LogPCM - 9.3 / Nonuniform Quantization 230 TABLE 9.2.1...

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9.3 / Nonuniform Quantization 230 TABLE 9.2.1 Optimum Step Sizes for Uniform Quantization of a Gaussian PDF Number of Step Size Minimum Mean- Levels (L) (AopJ Squared Error (D) SQNR(dB) 4 0.9957 0.1188 9.25 8 0.5860 0.03744 14.27 16 0.3352 0.01154 19.38 Source: J. Max, "Quantizing for Minirilum Distortion," IRE Trans.In/. Theory, © 1960 IEEE. 9.3 Nonuniform Quantization The most important nonuniform quantization method to date has been the logarithmic quantization used in the telephone network for speech digitization for over 20 years. The general idea behind this type of quantization is that for a fixed, uniform quantizer, an input signal with an amplitude less than full load will have a lower SQNR than a signal whose amplitude occupies the full dy- namic range of the quantizer (but without overload). This fact is illustrated by Example 9.2.2. Such a variation in performance (SQNR) as a function of quan. . tizer input signal amplitude is particularly detrimental for speech, since low- amplitude signals can be very important perceptually. There is the additional consideration for speech signals that low amplitudes are more probable than larger amplitudes, since speech is generally stated to have a gamma or Laplacian probability density, which is highly peaked about zero. Therefore, for speech signals a type of nonlinear quantization was invented called logarithmic companding. Initially, this scheme was implemented by pass- ing the analog speech signal through a characteristic of the form In [1 + IllslJ ( ) F ( ) = - 1 ::;; s ::;; 1, (9.3.1) II- S In [1 + IlJ sgn s, where s is the normalized speech signal and 11 is a parameter, usually selected to be 11 = 100, or more recently, 11 = 255. The function Fis) is shown in Fig. 9.3.1. Notice that F is) tends to amplify small amplitudes more than larger amplitudes whenever 11 > O. The output of F is) then served as input to a uni- form, n-bit quantizer. To resynthesize the speech signal, the quantizer output s was passed through the inverse function of Eq. (9.3.1) given by (9.3.2) where, of course, - 1 < s ~ 1. The performance in SQNR of this system for Il = 100 and n = 7 bits is shown in Fig. 9.3.2. It is evident from this figure that the SQNR is relatively flat over
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240 Pulse Code Modulation / 9 1.0 o o Input signal 1.0 FIGURE 9.3.1 Logarithmic compression characteristics. a wide dynamic range of input signal power (amplitudes), and hence low- amplitude signals are reproduced almost as well as higher-amplitude signals. In particular, we see from Eqs. (9.2.8) and (9.2.9) that for linear quantization, if we decrease the input signal power by 12 dB (from a peak value of V /2 to V /8), the output SQNR decreases by 12 dB. However, as we move along the "sine wave" curve in Fig. 9.3.2 from 0 to -12 dB on the input power axis, the SQNR decreases only by about 2 dB. The companding clearly improves the SQNR for low-amplitude signals. 40
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LogPCM - 9.3 / Nonuniform Quantization 230 TABLE 9.2.1...

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