# Delta modulation and Demodulation

DM is done on an over sampled message signal in its basic form, DM provides a stair case approximated signal to the over sampled version of message signal.

i.e, Delta Modulation (DM) is a Modulation scheme in which an incoming  message signal is over sampled (i.e, at a rate much higher than the Nyquist rate $f_{s}>&space;2f_{m}$) to purposely increase the correlation between adjacent samples of the signal. Over sampling is done to permit the use of a sample Quantizing strategy for constructing the encoded signal.

Signaling rate and Transmission Band Width are quite large in PCM. DM is used to overcome these problems in PCM .

DM transmits one bit per sample.

The process of approximation in Delta Modulation is as follows:-

The difference between the input ($x[nT_{s}]$) and the approximation ($x[(n-1)T_{s}]$) is quantized into only two levels $\pm&space;\Delta$ corresponding to Positive and negative differences.

i.e, If the approximation  ($x[(n-1)T_{s}]$) falls below the signal ($x[nT_{s}]$)at any sampling epoch(the beginning of a period)output signal level is increased by $\Delta$.

On the other hand the approximation   ($x[(n-1)T_{s}]$) lies above the signal ($x[nT_{s}]$) , output signal level is diminished by $\Delta$ provided that the input signal does not change too rapidly from sample to sample.

it is observed that the  change in stair case approximation lies with in  $\pm&space;\Delta$ .

This process can be illustrated in the following figure

Delta Modulated System:- The DM system consists of Delta Modulator and Delta Demodulator.

Delta Modulator:-

Mathematical equations involved in DM Transmitter are

error signal: $e[nT_{s}]=x[nT_{s}]-x_{q}[(n-1)T_{s}]$

Present sample of the (input) sampled signal: $x[nT_{s}]$

last sample approximation of stair case signal: $x_{q}[(n-1)T_{s}]$

Quantized  error signal( output of one-bit Quantizer): $e_{q}[nT_{s}]$

if         $x[nT_{s}]\geq&space;x_{q}[(n-1)T_{s}]&space;\Rightarrow&space;e_{q}[nT_{s}]&space;=&space;\Delta$.

and  $x[nT_{s}]<&space;x_{q}[(n-1)T_{s}]&space;\Rightarrow&space;e_{q}[nT_{s}]&space;=&space;-\Delta$.

encoding has to be done on the after Quantization that is when the output level is increased by $\Delta$ from its previous quantized level, bit ‘1’ is transmitted .

similarly when output is diminished by $\Delta$ from the previous level  a ‘0’ is transmitted.

from the accumulator $x_{q}[nT_{s}]=x_{q}[(n-1)T_{s}]+&space;e_{q}[nT_{s}]$

$e_{q}[nT_{s}]=e[nT_{s}]+&space;q[nT_{s}]$

where $q[nT_{s}]$ is the Quantization error.

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