Interpolation (779809), страница 5

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?P samplesbefore2Q+1samplesM missingsamplesP samplesafter2Q+1samplesFigure 10.11 A signal with M missing samples. P immediate samples each side ofthe gap and 2Q+1 samples a pitch period away are used for interpolation.the sample marked “?” coincides with the onset of an excitation pulse. Thissample is not well predictable from the P past samples, because they do notinclude a pulse event. The sample is more predictable from the 2Q+1samples a pitch period away, since they include the effects of a similarexcitation pulse. The predictor coefficients are estimated (see Chapter 7)using the so-called normal equations:−1c = R xxr xx(10.65)where Rxx is the autocorrelation matrix of signal x and rxx is the correlationvector.

In expanded form, Equation (10.65) can be written asr (1) a1   r (0) r (1)r (0) a2   a   r (2)r (1) 3    a  =r ( P − 2) P   r ( P − 1) p−Q   r (T + Q − 1) r (T + Q − 2) r (T + Q − 1) p−Q +1   r (T + Q)   p+ Q   r (T − Q − 1) r (T − Q − 2)r ( P − 1)r ( P − 2)r ( P − 3)r (0)r (T + Q − P)r (T + Q − P + 1)r (T− Q − P)r (T + Q − 1)r (T + Q − 2)r (T + Q − 3)r (T+ Q − P)r (T + Q)r (T + Q − 1)r (Tr (T+ Q − 2)+ Q − P + 1)r (0)r (1)r (1)r (0)r (2Q)r (2Q − 1)r (T − Q − 1) − 1 r (1)  r (T + Q − 2)   r (2) r (T + Q − 3)   r (3)   r (T + Q − P)r ( P)  r (2Q)   r (T + Q)  r (2Q − 1)   r (T + Q − 1)   TQr (0 )r−() (10.66)The modified AR model can be used for interpolation in the same way asthe conventional AR model described in the previous section.

Again, it isassumed that within a data window of N speech samples, a segment of Msamples commencing from the sample point k, xUk={x(k), x(k+1), ...,323Model-Based Interpolationx(k+M–1)} is missing. Figure 10.11 illustrates the interpolation problem.The missing samples are estimated using P samples in the immediatevicinity and 2Q+1 samples a pitch period away on each side of the missingsignal.

For the signal record of N samples, the modified AR equation(10.64) can be written in matrix form ase(T + Q) e(T + Q + 1) e( k − 1)e( k )e(k + 1)e( k + 2) e( k + M + P − 2)  e( k + M + P − 1)  e( k + M + P )  e(k + M + P + 1) e( N − 1)= x(T + Q − 1)x(T + Q)x ( k − 2)x(k − 1)xUK (k) xUK (k + 1 ) x(k + M + P − 3) x ( k + M + P − 2) x( k + M + P − 1) x( k + M + P)x ( N − 2)x(T + Q ) x(T + Q + 1) x( k − 1)xUK (k) xUK (k + 1)  xUK (k + 2)  x( k + M + P − 2)  x( k + M + P − 1)  x(k + M + P )  x(k + M + P + 1) x( N − 1)−x(T + Q − P )x(T + Q − P + 1)x ( 2Q )x(2Q + 1)x( k − P − 1)x ( k − P)x( k − P + 1)x( k − P + 2)x( k − T + Q − 1)x(k − T + Q)x(k − T + Q + 1)x ( k − T + Q + 2)xUK (k + M − 2)xUK (k + M − 1)x(k + M )x(k + M + 1)x( k + M + P − T + Q − 2)x( k + M + P − T + Q − 1)x( k + M + P − T + Q )x(k + M + P − T + Q + 1)x( N− P − 1)x( N− T + Q − 1)x( k − T − Q − 1)x(k − T − Q)x( k − T − Q + 1)x( k − T − Q + 2 )x( k + M + P − T − Q − 2) x( k + M + P − T − Q − 1) x(k + M + P − T − Q) x( k + M + P − T − Q + 1) x( N − T − Q − 1)x( 0)x(1) a1  a2  a  3  aP  p− Q p  +Q (10.67)where the subscript Uk denotes the unknown samples.

