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Unformatted text preview: a mapping of infinite precision value of to a finite precision representation picked from a finite set of symbols. [ ] [ ] February 11, 2012 Veton Kpuska 20 xi-1 < x[n] xi Scalar Quantization Quantization, therefore, is a mapping of a value x[n], xmin x xmax, to. The quantizer operator, denoted by Q(x), is defined by: where denotes one of L possible quantization levels where 1 i L and xi represents one of L +1 decision levels. ^ ^ x[ n] = xi = Q( x[ n]), xi-1 < x[n] xi The above expression is interpreted as follows; If , then x[n] is quantized to the quantization level and is considered quantized sample of x[n]. Clearly from the xi-1 < x[n] xi x ^ ^i x[ n] limited range of input values and finite number of symbols it follows that quantization is characterized by its quantization step size i defined by the difference of two consecutive decision levels: i = xi - xi -1
February 11, 2012 Veton Kpuska 21 Scalar Quantization Assume that a sequence x[n] was obtained from speech waveform that has been lowpassfiltered and sampled at a suitable rate with infinite amplitude precision. x[n] samples are quantized to a finite set of amplitudes denoted by ^ . x[n] Associated with the quantizer is a quantization step size . Quantization allows the amplitudes to be represented by finite set of bit patterns symbols. Encoding: Decoding Inverse process whereby transmitted sequence of codewords c'[n] is transformed back to a sequence of quantized samples (Figure 12.3b in the next slide). Mapping of to a finite set of symbols. This mapping yields a sequence of codewords denoted by c[n] (Figure 12.3a in the next slide). ^ x[n] February 11, 2012 Veton Kpuska 22 Scalar Quantization February 11, 2012 Veton Kpuska 23 Fundamentals Assume a signal amplitude is quantized into M levels. Quantizer operator is denoted by Q(x); Thus ^ ^ x[n] = xi = Q( x[n]) , xi-1 < x[n] xi Where denotes L possible reconstruction levels quantization levels, and ^ xi 1i L xi denotes L +1 possible decision levels with 0i L If xi1< x[n] < xi, then x[n] is quantized to the reconstruction level is quantized sample of x[n]. ^ x ^ x[n] i February 11, 2012 Veton Kpuska 24 Fundamentals Scalar Quantization Example: Assume there L=4 reconstruction levels. Amplitude of the input signal x[n] falls in the range of [0,1] Decision levels and Reconstruction levels are equally spaced: Figure 12.4 in the next slide. Decision levels are [0,1/4,1/2,3/4,1] Reconstruction levels assumed to be [0,1/8,3/8,5/8,7/8] February 11, 2012 Veton Kpuska 25 Example of Uniform 2bit Quantizer February 11, 2012 Veton Kpuska 26 Example Assume there are L = 24 = 16 reconstruction levels. Assuming that input values fall within the range [xmin=1, xmax=1] and that the each value in this range is equally likely. Decision levels and reconstruction levels are equally spaced; = i,= (xmax xmin)/L i=0, ..., L-1., Decision Levels: 15 13 11 9 7 5 3 3 5 7 9 11 13 15 - 2 ,- 2 ,- 2 ,- 2 ,- 2 ,- 2 ,- 2 ,- 2 ,+ 2 ,+ 2 ,+ 2 ,+ 2 ,+ 2 ,+ 2 ,+ 2 ,+ 2 Reconstruction Levels: [ - 8,-7,-6,-5,-4,-3,-2,-,+,+2,+3,+4,+5,+6,+7,+8] February 11, 20...
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