Quantization of audio signal.docx
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Quantization of audio signal.docx
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Quantizationofaudiosignal
Quantizationofaudiosignal
Ximengzhao
Abstract:
ThispaperismainlyfocusingontwobasiccharacteristicsofDSPthataresamplingandquantization.QuantizationrepresentsakeypartintheDSPprocess.Quantizationisalsoessentiallythemappingofananalogvalueintoacorrespondingbinarysequence.Quantizationistheprocessofmappingasetofcontinuousamplitudes(infinitenumberoflevels)intoafinitenumberofpossiblevalues.Whilesamplingrepresentsthetimeofcapturingasignal,quantizationistheamplitudecomponentofsampling.Thesamplesarequantizedbyananalogtodigitalconverterinordertoberepresentedindigitalform.Weloadaudiosignalafterdoingdown-sampledandQuantizethesampledsignalusing8bitand10bit.Afterthat,weproceedtocalculatequantizederrorbyusingthefollowingformulaandapplyamethodcalledVQ(vectorquantization)todealwiththesampledaudiosignal.
Introduction:
Thereiserrorinvolvedwithquantizationduetotheapproximationofthelevelbeingquantized.
Thedifferencebetweenthemisquantizationerror,thedefinedas:
Quantizationerror=Actual-DigitalRepresentation.Wemaythinkofthequantizedsignalasanoisyversionoftheoriginalsignalwithoutquantificationtowhichanoisecomponenthasbeenadded.
Thepurposeofproject:
•TocharacterizethefrequencycontentofanaudiosignalbycalculatingtheFouriertransform.
•Tosimulatesamplingaffectsbydown-samplingtheoriginalbandwidthataratioof1/8.
•DeterminetheMagnitudeandphaseoftheFouriertransformsofthesampledsignal.
•Quantizethesampledsignalusingtwodifferentresolutions:
8bitand10bit.
•ReconstructandanalyzethequantizedsignalfromitsfrequencycontentbycalculatingtheinverseFouriertransform.
•
Analyzetheerrorsoccurredduringsamplingandquantizationandcomparethetworesolutionsused.
Quantizederror=
•usetrainingalgorithmforvectorquantization
Experimentalprocess:
Figure
(1)showsfourimages,whichareoriginalaudiosignalwaveformattimedomain,frequencydomain,itsrealfrequencyandphase.
Figure
(1)
Figure
(2)showsfourimagesafterdown-sampling
Figure
(2)
Figure(3)showsfourimagesofreconstructingthesampledsignalusing8bit
Figure(3)
Figure(4)showsfourimagesofreconstructingthesampledsignalusing10bit
Figure(4)
Figure(5)showstwoImagesoferroranalysisfor8bitand10bit
Figure(5)
wegettheresult:
•Quantizedby8bit.Thequantizederroris0.00378,whichisapproximatelytothesamplingprecision.(1/2e-8=0.0039)
•Quantizedby10bit.Thequantizederroris9.7599e-004,whichisapproximatelytothesamplingprecision.(1/2e-10=9.7656e-004)
•Byreconstructingthetwosampledsignalsandcomparingtheirvariances,wecanseethatmorebitsweused,moreaccuracyweget.
Here,wegivethedefinitionofVQ,amethodforquantization.Thenumberofscalardatasetstoformavector,andthengivethewholequantizationinthewholevectorspacethatcompressthedatawithoutlosingmuchinformation.Encodedintheimagevectorquantizationistheadvancedmethodthatapplyingintheimageandspeechsignalcodingtechniques.
