Delta modulation encoder/decoder; PCM vs DM

Delta modulation (DM)

DM is a 1-bit DPCM: the signal is sampled much faster than the Nyquist rate and only one bit per sample is sent, telling whether the signal is above or below the previous staircase approximation. The staircase then steps up or down by a fixed $\Delta$.

Encoder

m(t)->[Sampler]-m[k]->(+)-e[k]->[1-bit Q]-+-> eq[k]
                       ^-                 | (+D/-D)
               mq[k-1] |                  v
                       +--[Delay Ts]<--(+)<
                             mq[k]      ^
                                        | mq[k-1]

Decoder

eq[k] -->(+)--mq[k]--> [LPF] --> m(t)
          ^       |
          +-[Delay Ts]   (accumulator)

Equations

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

The feedback loop (delay + adder) is an accumulator (integrator), so the transmitter tracks the input with a staircase. The decoder is the same accumulator followed by a LPF that smooths the staircase.

Errors

• Slope overload: if the signal slope exceeds $\Delta f_s$ the staircase cannot follow. For $m(t) = A\sin 2\pi f_m t$ we need

$$\Delta f_s \ge 2\pi f_m A \;\Rightarrow\; A_{max} = \frac{\Delta f_s}{2\pi f_m}$$

• Granular noise: for a flat signal the staircase hunts $\pm\Delta$; error is uniform in $(-\Delta, \Delta)$ with power $\Delta^2/3$. After the LPF of bandwidth $f_M$:

$$N_q = \frac{\Delta^2}{3}\frac{f_M}{f_s}, \qquad \text{SQNR}_{max} = \frac{3}{8\pi^2}\frac{f_s^3}{f_m^2 f_M}$$

So SQNR rises 9 dB for every doubling of $f_s$.

PCM vs DM