Is there a strong relationship between cepstrum of signal and 2nd derivative of signal?

In a recent thread about a very minimal voice recognition project, 
Rafael Deliano commented that the second derivative of the speech was 
used to characterize the signal.

My math is many decades in past, but I was wondering how cepstrum and 
second time derivative might be related.

On Jan 17, 2:21 am, Richard Owlett <rowl...@atlascomm.net> wrote:
> In a recent thread about a very minimal voice recognition project,
> Rafael Deliano commented that the second derivative of the speech was
> used to characterize the signal.
>
> My math is many decades in past, but I was wondering how cepstrum and
> second time derivative might be related.

Cepstrum is the Fourier Transform of the  Log of the Power Spectrum.
(nowadays it is often described as the inverse rather than the direct
Fourier Transform - this only alters the phase). The idea is that we
can convolve in the freq domain by adding since log(a.b) = log(a) +
log(b). We can separate (deconvolve) by subtraction since log(a/b) =
log(a)-log(b).

Now if we take the Cepstrum of a differentiated signal we need to
examine what the frequency somain representation of differentiation
is. For laplace it is multiplication by s and for fourier I suppose it
is jw (j omega)if we can get away with it.
The power spectrum would then only differ by w^2 from the original.

Taking logs we would get an extra term 2Log(w). We would then need to
Fourier transform the whole lot.
Maybe somebody can do the Maths.


H

> Is there a strong relationship between cepstrum of signal and 
> 2ndderivative of signal?
I doubt it. 

> minimal voice recognition project, 
> the second derivative of the speech was used to characterize the 
> signal.
Infinite peak clipping ( "1 Bit speech" ) is usually attributed to 
Licklider about 1946. The double differentiator as an improvement
in such systems comes in about 1970 by Thomas, Niederjohn. Many of 
these old papers are collected in: 
Lim "Speech Enhancement" Prentice Hall 1983  
Fits well in with Kryters Articulation Index 1962: 
http://www.embeddedforth.de/temp/dd2.pdf

During the 70ies many speech recognition systems used
double
http://www.embeddedforth.de/temp/dd1.pdf
and tripple differentiators
http://www.embeddedforth.de/temp/dd3.pdf
to better detect unvoiced sounds.

Interstate & clones ( 1 bit zero-crossing input ) 
only evaluate 3 states:
silence  ( low frequency )
voiced   ( medium  " )
unvoiced ( high  "  )
To get "silence" a small amount of threshold in
the KOP is required. For voiced / unvoiced separation one 
needs to enhance the weak unvoiced sounds. 

MfG JRD

Rafael Deliano wrote:
>>Is there a strong relationship between cepstrum of signal and 
>>2ndderivative of signal?
> 
> I doubt it. 
> 
> 
>>minimal voice recognition project, 
>>the second derivative of the speech was used to characterize the 
>>signal.
> 
> Infinite peak clipping ( "1 Bit speech" ) is usually attributed to 
> Licklider about 1946. The double differentiator as an improvement
> in such systems comes in about 1970 by Thomas, Niederjohn. Many of 
> these old papers are collected in: 
> Lim "Speech Enhancement" Prentice Hall 1983  
> Fits well in with Kryters Articulation Index 1962: 
> http://www.embeddedforth.de/temp/dd2.pdf
> 
> During the 70ies many speech recognition systems used
> double
> http://www.embeddedforth.de/temp/dd1.pdf
> and tripple differentiators
> http://www.embeddedforth.de/temp/dd3.pdf
> to better detect unvoiced sounds.
> 
> Interstate & clones ( 1 bit zero-crossing input ) 
> only evaluate 3 states:
> silence  ( low frequency )
> voiced   ( medium  " )
> unvoiced ( high  "  )
> To get "silence" a small amount of threshold in
> the KOP is required. For voiced / unvoiced separation one 
> needs to enhance the weak unvoiced sounds. 
> 
> MfG JRD

Thank you. Wish I could read German. There's enough English-German 
cognates in technical vocabulary to really tease me ;)

Googling for [ "articulation index" "speech recognition" -impaired ] 
yields an interesting collection of articles. Most are audiology 
related. Some are about artificial speech recognition. I get the 
impression that though "articulation index" has been replaced as method 
for speech recognition there is current research on using it as an aid 
in noisy environment. More reading ahead. Thanks again.

