# Delta Modulation

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This lecture will teach about Delta Modulation in a communication system, why we use delta modulation, the working principle of delta modulation, the advantages and disadvantages of Delta Modulation, slope overload distortion, granular noise, and the bit rate of delta modulator in every detail.

## Reason To use Delta Modulation

• We have observed in PCM that it transmits all the bits which are used to code a sample. Hence, signaling rate and transmission channel bandwidth are quite large in PCM. To overcome this problem, Delta modulation is used.

## Working Principle of Delta Modulation

• Delta Modulation transmits only one bit per sample. Hence, the present sample value is compared with the previous sample value a d this results in whether the amplitude is increased or decreased is transmitted.
• Input signal x(t) is approximated to step signal by the delta modulator. This step size is kept fixed. The difference between the input signal x(t) and the staircase approximated signal is confined to two levels, i.e. + \Delta and - \Delta. Now if the difference is positive, then the approximated signal is increased by one step. If the difference is negative, then the approximated signal is reduced by one step.
• When the step is reduced, ‘0’ is transmitted and if the step is increased, ‘1’ is transmitted. hence, for each sample, only one binary bit is transmitted.
• The below figure shows the analog signal x(t) and its staircase approximated signal by the delta modulator.

## Mathematical Expressions of Delta Modulation

• The principle of delta modulation can be explained with the help of few equations:
• The error between the sampled value of x(t) and last approximated sample is given as,

e(nT_s)=x(nT_s)-\hat{x}(nT_s)

where,

e(nT_s)= Error at present sample.

x(nT_s)= Sampled signal of x(t)

\hat{x}(nT_s)= last sample approximation of the staircase waveform.

• If we assume u(nT_s) as the present sample approximation of staircase output, then,

u[(n-1)]T_s=\hat{x}(nT_s)

• Let us define a quantity b(nT_s) in such a way that,

b(nT_s)=\Delta\; sgn[e(nT_s)]

• This means that depending on the sign of error e(nT_s), the sign of step size \Delta is decide.

## Generation of Delta Modulation

• The below figure shows the transmitter i.e., generation of Delta modulated signal.
• The summer in the accumulator adds quantizer output (\pm \Delta) with the previous sample approximation. This gives a present sample approximation.

u(nT_s)=u(nT_s-T_s) + [\pm \Delta]

or, u(nT_s)=u[(n-1)T_s] + b(nT_s)

• The previous sample approximation u[(n-1)T_s] is restored by delaying one sample period Ts. The sample input signal x(nT_s) and staircase approximated signal \hat{x}(nT_s) are subtracted to get error signal e(nT_s).
• Thus, depending on the sign of e(nT_s), one bit quantizer generates an output of +\Delta or -\Delta. If the step size is +\Delta, then binary ‘1’ is transmitted and if it is -\Delta, then binary ‘0’ is transmitted.

## The receiver of Delta Modulation

• At the receiver end, as shown in the figure, the accumulator and low pass filter (LPF) are used. The accumulator generates the staircase approximated signal output and is delayed by one sampling period Ts. It is then added to the input signal.
• If input is binary ‘1’ then is adds +\Delta step to the previous output. If input is binary ‘0’ then one step -\Delta is subtracted from the delayed signal.
• Also, the low pass filter has the cutoff frequency equal to the highest frequency in x(t). This low pass filter smoothness the staircase signal to reconstruct the original message signal x(t).

• The delta modulation has certain advantages over the PCM system:
1. Since the delta modulation transmits only one bit for one sample, therefore this signaling rate and transmission channel bandwidth are quite small for delta modulation compared to PCM.
2. The transmitter and receiver implementation are very much simple for delta modulation. There is no analog to digital converter required in delta modulation.

The delta modulation has two major drawbacks:

2. Granular Noise or Idle Noise.

Now let us discuss these two disadvantages in detail.

