DIP Project 1A.docx Sharath

DIPproject -1a

Lloyd Max Quantizer for Rayleigh Pdf

Submitted By :Beeram, Sharath Chandra ReddyUTA ID : 1001111692

LLYOD-MAX QUANTIZER FOR RAYLEIGH PDF

Rayleigh density function is defined in the problem 4.15 b (pp127). Develop quantization tables similar to tables 4.1 (pp104) and 4.2 (pp108) based on: (List also MSE, SNR(dB) and entropy)

0. Llyod-Max quantizer (see equations 4.34 and 4.35, pp101).0. Based on equations 4.36 and 4.37, pp101.0. Optimum mean square uniform quantizer (see section 4.8, pp115).

For L=2, 4, 8, 16, 32, 64 and 128

Rayleigh density function as given in problem 4.15 b (pp127) [1]:

Llyod-Max quantizer (equations 4.34 and 4.35, pp101) [1].

Equations 4.36 and 4.37, pp101 [1]

Fig. 4.17 Piecewise constant approximation of pu(u) [1]

Optimum mean square uniform quantizer [1]

Reference[1] A. K. Jain, Fundamentals of digital signal processing, Prentice Hall, 1989

Code for Lloyd Max Quantizer:

clc;clear all;

%Set desired L value eg:2,4,8,16,32,64,128% L=2;% L=4;% L=8for step = 1:7L=2^step;err=1;Rk=zeros(1,L);

Tk(1)=0;for i=2:LTk(i)=Tk(i-1)+(10/(L-1));endTk(L+1)=20;

G1 = @(x) (x./1.5^2).*exp(-power(x,2)./(2*(1.5^2)));G2 = @(x) x.*(x./1.5^2).*exp(-power(x,2)./(2*(1.5^2)));

%Minimising errorwhile (err>0.1)err=0;entropy=0;

for k=1:1:Lnumerator=quad(G2,Tk(k),Tk(k+1));denominator=quad(G1,Tk(k),Tk(k+1));R(k)=numerator/denominator;entropy=entropy-(denominator*log2(denominator));end

%Determining the error of received samplesfor i=1:Lerr=err+abs((R(i)-Rk(i)));endRk=R; %Calculating new tk using rk for k=2:LTk(k)=(R(k)+R(k-1))/2;endend

%Calculating Mean square errorMean=0;entropyfor k=1:LG=@(v) (((v-R(k)).^2).*(v./1.5^2).*exp(-(v.^2)./(2*(1.5^2))));Mean=Mean+quad(G,Tk(k),Tk(k+1));endMean%Calculating SNRsnr=-10*log10(Mean)Tk=Tk';Rk=R'len_1 = length(Tk);Tk1 = Tk(1:len_1-1) end

Results :

Lloyd Max Quantizer

Optimum Mean Square Uniform Quantizer

Code:

function[]=dip1a_quantizer2()clc;clear all;close allfor state = 1:7G=2^state[Tk Rk]=optimum_TkRk(G);[Mean SNR entropy]=optpara(G)Tk=Tk';Rk=Rk';end

%Functions:function[Tk,Rk]=optimum_TkRk(G)Rk=zeros(1,G);Tk=zeros(1,G);Tk(G+1)=10;A=Tk(G+1)-Tk(1);

for k=1:GH=@(x) power((x/1.5^2).*exp(-(x.^2)/(2*(1.5^2))),(-1/3)); y=(k/G)*A+Tk(1); c=quad(H,Tk(1),y+Tk(1)); numerator=A*c; denominator=quad(H,Tk(1),Tk(G+1)); Tk(k+1)=(numerator/denominator)+Tk(1);end

for k=1:GRk(k)=(Tk(k)+Tk(k+1))/2;endend

function[Mean SNR entropy]=optpara(G)[Tk Rk]=optimum_TkRk(G);% MeanH1=@(x) power((x/1.5^2).*exp(-(x.^2)/(2*(1.5^2))),(1/3));p=quad(H1,Tk(1),Tk(G+1));Mean=(1/(12*(G^2))*(p^3));%SNRSNR=-10*log10(Mean); %Entropyentropy=0;H2=@(x) (x/1.5^2).*exp(-(x.^2)/(2*(1.5^2)));for k=1:G c=quad(H2,Tk(k),Tk(k+1));entropy=entropy-(c*log2(c));endend end

Results :

Description :

Optimum Mean Square or Lloyd Max Quantizer : This Quantizer minimises the mean square error for a given number of quantisation levels.

