3. HEPOHTTYPE AND DATES COVERED
`Professional Paper
`5. FUNDING NUMBERS
`
`4, TITLEAND SUBTITLE
`
`EFFICIENT CALCULATION OF SPECTRAL TILT FROM VARIOUS LPC
`PARAMETERS
`6. AUTHOR(S)
`
`PR. CE77
`PE: 0603013N
`WU: DN302104
`
`V. Goncharoff, E. VonCoIln. and R. Morris
`
`7. PERFORMING ORGANIZATION NAME(S) AND ADDRESSIES)
`Naval Command. Control and Ocean Surveillance Center (NCCOSC)
`RDT&E Division
`
`San Diego, CA 92152—5001
`9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)
`
`Naval Engineering Logistics Office
`P. O. Box 2304, Eads Street
`Arlington, VA 22201
`
`1i. SUPPLEMENTARY NOTES
`
`8. PERFORMING ORGANIZATION
`REPORT NUMBER
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`10. SPONSORING/MONITORING
`AGENCY REPORT NUMBER
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`5%] 258307370188
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`1. AGENCY USE ONLY {Leave blank)
`2. REPORT DATE
`May 1996
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`DTIC QUALITY INSPECTED 1
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`12a. DISTRIBUTION/AVAILABILITY STATEMENT
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`12b, DISTRIBUTION CODE
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`Approved for public release; distribution is unlimited.
`
`13. ABSTRACT (Maximum 200 words)
`
`A coarse measure of a discrete signal’s power distribution vs. frequency is its “spectral tilt.” which we define as the slope of least-
`squares linear fit to the log power spectrum (0 S OJ s It rads/sec). One application of calculating spectral tilt is to help discriminate between
`voiced and unvoiced/silent time segments while processing sampled speech signals. When linear predictive coding (LPC) of speech is used
`to model its short-time spectrum, it is computationally more efficient to calculate spectral tilt directly from the LPC model parameters instead
`of from the log power spectrum. In this paper we present methods for calculating spectral tilt from cepstral coefficients, pole values, and
`polynomial coefficients. These methods are all based on calculating spectral tilt as‘a weighted sum of cepstral coefficients.
`
`14. SUBJECT TERMS
`
`Mission Area: Speech Technology
`Spectral Tilt
`Linear Predictive Coding (LPC)
`Polynomial Coefficients
`17. SECURITY CLASSIFICATION
`OF REPORT
`
`i8. SECURITV CLASSIFICATION
`OF THIS PAGE
`
`Ccpstral Coefficients
`Pole Values
`
`15. NUMBER OF PAGES
`
`I6. PRICE CODE
`
`19. SECURITY CLASSIFICATION
`OF ABSTRACT
`
`20. LIMITATION OF ABSTRACT
`
`UNCLASSIFIED
`
`UNCLASSIFIED
`
`UNCLASSIFIED
`
`SAME AS REPORT
`
`NSN 7540—01-2805500
`
`Standard Iorm 298 (FRONT)
`
`Ex. 1032 / Page 1 of 6
`Apple v. Saint Lawrence
`
`

`

`UNCLASSIFIED
`
`21a. NAME OF RESPONSIBLE INDIVIDUAL
`
`21b. TELEPHONE (include Area Code)
`
`210. OFFICE SYMBOL
`
`E. VonColln
`
`(619) 553—3655
`
`Code 44213
`
`
`
`NSN 7540014806500
`
`Ex. 1032 / Page 2 of 6
`Standard form 298 (BACK)
`
`UNCLASSIFIED
`
`
`
`Ex. 1032 / Page 2 of 6
`
`

