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\ signal\ power}{Normalized\ Noise \ power}.

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

The maximum value of RMS signal power is P_{rms} = \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 \frac{\Delta}{2\pi f_{m}T_{s}}.

let A_{m}= \frac{\Delta}{2\pi f_{m}T_{s}}

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

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

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

if uniform (or) linear Quantization is used in DM 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)   (approximation process)

In discrete time domain e(nT_{S}) = 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 ,-\Delta )

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

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

Mean square value of this random variable is with zero mean

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

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

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

Mean Square value= Quantization Noise Power.

\therefore N_{q} = \frac{\Delta ^{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 f_{m} \ and \ f_{M}< < 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 ^{2}}{3} .

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

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

\therefore SQR \ in \ DM system=\frac{Signal \ power}{Noise \ power} .

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

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

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Author: vikramarka

Completed M.Tech in Digital Electronics and Communication Systems and currently working as a faculty.