• This distortion arises because of the large dynamic range of the input signal.
• As can be observed from the figure below, the rate of rising of input signal x(t) is so high that the staircase signal cannot approximate it, and the step \Delta becomes too small for the staircase signal u(t) to follow the step segment of x(t).
• hence, there is a large error between the staircase approximated signal and the original input signal x(t). This error or noise is known as Slope Overload Distortion.
• To reduce this error, the step size must be increased when the slope of signal x(t) is high. Since the step size of the delta modulator remains fixed, its maximum or minimum slopes occur along staircase lines. Therefore, this modulator is knowns as Linear Delta Modulator (LDM).

### 2. Granular Noise:

Granular Noise or Idle Noise occurs when the step size is too large compared to small variations in the input signal. This means that for very small variations in the input signal, the staircase signal is changed by a large amount because of the large step size.

The figure shows that when the input single is almost flat, the staircase signal u(t) keeps on oscillating by \pm +\Delta around the signal.

The error between the input and approximated signal is called granular noise. The solution to this problem is to make the step size small.

Important Point: Therefore, a large step size is required to accommodate a wide dynamic range of the input signal (to reduce slope overload distortion), and small steps are required to reduce granular noise. in fact, Adaptive Delta modulation is the modification to overcome these errors.

## The bit rate of Delta Modulator

Delta modulation bit rate (r) = Number of bits transmitted/second

= Number of samples/sec X Number of bits/sample

Therefore, the delta modulation bit rate is (1/N) times the bit rate of a PCM system, where N is the number of bits per transmitted PCM codeword. Hence we can say that the channel bandwidth for the delta modulation system is reduced to a great extent as compared to that for the PCM system.

## Derivation for Slope Overload Distortion

Example: Given a sine wave of frequency fm and amplitude Am applied to a delta modulator having step size \Delta. Shows that the slope overload distortion will occur if A_m > \frac{\Delta}{2\pi f_m T_s }. Here Ts is the sampling period.

Solution: Let us consider that the sine wave is represented as,

x(t) = A_m \sin(2\pi f_mt)

It may be noted that the slope of x(t) will be maximum when the derivative of x(t) with respect to t will be maximum. The maximum slope of the delta modulator may be given as,

Maximum\; slope= \frac{Step\;size}{Sampling\;Period}=\frac{\Delta}{T_s}

We know that slope overload distortion will take place if the slope of the sine wave is greater than the slope of the delta modulator i.e.,

max\left | \frac{d}{dt}x(t) \right | > \frac{\Delta}{T_s}

or, max\left | \frac{d}{dt}A_m\sin (2\pi f_mt) \right | > \frac{\Delta}{T_s}

or, max\left | A_m 2\pi f_m\sin (2\pi f_mt) \right | > \frac{\Delta}{T_s}

or, A_m 2\pi f_m> \frac{\Delta}{T_s}

or, \boxed{A_m>\frac{\Delta}{2\pi f_mT_s}} . Hence proved.

## Frequently Asked Questions on Delta Modulator

1. ### What is delta modulation and how does it work?

Answer: Delta modulation process compares the present sample value to the previous sample value. Based upon the difference amplitude is going to be increased or decreased by step signal. If the amplitude is increased then step size increases by one step i.e., +Δ and bit 1 are generated.

2. ### What is the use of a delta modulator?

Answer: A delta modulation (DM or Δ-modulation) is an analog-to-digital and digital-to-analog signal conversion technique used for the transmission of voice information where quality is not of primary importance.

3. ### What are the advantages of the delta modulator?

Answer: It is a cost-effective technique and can be used if financial problems are there. Delta modulation transmits only one bit for one sample, therefore the signaling rate and transmission channel bandwidth are quite small for the delta modulation as compared to the pulse code modulation.

4. ### What is the difference between delta modulation and Adaptive delta modulation?

Answer: In Delta Modulation step size is fixed for the whole signal. Whereas in Adaptive delta modulation, the step size varies depending upon the input signal. The slope overload and granular noise errors which are present in delta modulation are not seen in this modulation.

5. ### What are the two noises present in delta modulation?

Answer: Quantizing noise in delta modulation systems falls into two categories, granular noise, and slope overload noise. Granular noise exists because the decoded output signal can assume only a specified number of levels in the range of interest.

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