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Time domain conversion can also be based on various numerical methods, e.g., on polynomial<br />

interpolation. Lagrange interpolation or spline interpolation can effectively be used [42,53]. In the case<br />

of an Nth-order spline with a function defined in interval [x k,…,x k+m] as<br />

where<br />

© 2002 by CRC Press LLC<br />

(27.52)<br />

a 6th-order interpolation is used with a simple FIR filter to compensate sinc 7 -distortion in the frequency<br />

domain caused by the spline interpolator [53].<br />

27.8 Lossless Audio Coding<br />

Pulse Code Modulation<br />

The most typical digital waveform coding is the pulse code modulation (PCM), in which a stream<br />

uniformly distributed digitally coded samples, which represent a given analog continuous-time signal is<br />

used. Basic PCM coder consists of an antialiasing filter, sample device and a quantizer. In practice, in<br />

order to improve the subjective audio quality, the quantizer should have a nonlinear (logarithmic)<br />

characteristic based on, e.g., a 13-segment A-law and a 15-segment µ-law used in telephone systems. The<br />

normalized characteristics are given by [2]<br />

• A-law<br />

• µ-law<br />

φ i x<br />

N<br />

Mk ( x)<br />

= aiφ i( x)<br />

N<br />

(27.53)<br />

(27.54)<br />

For the compression from 16 bits to 8 bits typical values of the coefficients are A = 87.6 and µ = 255.<br />

In most cases, PCM bit stream has highly redundant information. Thus, using a number previous<br />

samples of the input signal, we can predict the next sample with a relatively small error. This feature is<br />

used in differential pulse code modulation (DPCM), in which a difference between input sample and its<br />

estimation is coded. The prediction is realized with appropriate FIR filter. In these case in which the<br />

statistics of the input signal changes in time or is unknown, the prediction should be made adaptive.<br />

An adaptive coding is realized in an adaptive difference pulse code modulation (ADPCM). The respective<br />

schemes, i.e., those of the ADPCM encoder and the ADPCM decoder are shown in Fig. 27.27.<br />

A special case of the DPCM approach is delta modulation (DM). The DM encoder is very simple to<br />

implement because it uses a 1-bit quantizer and a first order predictor (cf. Fig. 27.28). The encoder is so<br />

strongly simplified, so high sampling rates are required. Among the disadvantages of DM are possible<br />

k+m<br />

∑<br />

i=k<br />

( ) = ( x– xi) + =<br />

f( x)<br />

f( x)<br />

=<br />

⎧ 0<br />

⎨<br />

⎩(<br />

x– xi) ⎧<br />

Ax<br />

1<br />

------------------ 0 ≤ x ≤ ---<br />

⎪1+<br />

lnA A<br />

⎨<br />

⎪ 1 + lnAx<br />

1<br />

--------------------- --- ≤ x ≤ 1<br />

⎩1<br />

+ log A A<br />

x< xi x≥xi ln(<br />

1 + µx)<br />

=<br />

------------------------- for 0 ≤ x ≤ 1<br />

ln(<br />

1 + µ )

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