---
title: "Delta and adaptive delta modulation"
subject: "Communication Systems"
chapter: 4
key: "semester_6/communication_systems:dm-adm-encoder-decoder"
source: https://www.bctengineer.com/q/semester_6/communication_systems/dm-adm-encoder-decoder
site: BCT Engineering
---

# Delta and adaptive delta modulation

Communication Systems · Chapter 4: Pulse Modulation

- Asked in: 2080 Chaitra, 6+2 marks
- Video: [DELTA MODULATION AND ADM](https://www.youtube.com/watch?v=77CAu25ICS0) — Prof. Barapate's Tutorials
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/dm-adm-encoder-decoder
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-4

## Question

Explain the Delta and Adaptive Delta Modulations encoders and decoders with their derivations and diagram. List out their merits and demerits.

## Short answer

In delta modulation, a 1-bit quantizer outputs ±Δ based on whether the input is above or below the previous staircase approximation; the decoder integrates these steps and low-pass filters. Adaptive DM varies step size using Jayant's algorithm: Δ[n] = KΔ[n-1] if bits equal, else Δ/K, reducing both slope overload and granular noise. Merits: simplicity; demerits: slope overload, granular noise.

## Answer

#### Delta modulation (DM)

**Delta modulation** is 1-bit DPCM: the signal is oversampled, and only one bit per sample tells whether the signal is **above or below** the previous staircase approximation, which then moves up or down by a fixed step $\Delta$.

**Equations:**

$$
\begin{aligned}
e[n] &= m[n] - m_q[n-1] \\
e_q[n] &= \Delta\,\text{sgn}(e[n]) \quad (\text{bit 1 for } +\Delta,\ 0 \text{ for } -\Delta) \\
m_q[n] &= m_q[n-1] + e_q[n] = \sum_{i=1}^{n} e_q[i]
\end{aligned}
$$

So the receiver output is the **accumulated** (integrated) sum of $\pm\Delta$ steps.

**DM transmitter**

```text
m(t)->[Sampler]->m[n]->(+)--e[n]-->[1-bit    ]--+--> bits
                        ^ -        [quantizer]  |
                        |                       v
                  mq[n-1]                     (+)
                        |                       |
                        +--[Delay z^-1]<--mq[n]-+
                         (accumulator)
```

**DM receiver**

```text
bits->[Decoder +/-D]->(+)--+--> mq[n] --> [LPF] --> m(t)
                       ^   |
                       +-[z^-1]  (accumulator)
```

**Noise in DM:**

- **Slope overload:** if the signal slope exceeds $\Delta/T_s$, the staircase cannot follow. No overload needs $\left|\frac{dm}{dt}\right|_{max} \le \Delta f_s$; for a tone, $A_m \le \frac{\Delta f_s}{2\pi f_m}$.
- **Granular noise:** when the signal is nearly flat, the staircase hunts by $\pm\Delta$.
- SQNR (no overload, single tone): $\text{SQNR} = \frac{3f_s^3}{8\pi^2 f_m^2 W}$.

#### Adaptive delta modulation (ADM)

In ADM the step size is **varied according to the signal slope**: large steps for steep parts (to avoid slope overload) and small steps for flat parts (to reduce granular noise).

A common rule (Jayant's algorithm):

$$
\Delta[n] = \begin{cases} K\,\Delta[n-1], & b[n] = b[n-1] \\ \Delta[n-1]/K, & b[n] \ne b[n-1] \end{cases} \qquad (K \approx 1.5)
$$

with limits $\Delta_{min} \le \Delta[n] \le \Delta_{max}$. Consecutive equal bits mean the staircase is lagging (step grows); alternating bits mean granular hunting (step shrinks).

```text
m(t)->(+)->[1-bit Q]--+--------------> bits
       ^-             |
       |        [Step-size logic]
       |              | D[n]
       |              v
       +--[Accum.]<-[x]
Receiver: same step logic + accumulator + LPF
```

The receiver uses the same step-size logic on the received bits, so no side information is needed.

#### Merits and demerits

| | DM | ADM |
|---|---|---|
| Merits | Very simple 1-bit codec; no framing needed; robust to channel errors | Reduces both slope overload and granular noise; wider dynamic range; lower bit rate for same quality |
| Demerits | Slope overload and granular noise; needs high oversampling, so high bit rate | More complex step-size logic; errors can disturb step adaptation |

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is transmitted in delta modulation?
   A. The sample difference with 8 bits
   B. The full quantised sample
   C. The step size each time
   D. One bit per sample showing whether the signal is above or below the staircase
   Answer: D. A 1-bit quantizer outputs sgn(e[n]) and the staircase moves by +-Delta. (Key point: DM principle)

2. What does the DM receiver do?
   A. Accumulates the +-Delta steps then low-pass filters
   B. Differentiates the bits
   C. Multiplies bits by the carrier
   D. Squares the bits
   Answer: A. mq[n] is the running sum of eq, smoothed by an LPF. (Key point: Receiver)

3. In ADM, how does the step change for consecutive equal bits?
   A. Step resets to zero
   B. Step stays fixed
   C. Step grows (multiplied by K)
   D. Step shrinks
   Answer: C. Equal bits mean the staircase lags the signal, so the step increases to avoid slope overload. (Key point: ADM rule)

4. What is an advantage of ADM over DM?
   A. Needs no step-size logic
   B. Needs a lower sampling rate than Nyquist
   C. Is immune to channel errors
   D. Reduces both slope overload and granular noise
   Answer: D. Adaptive step: large for steep parts, small for flat parts. (Key point: Merit of ADM)

## Related in Communication Systems

- [Short note: Adaptive delta modulation](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ashort-note-adm) — Communication Systems, chapter 4
- [T1 digital carrier system; adaptive delta modulation](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3At1-hierarchy-adm) — Communication Systems, chapter 5
- [Delta modulation encoder/decoder; PCM vs DM](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adm-encoder-decoder-pcm-vs-dm) — Communication Systems, chapter 4
- [Merits and demerits of flat-top sampling](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aflat-top-sampling-merits) — Communication Systems, chapter 4
- [SQNR of delta modulation](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adm-sqnr-derive) — Communication Systems, chapter 4, asked 2×
- [SQNR of delta modulation for a single-tone signal](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adm-sqnr-single-tone) — Communication Systems, chapter 4

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