Properties of Optimum Mean Square Quantizer :

The quantizer output is an Unbiased estimate of the input The quantisation error is orthogonal to quantizer output It is sufficient to design mean square quantisers for zero mean unity variance distributions Sheet1L24816L3264128MSE0.330300.104200.031700.01030MSE0.000220.000060.00002SNR4.81089.822314.984219.8708SNR26.535931.951836.8754Entropy0.93981.81352.67693.4850Entropy4.60595.60536.6035tkrktkrktkrktkrktkrktkrktkrk2.11851.28691.36460.88220.90110.59310.00000.40540.18160.12080.10870.07240.06810.04542.95002.65431.84711.48571.20910.49230.81820.29460.24230.17600.14490.11020.09072.89702.81042.02941.76230.66411.17910.39480.34690.23540.20710.14720.12964.01132.56672.29660.82751.51550.48820.44280.29030.26370.18140.16493.12452.83670.98751.83790.57710.53350.34210.31680.21350.19793.74653.41221.14712.15180.66300.62070.39170.36740.24420.22924.54814.08081.30912.46090.74680.70530.43950.41600.27370.25925.01551.47592.76780.82930.78830.48600.46310.30220.28811.65053.07530.91090.87020.53130.50890.32990.31621.83653.38660.99200.95150.57580.55380.35690.34362.03893.70601.07311.03250.61950.59780.38340.37032.26534.04061.15451.11370.66250.64120.40930.39642.52874.40201.23651.19530.70510.68390.43470.42212.85554.81251.31951.27770.74720.72620.45980.44733.31855.32171.40381.36130.78890.76810.48440.47226.08081.48971.44630.83040.80970.50880.49671.57771.53310.87170.85110.53280.52091.66821.62220.91280.89220.55660.54481.76161.71410.95380.93330.58010.56841.85871.80920.99470.97420.60340.59181.96011.90821.03561.01510.62650.61502.06682.01201.07661.05610.64940.63802.17982.12151.11761.09700.67210.66082.30072.23801.15871.13810.69470.68342.43142.36331.20001.17930.71710.70592.57482.49951.24161.22070.73940.72832.73522.65021.28331.26240.76150.75052.91922.82031.32541.30430.78360.77263.13853.01811.36781.34650.80550.79463.41723.25891.41051.38900.82740.81653.82203.57561.45371.43200.84910.83834.06851.49741.47540.87080.86001.54161.51930.89240.88161.58641.56380.91400.90321.63181.60890.93550.92471.67791.65470.95690.94621.72481.70120.97830.96771.77261.74850.99970.98901.82121.79671.02111.01041.87091.84581.04241.03171.92181.89611.06371.05311.97381.94751.08501.07442.02722.00021.10631.09562.08212.05431.12761.11692.13872.11001.14891.13822.19712.16741.17021.15952.25752.22671.19151.18082.32022.28821.21281.20222.38552.35221.23421.22352.45372.41881.25561.24492.52532.48861.27701.26632.60072.56191.29851.28772.68052.63941.32001.30922.76552.72161.34161.33082.85692.80951.36321.35242.95572.90421.38491.37403.06403.00721.40661.39573.18423.12071.42851.41753.32023.24771.45041.43943.47803.39271.47231.46133.66853.56341.49441.48343.91413.77371.51661.50554.27664.05441.53891.52774.49891.56131.55001.58381.57251.60641.59501.62911.61771.65201.64061.67511.66351.69831.68661.72161.70991.74511.73331.76881.75691.79271.78071.81671.80471.84101.82881.86551.85321.89021.87781.91521.90271.94041.92771.96591.95311.99161.97872.01772.00462.04402.03082.07072.05732.09772.08412.12502.11132.15282.13882.18092.16672.20952.19512.23852.22392.26792.25312.29792.28282.32842.31302.35942.34382.39112.37512.42342.40712.45632.43972.49002.47292.52442.50702.55962.54182.59582.57752.63282.61402.67092.65162.71012.69022.75052.73002.79222.77102.83532.81342.88002.85732.92642.90282.97472.95003.02512.99933.07783.05083.13323.10483.19163.16163.25343.22163.31923.28533.38973.35323.46563.42613.54823.50513.63893.59133.73993.68653.85453.79343.98743.91564.14724.05924.35044.23524.63834.46564.8109