On Jan 17, 9:12 pm, Richard Owlett <rowl...@atlascomm.net> wrote:
> Rafael Deliano wrote:
> >>Is there a strong relationship between cepstrum of signal and
> >>2ndderivative of signal?
>
> > I doubt it.
>
> >>minimal voice recognition project,
> >>the second derivative of the speech was used to characterize the
> >>signal.
>
> > Infinite peak clipping ( "1 Bit speech" ) is usually attributed to
> > Licklider about 1946. The double differentiator as an improvement
> > in such systems comes in about 1970 by Thomas, Niederjohn. Many of
> > these old papers are collected in:
> > Lim "Speech Enhancement" Prentice Hall 1983
> > Fits well in with Kryters Articulation Index 1962:
> >http://www.embeddedforth.de/temp/dd2.pdf
>
> > During the 70ies many speech recognition systems used
> > double
> >http://www.embeddedforth.de/temp/dd1.pdf
> > and tripple differentiators
> >http://www.embeddedforth.de/temp/dd3.pdf
> > to better detect unvoiced sounds.
>
> > Interstate & clones ( 1 bit zero-crossing input )
> > only evaluate 3 states:
> > silence  ( low frequency )
> > voiced   ( medium  " )
> > unvoiced ( high  "  )
> > To get "silence" a small amount of threshold in
> > the KOP is required. For voiced / unvoiced separation one
> > needs to enhance the weak unvoiced sounds.
>
> > MfG JRD
>
> Thank you. Wish I could read German. There's enough English-German
> cognates in technical vocabulary to really tease me ;)
>
> Googling for [ "articulation index" "speech recognition" -impaired ]
> yields an interesting collection of articles. Most are audiology
> related. Some are about artificial speech recognition. I get the
> impression that though "articulation index" has been replaced as method
> for speech recognition there is current research on using it as an aid
> in noisy environment. More reading ahead. Thanks again.

Hello Richard, I will put my two cents on it.

Actually speech recognition is done via Hidden Markov Models (HMMs),
which are 'fed' with a cepstral coefficients obtained by framing the
voice signal. In has been known for quite a long time that speech
dynamics, that is the rate of variation of voice characteristics,
yields very important information but including this in directly in
HMMs framework is not possible. So, what has been done is to obtain
directly the first derivatives (or delta) coefficients of the cepstral
vector, and obtain the second derivatives (delta-delta) coefficients.
This is added to the features vector, speech recognition results have
improved and this has become standard in today's speech recognition
systems.

The articulation index is a measure of speech intelligibility made by
Fletcher in early experiments on AT&T for telephone communications.
You can find more about it in the paper HOW DO HUMANS PROCESS AND
RECOGNIZE SPEECH? by J.B Allen appeared in IEEE Transactions on Speech
and Audio Processing,  volume 2,  number 4,1994.

Regards

Juan Pablo

On Jan 16, 8:21&#4294967295;am, Richard Owlett <rowl...@atlascomm.net> wrote:
> In a recent thread about a very minimal voice recognition project,
> Rafael Deliano commented that the second derivative of the speech was
> used to characterize the signal.
>
> My math is many decades in past, but I was wondering how cepstrum and
> second time derivative might be related.

Hi Richard,

Complex cepstrum (includes effect of FFT phase) or real cepstrum (no
FFT phase used)?