`

`
`
`Efficient Calculation of Spectral Tilt from Various
`LPC Parameters
`
`Vladimir Goncharoft’“, Eric VonColln and Robert Morris
`
`*University of Illinois at Chicago
`College of Engineering, EECS Dept, Mail Code 154
`851 S. Morgan St., Chicago, IL 60607-7053
`Internet: goncharo@eecs.uic.edu
`
`NCCOSC
`
`RDT&E Division, Code 44213
`53245 Patterson Rd., San Diego, CA 92152
`Internet: voncolln@cod.nosc.mil
`
`E is minimum when b=bopt and m=mopt.
`In the continuous-frequency case, calculating
`spectral tilt using this formula is difficult ex-
`cept for the simplest |H(ej°’ )[2. An approxi-
`mation to the spectral tilt is usually found
`fi'om M samples of 1n[H(eJ"°)|2, taken uni—
`forme over the range 0 S co S n . Estimating
`spectral tilt in this manner falls within the
`general class of least-squares polynomial in-
`terpolation [3], and estimation error is re-
`duced to zero as M —>°°. When H(z) is in
`the form of an all-pole model,
`
`H(z) =
`
`G
`
`(1" p12"1)(1- 1022-1)‘ ' '(1- FIVE—‘1)
`
`’
`
`it is advantageous to find map, from the N
`pole values directly, without first having to
`calculate samples of lan(eJ"°)[2. This not
`only reduces the number of calculations re-
`quired, but also provides an exact measure of
`spectral tilt of the LPC log power spectrum.
`The rest of this paper deals with estimating
`map,
`from various LPC filter parameters,
`such as the pole values {p1, p2,
`, pN}.
`
`Background
`of
`tilt
`spectral
`the
`Let
`us
`represent
`S(co)=ln|H(e"“°)|2 using functional notation:
`m= Tilt{S(co)}. The underlying principle to
`our method is the linearity property of least-
`squares spectral tilt calculation.
`
`i i
`
`Abstract
`
`A coarse measure of a discrete signal's power
`distribution vs. frequency is its "spectral tilt",
`which we define as the slope of least-squares
`linear
`fit
`to the
`log power
`spectrum
`(0 S (o S 11: rads/sec). One application of cal-
`culating spectral tilt is to help discriminate
`between voiced and unvoiced/silent
`time
`segments while processing sampled speech
`signals. When linear predictive coding (LPC)
`of speech is used to model
`its .short-u'me
`spectrum, it is computationally more efficient
`to calculate spectral tilt directly from the LPC
`model parameters instead of from the log
`power spectrum.
`In this paper we present
`methods for calculating spectral
`tilt
`from
`cepstral coefficients, pole values, and poly-
`nomial coefficients. These methods are all
`based on calculating spectral
`tilt
`as
`a
`weighted sum of cepstral coefficients.
`
`Introduction
`
`Calculation of linear spectral tilt has been of
`interest to scientists in various disciplines [1-
`2], as it provides a coarse measure of a sig-
`nal's power distribution vs. frequency. Given
`z-domain transfer function H(2) we define
`spectral
`tilt
`to be mow,
`the slope of line
`yloa) = moptw+bopt
`that best
`fits the log
`power spectrum lan(ej‘”)l2 over the range
`0S0) Sr: , by the method of least-squares.
`That is, when error E is defined as
`
`E =3 {mco +b-'1n|111(ei‘”)l2 }2dco,
`
`
`
`Ex. 1032 / Page 3 of6
`
`Ex. 1032 / Page 3 of 6
`
`

`

`That is, if 5(0)) = alSl(co)+a2S2(co), then .
`
`Tilt{S(co)} = Tilt{alSl (0)) + a2 52 (03)}
`
`= asz'lt{S1 (0.))}+ azTilt{Sz (co)}.
`
`Thus when S(on) = 25,403). spectral tilt may
`be found as the sum of tilts from each term
`Sicko):
`
`m= ZTilt{Sk(co)}= ka
`k
`k
`
`To prove that the linearity property holds in
`this case, consider the following: when opti-
`mally fitting samples of 5(a))
`(OSm Sn)
`with a first-order polynomial by the method
`of least squares, [70pt and map, may be found
`by solving the following system of normal
`equations Ax = y:
`
`M4
`
`[gawk
`
`M-1
`Zoo]?
`k=O
`
`M
`
`M—1
`203k
`k=0
`
`[bowJ
`
`m
`0’”
`
`=
`
`M4
`
`1;)5(mk)
`
`M—l
`Zwkstokl
`=0
`
`is
`defined
`being
`polynomial
`the
`'Here
`ylco) = moptm+bopt,
`and frequency values
`co)t = lcrr/(M—l) (k =0,1,---,M—1) are M
`uniformly-spaced samples of (0 over the in-
`terval 030) Sn . Vector y may be ex-
`pressed as y = V’s, where:
`
`r5(a))1
`
`5(01)
`
`hall
`
`F1
`
`'1
`
`V=l
`5
`
`we
`
`031
`
`(02
`
`1
`
`1
`
`Since matrix multiplication is a linear opera-
`tion, replacing S with a sum of terms yields:
`
`N
`N
`N
`x= DES]. = ZDS‘. = in
`i-l
`i-l
`i-l
`_[ml +m2 +---+mN]— mop,
`
`b1+b2+---+bN
`
`bop,
`
`Thus we have proven that for polynomial fit-
`ting of sampled points by the method of least-
`squares, the sum of polynomial coefficients
`calculated to optimally fit each additive com-
`ponent of s equals the polynomial coeffi-
`cients calculated to optimally fit 3. The true
`spectral tilt of S(cn) is approached when the
`number of sample pointsM —> 0°, so this
`property is also valid in the continuous fre-
`quency domain.
`
`Spectral Tilt from Cepstral Coefficients
`The log power spectrum 5(a)) may be ex-
`pressed as a summation of cosine functions,
`weighted by the cepstral coefficients [4]:
`
`5(a)) = lan(ej°’)l2 = Zhl'H(ejm)'
`= Zick cos(koa) = iSk(cu)
`
`k-l
`
`k-l
`
`Thus 5(a)) is the sum of infinitely many co-
`sine terms. Using the linearity property dis-
`cussed above, we may express spectral tilt in
`terms of the cepstral coefficients:
`
`m= gmk = gTz’leck cos(kw)}
`=2ickTilt{cos(kco)}=-_fl 2 £15-
`
`k-1
`
`3
`
`75
`
`k-1.3.5,---
`
`, 2
`1»
`
`Vector x may be found after a single matrix
`multiplication: x = A-IV Ts = Ds
`(A’1 will
`always exist for unique sampled values of co).
`Note that matrix D is independent of 5(a)).
`
`Because the even-indexed cepstral coeffi-
`cients are the weights for cosine waves hav-
`ing an integer number of periods in the range
`0 S co S 1: rads/sec, there is zero contribution
`to spectral tilt from these terms.
`
`
`
`Ex. 1032 / Page 4 of6
`
`Ex. 1032 / Page 4 of 6
`
`

