REAL AND COMPLEX CEPSTRUM

MANJUNATH.V.SHETH MTECH I SEM,DEC NMAMIT

CONTENTS1. 2. 3. 4. 5. 6. 7.

INTRODUCTION CLASSIFICATION PROPERTIES COMPARISION APPLICATIONS CONCLUSION REFERENCES

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INTRODUCTION

Proposed by TUKEY Cepstrum is anagram of spectrum Defination: Result of taking fourier transform of the decibel spectrum Purpose of Cepstrum: Reconstruction of Original Signal

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INTROCepstral Analysis

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INTRO

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CLASSIFICATION

Real Cepstrum:The logarithm function defined for real values. Only uses the information of the magnitude of the spectrum. The real cepstrum is used to design an arbitrary length minimum-phase FIR filter.

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REAL CEP ..

Inverse Fourier transfrom of the logarithm of the magnitude of the Fourier transform

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REAL CEP ..Then real cepstum is found by using the formula Cs(n)=F 1[log|F[S(n)]|] 1[log|F[S(n)]|] =1/2 log |s(w)|e^jwn dw

Cs(w)=log |s( )|=log |E( ) ( )| =log|E( )|+log| ( ) )|+log| Cs(w)=Ce( Cs(w)=Ce( )+C ( ) Cs(n)=1/2 log |Cs( )|e^j n d )|e^j

Cs(n) is the cepstrum Cs(w) is the spectrum3/10/2011 REAL AND COMPLEX CEPSTRUM 8

REAL CEP

The full real cepstrum. Calculated using Matlab: ifft(log(abs(fft(hamming(512) .* sig))))

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COMPLEX CEPSTRUM

The complex cepstrum holds information about magnitude and phase of the initial spectrum. The term complex cepstrum refers to the use of the complex logarithm, not to the sequence. The complex cepstrum of a real sequence is also a real sequence.

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COMPLEX CEP

Cepstrum analysis

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COMPLEX CEP ..

Complex cep formula

arg is the continuous (unwrapped) phase function.REAL AND COMPLEX CEPSTRUM 12

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COMPLEX CEPSThe following example uses cceps to show an echo. Fs = 100; t = 0:1/Fs:1.27; % 45Hz sine sampled at 100Hz s1 = sin(2*pi*45*t); % Add an echo with half the amplitude and 0.2 second later s2 = s1 + 0.5*[zeros(1,20) s1(1:108)]; c = cceps(s2); plot(t,c) Notice the echo at 0.2 second.

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COMPLEX CEPS..

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PROPERTIES OF CEPS1) The complex cepstrum decays at least as fast as 1/|n| 2) It has infinite duration, even if x[n] has finite duration 3) It is real if x[n] is real (poles and zeros are in complex conjugate pairs)

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PROPERTIES.4) The complex cepstrum is causal (0 for all n0 if and only if x[n] is maximum phase, i.e. X(z) has all poles and zeros outside the unit circle.

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COMPARISION OF CEPSREAL CEPSTRUM Discards Phase Easy to compute Speech Analysis Recognition COMPLEX CEPSTRUM Retains Phase Difficult to compute Vocoders and speech coding

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MATLAB CODEGet a section of vowel [x,fs]=wavread('six.wav',[24120 25930]); ms1=fs/1000; % maximum speech Fx at 1000Hz ms20=fs/50; % minimum speech Fx at 50Hz % plot waveform t=(0:length(x)t=(0:length(x)-1)/fs; % times of sampling instants subplot(3,1,1); plot(t,x); legend('Waveform'); xlabel('Time (s)'); ylabel('Amplitude'); % do fourier transform of windowed signal Y=fft(x.*hamming(length(x))); % plot spectrum of bottom 5000Hz hz5000=5000*length(Y)/fs; f=(0:hz5000)*fs/length(Y);

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subplot(3,1,2); plot(f,20*log10(abs(Y(1:length(f)))+eps)); legend('Spectrum'); xlabel('Frequency (Hz)'); ylabel('Magnitude (dB)'); % cepstrum is DFT of log spectrum C=fft(log(abs(Y)+eps)); % plot between 1ms (=1000Hz) and 20ms (=50Hz) q=(ms1:ms20)/fs; subplot(3,1,3); plot(q,abs(C(ms1:ms20))); legend('Cepstrum'); xlabel('Quefrency (s)'); ylabel('Amplitude');

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FIGURE

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APPLICATIONS

Processing signals containing echoes seismology, measuring properties of reflecting surfaces, loudspeaker design,dereverberation, restoration of acoustic recordings Speech processing estimating parameters of the speech model

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APPLICATIONS.

Machine diagnostics detection of families of harmonics and sidebands, for example in gearbox and turbine vibrations. Calculating the minimum phase spectrum corresponding to a given log amplitude spectrum with hilbert transform.

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CONCLUSION

New concepts of cepstrum are useful for computing and variety of applications. Effective method to construct minimum phase FIR filter from mixed phase. We require only two ffts and recursive procedure to compute impulse response hence reduce complexity.

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CONCLUSION.

The new concept of differential cepstrum is useful in computing four times faster than other methods.

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REFERENCES

Rabiner and Juang Fundamentals Of Speech Recognition John G Proakis Digital Signal Processing www.wikipedia.org

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THANK YOU

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