Sheet2Uniform Quantizer for large levels (Assume Standard Deviation=1.5)L24816L3264128MSE0.532400.130800.032700.00820MSE0.002000.000510.00012SNR2.81178.832314.852920.8735SNR26.894132.914738.9353Entropy0.04060.48441.39442.0440Entropy2.98033.95994.9546tkrktkrktkrktkrktkrktkrktkrk0.14040.07020.05370.02680.03080.01540.01890.00950.01190.00590.00750.00370.00470.00245.07020.14040.09710.05370.04220.03080.02490.01890.01540.01190.00970.00750.00610.67390.40720.08450.06910.04190.03630.02510.02200.01560.01370.00980.00865.33690.14040.11250.05370.04780.03080.02790.01890.01730.01190.01080.27240.20640.06740.06060.03630.03350.02210.02050.01380.01280.67390.47320.08450.07600.04190.03910.02510.02360.01560.01472.23681.45540.10750.09600.04760.04480.02790.02650.01730.01646.11840.14040.12400.05370.05070.03080.02930.01890.01810.19070.16560.06020.05700.03350.03210.02050.01970.27240.23160.06740.06380.03630.03490.02210.02130.41380.34310.07540.07140.03910.03770.02360.02280.67390.54390.08450.08000.04190.04050.02510.02431.18210.92800.09510.08980.04470.04330.02650.02582.23681.70950.10750.10130.04760.04620.02790.02724.56153.39920.12240.11500.05060.04910.02940.02867.28070.14040.13140.05370.05220.03080.03010.16270.15160.05690.05530.03210.03140.19070.17670.06020.05860.03350.03280.22630.20850.06370.06200.03490.03420.27240.24940.06740.06560.03630.03560.33300.30270.07130.06940.03770.03700.41380.37340.07540.07340.03910.03840.52330.46860.07980.07760.04050.03980.67390.59860.08450.08220.04190.04120.88410.77900.08960.08710.04330.04261.18211.03310.09510.09240.04470.04401.61081.39640.10110.09810.04620.04542.23681.92380.10750.10430.04760.04693.16502.70090.11460.11110.04910.04844.56153.86320.12240.11850.05060.04996.69425.62780.13100.12670.05210.05148.34710.14040.13570.05370.05290.15100.14570.05530.05450.16270.15690.05690.05610.17590.16930.05850.05770.19070.18330.06020.05940.20740.19910.06200.06110.22630.21690.06370.06280.24790.23710.06550.06460.27240.26020.06740.06650.30060.28650.06930.06840.33300.31680.07130.07030.37050.35170.07330.07230.41380.39210.07540.07440.46430.43910.07760.07650.52330.49380.07980.07870.59250.55790.08210.08100.67390.63320.08450.08330.77010.72200.08700.08580.88410.82710.08960.08831.01990.95200.09230.09101.18211.10100.09510.09371.37661.27930.09800.09661.61081.49370.10110.09951.89371.75220.10420.10262.23682.06530.10750.10592.65462.44570.11100.10933.16502.90980.11460.11283.79093.47790.11840.11654.56154.17620.12240.12045.51355.03750.12660.12456.69426.10380.13100.12888.16387.42900.13560.13339.08190.14040.13800.14560.14300.15100.14830.15670.15380.16270.15970.16910.16590.17590.17250.18310.17950.19070.18690.19880.19480.20740.20310.21660.21200.22630.22150.23680.23150.24790.24230.25970.25380.27240.26610.28600.27920.30060.29330.31620.30840.33300.32460.35110.34200.37050.36080.39130.38090.41380.40260.43810.42600.46430.45120.49270.47850.52330.50800.55650.53990.59250.57450.63150.61200.67390.65270.72000.69690.77010.74500.82460.79740.88410.85440.94900.91661.01990.98441.09731.05861.18211.13971.27491.22851.37661.32571.48821.43241.61081.54951.74551.67811.89371.81962.05691.97532.23682.14692.43542.33612.65462.54502.89692.77573.16503.03093.46183.31343.79093.62644.15603.97354.56154.35875.01214.78685.51355.26286.07185.79276.69426.38307.38857.04138.16387.77629.03058.59729.5152

