CEPSTRAL ANALYSIS AND HOMOMORPHIC DECONVOLUTION

CHAPTER 1: INTRODUCTION

 A signal coming out from a system is due to the input excitation and also the response of the system. From the signal processing point of view, the output of a system can be treated as the convolution of the input excitation with the system response. At times, we need each of the components separately for study and/or processing. The process of separating the two components is termed as deconvolution.

 In the first case, if we knew the input excitation, then the system component can be separated/ constructed by exciting the system with the inputs and collecting its responses. This is what is done in same channel  estimation problems. In the second case, if we knew the system response, then the input excitation can be recovered using the inverse filter theory concept. For instance, Linear Prediction(LP) analysis of speech to recover excitation. There is yet another type of deconvolution, where the assumption is both input excitations as well as system responses are unknown. The present study of cepstral analysis of speech comes under this category.

Speech is composed of excitation source and vocal tract system components. In order to analyze and model the excitation and system components of the speech independently and also use that in various speech processing applications, these two components have to be separated from the speech. The objective of cepstral analysis is to separate the speech into its source and system components without any a priori knowledge about source and / or system. According to the source filter theory of speech production, voiced sounds are produced by exciting the time varying system characteristics with periodic impulse sequence and unvoiced sounds are produced by exciting the time varying system with a random noise sequence. The resulting speech can be considered as the convolution of respective excitation sequence and vocal tract filter characteristics. If e(n) is the excitation sequence and h(n) is the vocal tract filter sequence, then the speech sequence s(n) can be

expressed as follows:S(n)=e(n)*h(n)          (1)

 This can be represented in frequency domain as,S(w)=E(w)*H(w)                  (2)

The Eqn. (2) indicates that the multiplication of excitation and system components in the frequency domain for the convolved  sequence of the same in the time domain. The speech sequence has to be deconvolved into the excitation  and vocal tract components in the time domain. For this, multiplication of the two components in the frequency domain has to be converted to a linear combination of the two components. For this purpose cepstral analysis is used for transforming the multiplied source and system components in the frequency domain

to linear combination of the two components in the cepstral domain.

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Basic Principles of Cepstral Analysis:

From the Eqn. (2) the magnitude spectrum of given speech sequence can be represented as,

                       

To linearly combine the E(ω) and H(ω) in the frequency domain, logarithmic representation is used. So the logarithmic representation of Eqn. (3) will be,

        

As indicated in Eqn. (4), the log operation transforms the magnitude speech spectrum where the excitation component and vocal tract component are multiplied, to a linear combination (summation) of these components i.e. log operation converted the "*" operation into "+" operation in the frequency domain. The separation can be done by taking the inverse discrete fourier transform (IDFT) of the linearly combined log spectra of excitation and vocal tract system components. It should be noted that IDFT of linear spectra transforms back to the time domain but the IDFT of log spectra transforms to quefrency domain or the cepstral domain which is similar to time domain. This is mathematically explained in Eqn. (5). In the quefrency domain the vocal tract components are represented by the slowly varying components concentrated near the lower quefrency region and excitation components are represented by the fast varying components at the higher quefrency region.

 

Fig 1 details the various steps involved in converting the given short term speech signal to its cepstral domain representation. The output obtained at different stages of cepstrum computation as described in Figure 1, is given in Figure2. In Fig 2, s(n) is the voiced frame considered and x(n) is the windowed frame. Here s(n) is multiplied by a hamming window to get x(n). |x(ω)| in Fig 2 represent the spectrum of the windowed sequence x(n). As the spectrum of the given frame is symmetric, only one half of the spectral components is plotted. The  log|x(ω)| represents the log magnitude spectrum obtained by taking logarithm of the |x(ω)|. c(n) of Fig 2 shows the computed spectrum for the voiced frame s(n). The obtained cepstrum contains vocal tract components which are linearly combined according Eqn.(5). As the cepstrum is derived from the log magnitude of the linear spectrum, it is also symmetrical in the quefrency domain. Here also only one symmetric part of the cepstrum

 is used for plotting.