Interpolation (779809), страница 4

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The excitation may be a random signal, a quasiperiodic impulse train, or a mixture of the two. The AR coefficients, ak,model the correlation structure or equivalently the spectral patterns of thesignal.Assume that we have a signal record of N samples and that within thisrecord a segment of M samples, starting from the sample k, xUk={x(k), ...,x(k+M–1)} are missing. The objective is to estimate the missing samplesxUk, using the remaining N–k samples and an AR model of the signal.Figure 10.8 illustrates the interpolation problem. For this signal record of Nsamples, the AR equation (10.53) can be expanded to form the followingmatrix equation:e(P )eP+(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(P)xP+(1)x ( k − 1)x Uk ( k ) x Uk ( k + 1 ) xk=  Uk ( + 2 )  x (k + M + P − 2)  x ( k + M + P − 1)  x(k + M + P )  x ( k + M + P + 1) x ( N − 1)x ( P − 1)x(P)x( k − 2)x ( k − 1)xUk ( k )x k−  Uk ( + 1 ) x ( k + M + P − 3) x (k + M + P − 2) x ( k + M + P − 1) x(k + M + P )x ( N − 2)x( P − 2)x ( P − 1)x ( k − 3)x(k − 2)x(k− 1)xUk ( k )x(k + Mx(k + Mx(kx(k+ P − 2)+ P − 1)+ M + P)+ M + P + 1)x(N− 3)x ( k − P − 1) x(k − P)a x ( k − P + 1)   1  ax ( k − P + 2 )   2 a 3 x Uk ( k + M − 2 )    aP x Uk ( k + M − 1 ) x(k + M ) x ( k + M + 1) x ( N − P − 1) x (0)x (1)(10.54)where the subscript Uk denotes the unknown samples.

Equation (10.54) canbe rewritten in compact vector notation as318Interpolatione ( x Uk , a ) = x − Xa(10.55)where the error vector e(xUk, a) is expressed as a function of the unknownsamples and the unknown model coefficient vector. In this section, theoptimality criteriobbn for the estimation of the model coefficient vector aand the missing samples xUk is the minimum mean square error given bythe inner vector producte T e ( x Uk , a ) = x T x +a T X T Xa − 2 a T X T x(10.56)The squared error function in Equation (10.56) involves nonlinear unknownterms of fourth order, aTXTXa, and cubic order, aTXTx.

The least squareerror formulation, obtained by differentiating eTe(xUk ,a), with respect to thevectors a or xUk, results in a set of nonlinear equations of cubic order whosesolution is non-trivial. A suboptimal, but practical and mathematicallytractable, approach is to solve for the missing samples and the unknownmodel coefficients in two separate stages.

This is an instance of the generalestimate-and-maximise (EM) algorithm, and is similar to the linearpredictive model-based restoration considered in Section 6.7. In the firststage of the solution, Equation (10.54) is linearised by either assuming thatthe missing samples have zero values or discarding the set of equations in(10.54), between the two dashed lines, that involve the unknown signalsamples. The linearised equations are used to solve for the AR modelcoefficient vector a by forming the equation(Taˆ = X KnX Kn)−1 (X KnT x Kn )(10.57)where the vector is an estimate of the model coefficients, obtained from theavailable signal samples.The second stage of the solution involves the estimation of theunknown signal samples xUk.

For an AR model of order P, and an unknownsignal segment of length M, there are 2M+P nonlinear equations in (10.54)that involve the unknown samples; these areaˆ319Model-Based InterpolationxUk ( k )x( k − 1)e( k )    ++(1)kxxUk ( k )(1)ke Uk xUk (k + 1)e( k + 2)xUk ( k + 2)−=   e( k + M + P − 2)   x (k + M + P − 2)   x ( k + M + P − 3)  Uk   e( k + M + P − 1)   x(k + M + P − 1)   xUk ( k + M + P − 2)x ( k − 2)x( k − 1)xUk ( k )xUk ( k + M + P − 4)xUk ( k + M + P − 3)x( k − p )   a1 x( k − p + 1)   a2 x( k − p + 2)   a3 xUk (k + M − 2)   aP -1 xUk (k + M − 1)   aP (10.58)The estimate of the predictor coefficient vector , obtained from the firststage of the solution, is substituted in Equation (10.58) so that the onlyremaining unknowns in (10.58) are the missing signal samples.

