# Drawbacks in Delta Modulation (or) Errors in DM

Drawbacks in Delta Modulation (or) Errors in Delta Modulation:-

Delta Modulation is subject to two types of Quantization errors

1. Slope Overload Distortion (SOD) / SOD error.
2. Granular Noise /Granular error.

During the process of digital equivalent integration of that is approximating $x(t)$  with  $\widehat{x}(t)$  there exists an error called Quantization error as shown by

$\widehat{x}(nT_{s})={x}(nT_{s})+{q}(nT_{s})$.

$\widehat{x}(n)={x}(n)+{q}(n)$.

If time instance is  $(n-1)^{th}$   instance.

The Quantization error  is of two types in Delta Modulation.

If the rate of rise  $(\frac{dx(t)}{dt})$ of input signal is so high ( i.e, the slope of the signal is so high) so that the stair case signal cannot approximate to $x(t)$ .

As    $(\frac{dx(t)}{dt})$     is large enough    $\rightarrow&space;\widehat{x(t)}&space;\neq&space;x(t)$ .

In this case the step size  $'\Delta&space;'$ becomes too small for the stair case approximation to follow a steep segment of the input waveform  $x(t)$ with the result that $\widehat{x(t)}$    falls behind  which can be clearly visible in the figure.

The Quantization error that exists between  $x(t)$   and  $\widehat{x(t)}$  in this condition is called as slope overload Distortion.

Generally Delta Modulation is often referred as a linear Delta Modulator because the step size   $'\Delta&space;'$ is fixed during approximation process and also it’s maximum (or) minimum slopes occur straight lines.

To avoid slope overload distortion , step size    $'\Delta&space;'$  must be increased.

Granular Noise (or) Idle Noise:-

In contrast to slope overload distortion, Granular noise occurs when the step size   $'\Delta&space;'$  is  too large relative to the local slope characteristic of the input wave form  $x(t)$ .

This large value of  $'\Delta&space;'$  causes that the stair case approximation  $\widehat{x(t)}$ to hunt around a flat segment of the input waveform as shown in the above figure

i.e,    $\widehat{x(t)}$  Oscillates between    $\pm&space;\Delta$   when  $x(t)$  is almost straight.

$\therefore$  The error between $\widehat{x(t)}$   and  $x(t)$  in this condition is called as Granular Noise (or) Idle Noise.

To eliminate their error is to make the step size  $\Delta$  small.

Granular Noise occurs that for a very small (change) variations in the approximated signal  $\widehat{x(t)}$.

Thus we see that there is a need to have a large step size to accommodate a wide dynamic range of input signal

and a small step size is required to accurate representation of relatively low-level signals.

And a small step size is required to reduce slope overload distortion and small step size is required to reduce Granular noise.

It is clear that the choice of the optimum step size that minimizes the mean square value of the Quantization error in a Linear Delta Modulator will be the result of a compromise between slope overload distortion and granular noise.

To satisfy such requirement, we need to make the Delta Modulator “Adaptive”  in the sense that the step size is made to vary in accordance with the input signal.

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