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Unformatted text preview: represents quantized version L=8, c) represents quantization error e[n] for B=3 bits (L=8 quantization levels), and d) is quantization error for B=8 bits (L=256 quantization levels). February 11, 2012 Veton Kpuska 36 Quantization Noise Overload Distortion Maximumvalue constant: xmax = 4x (4xx[n]4x) For Laplacian pdf, 0.35% of the speech samples fall outside the range of the quantizer. Clipped samples incur a quantization error in excess of /2. Due to the small number of clipped samples it is common to neglect the infrequent large errors in theoretical calculations. February 11, 2012 Veton Kpuska 37 Quantization Noise Statistical Model of Quantization Noise Desired approach in analyzing the quantization error in numerous applications. Quantization error is considered an ergodic whitenoise random process. The autocorrelation function of such a process is expressed as: re [m] = E (e[n]e[n + m]) e2 re [m] = 0
February 11, 2012 Veton Kpuska , m = 0 , m 0
38 Quantization Error Previous expression states that the process is uncorrelated. Furthermore, it is also assumed that the quantization noise and the input signal are uncorrelated, i.e., Final assumption is that the pdf of the quantization noise is uniform over the quantization interval:
1 0 , e 2 2 E(x[n]e[n+m])=0, m. pe ( e ) =
February 11, 2012 , otherwise
39 Veton Kpuska Quantization Error Stated assumptions are not always valid. Consider a slowly varying linearly varying signal then e[n] is also changing linearly and is signal dependent (see Figure in the next slide). Correlated quantization noise can be annoying. When quantization step is small then assumptions for the noise being uncorrelated with itself and the signal are roughly valid when the signal fluctuates rapidly among all quantization levels. Quantization error approaches a whitenoise process with an impulsive autocorrelation and flat spectrum. One can force e[n] to be whitenoise and uncorrelated with x[n] by adding whitenoise to x[n] prior to quantization. February 11, 2012 Veton Kpuska 40 Example of Quantization Error due to Correlation a) b) c) d) Example of slowly varying signal that causes quantization error to be correlated. Plot represents sequence x[n] with infinite precision, represents quantized version , represents quantization error e[n] for B=3 bits (L=9 quantization levels), and is quantization error for B=8 bits (L=256 quantization levels). Note reduction in correlation level with increase of number of quantization levels which implies degrease of step size . February 11, 2012 Veton Kpuska 41 Quantization Error Process of adding white noise is known as Dithering. This decorrelation technique was shown to be useful not only in improving the perceptual quality of the quantization noise but also with image signals. SignaltoNoise Ratio A measure to quantify severity of the quantization noise. Relates the strength of the signal to the strength of the quantization noise.
Veton Kpuska 42 February 11, 2012 Quantization Error SNR is defined as: 1 N 1 2 x [ n] x2 E ( x 2 [n]) N SNR = 2 = n =01 e E ( e 2 [n]) 1 N 2 e [ n] N n =0 Gi...
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This note was uploaded on 02/10/2012 for the course ECE 3552 taught by Professor Staff during the Fall '10 term at FIT.
 Fall '10
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