Sheet3

Sheet1Lloyd-Max Quantizer(Assume Standard Deviation=1)L24816L3264128MSE0.146300.044700.012600.00340MSE0.003900.001300.00043SNR8.347813.494318.992324.7195SNR24.144328.982533.6808Entropy0.96391.89232.83283.7939Entropy4.19534.99565.7580tkrktkrktkrktkrktkrktkrktkrk1.37490.82930.82190.52930.49930.32870.30200.20040.18160.12080.10870.18440.06810.11151.92051.41971.11450.82540.66990.49230.40350.29460.24230.17600.36970.11020.22332.12701.72481.13550.98080.66410.58100.39480.34690.23540.52900.14720.31922.52921.45361.29020.82750.74710.48820.44280.29030.67490.18140.40681.80051.61700.98750.90790.57710.53350.34210.81220.21350.48912.20871.98401.14711.06700.66300.62070.39170.94350.24420.56772.76012.43341.30911.22720.74680.70530.43951.07030.27370.64373.08681.47591.39090.82930.78830.48601.19380.30220.71761.65051.56090.91090.87020.53131.31460.32990.78991.83651.74020.99200.95150.57581.43320.35690.86112.03891.93291.07311.06070.63910.59780.38340.37032.26532.14491.15451.15130.69020.64120.40930.39642.52872.38561.23651.24190.74140.68390.43470.42212.85552.67171.31951.27770.74720.72620.45980.44733.31853.03931.40381.36130.78890.76810.48440.47223.59771.48971.44630.83040.80970.50880.49671.57771.53310.87170.85110.53280.52091.66821.62220.91280.89220.55660.54481.76161.71410.95380.93330.58010.56841.85871.80920.99470.97420.60340.59181.96011.90821.03561.01510.62650.61502.06682.01201.07661.05610.64940.63802.17982.12151.11761.09700.67210.66082.30072.23801.15871.13810.69470.68342.43142.36331.20001.17930.71710.70592.57482.49951.24161.22070.73940.72832.73522.65021.28331.26240.76150.75052.91922.82031.32541.30430.78360.77263.13853.01811.36781.34650.80550.79463.41723.25891.41051.38900.82740.81653.82203.57561.45371.43200.84910.83834.06851.49741.47540.87080.86001.54161.51930.89240.88161.58641.56380.91400.90321.63181.60890.93550.92471.67791.65470.95690.94621.72481.70120.97830.96771.77261.74850.99970.98901.82121.79671.02111.01041.87091.84581.04241.03171.92181.89611.06371.05311.97381.94751.08501.07442.02722.00021.10631.09562.08212.05431.12761.11692.13872.11001.14891.13822.19712.16741.17021.15952.25752.22671.19151.18082.32022.28821.21281.20222.38552.35221.23421.22352.45372.41881.25561.24492.52532.48861.27701.26632.60072.56191.29851.28772.68052.63941.32001.30922.76552.72161.34161.33082.85692.80951.36321.35242.95572.90421.38491.37403.06403.00721.40661.39573.18423.12071.42851.41753.32023.24771.45041.43943.47803.39271.47231.46133.66853.56341.49441.48343.91413.77371.51661.50554.27664.05441.53891.52774.49891.56131.55001.58381.57251.60641.59501.62911.61771.65201.64061.67511.66351.69831.68661.72161.70991.74511.73331.76881.75691.79271.78071.81671.80471.84101.82881.86551.85321.89021.87781.91521.90271.94041.92771.96591.95311.99161.97872.01772.00462.04402.03082.07072.05732.09772.08412.12502.11132.15282.13882.18092.16672.20952.19512.23852.22392.26792.25312.29792.28282.32842.31302.35942.34382.39112.37512.42342.40712.45632.43972.49002.47292.52442.50702.55962.54182.59582.57752.63282.61402.67092.65162.71012.69022.75052.73002.79222.77102.83532.81342.88002.85732.92642.90282.97472.95003.02512.99933.07783.05083.13323.10483.19163.16163.25343.22163.31923.28533.38973.35323.46563.42613.54823.50513.63893.59133.73993.68653.85453.79343.98743.91564.14724.05924.35044.23524.63834.46564.8109


