The Stochastically Excited Linear Prediction (SELP) algorithm is described as a speech coding method employing a two-stage vector quantization. The first stage uses an adaptive codebook which efficiently encodes the periodicity of voiced speech, and the second stage uses a stochastic codebook to encode the remainder of the excitation signal. The adaptive codebook performs well when the pitch period of the speech signal is larger than the frame size. An extension is introduced, which increases its performance for the case that the frame size is longer than the pitch period. The performance of the stochastic stage, which improves with frame length, is shown to be best in those sections of the speech signal where a high level of short-term correlations is present. It can be concluded that the SELP algorithm performs best during voiced speech where the pitch period is longer than the frame length

Ketchum, R. H.

Kleijn, W. B.

Krasinski, D. J.

English

NASA Technical Reports Server

5/88-25756
ANALYSIS AND IMPROVEMENT O1_ THE VECTOR QUANTIZATION IN SELP
W. B. KLEIJN, D. J. KRASINSKI, and R. H. KETCHUM
AT&T BELL LABORATORIES
200 Park Plaza
Naperville, IL 60510, USA
ABSTRACT
The SELP algorithm is described as a speech coding method
employing a two-stage vector quantization. The first stage uses an
adaptive codebook which efficiently encodes the periodicity of voiced
speech, and the second stage uses a stochastic codebook to encode
the remainder of the excitation signal. The adaptive codebook
performs well when the pitch period of the speech signal is larger
than the frame size. An extension is introduced, which increases its
performance for the case that the frame size is longer than the pitch
period. The performance of the stochastic stage, which improves
with frame length, is shown to be best in those sections of the
speech signal where a high level of short-term correlation is present.
It can be concluded that the SELP algorithm performs best during
voiced speech where the pitch period is longer than the frame
length.
1. INTRODUCTION
The Stochastically Excited Linear Prediction (SELP) algorithm for speech coding,
originally proposed by Atal and Schroeder [1], is a method of vector quantizing the sampled
speech signal on a frame-by-frame basis. The algorithm vector quantizes a signal which,
ideally, contains no short-term correlation. When the quantized signal is filtered through
an appropriate linear filter (determined with LPC methods) to provide the correct short-
term correlation, synthetic speech results. The signal can be synthesized at the receive end
of the speech coder by transmitting both an index to the particular excitation vector
selected from the codebook and a description of the filter.
The effect of the present frame excitation on the synthetic speech in the present frame
can be expressed as a convolution of the excitation with the filter impulse response. This
convolution can be implemented as a matrix multiplication of the excitation vector with a
lower-triangular Toeplitz matrix H containing the filter impulse response along the first
column. The synthetic speech signal y produced by a candidate excitation vector r is now:
y = + (1)
where p is a scaling factor, and z represents the zero-input response of the filter describing
the present frame response to the preceding excitation.
Ideally the excitation vector r results in a synthetic speech signal identical to the
original speech. It is this ideal "target" excitation vector, t, which is vector quantized in
the SELP algorithm. The squared error between the original speech signal and the
synthetic speech signal can be used as the quantization error criterion. Expressed in terms
of excitation vectors, it becomes:
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https://ntrs.nasa.gov/search.jsp?R=19880016372 2020-03-20T06:50:50+00:00Z
= (t-_r)THTH(t-_r). (2)
The error criterion described represents the squared error of the synthetic speech signal. It
does not consider that uniform matching of the speech signal over its entire spectrum does
not minimize the perceived noise level. The error signal is less perceptible in those sections
of the spectrum which contain relatively high energy (the formants). Thus, it is
advantageous to move more of the error signal under the formants, while reducing the error
