Zoom out Search Issue

ManualsBrandsContents Manualsaudio & home theatreZoom in

IEEE SIGNAL PROCESSING MAGAZINE [45] MARCH 2015

is nonnegative and cannot be larger than the entropy .()H M

Thus, the difference (, ) () (; )DHIMM M MM

LT T LT

˘˘ ˘˘˘

=- is non-

negative and can be interpreted as a distortion. It can be written

as a general distortion measure operating on

|LT

for a given

talker message distribution :p

( , ) ( ) ( ( | )),DpMdpMMMM

LT L T

˘˘

{{{

{{

(2)

where d is a nonnegative function of ( | ).pMM

|LT L T

{{

For the

mutual information based distortion measure ((|))dp M M

|LT L T

{{

( | ) ( ( )/ ( , )),logpMM pMpMM

|,LTL T LLLTLT

{{ { {{

where we note that the

argument of the logarithm can be written in terms of (|)pMM

|LT L T

{{

and the given ()pM

{

only. The intelligibility enhancement

problem is to find the p

|LT

that minimizes the distortion (2)

subject to the constraints set by the scenario.

An alternative to the mutual information based distortion

measure can be based on the hit-or-miss distortion,

( (|)) (|)( ),dp M M p M M 1

|| ,LT L T LT L T M M

d=-

{{ {{

{{

where

,MM

{{

is a

Kronecker delta function. In this case (2) becomes

(, ) (, ) [ ( | )].EDpMMp11MM M M

|LT

TT TLT

˘˘ ˘ ˘

=- =-

{{

{

(3)

The conditional probability (|)pMM

|LT T T

{{

in (3) corresponds to

the probability that the message is interpreted correctly. Thus, an

alternative to maximizing the mutual information of the conveyed

and received message is to maximize the expected probability of

correct message interpretation,

[ ( | )],pE MM

|TLT

˘˘

where the

expectation is over the conveyed messages, .M

We will discuss

the practical use of this high-level measure in the section “Mea-

sures Operating on a Word Sequence.”

While the measures (1) and (3) are general, they cannot be

used directly. Either the description of the message or the

human cognitive system must be approximated such that the

measures can be applied to observable signals. The paradigm

shows where such approximations are made, but it does not

show their quantitative impact. Thus, experiments must be used

to verify the validity of the resulting system.

Next, we consider how to derive a low-level, acoustics-based

measure from a high-level, message-based measure. For this it is

convenient to consider the message as a continuous variable. A

conveyed speech message

is rendered in the form of an acous-

tic signal, which we represent by an acoustic sequence a

. The

sequence a

can, for example, consist of signal samples or short-

term spectral descriptions, such as cepstral vectors. This sequence

is rendered in a noisy environment and the listener observes a cor-

rupted sequence

which is then interpreted as a message .M

The communication process thus forms a Markov chain

.MaaM

TTL L

""" It is natural that environmental noise

makes the mapping aa

" stochastic.

Upon reflection, it is clear that the mappings Ma

" and

" are also stochastic: a message is generally not formulated

and never articulated in precisely the same manner, and the inter-

pretation of the acoustic sequence

is subject to random varia-

tions during the human cognitive process. Anticipating the

discussions in the section “Measures Operating on a Word

Sequence,” it can be argued that these variations are captured by

the statistical modeling of modern automatic speech recognition

(ASR) algorithms. If we assume the message formulation is perfect,

a simple but effective model of the production and interpretation

processes is that they are subject to additive noise components [15],

which we will refer to as, respectively, production noise and inter-

pretation noise. For example, variability in articulation across differ-

ent persons may be approximated as additive noise in a

representation based on cepstral or log spectral vectors.

For convenience let us define auxiliary bijective mappings

) and ,Ms

) where s

and s

are realizations of ran-

dom acoustic sequences. We have

asv

TTT

aav

LTE

=+ (4)

,sav

LLL

where ,v

and v

are additive noise processes, modeling the

production noise, environmental noise, and interpretation noise,

respectively. Note that the system model differs from the stan-

dard system model in communication theory, which does not

include production noise and interpretation noise.

To facilitate analysis, let us assume the sequences

and v

to be jointly Gaussian processes. Furthermore, we denote

t the correlation coefficient of (the samples of the) processes

s and a and write .

0 sa as

ttt=

TT LL

Let us first consider the case

where the signals are white. Exploiting that mutual information is

invariant under reparametrization of the marginal variables, it is

then easy to see that [15]

(; ) (;)

()

,logII

MM ss

LT TL

==-

(5)

where (/)

pvv=

is the signal-to-noise ratio (SNR) of the

acoustic channel ,aa

" and

and

are the variances of

processes a

and ,v

respectively. An important and intuitive con-

clusion that can be drawn from (5) is that if the environmental

noise variance is small compared to the production and interpreta-

tion noise variances, then the mutual information between talker

and listener is not affected significantly by the environmental noise.

The spectral coloring of the acoustic content can be accounted

for by splitting the signal into spectral bands such that each band

can be approximated as white. If we assume the signals to be sta-

tionary, the frequency bands are independent and the mutual

information can be written as the sum of the mutual informations

in the bands

(; )

()

,logI

(6)

where i is the band index and where ( / )

pvv=

,,Ti Ei

is the SNR

of the acoustic channel in band .i Note that the SNR in (6) is com-

puted on whichever representation is used for the acoustic fea-

tures. Also note that the variances

,Ti

and

,Ei

are generally

unknown and must be estimated in practice. For example, if the

THE WORLD’S NEWSSTAND

THE WORLD’S NEWSSTAND