# SNR in DM (or) Signal-to-Quantization Noise Ratio in Delta Modulation system

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

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

let $x(t)$ is a given signal which is a single tone signal $x(t)&space;=A_{m}&space;\cos&space;2\pi&space;f_{m}t&space;\&space;where&space;\&space;\omega&space;_{m}&space;=&space;2\pi&space;f_{m}$.

The maximum value of RMS signal power is $P_{rms}&space;=&space;\frac{A_{m}^{2}}{2R}$.

Normalized signal power $P=\frac{A_{m}^{2}}{2}$    with R=1.

we know that slope overload distortion can be eliminated if and only if $A_{m}\leq&space;\frac{\Delta}{2\pi&space;f_{m}T_{s}}$.

let $A_{m}=&space;\frac{\Delta}{2\pi&space;f_{m}T_{s}}$

By substituting $A_{m}$ value in P  then the power results to be $P=&space;\frac{(\frac{\Delta}{2\pi&space;f_{m}T_{s}})^{2}}{2}$ .

$P=&space;\frac{\Delta^{2}}{8\pi^{2}&space;f_{m}^{2}T_{s}^{2}}----EQN(1)$

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

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

$x(t)&space;\approx&space;x_{q}(t)$   (approximation process)

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

Quantization error = Quantized signal- original signal.

Now to find out Quantization noise , assume it is uniformly distributed random variable $(+\Delta&space;,-\Delta&space;)$

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

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

Mean square value of this random variable is with zero mean

$E(\epsilon&space;^{2})&space;=&space;\int_{-\infty&space;}^{\infty&space;}&space;\epsilon&space;^{2}f_{\epsilon&space;}(\epsilon&space;)d\epsilon$.

$E(\epsilon&space;^{2})&space;=&space;\int_{\Delta&space;}^{\Delta&space;}\epsilon&space;^{2}&space;\frac{1}{2\Delta&space;}d\epsilon$ .

simplification gives $E(\epsilon&space;^{2})&space;=&space;\frac{\Delta&space;^{2}}{3}$.

Mean Square value= Quantization Noise Power.

$\therefore&space;N_{q}&space;=&space;\frac{\Delta&space;^{2}}{3}----EQN(2)$.

The M signal is passed through a reconstruction Low pass Filter at the output of a DM Receiver . The Band width of this filter is $f_{M}$  in such a way that $f_{M}\geq&space;f_{m}&space;\&space;and&space;\&space;f_{M}<&space;<&space;f_{s}$.

where $f_{s}$  is the sampling frequency of the signal.

now assume that Quantization noise is distributed over a frequency band up to $f_{s}$  and is given by $\frac{\Delta&space;^{2}}{3}$ .

then the noise power $N_{q}^{'}$  distributed over $f_{M}$   will be

$N_{q}^{'}&space;=&space;\frac{f_{M}}{f_{s}}\frac{\Delta&space;^{2}}{3}---EQN(3)$ .

$\therefore&space;SQR&space;\&space;in&space;\&space;DM&space;system=\frac{Signal&space;\&space;power}{Noise&space;\&space;power}$ .

$\frac{S}{N}&space;=&space;\frac{S}{N_q^{'}}&space;=&space;SQR=\frac{\frac{\Delta^{2}}{8\pi^{2}&space;f_{m}^{2}T_{s}^{2}}}{\frac{f_{M}}{f_{s}}\frac{\Delta&space;^{2}}{3}}$ .

$SQR_{DM}&space;=&space;\frac{3}{8\pi&space;^{2}f_{m}^{2}T_{s}^{3}f_{M}}&space;\&space;where&space;\&space;T_{s}=\frac{1}{f_{s}}$.

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