Dirk

dbell wrote:
> On Jan 16, 8:21 am, Richard Owlett <rowl...@atlascomm.net> wrote:
> 
>>In a recent thread about a very minimal voice recognition project,
>>Rafael Deliano commented that the second derivative of the speech was
>>used to characterize the signal.
>>
>>My math is many decades in past, but I was wondering how cepstrum and
>>second time derivative might be related.
> 
> 
> Hi Richard,
> 
> Complex cepstrum (includes effect of FFT phase) or real cepstrum (no
> FFT phase used)?
> 
> Dirk

Guess I know even less than I thought ;)
So I suspect the appropriate answer would be "both/either".
My goal is to know what people are talking about than being able to 
apply it. I had ~2 years of college math >40 years ago, and then 
actually used little of it.

On Jan 17, 1:47&#4294967295;pm, Richard Owlett <rowl...@atlascomm.net> wrote:
> dbell wrote:
> > On Jan 16, 8:21 am, Richard Owlett <rowl...@atlascomm.net> wrote:
>
> >>In a recent thread about a very minimal voice recognition project,
> >>Rafael Deliano commented that the second derivative of the speech was
> >>used to characterize the signal.
>
> >>My math is many decades in past, but I was wondering how cepstrum and
> >>second time derivative might be related.
>
> > Hi Richard,
>
> > Complex cepstrum (includes effect of FFT phase) or real cepstrum (no
> > FFT phase used)?
>
> > Dirk
>
> Guess I know even less than I thought ;)
> So I suspect the appropriate answer would be "both/either".
> My goal is to know what people are talking about than being able to
> apply it. I had ~2 years of college math >40 years ago, and then
> actually used little of it.- Hide quoted text -
>
> - Show quoted text -

Richard,

By "strong relationship" do you mean are they similar or that one is
in some (possibly complex) way related to the other?

Dirk

dbell wrote:

> On Jan 17, 1:47 pm, Richard Owlett <rowl...@atlascomm.net> wrote:
> 
>>dbell wrote:
>>
>>>On Jan 16, 8:21 am, Richard Owlett <rowl...@atlascomm.net> wrote:
>>
>>>>In a recent thread about a very minimal voice recognition project,
>>>>Rafael Deliano commented that the second derivative of the speech was
>>>>used to characterize the signal.
>>
>>>>My math is many decades in past, but I was wondering how cepstrum and
>>>>second time derivative might be related.
>>
>>>Hi Richard,
>>
>>>Complex cepstrum (includes effect of FFT phase) or real cepstrum (no
>>>FFT phase used)?
>>
>>>Dirk
>>
>>Guess I know even less than I thought ;)
>>So I suspect the appropriate answer would be "both/either".
>>My goal is to know what people are talking about than being able to
>>apply it. I had ~2 years of college math >40 years ago, and then
>>actually used little of it.- Hide quoted text -
>>
>>- Show quoted text -
> 
> 
> Richard,
> 
> By "strong relationship" do you mean are they similar or that one is
> in some (possibly complex) way related to the other?
> 
> Dirk

I do not know.
I do not know even what questions I should be asking.
I hope others would chime in with questions I _should_ *ASK*
    [hint to BCC recipients ;]

> I get the impression that though "articulation index" has been 
> replaced as method for speech recognition there is current
> research on using it as an aid in noisy environment. 
It wasn�t a method of speech recognition, but used to a measure
speech intelligibility in noise. Old ANSI-standard, about 1969, not
so common anymore. But it shows which parts of the spectrum are 
important. 
It should be compared to the long term spectrum of speech:
http://www.embeddedforth.de/temp/hoth.pdf
( > Wish I could read German.
  Looking at the pictures is usually sufficient ... )
Normally there is not much energy in the unvoiced parts of speech.  
  The hoth noise mentioned there is a simulation of the noise 
in domestic/office environment. Its low frequency.
Both reasons for the (double-)differentiator. 

Thats the original circuit for the VCP200 ( plus much German text ): 
http://www.embeddedforth.de/temp/vcp1.pdf

The very first speech recognition unit 
1952 Davis, Biddulp, Balashek
did use a different circuit with two channels: 
http://www.embeddedforth.de/temp/vcp2.pdf
Resurfaced in the 70ies again. 

What do you want to build ?

MfG  JRD