STEM Problem Detail

Problem #96

Computer Science Hard

Problem Statement

Prompt

Please provide me step by step solution for the given Example - 8.1

Solution Rendering...

From Figure E8.1, the pdf of the input signal can be recognized as:

\[
p(x) = \frac{x}{32}
\qquad 0 \le x \le 8
\]

\[
p(x) = 0
\qquad \text{elsewhere}
\]

Given the quantization levels to be \( \{1, 3, 5, 7\} \), we can define the quantization
boundaries as \( \{0, 2, 4, 6, 8\} \).

Mean square error distortion \( = D \)

\[
D = \int_{-\infty}^{\infty} (x - f_{Q}(x))^{2} p(x)\,dx
\]

\[
D = \int_{0}^{2} (x-1)^{2} p(x)\,dx
+ \int_{2}^{4} (x-3)^{2} p(x)\,dx
+ \int_{4}^{6} (x-5)^{2} p(x)\,dx
+ \int_{6}^{8} (x-7)^{2} p(x)\,dx
\]

This expression evaluates to \( 0.333 \).

Signal Power \( = E[x^{2}] \)

\[
E[x^{2}] = \int_{0}^{8} p(x)x^{2}\,dx
= \int_{0}^{8} \frac{1}{32}(x \cdot x^{2})\,dx = 32
\]

Signal to distortion ratio

\[
= 10\log[E[x^{2}]/D]
= 10\log(32/0.333)
= 19.82\ \text{dB}
\]

To minimize the distortion, we need to concentrate the quantization levels in regions
of higher probability. Since the input signal has a greater probability of higher
amplitude levels than lower amplitudes, we need to place the quantization levels closer
(i.e., more quantization levels) at amplitudes close to eight and farther (i.e., less
quantization levels) at amplitudes close to zero.

Since this quantizer has quantization levels uniformly distributed, this would be
optimal for an input signal with a uniform pdf.

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