SNR in PCM (or) Signal-to-Quantization Noise Ratio in PCM system

Advertisements

Advertisements3

Advertisements4

Advertisements6

Advertisements66

we know that signal-to-Noise Ratio is defined as

$SNR=\frac{S}{N}=\frac{Normalized\ signal\ power}{Normalized\ Noise \ power}$ .

let $x(t)$ is a given signal with which is an arbitrary signal with Normalized signal power P watts.

and this $x(t)$ is some arbitrary signal oscillating between $+x_{max}$ and $-x_{max}$ .

then the step size used in the Quantization process is $\Delta =\frac{2 . x_{max}}{L}$ .

and $L=2^{n}$ .

where L- is the number of Quantization levels.

n- no.of bits required to encode each Quantization level.

$\therefore \Delta =\frac{2 . x_{max}}{2^{n}} ----EQN(1)$ .

Quantization Noise Power $\bg_black N \ (or ) \ N_{Q}$ :-

if uniform (or) linear Quantization is used in PCM system, during the approximation process of $x(t)$ with $x_{q}(t) \ (or) \ \widehat{x}(t)$ there exists some error between these two signals . This error is called as Quantization error (or) noise.

$x(t) \approx x_{q}(t)$

In discrete time domain $e(nT_{S}) = x_{q}(nT_{S})-x(nT_{S})$ .

Quantization error = Quantized signal- original signal.

we know that step size is $\Delta =\frac{2 . x_{max}}{L}$ .

Now to find out Quantization noise , assume it is uniformly distributed random variable .

now the Probability density function of this uniformly distributed random variable is $f_{\epsilon }(\epsilon )$

$f_{\epsilon }(\epsilon ) = \left\{\begin{matrix} \frac{1}{\Delta } , \frac{-\Delta }{2}\leq0\leq \frac{\Delta }{2}.\\ 0,otherwise. \end{matrix}\right.$