DELTA MODULATION (DM)

12.4 DELTA MODULATION (DM)

Delta modulation may be viewed as a simplified form of DPCM in which

a 2-level (1-bit) quantizer is used in conjunction with a fixed 1st-order predictor. The block diagram of a DM encoder-decoder is shown in Figure 12.8. We note that

-˜s(n) = ˜s(n − 1) = -˜s(n − 1) + ˜e(n − 1)

Since

q(n) = ˜ e(n) − e(n) = ˜ e(n) − s(n) − -˜ s(n)

Delta Modulation (DM) 619

FIGURE 12.8 Block diagram of a delta modulation system

it follows that

(12.22) Thus the estimated (predicted) value of s(n) is really the previous sam-

-˜s(n) = s (n − 1) + q (n − 1)

ple s (n − 1) modified by the quantization noise q (n − 1). We also note that the difference equation in (12.21) represents an integrator with an input ˜ e(n). Hence an equivalent realization of the 1-step predictor is an accumulator with an input equal to the quantized error signal ˜ e(n). In gen-

eral, the quantized error signal is scaled by some value, say ∆ 1 , which is called the step size. This equivalent realization is illustrated in Figure 12.9. In effect, the encoder shown in Figure 12.9 approximates a waveform s (t)

FIGURE 12.9 An equivalent realization of a delta modulation system

Chapter 12

APPLICATIONS IN COMMUNICATIONS

FIGURE 12.10 Two types of distortion in the DM encoder

by a linear staircase function. In order for the approximation to be rela- tively good, the waveform s (t) must change slowly relative to the sampling rate. This requirement implies that the sampling rate must be several (a factor of at least 5) times the Nyquist rate. A lowpass filter is usually incorporated into the decoder to smooth out discontinuities in the recon- structed signal.

12.4.1 ADAPTIVE DELTA MODULATION (ADM) At any given sampling rate, the performance of the DM encoder is limited by two types of distortion as shown in Figure 12.10. One is called slope-

overload distortion. It is due to the use of a step size ∆ 1 that is too small to follow portions of the waveform that have a steep slope. The second type of distortion, called granular noise, results from using a step size that is too large in parts of the waveform having a small slope. The need to minimize both of these two types of distortion results in conflicting requirements in

the selection of the step size ∆ 1 .

An alternative solution is to employ a variable size that adapts itself to the short-term characteristics of the source signal. That is, the step size is increased when the waveform has a steep slope and decreased when the waveform has a relatively small slope.

A variety of methods can be used to set adaptively the step size in every iteration. The quantized error sequence ˜ e(n) provides a good indica- tion of the slope characteristics of the waveform being encoded. When the quantized error ˜ e(n) is changing signs between successive iterations, this is an indication that the slope of the waveform in the locality is relatively small. On the other hand, when the waveform has a steep slope, successive values of the error ˜ e(n) are expected to have identical signs. From these ob- servations it is possible to devise algorithms that decrease or increase the step size, depending on successive values of ˜ e(n). A relatively simple rule devised by [13] is to vary adaptively the step size according to the relation

˜ e (n)˜ e ∆(n) = ∆ (n − 1) K (n−1) , n = 1, 2, . . . (12.23)

Delta Modulation (DM) 621

FIGURE 12.11 An example of a delta modulation system with adaptive step size

where K ≥ 1 is a constant that is selected to minimize the total distortion.

A block diagram of a DM encoder-decoder that incorporates this adaptive algorithm is illustrated in Figure 12.11.

Several other variations of adaptive DM encoding have been inves- tigated and described in the technical literature. A particularly effective and popular technique first proposed by [6] is called continuously variable slope delta modulation (CVSD). In CVSD the adaptive step size parame- ter may be expressed as

∆(n) = α∆ (n − 1) + k 1 (12.24) if ˜ e(n), ˜ e (n − 1), and ˜ e(n − 2) have the same sign; otherwise ∆(n) = α∆ (n − 1) + k 2 (12.25) The parameters α, k 1 , and k 2 are selected such that 0 < α < 1 and

k 1 >k 2 > 0. For more discussion on this and other variations of adaptive

Chapter 12

APPLICATIONS IN COMMUNICATIONS

DM, the interested reader is referred to the papers by Jayant [14] and Flanagan et al. [4] and to the extensive references contained in these papers.

12.4.2 PROJECT 12.4: DM AND ADM The purpose of this project is to gain an understanding of delta modula- tion and adaptive delta modulation for coding of waveforms. This project involves writing MATLAB functions for the DM encoder and decoder as shown in Figure 12.9, and for the ADM encoder and decoder shown in Figure 12.11. The lowpass filter at the decoder can be implemented as a linear-phase FIR filter. For example, a Hanning filter that has the impulse response

1 2πn

h(n) =

1 − cos

, 0≤n≤N−1

2 N−1

may be used, where the length N may be selected in the range 5 ≤ N ≤ 15. The input to the DM and ADM systems can be supplied from the waveforms generated in Project 12.1 except that the sampling rate should

be higher by a factor of 5 to 10. The output of the decoder can be plotted. Comparisons should be made between the output signal from the DM and ADM decoders and the original input signal.