signal in other regions [2]. This can be accomplished by moving the poles of the filter
inward by a factor % which is chosen in the range between 0.7 and 0.9. An added
advantage of this _noise-weighting" procedure is that it dampens the impulse response
significantly, so that it can be truncated at a relatively short length. From here on we
assume that the matrix H includes noise weighting.
The development of fast algorithms benefits from increased symmetry. By truncating
the impulse response after R samples, and considering this entire response length for each
sample of the excitation vector in the error criterion, a more symmetric spectral-weighting
matrix HTH can be created [3]. This modified error criterion compares the response of the
excitation vector to that of the target excitation vector in the present as well as subsequent
frames. The resulting HTH matrix is still NxN but is now a Toeplitz band matrix. A fast
algorithm which takes advantage of this symmetry is described in [3].
Two codebooks axe used in our SELP implementation. The first codebook is adaptive,
adjusting to the periodicity of the speech signal. The method is similar to procedures
described in [4] and [5]. The adaptive codebook is constructed by buffering the synthetic
excitation. If M samples of past excitation are stored, and the frame length is N samples,
then the adaptive codebook contains M-N-}-1 distinct candidate vectors. Neighboring
candidate vectors differ by a shift of one sample.
Because of the lineaxity of the filtering procedure, the excitation vector selected from
the adaptive codebook can be subtracted from the taxget excitation vector t to obtain a
new target vector for the second vector quantization procedure. Since the LPC filter
models the short-term correlation, and the adaptive codebook the long-term correlation of
the speech signal, the new target vector has a noisy character. The stochastic codebook
consists of a set of vectors containing normally distributed samples. It is often center
clipped to increase computational speed [3], [6], [7].
During the vector quantization procedures of both the adaptive and stochastic
codebooks all candidate vectors axe scaled by optimal scaling factors. This allows for the
large dynamic range present in speech. It is easily shown from equation (2) that the
optimal scaling factor p for a candidate vector r takes the value rTHTHt/r THTHr. The
scaling factors also must be transmitted to the synthesizer.
2. ANALYSIS AND IMPROVEMENT OF SELP
2.1 The Adaptive Codebook
The adaptive codebook procedure performs best in voiced sections of speech. The
codebook vector selected usually contains a section of synthetic excitation that had been
applied one or more integer number of pitch periods prior to the present frame. Figs. la
and lc illustrate the probability density function for the delay (in samples) between the
selected candidate vector and the present vector for speech from a female and a male
speaker, respectively. The probability density function for each speaker was approximated
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from an analysis of 10000 frames of 8 kHz sampled speech using a SELP coder with two
codebooks of 256 vectors and a frame length of 40 samples. The pitch period of the
speakers is clearly displayed in these distributions. For the male speaker a delay of one
pitch period is most prevalent, but two and three pitch delays contribute significantly also.
For the female speaker, a delay of one pitch period cannot be chosen, because the frame
length is longer than the pitch period. Instead, delays of two through five integer pitch
periods are prevalent.
The subjective performance of the SELP coder is better for the speech signal with a
pitch period longer than the fra_m__elength. However, we have found that, at fixed _,_ot,,_l,
size, the segmental signal to noise ratio decreases smoothly as a function of increasing frame
length. Thus, the modeling accuracy is not affected significantly by the frame length being
longer or shorter than the pitch period. However, only if the frame length is shorter than
the pitch period will noise resulting from inaccurate modeling be repeated at the same rate
as the pitch pulses. This leads to desirable harmonics in spectral regions which are
numerically unimportant in the evaluation of the error criterion. The harmonics at the