Sheet3

Sheet1Lloyd-Max Quantizer(Assume Standard Deviation=1)L24816L3264128MSE0.146300.044700.012600.00340MSE0.000870.000220.00006SNR8.347813.494318.992324.7195SNR30.581836.396841.8328Entropy0.96391.89232.83283.7939Entropy4.77095.75846.7254tkrktkrktkrktkrktkrktkrktkrk1.37490.82930.82190.52930.49930.32870.30200.20040.18160.12080.10870.07240.06810.04541.92051.41971.11450.82540.66990.49230.40350.29460.24230.17600.14490.11020.09072.12701.72481.13550.98080.66410.58100.39480.34690.23540.20710.14720.12962.52921.45361.29020.82750.74710.48820.44280.29030.26370.18140.16491.80051.61700.98750.90790.57710.53350.34210.31680.21350.19792.20871.98401.14711.06700.66300.62070.39170.36740.24420.22922.76012.43341.30911.22720.74680.70530.43950.41600.27370.25923.08681.47591.39090.82930.78830.48600.46310.30220.28811.65051.56090.91090.87020.53130.50890.32990.31621.83651.74020.99200.95150.57580.55380.35690.34362.03891.93291.07311.03250.61950.59780.38340.37032.26532.14491.15451.11370.66250.64120.40930.39642.52872.38561.23651.19530.70510.68390.43470.42212.85552.67171.31951.27770.74720.72620.45980.44733.31853.03931.40381.36130.78890.76810.48440.47223.59771.48971.44630.83040.80970.50880.49671.57771.53310.87170.85110.53280.52091.66821.62220.91280.89220.55660.54481.76161.71410.95380.93330.58010.56841.85871.80920.99470.97420.60340.59181.96011.90821.03561.01510.62650.61502.06682.01201.07661.05610.64940.63802.17982.12151.11761.09700.67210.66082.30072.23801.15871.13810.69470.68342.43142.36331.20001.17930.71710.70592.57482.49951.24161.22070.73940.72832.73522.65021.28331.26240.76150.75052.91922.82031.32541.30430.78360.77263.13853.01811.36781.34650.80550.79463.41723.25891.41051.38900.82740.81653.82203.57561.45371.43200.84910.83834.06851.49741.47540.87080.86001.54161.51930.89240.88161.58641.56380.91400.90321.63181.60890.93550.92471.67791.65470.95690.94621.72481.70120.97830.96771.77261.74850.99970.98901.82121.79671.02111.01041.87091.84581.04241.03171.92181.89611.06371.05311.97381.94751.08501.07442.02722.00021.10631.09562.08212.05431.12761.11692.13872.11001.14891.13822.19712.16741.17021.15952.25752.22671.19151.18082.32022.28821.21281.20222.38552.35221.23421.22352.45372.41881.25561.24492.52532.48861.27701.26632.60072.56191.29851.28772.68052.63941.32001.30922.76552.72161.34161.33082.85692.80951.36321.35242.95572.90421.38491.37403.06403.00721.40661.39573.18423.12071.42851.41753.32023.24771.45041.43943.47803.39271.47231.46133.66853.56341.49441.48343.91413.77371.51661.50554.27664.05441.53891.52774.49891.56131.55001.58381.57251.60641.59501.62911.61771.65201.64061.67511.66351.69831.68661.72161.70991.74511.73331.76881.75691.79271.78071.81671.80471.84101.82881.86551.85321.89021.87781.91521.90271.94041.92771.96591.95311.99161.97872.01772.00462.04402.03082.07072.05732.09772.08412.12502.11132.15282.13882.18092.16672.20952.19512.23852.22392.26792.25312.29792.28282.32842.31302.35942.34382.39112.37512.42342.40712.45632.43972.49002.47292.52442.50702.55962.54182.59582.57752.63282.61402.67092.65162.71012.69022.75052.73002.79222.77102.83532.81342.88002.85732.92642.90282.97472.95003.02512.99933.07783.05083.13323.10483.19163.16163.25343.22163.31923.28533.38973.35323.46563.42613.54823.50513.63893.59133.73993.68653.85453.79343.98743.91564.14724.05924.35044.23524.63834.46564.8109


