Derive the expression for signal to quantization noise ratio in PCM.

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teamques10 ★ 64k

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Signal-to-Quantization-Noise Ratio (SQNR or SNqR) is widely used quality measure in analysing digitizing schemes such as PCM (pulse code modulation) and multimedia codecs.
The SQNR reflects the relationship between the maximum nominal signal strength and the quantization error (also known as quantization noise) introduced in the analog-to-digital conversion.
The SQNR formula is derived from the general SNR (Signal-to-Noise Ratio) formula for the binary pulse-code modulated communication channel:

$\mathrm{SNR}=\frac{3 \times 2^{2n}}{1+4P_e \times (2^{2n} - 1)} \frac{m_m(t)^2}{m_p(t)^2}$

where

$ P_{e}$ is the probability of received bit error

$m_{p}(t)$ is the peak message signal level

$m_{m}(t) $ is the mean message signal level
As SQNR applies to quantized signals, the formulae for SQNR refer to discrete-time digital signals.
Instead of $m_{t}$, we will use the digitized signal $ x{_n}$. For $N$ quantization steps, each sample, $ X $requires $\nu =\log _{2}N$ bits. The probability distribution function (pdf) representing the distribution of values in $ X$ and can be denoted as $f(x)$. The maximum magnitude value of any {\displaystyle x} x is denoted by $x_{max}$ - As SQNR, like SNR, is a ratio of signal power to some noise power, it can be calculated as: $\mathrm{SQNR} = \frac{P_{signal}}{P_{noise}} = \frac{E[x^2]}{E[\tilde{x}^2]}$ - The signal power is: $\overline{x^2} = E[x^2] = P_{x^\nu}=\int_{}^{}x^2f(x)dx$ - The quantization noise power can be expressed as: $E[\tilde{x}^2] = \frac{x_{max}^2}{3\times4^\nu}$ Giving: $\mathrm{SQNR} = \frac{3 \times 4^\nu\times \overline{x^2}}{x_{max}^2}$ - When the SQNR is desired in terms of Decibels (dB), a useful approximation to SQNR is: $\mathrm{SQNR}|_{dB}=P_{x^\nu}+6\nu+4.8$ where $\nu$ is the number of bits in a quantized sample, and $P_{x^{\nu }}$ is the signal power calculated above. Note that for each bit added to a sample, the SQNR goes up by approximately 6dB $ (20\times log_{{10}}(2))$.

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