high end of the spectrum, which contains less energy, are often created from noise signals.
This effect is eliminated when the pitch period is shorter than the frame length.
Since (at constant codebook size) the bit rate of the SELP coder is inversely related to
frame length, it is desirable to minimize the negative impact of increased frame length.
This can be accomplished with an extension of the adaptive codebook which results in
enhanced periodicity [3]. In the conventional method, the candidate vector containing the
most recent synthetic excitation starts exactly N samples before the present frame. A
candidate vector starting at a more recent sample is incomplete. However, if the existing
portion of such a vector is presented in the first part of the present frame, it will create a
periodicity. If the vector starts p samples previous to the present frame, then the
periodicity will be p samples. This periodicity can be maintained by repeating the existing
portion of the vector again, starting p samples into the present frame. Thus, the adaptive
codebook can be extended with nvirtual" vectors with a period equal to the length of the
existing portion.
°f
d
0
G
d
0
oi
d
0
°f
d
0
50 IO0 150 200 2.50 300
a) female speaker
50 1O0 150 200 250
b) female speaker, extended codebook
300
50 1O0 150 200 250
c) male speaker
300
A
50 1O0 1,50 200 250
d) male speoker, extended codebook
300
Fig. 1. Probability Density of Delay
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At an equal number of candidateentries,the extended adaptivecodebook resultsin an
improved subjectiveperformance when the frame length is longer than a singlepitch
period. Fig. lb displaysthe probabilitydensityfunction for the delay of the selected
vectorsforthe extended codebook case. The candidatevectorschosen most oftenarethose
with a delay of one pitch period,which were completed in the manner describedabove.
This means that in a largenumber of frames,the newly created periodicentriesin the
codebook provided the best match to the targetexcitationvector t. In Fig. 2 spectra
generated with and without the extensionare shown. The virtualcandidatescreate a
periodicity,resultingin enhanced harmonicityofthe spectrum,especiallybetween 1200 and
3000 Hz. It is thisenhanced periodicitywhich improves the subjectivespeech quality.
However, the associatedsegmental signalto noiseratioincreasedonly about 0.1 dB. The
improvement of the subjectivespeech qualityfor the female speaker isin contrastto the
resultsfor the male speaker (Fig.ld),for whom the revisedcodebook with the modified
range of availabledelayshas no noticeableffecton overallperformance.
"O ,_"
c-
crt
a !
v=.
0
c
I ! ! t
lOOO 2000 3000 --> Hz 4.000
Fig. 2. Spectral Characteristics
a: original, b: extended codebook, c: conventional codebook
2.2 The Stochastic Codebook.
The performance of the stochastic codebook can be evaluated by assuming that the part
of the excitation signal which remains after the first vector quantization stage contains
normally distributed random samples. Its error criterion is simplified when expressed in the
eigenspace of the HTH matrix. The diagonal transform of the spectral weighting matrix is
A=UHTHU T, where U is a unitary matrix. The transforms of the excitation vectors are
r----Ut and p--Ur respectively. Since the vectors t and r were assumed to be sequences of
normally distributed independent samples, and since HTH is positive definite, r and p are
also normally distributed independent sequences. Equation (2) can now be written as:
= (3)
The behavior of the error criterion _ is a function of the framelength N, the eigenvalues
described by A, and codebook size.
When trying to match noise signals, the matrix HTH tends to be close to the identity
matrix. Fig. 3 shows the dependence of the minimized error criterion e, normalized to a per
sample basis, on the frame length for a weighting matrix whose eigenvalues are all equal.
Six error curves are shown for codebooks of various sizes. In addition, the error as a
function of frame size under the constraint of constant bit rate for the codebook index is
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shown. From the lattercurvesitappearsthatthe coOingef[iciencyisidenticalat allframe
lengths.This isnot the case,however, sinceadditionalbitsmust be spent on the encoOing
