Companding; DPCM with derivation and block diagram

Companding

Companding (compressing + expanding) is a method of achieving non-uniform quantization. A compressor at the transmitter boosts weak signal amplitudes relative to strong ones, the result is uniformly quantized, and an expander at the receiver applies the inverse law. Weak signals thus get small effective steps and strong signals large steps, keeping SQNR almost constant over a wide range of speech levels. Standard laws are the µ-law ($\mu = 255$) and A-law ($A = 87.6$).

Differential PCM (DPCM)

Speech/video samples are strongly correlated, so DPCM transmits the difference between the current sample and its predicted value rather than the sample itself. The difference has a smaller range, so fewer bits are needed for the same SQNR.

Transmitter
m[k]-->(+)--d[k]-->[Quantizer]--dq[k]-->[Encoder]--> out
        ^-                       |
   m^[k]|                        v
        +---[Predictor]<--(+)<---+
                           ^ m^[k]
Receiver
in-->[Decoder]--dq[k]-->(+)--mq[k]--> LPF --> m(t)
                         ^    |
                         +-[Predictor]

Operation and derivation

1. Prediction error: $d[k] = m[k] - \hat m[k]$, where $\hat m[k]$ is predicted from past reconstructed samples.
2. Quantized error: $d_q[k] = d[k] + q[k]$, with $q[k]$ the quantization error.
3. Predictor input:

$$\begin{aligned} m_q[k] &= \hat m[k] + d_q[k] = \hat m[k] + d[k] + q[k] \\ &= m[k] + q[k] \end{aligned}$$

So the reconstructed sample differs from the true one only by the quantization error of $d[k]$; errors do not accumulate. 4. The receiver has the same predictor, adds $d_q[k]$ to its prediction and gets $m_q[k]$, then low-pass filters it. 5. SNR gain: since $d[k]$ is quantized instead of $m[k]$,

$$\text{SNR}_{DPCM} = \frac{\overline{m^2}}{\overline{d^2}}\cdot\frac{\overline{d^2}}{\overline{q^2}} = G_p \times \text{SNR}_q$$

where $G_p = \overline{m^2}/\overline{d^2}$ is the prediction gain (4–11 dB for speech), equivalent to saving 1–2 bits/sample.