In compact matrixnotation, this set of equation can be written in the forme( x Uk , c ) = x + Xc(10.68)As in Section 10.3.2, the interpolation problem is solved in two stages:(a) In the first stage, the known samples on both sides of the missingsignal are used to estimate the AR coefficient vector c.(b) In the second stage, the AR coefficient estimates are substituted inEquation (10.68) so that the only unknowns are the data samples.The solution follows the same steps as those described in Section 10.3.2.10.3.5 LSAR Interpolation ErrorIn this section, we discuss the effects of the signal characteristics, the modelparameters and the number of unknown samples on the interpolation error.The interpolation error v(m), defined as the difference between the originalsample x(m) and the interpolated sample xˆ (m) , is given byxˆ(m)324Interpolationv(m) = x(m) − xˆ (m)(10.69)A common measure of signal distortion is the mean square error distancedefined as1 M −1D(c, M ) = E  ∑ [x(k + m)− xˆ (k + m)]2 (10.70)M  m =0where k is the beginning of an M-samples long segment of missing signal,and E [.] is the expectation operator.

In Equation (10.70), the averagedistortion D is expressed as a function of the number of the unknownsamples M, and also the model coefficient vector c. In general, the qualityof interpolation depends on the following factors:(a) The signal correlation structure. For deterministic signals such assine waves, the theoretical interpolation error is zero. Howeverinformation-bearing signals have a degree of randomness that makesperfect interpolation with zero error an impossible objective.(b) The length of the missing segment. The amount of information lost,and hence the interpolation error, increase with the number ofmissing samples.

Within a sequence of missing samples the error isusually largest for the samples in the middle of the gap. Theinterpolation Equation (10.63) becomes increasingly ill-conditionedas the length of the missing samples increases.(c) The nature of the excitation underlying the missing samples. TheLSAR interpolation cannot account for any random excitationunderlying the missing samples. In particular, the interpolationquality suffers when the missing samples coincide with the onset ofan excitation pulse. In general, the least square error criterion causesthe interpolator to underestimate the energy of the underlyingexcitation signal. The inclusion of long-term prediction and the useof quasi-periodic structure of signals improves the ability of theinterpolator to restore the missing samples.(d) AR model order and the method used for estimation of the ARcoefficients.

The interpolation error depends on the AR model order.Usually a model order of 2–3 times the length of missing datasequence achieves good result.325Model-Based Interpolation2500100020005001500100005000-500-500-1000-1000-1500-2000-2500050100150200250300350400-1500050100150200250300350400250300350400(a)(a)1000250020005001500100005000-500-500-1000-1000-1500-2000-2500050100150200250300350400(b)-1500050100150200(b)Figure 10.12 (a) A section of speech Figure 10.13 (a) A section of speechshowing interpolation of 60 samplesstarting from the sample point 100 (b)Interpolation using short and long-termcorrelations. Interpolated samples areshown by the light shaded line.showing interpolation of 50 samplesstarting from the sample point 175 (b)Interpolation using short and long-termcorrelations.

Interpolated samples areshown by the light shaded line.The interpolation error also depends on how well the AR parameterscan be estimated from the incomplete data. In Equation (10.54), in the firststage of the solution, where the AR coefficients are estimated, two differentapproaches may be employed to linearise the system of equations. In thefirst approach all equations, between the dashed lines, that involvenonlinear terms are discarded.

This approach has the advantage that noassumption is made about the missing samples. In fact, from a signalensemble point of view, the effect of discarding some equations is326Interpolationequivalent to that of having a smaller signal record. In the second method,starting from an initial estimate of the unknown vector (such as xUk=0),Equation (10.54) is solved to obtain the AR parameters. The ARcoefficients are then used in the second stage of the algorithm to estimatethe unknown samples. These estimates may be improved in furtheriterations of the algorithm. The algorithm usually converges after one ortwo iterations.Figures 10.12 and 10.13 show the results of application of the leastsquare error AR interpolation method to speech signals.

The interpolatedspeech segments were chosen to coincide with the onset of an excitationpulse. In these experimental cases the original signals are available forcomparison. Each signal was interpolated by the AR model of Equation(10.53) and also by the extended AR model of Equation (10.64). The lengthof the conventional linear predictor model was set to 20. The modifiedlinear AR model of Equation (10.64) has a prediction order of (20,7); thatis, the short-term predictor has 20 coefficients and the long-term predictorhas 7 coefficients. The figures clearly demonstrate that the modified ARmodel that includes the long-term as well as the short-term correlationstructures outperforms the conventional AR model.10.3.6 Interpolation in Frequency–Time DomainTime-domain, AR model-based interpolation methods are effective for theinterpolation of a relatively short length of samples (say less than 100samples at a 20 kHz sampling rate), but suffer severe performancedegradations when used for interpolation of large sequence of samples.

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