of the scalingfactorused in the quantizationprocedure. Since this overhead becomes
smMler with increasingframe length,the encoding becomes more ef[icientat largerframe
lengths.
A high levelof short-termcorrelationresultsin a largerange forthe magnitudes of the
eigenvalues of the spectralweighting matrix. Then, only a small fraction of the
components of the target vector r (those associatedwith large eigenvalues)are of
importance in the evaluation of the errorcriterion.It is illuminatingto considerthe
extreme instance where only one eigenvalue is nonzero. In this case only a single
component of the targetvectorr must be matched. Using an optimal scalingfactor,any
vector not orthogonal to this component will result in zero error. However, the
quantizationofthisscalingfactorwillintroduceerrorin an actualcooer.
Using these two extreme cases as guideline,itisseen that the stochasticquantization
procedure can be expected to perform best in voiced speech, which has pronounced
formants. Thus, the stochasticcooebook search willperform best in the same voiced
regionswhere the adaptive coOebook searchperforms best. These resultsare consistent
with intuition;the speech codersperform bestwhere the signalhas the most structure(low
entropy,low informationcontent),and worst where the signalhas the leaststructure (high
entropy, high informationcontent). These resultslead to segmental signalto noise ratios
which arehigh duringspeech and low duringbackground noise.
CD
C3
CD
C_
v==
t.
_3
g
_r
c}
2400 _s
$4800 b/. s
_" , . squared, error p_er somple for optimally scoled
_" codebooks with 64, 128, 256,'512, 1024, 2048 vectors
I | I | I I I
o _o 4o so so 1oo 12o
framelencjth (samples)
Fig.3.Squared ErrorforNoiseSignal
3.CONCLUSION
The encoding of the periodicityof the excitationsignalof SELP coders deteriorates
when the frame lengthexceedsthe lengthof a pitchperiod. This effectcan be diminished
by extending the adaptive cooebook with vectorsof progressivelyshorterperiodicity.The
performance of the stochasticooebook improveswhen the frame lengthisincreased.Both
the adaptive and stochasticcooebook quantizationprocedure perform best during voiced
speechwhere both short-termand long-termcorrelationarehighest.
Itisnaturalthatthe adaptivecodebook quantizationisperformed beforethe stochastic
quantization,with the latteractingas a correctionto an alreadygood excitationsequence.
Because of the recursivecharacterof the adaptive codebook, it is logicalto apply the
stochasticcodebook correctionbeforeenteringthe excitationinto the adaptive cooebook.
531
This means that the stochasticodebook frame lengthmust fitan integernumber of times
in the frame lengthused forthe adaptivecodebook. This and the factthat the stochastic
codebook performs better with increasinglength means that the frame length of the
stochasticodebook isbestchosen to be equal to thatof the adaptivecodebook.
REFERENCES
[1] Atal B.S. and M.R. Schroeder, "StochasticCoding of Speech at Very Low Bit
Rates",Proc.ofICG, Amsterdam, 1610-1613,1984.
[2] Atal, B.S. and M.R. Schroeder, "Predictive Coding of Speech Signals and
Subjective Error Criteria" IEEE Trans. Speech Signal Proc. Vol. ASSP-_7, no 8,
247-254, 1979.
[3] Kleijn, W.B., D.J. Krasinski, and R.H. Ketchum, "Improved Speech Quality and
Efficient Vector Quantization in SELP" Proc. Int. Conf. Acoust., Speech and Sign.
Process., New York, 1988.
[4] Singhal S. and B.S. Atal, "Improving Performance of Multi-Pulse LPC Coders at
Low Bit Rates", Proc. Int. Conf. Acoust., Speech and Sign. Process., San Diego,
1.3.1-1.3.4, 1984.
[5] Rose, R.C. and T.P. Barnwell, "Quality Comparison of Low Complexity 4800bps
Self Excited and Code Excited Vocoders", Proc. Int. Conf. Acoust., Speech and
Sign. Process., Dallas, 1637-1640, 1987.
[6] Lin, D., "New approaches to Stochastic Coding of Speech Sources at Very Low Bit
Rates", in Signal Processing III: Theories and Applications, I.T. Young et al. eds.,
Elsevier, 445-447, 1986.
[7] Davidson, G. and A. Gersho, "complexity Reduction Methods for Vector
Excitation Coding", Proc. Int. Conf. Acoest., Speech and Sign. Process., Tokyo,
3055-3058, 1986.
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Analysis and improvement of the vector quantization in SELP (Stochastically Excited Linear Prediction)

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