# Delta modulation

The **delta modulation** , also known as *Δ-modulation* or *Δ-M* referred to, is intended for **pulse frequency modulation** (PFM) and **pulse density modulation** (PDM) and is a variant of the Differential Pulse Code Modulation (DPCM), wherein the pulse coded signal only two in the transmission channel logic level can take .

The method converts an analog signal course into a signal which, like a digital signal, can only assume two discrete values, but its pulse spacing can be continuously changed, ie not time-discrete. The larger the input signal to be converted, the more pulses of constant duration are generated per time period. Pulse width modulation (PWM) is related to the process , although the pulse width is varied at a constant frequency.

Delta modulation is used, among other things, in measurement technology and to control switching regulators and DC voltage converters , and in this context it is usually referred to as pulse frequency modulation. In the field of signal processing , the same method is called *delta modulation* and is used to convert analog signal curves into a binary sequence. Delta-sigma modulation (ΣΔ modulation) is an extension of delta modulation , which is used in addition to digital filters in analog-to-digital converters (ADCs).

## General

In the case of delta modulation, the analog signal is sampled at regular intervals, and one sample is saved and compared with the previous one. If the second sample is greater than the first, the delta modulator generates a 1 signal. If the second sample is smaller, a 0 signal is generated.

In order to achieve a corresponding dynamic range with delta modulation, a higher sampling frequency is used than with pulse code modulation . As with pulse code modulation, the sampling frequency must be selected to be at least high enough that the Nyquist-Shannon sampling theorem is fulfilled. With delta modulation, this corresponds to the dynamic range of one bit. In the case of delta modulation, the sampling frequency is therefore selected to be significantly higher than this lower value, for a dynamic range of *n* bits by a factor of 2 ^{n} higher, since otherwise slope overload distortion can occur.

## Mathematical description

The pulse frequency modulation can be described as a sequence *a _{n of}* the values {0, 1}. The logical value 1 of a bit is usually assigned to the positive signal value

*+ A*, the logical value 0 to the value

*−A*. The two values form the bipolar value sequence

*x*which is formed by the PFM or PDM.

_{n}The bipolar pulse rate modulation can thus be expressed as:

or as

A sequence of constant values 1 results in a constant signal with the value *+ A* as the positive extreme value, a sequence of constant values 0 results in the constant signal with the negative extreme value *−A* . Signal values between *+ A* and *−A* are formed by different sequences with different frequencies from the values {0, 1}. By low-pass filtering the sequence *x _{n}* , an analog signal curve can be formed.

## literature

- John G. Proakis, Masoud Salehi: Communication Systems Engineering . 2nd Edition. Prentice Hall, 2002, ISBN 0-13-095007-6 .
- Djuro G. Zrilic: Circuits and Systems Based on Delta Modulation: Linear, Nonlinear and Mixed Mode Processing . Springer, 2005, ISBN 3-540-23751-8 .
- Joshua D. Reiss: Understanding Sigma-Delta Modulation: The Solved and Unsolved Issues . Volume 56, No. 1/2 . Journal of the Audio Engineering Society, 2008, ISSN 0004-7554 ( Online [PDF; 6.3 MB ]).