`

`Spectral Tilt from Linear Predictor Poles
`The log power spectrum of an N ~th order
`LPC filter may also be expressed in terms of
`its N pole values and gain constant G:
`
`N
`
`.
`
`-
`
`2
`
`
`
`S(co) = lan(ej°’)
`
`
`
`k-l
`
`2 =1ninIl(1— pke'”) 1
`=1nG2 ~2filnll—pke'jml
`
`floating-point precision). Another LPC pole-
`based method is to pre-calculate spectral tilt
`for various combinations of pole magnitude
`and angle, then store the results in a look-up
`table. We obtained good quantization levels
`by uniform sampling of 6 vs. ln(l—r), where
`p = re!“ .
`
`Spectral Tilt from Linear Predictor
`Coefficients
`
`The log power spectrum of an N -th order
`LPC filter may also be expressed in terms of
`its linear predictor coefficients and gain con-
`stant G:
`
`5(0)) =ln,H(ej°’)'2 =lnG2—l_nll—kglake-jmk
`
`2
`
`
`
`In this form 5(a)) may not be expressed as a
`sum of terms, where each term depends on a
`coefficient ark. The approach we take is to
`take advantage of a recursive solution for the
`cepstral coefficients from the LPC coeffi-
`cients [6]:
`'
`
`k—l
`ock +%znc,,ak_n,
`n=l
`k—l
`
`Ck =
`
`Is k S N;
`
`l-kL z ncnock_n,
`
`n=k-N
`
`k>N.
`
`The cepstral coefficients are then used to cal-
`culate spectral tilt from the formula previ-
`ously derived. In most cases this method Will
`be more computationally efficient than factor-
`ing the LPC polynomial to obtain the pole
`values, and solving for spectral tilt using the
`pole equations.
`
`Experimental Results
`We have applied the equations derived above
`to measuring the spectral
`tilt of sampled
`speech data: a 30 msec harming window was
`stepped across the speech samples with 0.1
`msec steps, the windowed data zero-padded
`to 1024 points prior to calculating its FFT.
`
`In this form 5(a)) is the sum of N+1 terms.
`The spectral
`tilt of the term lnG2 equals
`zero,
`leaving the following expression for
`spectral tilt of S(co) in terms of its N pole
`values;
`
`m= Tilt{S(w)} = —2€3Ti1r{1nl1— pke‘jwl}
`
`k=1
`
`We were not able to solve the integral and
`express m in terms of
`the pole values
`{p1, p2,... , pN} directly from this equation.
`An alternate solution method exists, how-
`ever:
`to first calculate cepstral coefficients
`from the pole values, then solve for spectral
`tilt by the formula derived in the previous
`section.
`The cepstral coefficients corre-
`sponding to the N —th order all-pole log
`power spectrum may be found as follows [5]:
`
`Combining this result with the previous ex-
`pression for spectral tilt in terms of the cep-
`stral coefficients, we obtain the following in-
`finite summation:
`
`W
`
`mi? 2 {firm}
`
`7‘
`
`k=l,3,5,-~~
`
`n=1
`
`the
`Fortunately, due to the pk /k3 terms,
`solution for spectral tilt m converges rapidly
`with increasing k (typically, only fifteen or so
`terms yield a solution to full lEEE double
`
`
`
`Ex. 1032 / Page 5 of6
`
`Ex. 1032 / Page 5 of 6
`
`

