*+ (0.2 -0.125) + (0.15-0.125 St (0.25 +0.125 Ep. 0.0156+ Tood + i + + Thoot by] F0.028+ 0.0156] 1600 16.00 = = [0.0436] ray = 0.00726 Mean of givey samples y Mean = 5 0.5-0.4 -0,7 +0.2 +0.15-0.257 = -0.08

orxanh (xrm) - a mean)? - (0.577 (0.427(0-3)*(0.21Y +00.158+00.2513 => [0:25+ 0.16 +0:49+ 0.04 + 0.0225+ 0.0625] = % (1.025] x = 0.1908 SNR = 1olog10 ( ) = 13.41 dB Sir= 13.71 dB)

The giver sequence in 20.5,-0.47-0.7,0.2,0-15-0.25 ---} quantized sequeill us (part 2) v Ž 0.467 -0.467-0:50.12) 0.12,; -0.31} quantization allory 89% = 160.50.46777 (0.4 -0.467" +(0.7 0.54 (0.2-0.12) + (0.15-0.12)? + (0.15-0.3107] 9 2500 9 + + Toooo Ś (0.0561] loor 50.009357 Mean of giver Sequence is : Meau = {0.5-0.4-0.7 +0.2 +0.15-0.25% . 6 -0.08

Meau = 0, rx? = 2 / 4 € (Xins - mean) = ☆ [1.025] x = 0.1708 Save: 1odogin ] = 10l05o .Dones] 1 SUR = 120616 dB SAR= 12.616 dB For this given sequence Mu-law quantizer is not very good, because an Sequence there wie' equal number of Small and large values, So, Uniforms quantizer produres more accurate result