Equation(10.58) may be partitioned and rearranged in vector notation in thefollowing form:aˆe( k )e( k + 1)+ek(2)e( k + 3)e ( k + 4) e( k + P − 1)  =e( k + P ) e( k + P + 1)  e ( k + M + P − 2)  e(k + M + P − 1)  − aP 0 0 0 0 0 0 0 0 0− a P −1− aP00000000 1−a− 1 a2 − a3 − a4 − aP 0 0 0 001− a1− a2− a3001− a1− a2− aP −1− aP0− aP − 2− aP −1− aP0000 0 0 0 0 0 0−a 0−a−a 0 − a0 −a0P −3P −2P −1 −a 0 0 −a 0 0 −a 0 0 −a 000000 0000 1 0 0 −a0 0 0 −a0 0 0 −a0 −a000− aP−2− a P −1− aP  xUk ( k )   x ( k + 1)   Uk  xUk ( k + 2) +  xUk ( k + 3)   xUk ( k + M − 1) P −1 P 0001− a112300030001− a1− a2P −1− a P −2P12 0 0 0 00000 000 01 0−a 00001− a P −3  x(k − P )   x ( k − P + 1)   x (k − P + 2) x ( k − 1) 0  x(k + M )   x( k + M + 1)   x(k + M + 2) ++−1)kMP(x− a1  (10.59)In Equation (10.59), the unknown and known samples are rearranged andgrouped into two separate vectors.

In a compact vector–matrix notation,Equation (10.58) can be written in the forme = A1 x Uk + A2 x Kn(10.60)320Interpolationwhere e is the error vector, A1 is the first coefficient matrix, xUk is theunknown signal vector being estimated, A2 is the second coefficient matrixand the vector xKn consists of the known samples in the signal matrix andvectors of Equation (10.58). The total squared error is given bye T e = ( A1 x Uk + A2 x Kn )T ( A1 x Uk + A2 x Kn )(10.61)The least square AR (LSAR) interpolation is obtained by minimisation ofthe squared error function with respect to the unknown signal samples xUk:∂ eTe= 2 A1T A1 x Kn + 2 A1T A2 x Kn = 0∂ x Uk(10.62)From Equation (10.62) we have(LSAR= − A1T A1xˆ Uk)−1 (A1T A2 )x Kn(10.63)LSARThe solution in Equation (10.62) gives the x̂ Uk , vector which is the leastsquare error estimate of the unknown data vector.10.3.4 Interpolation Based on Long-Term and Short-termCorrelationsFor the best results, a model-based interpolation algorithm should utilise allthe correlation structures of the signal process, including any periodicstructures.

For example, the main correlation structures in a voiced speechsignal are the short-term correlation due to the resonance of the vocal tractand the long-term correlation due to the quasi-periodic excitation pulses ofthe glottal cords. For voiced speech, interpolation based on the short-termcorrelation does not perform well if the missing samples coincide with anunderlying quasi-periodic excitation pulse. In this section, the ARinterpolation is extended to include both long-term and short-termcorrelations. For most audio signals, the short-term correlation of eachsample with the immediately preceding samples decays exponentially withtime, and can be usually modelled with an AR model of order 10–20.

Inorder to include the pitch periodicities in the AR model of Equation (10.53),321Model-Based Interpolation?m2Q+1 samples apitch period awayP past samplesFigure 10.10 A quasiperiodic waveform. The sample marked “ ? ” is predicted usingP immediate past samples and 2Q+1 samples a pitch period away.the model order must be greater than the pitch period.

For speech signals,the pitch period is normally in the range 4–20 milliseconds, equivalent to40–200 samples at a sampling rate of 10 kHz. Implementation of an ARmodel of this order is not practical owing to stability problems andcomputational complexity.A more practical AR model that includes the effects of the long-termcorrelations is illustrated in Figure 10.10. This modified AR model may beexpressed by the following equation:Px ( m) = ∑ a k x ( m − k ) +k =1Q∑p k x ( m − T − k ) + e( m )(10.64)k =−QThe AR model of Equation (10.64) is composed of a short-term predictorΣak x(m-k) that models the contribution of the P immediate past samples,and a long-term predictor Σpk x(m–T–k) that models the contribution of2Q+1 samples a pitch period away.

The parameter T is the pitch period; itcan be estimated from the autocorrelation function of x(m) as the timedifference between the peak of the autocorrelation, which is at thecorrelation lag zero, and the second largest peak, which should happen apitch period away from the lag zero.The AR model of Equation (10.64) is specified by the parameter vectorc=[a1, ..., aP, p–Q, ..., pQ] and the pitch period T. Note that in Figure 10.10322Interpolationx Kn1x Kn2x Uk? ...

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