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Unformatted text preview: ven assumptions for Quantizer range: 2xmax, and Quantization interval: = 2xmax/2B, for a Bbit quantizer Uniform pdf, it can be shown that: 2 xmax 2 2 e = = 12 12
2 ( B 2 ) 2 xmax = ( 3) 22 B February 11, 2012 Veton Kpuska 43 Quantization Error Thus SNR can be expressed as: ( 3) 2 2 B ( 3) 22 B x2 2 SNR = 2 = x 2 = 2 x (x e x) max max 2 x x 2 = 10( log10 3 + 2 B log10 2 )  20 log10 max SNR ( dB ) = 10 log10 x e xmax 6 B + 4.77  20 log10 x Or in decibels (dB) as: Because xmax = 4x, then SNR(dB)6B7.2
February 11, 2012 Veton Kpuska 44 Quantization Error Presented quantization scheme is called pulse code modulation (PCM). Bbits per sample are transmitted as a codeword. Advantages of this scheme: Disadvantages: It is instantaneous (no coding delay) Independent of the signal content (voice, music, etc.) It requires minimum of 11 bits per sample to achieve "toll quality" (equivalent to a typical telephone quality) For 10,000 Hz sampling rate, the required bit rate is: B=(11 bits/sample)x(10000 samples/sec)=110,000 bps=110 kbps For CD quality signal with sample rate of 20,000 Hz and 16 bits/sample, SNR(dB) =967.2=88.8 dB and bit rate of 320 kbps. February 11, 2012 Veton Kpuska 45 Nonuniform Quantization Uniform quantization may not be optimal (SNR can not be as small as possible for certain number of decision and reconstruction levels) Consider for example speech signal for which x[n] is much more likely to be in one particular region than in other (low values occurring much more often than the high values). This implies that decision and reconstruction levels are not being utilized effectively with uniform intervals over xmax. A Nonuniform quantization that is optimal (in a leastsquared error sense) for a particular pdf is referred to as the Max quantizer. Example of a nonuniform quantizer is given in the figure in the next slide. February 11, 2012 Veton Kpuska 46 Nonuniform Quantization February 11, 2012 Veton Kpuska 47 Nonuniform Quantization Max Quantizer Problem Definition: For a random variable x with a known pdf, find the set of M quantizer levels that minimizes the quantization error. Therefore, finding the decision and boundary levels x i and xi, ^ respectively, that minimizes the meansquared error (MSE) distortion measure: Edenotes expected value and x is the quantized version of x. It turns out that optimal decision level xk is given by: D=E[(xx)2] ^ ^ ^ ^ xk +1 + xk xk = , 2 1 k M1
48 February 11, 2012 Veton Kpuska Nonuniform Quantization Max Quantizer (cont.) ^ The optimal reconstruction level xk is the centroid of px(x) over the interval xk1 x xk: xk xk px ( x ) ^ xk = x k xdx = ~x ( x ) dx p xk 1 xk 1 p x ( x ) dx xk 1 It is interpreted as the mean value of x over interval x k1 x ~ xk for the normalized pdf p(x). ^ Solving last two equations for xk and xk is a nonlinear problem in these two variables. Iterative solution which requires obtaining pdf (can be difficult).
Veton Kpuska 49 February 11, 2012 Nonuniform Quantization February 11, 2012 Veton Kpuska 50 Companding
A fixed nonuniform quantizer Companding Alternative to the nonuniform quantizer is companding. It is based on the fa...
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