Audio Enhancement and Robustness - Intelligent Audio Analysis - page 155

non-linguistic vocalisations [ 51 ] or the segmentation of meeting speech [ 52 ]. A par-
ticular strength is the possibility to use arbitrary functions for the observations without
complication of the parameter learning.
The HCRF models the conditional probability of a class c , given the sequence of
observations X = x 1 , x 2 ,..., x T :

1

e λ f ( c , Seq , X ) ,

p ( c | X ,λ) = (9.24)

z ( X ,λ)

Seq ∈ c

where λ is the parameter vector and f the 'vector of sufficient statistics', and Seq =

s 1 , s T is the hidden state sequence run through during the computation of

this conditional probability. The probability is normalised by the 'partition function'
z

s 2 ,...,

( X ,λ) to ensure a properly normalised probability [ 15 ]:

e λ f ( c , Seq , X ) .

z ( X ,λ) = (9.25)

c Seq ∈ c

The vector f determines the probability to model. With a suited f a left-right
HMM can be imitated [ 15 ]. Let us now now restrict the HCRF to a Markov chain,
but without the requirements of the transition probabilities to sum to one and real
probability densities for the observations. In analogy to a HMM a parametrisation
by transition scores a i , j and observation scores b j (

x t ) can then be reached with the

parameters , where and i and j are states of the model (cf. Sect. 7.3.2 ) . Forward

and backward recursions (cf. Sect. 7.3.1 ) as for a HMM can then further be used.

λ

9.3.3 Audio Modelling in the Time Domain

Modelling of the raw signal in the time domain is a sparsely pursued option, but can
offer easy explicit noise modelling [ 16 ]. We will look at SAR-HMMs to this end
first, and then at the extension to SLDS.

9.3.3.1 Switching Autoregressive Hidden Markov Models

The SAR-HMM models the audio signal of interest as an autoregressive (AR)
process. The non-stationarity is realised by switching between different AR parame-
ter sets [ 17 ] by a discrete switch variable s t similar to the HMM states. At a time
step t —referring to the sample-level in this case—, exactly one out of S states is
occupied. The state at time step t depends exclusively on its predecessor with the
transition probability p v t at this time step is assumed as a

linear combination of its R preceding samples superposed by a Gaussian distributed
'innovation'

( s t | s t − 1 ) .Thesample

η( s t ) . η( s t ) and the AR weights c r ( s t ) are the parameter set given by

the state s t :