Homomorphic filtering and speech processing using cepstrum analysis

OutlineIntroduction of Homomorphic

filteringHomomorphic SystemsZ-transform in HomomorphicApplication in speech processingVoiced and unvoiced speechCepstral Analysis of windowsConclusion

IntroductionFiltering is a non-linear transformation

Applied to the image and speech processing

Used to convert a signal from a convolution of two original signal into the sum of two signals

Consider some linear transformation L

L is a linear system it will satisfy the principle of superposition

Define a class where addition is replaced by convolution

System having this property are known as Homomorphic systems for convolution

Homomorphic system has a important property is that they can be viewed as a cascade of three Homomorphic systems

Homomorphic Systems The first system takes inputs

combined by convolution and transforms them into an additive combination of the corresponding outputs

D* is a Homomorphic system in which convolution is converted in to the addition

Contd………. The second system is a conventional linear

system that obeys the principle of superposition

Some linear System

The third system is the inverse of the first system: it transforms signals combined by addition into signals combined by convolution

Some inverse Homomorphic system in which addition is converted in to convolution

This is important because design of such systems reduces to the design of linear system

Z-transformZ-transform of two convolved signals

is the product of their z-transforms

Then take logs to obtain

so log of Z-transform is viewed as a Homomorphic systems

The frequency domain representation of a Homomorphic system for deconvolution can be represented as

Represent signals as sequence rather than in the frequency domain, then the systems ∗[ ]and 𝐷

∗𝐷 −1[ ] can be represented as

speech processing Homomorphic systems are very

frequently used in speech processing applications

We have to separate the excitation from the vocal tract filter h(n) by using a Homomorphic transformation

Do so easily as the filter parameters usually reside in the lower quefrencies

While the excitation parameters have higher quefrencies

We have to recover filter’s response from a periodic signal ( such as a voiced signal excitation)

The filter response can be recovered if we can separate the output of the Homomorphic transformation using a simple filter

Deconvolution of speech

Cepstrum of a generic voiced signal

Contributions to the cepstrum due to periodic excitation will occur at integer multiples of the fundamental period. NOTE that for children and high-pitch women we might have a problem

Contributions due to parameters usually modeled by the filter will concentrate in the low quefrequency region and will decay quickly with n

Cepstral analysis of speech (voiced signals)

Cepstral analysis of speech (unvoiced signal)

Cepstral analysis of vowel (rectangular window)

Cepstral analysis of vowel (tapering window)

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