Newsgroups: comp.speech
Path: lyra.csx.cam.ac.uk!pipex!uknet!cf-cm!cybaswan!eerichar
From: eerichar@uk.ac.swan.pyr (h b richards)
Subject: Re: Q : LPC - Spectrum
Message-ID: <1994Apr21.184043.3229@uk.ac.swan.pyr>
Organization: Swansea University College
References: <kordmann.766763181@ldv01> <2p2poa$bol@infa.central.susx.ac.uk> <2p3fpb$bd@dr-pepper.East.Sun.COM>
Date: Thu, 21 Apr 1994 18:40:43 GMT
Lines: 44

In article <2p3fpb$bd@dr-pepper.East.Sun.COM> mscordilis@antonia.East.Sun.COM ( Michael Scordilis - Contractor ) writes:
>In article <2p2poa$bol@infa.central.susx.ac.uk> tapj6@central.susx.ac.uk (Rod Dorrell) writes:
>>and updated periodically as the frame shifts in time. To compute the
>>vocal tract spectrum from the LPC coefficients, you could determine the
>>impulse response of the above mentioned IIR filter and then perform an
>>FFT on this response. 
>
>This is an unnecessary complication.
>Just express the transfer function of the system in terms of the LPC coeffs.
>(constant-gain numerator, nth order polynomial denominator).
>Then evaluate the function on the unit circle (z=cos(a) +jsin(a)), varying
>angle a from 0 to pi.  The magnitude of the function is the magnitude
>of the desired spectrum. Angle pi corresponds to analog frequency Fs/2, 
>where Fs is the sampling rate.
>
>>Rod Dorrell (Bio-medical Engineering, University of Sussex, Brighton,
>>UK)
>
>Michael Scordilis
>

I think the most efficient way of evaluating the spectrum of an LPC
filter is to first calculate the spectrum of the *whitening* filter.
As this is an FIR filter the impulse response will only be a few
samples long and these samples are equal to the LPC coefficients,
i.e.  1, a(1), a(2), ... a(n), 0, 0, 0....
The FFT can then be used to calculate the spectrum to the most
convenient power of two giving sufficient spectral resolution.

As this gives the spectrum of the whitening filter however, the
response of the vocal tract model can be obtained by calculating the
inverse of the power spectrum, or more conveniently negating the
log spectrum which has been derived from the FFT.

All of the zeros in the impulse response also means that the FFT
calculation can be speeded up if you're keen enough to write the
software...
Anyway, a look at Markel and Grey "Linear Prediction of Speech" 1976
page 159 might be useful if you've got it to hand.

H.B.Richards.

eerichar@uk.ac.swan.pyr