Theprincipleoftrainingalgorithmforvectorquantizationis:
•Pickasamplepointatrandom
•Movethenearestquantizationvectorcentroidtowardsthissamplepoint,byasmallfractionofthedistance
•Eachvectoriscomparedwithacollectionofrepresentativecodevectors
•Usingaminimumdistortionruletochoosebestcodevector
Thefollowingtwoimagesshowthattherelationshipbetweenerrorandbothdimensionsizeandcodebooksize:
Fromtheabovetwoimages,wecanseethattheimageontheleftreflectstherelationshipbetweenerrorandcodebooksize.Ontheotherhand,theimageontherightreflectstherelationshipbetweenerroranddimensionsize.Tosumup,withcodebooksizeincreasedanddimensionsizefixed,theerrordecreases.Onthecontrary,withdimensionsizeincreasesandcodebooksizefixed,theerroralsoturnstohigher.
Wecanseemoreclearlyfromthefollowingimage,whichreflectsdifferentdimensionsizesinoneCoordinatesystemwithcodebooksizeincreasesandanalyzetheerror.
Thehorizontalordinateofthisimageiscodebooksizeandtheverticalordinateofthisimageiserror.Threedifferentcolorlinesshowsthedimensionsize,whichis4,8,16.
Conclusion:
1:
Intheprocessofquantization,byreconstructingthetwosampledsignalsandcomparingtheirvariances,wecanseethatmorebitsweused,moreaccuracyweget.
2:
Intheprocessofvectorquantization,wedrawtheconclusionthatwithcodebooksizeincreasedanddimensionsizefixed,theerrordecreases.Onthecontrary,withdimensionsizeincreasesandcodebooksizefixed,theerroralsoturnstohigher.
References:
[1]:
Dr.BrainVijay”quantizationdspslide”.
[2]:
AlanV.OppenheimRonaldW.Schafer”Discrete-TimeSignalProcessing”ThirdEdition,chapter4.8.
[3]:
[4]:
http:
//en.wikipedia.org/wiki/Vector_quantization
MatlabCode:
%Project2
clear
loadwav
wavplay(wav,fs);
%%
%originalsignalanalysis
figure
(1)
L=length(wav);
subplot(221)%waveformtimedomain
plot((1:
L)/L,wav);
xlabel('time(s)')
ylabel('Amplitude(v)')
gridon
axis([0,1,-0.1,0.15])
subplot(222)%frequencydomain
yf=fft(wav);
plot(yf);
xlabel('Realpart')
ylabel('Imaginarypart')
gridon
subplot(223)%realfrequency
MagY=abs(yf);
%MagY=20*log10(MagY/max(MagY));
AngY=angle(yf);
plot((1:
L)/L*fs,MagY)
xlabel('Frequency(Hz)')
ylabel('Amplitude')
gridon
axis([0,L,040])
subplot(224)%phase
plot(AngY);
xlabel('Samples')
ylabel('Phase')
gridon
%%
%downsamplinganalysis
downS=8;
wav1=wav(1:
downS:
L,:
);
L1=length(wav1);
yf1=fft(wav1);
MagY1=abs(yf1);
angY1=angle(yf1);%MagY=20*log10(MagY/max(MagY));
figure
(2)
subplot(221)
plot((1:
L1)/L1,wav1);
xlabel('Time(s)')
ylabel('Amplitude(V)')
gridon
subplot(222)%frequencydomain
plot(yf1);
xlabel('Realpart')
ylabel('Imaginarypart')
gridon
subplot(223)%realfrequency
plot((1:
L1)/L1*fs,MagY1)
xlabel('Frequency(Hz)')
ylabel('Amplitude')
gridon
axis([0,L,05])
subplot(224)%phase
plot(angY1);
xlabel('Samples')
ylabel('Phase')
gridon
%%
%quantization
na1=10;%quantizeresolution8bits
na2=8;
wav8=quant(wav1,na2);
wav10=quant(wav1,na1);
yf8=fft(wav8);
yf10=fft(wav10);
MagY8=abs(yf8);
Ang8=angle(yf8);
MagY10=abs(yf10);
Ang10=angle(yf10);
figure(3)
subplot(221)
plot((1:
L1)/L1,wav8);
xlabel('Time(s)')
ylabel('Amplitude(V)')
gridon
subplot(222)%frequencydomain
plot(yf8);
xlabel('Realpart')
ylabel('Imaginarypart')
gridon
subplot(223)%realfrequency
plot((1:
L1)/L1*fs,MagY8)
xlabel('Frequency(Hz)')
ylabel('Amplitude')
gridon
axis([0,L,05])
subplot(224)%phase
plot(Ang8);
xlabel('Samples')
ylabel('Phase')
gridon
%%
figure(4)
subplot(221)
plot((1:
L1)/L1,wav10);
xlabel('Time(s)')
ylabel('Amplitude(V)')
gridon
subplot(222)%frequencydomain
plot(yf10);
xlabel('Realpart')
ylabel('Imaginarypart')
gridon
subplot(223)%realfrequency
plot((1:
L1)/L1*fs,MagY10)
xlabel('Frequency(Hz)')
ylabel('Amplitude')
gridon
axis([0,L,05])
subplot(224)%phase
plot(Ang10);
xlabel('Samples')
ylabel('Phase')
gridon
%%
%erroranalysis
yf8=[yf8;zeros((downS-1)*length(yf8),1)];%upsampling
yf10=[yf10;zeros((downS-1)*length(yf10),1)];
wav8=ifft(yf8);
wav10=ifft(yf10);
error8=(real(wav8)-wav).^2;
er8=sum(error8)
error10=(real(wav10)-wav).^2;
er10=sum(error10)
figure(5)
subplot(211)
plot(1:
L,error8)
xlabel('Samples')
ylabel('Meansquareerror')
title('Quantizedby8bit')
subplot(212)
plot(1:
L,error10)
xlabel('Samples')
ylabel('Meansquareerror')
title('Quantizedby10bit')
%%
function[disdistortion]=VQ(dimension,codebook_size)
loadwav
Y=wav;
a_seq=downsample(Y(:
1)',8);
%dimension=2^ii
a_seq=[a_seqzeros(1,ceil(length(a_seq)/dimension)*dimension-length(a_seq))];
%codebook_size=2^4;
%codebook_size=2^ii;
tolerance=10^(-8);
m=round(length(a_seq)/dimension);
if(m*dimension-length(a_seq)<0)
error('')
end
initial=a_seq(1:
dimension*codebook_size);
initialcodebook=(reshape(initial,dimension,codebook_size))';
updated_codebook=initialcodebook;
newdistortion=0;
distortion=0;
fori=1:
m;
a_seq_block=a_seq((i-1)*dimension+1:
i*dimension);
training_matrix(i,:
)=a_seq_block;
distortion_block=[];
forj=1:
codebook_size;
distort=sum((a_seq_block-updated_codebook(j,:
)).^2);
distortion_block=[distortion_blockdistort];
end
[distortion_min,ind]=min(distortion_block);
newdistortion=newdistortion+distortion_min;
index(i)=ind;
end
forl=1:
codebook_size;
partition=(index==l);
ifsum(partition)>0
updated_codebook(l,:
)=partition*training_matrix/sum(partition);
end
end
newdistortion=newdistortion/m;
while(abs(distortion-newdistortion)/newdistortion>tolerance)
distortion=newdistortion;
newdistortion=0;
fori=1:
m;
a_seq_block=a_seq((i-1)*dimension+1:
i*dimension);
training_matrix(i,:
)=a_seq_block;
distortion_block=[];
forj=1:
codebook_size;
distort=sum((a_seq_block-updated_codebook(j,:
)).^2);
distortion_block=[distortion_blockdistort];
end
[distortion_min,ind]=min(distortion_block);
newdistortion=newdistortion+distortion_min;
index(i)=ind;
end
forl=1:
codebook_size;
partition=(index==l);
ifsum(partition)>0
updated_codebook(l,:
)=partition*training_matrix/sum(partition);
end
end
newdistortion=newdistortion/m;
end
codebook=updated_codebook;
disdistortion=newdistortion/dimension;
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