Sheet3

Sheet1Lloyd-Max Quantizer(Assume Standard Deviation=1)L248163264128MSE0.146300.044700.012600.003400.000870.000220.00006SNR8.347813.494318.992324.719530.581836.396841.8328Entropy0.96391.89232.83283.79394.77095.75846.7254tkrktkrktkrktkrktkrktkrktkrk1.37490.82930.82190.52930.49930.32870.30200.20040.18160.12080.10870.07240.06810.04541.92051.41971.11450.82540.66990.49230.40350.29460.24230.17600.14490.11020.09072.12701.72481.13550.98080.66410.58100.39480.34690.23540.20710.14720.12962.52921.45361.29020.82750.74710.48820.44280.29030.26370.18140.16491.80051.61700.98750.90790.57710.53350.34210.31680.21350.19792.20871.98401.14711.06700.66300.62070.39170.36740.24420.22922.76012.43341.30911.22720.74680.70530.43950.41600.27370.25923.08681.47591.39090.82930.78830.48600.46310.30220.28811.65051.56090.91090.87020.53130.50890.32990.31621.83651.74020.99200.95150.57580.55380.35690.34362.03891.93291.07311.03250.61950.59780.38340.37032.26532.14491.15451.11370.66250.64120.40930.39642.52872.38561.23651.19530.70510.68390.43470.42212.85552.67171.31951.27770.74720.72620.45980.44733.31853.03931.40381.36130.78890.76810.48440.47223.59771.48971.44630.83040.80970.50880.49671.57771.53310.87170.85110.53280.52091.66821.62220.91280.89220.55660.54481.76161.71410.95380.93330.58010.56841.85871.80920.99470.97420.60340.59181.96011.90821.03561.01510.62650.61502.06682.01201.07661.05610.64940.63802.17982.12151.11761.09700.67210.66082.30072.23801.15871.13810.69470.68342.43142.36331.20001.17930.71710.70592.57482.49951.24161.22070.73940.72832.73522.65021.28331.26240.76150.75052.91922.82031.32541.30430.78360.77263.13853.01811.36781.34650.80550.79463.41723.25891.41051.38900.82740.81653.82203.57561.45371.43200.84910.83834.06851.49741.47540.87080.86001.54161.51930.89240.88161.58641.56380.91400.90321.63181.60890.93550.92471.67791.65470.95690.94621.72481.70120.97830.96771.77261.74850.99970.98901.82121.79671.02111.01041.87091.84581.04241.03171.92181.89611.06371.05311.97381.94751.08501.07442.02722.00021.10631.09562.08212.05431.12761.11692.13872.11001.14891.13822.19712.16741.17021.15952.25752.22671.19151.18082.32022.28821.21281.20222.38552.35221.23421.22352.45372.41881.25561.24492.52532.48861.27701.26632.60072.56191.29851.28772.68052.63941.32001.30922.76552.72161.34161.33082.85692.80951.36321.35242.95572.90421.38491.37403.06403.00721.40661.39573.18423.12071.42851.41753.32023.24771.45041.43943.47803.39271.47231.46133.66853.56341.49441.48343.91413.77371.51661.50554.27664.05441.53891.52774.49891.56131.55001.58381.57251.60641.59501.62911.61771.65201.64061.67511.66351.69831.68661.72161.70991.74511.73331.76881.75691.79271.78071.81671.80471.84101.82881.86551.85321.89021.87781.91521.90271.94041.92771.96591.95311.99161.97872.01772.00462.04402.03082.07072.05732.09772.08412.12502.11132.15282.13882.18092.16672.20952.19512.23852.22392.26792.25312.29792.28282.32842.31302.35942.34382.39112.37512.42342.40712.45632.43972.49002.47292.52442.50702.55962.54182.59582.57752.63282.61402.67092.65162.71012.69022.75052.73002.79222.77102.83532.81342.88002.85732.92642.90282.97472.95003.02512.99933.07783.05083.13323.10483.19163.16163.25343.22163.31923.28533.38973.35323.46563.42613.54823.50513.63893.59133.73993.68653.85453.79343.98743.91564.14724.05924.35044.23524.63834.46564.8109


Sheet3



Sheet3



Sheet3

Documents

DIP Project 1A.docx Sharath