`

`
`
`The MATLAB "polyfit" routine was used to
`estimate the spectral tilt of 513 samples of
`1n|H(eJ"”)|2, spaced uniformly between 0 and
`5 KHz. A 180 ms section of the speech
`waveform used for this experiment is plotted
`as Figure 1(a). The corresponding spectral
`tilt, calculated directly from ln|H(ej“’)|2,
`is
`plotted as the solid line in Figure 1(b).
`(The
`slope value is expressed in dB/kHz.) This
`plot shows interesting spectral slope features
`corresponding to silence, onset and end of
`voiced speech, sibilant and plosive sounds.
`
`Conclusion
`
`From the equations presented here one may
`calculate spectral tilt directly from various
`LPC model parameters, avoiding the neces-
`sity and computational expense of first calcu-
`lating the log power spectrum. We rely on
`the linearity property of least-squares poly-
`nomial data fitting to simplify our expres-
`sions. The results may be applied to effi-
`ciently estimate the short-time spectral tilt of
`speech signals to help discriminate between
`various speech waveform components.
`
`0
`
`200
`
`400
`
`600
`
`800
`
`1000
`
`1200
`
`1400
`
`1600
`
`1800
`
` 0
` 0
`
`200
`
`400
`
`600
`
`300
`
`1000
`
`1200
`
`1400
`
`1600
`
`1800
`
`200
`
`400
`
`600
`
`800
`
`1000
`
`1200
`
`1400
`
`1600
`
`1800
`
`Figure 1(a). A 180 ms section of female speech se-
`lected for analysis.
`'
`
`Figure 1(b). Solid line: exact spectral tilt. calculated
`from the log-magnitude spectrum. Dashed line:
`spectral tilt estimated from the poles of the corre-
`sponding LPC- 14 spectral model.
`
`Next we calculated LPC-l4 coefficients cor-
`
`spectrum
`responding to each short-time
`(autocorrelation method), factored the de-
`nominator polynomial to obtain pole values
`for each frame, and applied the formula pre-
`sented above that calculated LPC spectral tilt
`from the pole values. The result is plotted as
`the dashed line in Figure 1(b). We also found
`that lower LPC orders provide reasonable
`estimates of spectral tilt, as shown in Fig. 2.
`
`Figure 2. Solid line: the LPC-l4 spectral tilt firom
`Figure 1(b). Dashed line: LPC-2 spectral
`tilt for
`comparison.
`
`[1] Y. Qi and B. Winberg, "Spectral slope of vowels pro—
`duced by tracheoesophageal speakers."
`Journal of
`Speech and Hearing Research, 342243, 1991.
`[2] R. Trawinski,
`I. Domyslawska, and R. Ciurylo,
`"Method of accurately measuring the spectral tilt in
`the Ebert spectrograph." Applied Optics, 32:4828,
`1993.
`
`[3] W. Press, B. Flannery, S. Teukolsky, and W. Vetter-
`ling, Numerical Redgs: the art of scientific comput-
`ing (FORTRAN version].
`Cambridge University
`Press, Cambridge, 1990, ISBN 0-521-38330-7.
`J. Deller, Jr., J. Proakis, and J. Hansen, Discrete-Time
`Processing of Speech Simals. Macmillan Publishing
`Co., New York. 1993.
`
`[4]
`
`[5] K. Assaleh, "Supplementary orthogonal cepstral fea-
`tures." Proc. IEEE Int. Conf. Acoust, Speech and
`Signal Process. 1995 (Detroit), 1:413.
`[6] M. Schroeder, "Direct (nonrecnrsive) relations be-
`twpen cepstrum and predictor coefficients." Trans.
`Acoust., Speech and Signal Process., 29:297. April
`1981.
`
`
`
`Ex. 1032 / Page 6 of6
`
`Ex. 1032 / Page 6 of 6
`
`