## Differential Pulse Code Modulation

### Section Overview

The quantization in the section Quantizing a Signal requires no *a
priori* knowledge about the transmitted signal. In practice, you can
often make educated guesses about the present signal based on past signal
transmissions. Using such educated guesses to help quantize a signal is known as
*predictive quantization*. The most common predictive
quantization method is differential pulse code modulation (DPCM).

The functions `dpcmenco`

, `dpcmdeco`

, and `dpcmopt`

can help you implement a
DPCM predictive quantizer with a linear predictor.

### DPCM Terminology

To determine an encoder for such a quantizer, you
must supply not only a partition and codebook as described in Represent Partitions
and Represent Codebooks, but also a *predictor*. The
predictor is a function that the DPCM encoder uses to produce the educated guess at
each step. A linear predictor has the form

y(k) = p(1)x(k-1) + p(2)x(k-2) + ... + p(m-1)x(k-m+1) + p(m)x(k-m)

where x is the original signal, `y(k)`

attempts to predict the
value of `x(k)`

, and `p`

is an
`m`

-tuple of real numbers. Instead of quantizing
`x`

itself, the DPCM encoder quantizes the *predictive
error*, x-y. The integer `m`

above is called the
*predictive order*. The special case when
`m = 1`

is called *delta
modulation*.

### Represent Predictors

If the guess for the `k`

th value of the signal
`x`

, based on earlier values of `x`

, is

y(k) = p(1)x(k-1) + p(2)x(k-2) +...+ p(m-1)x(k-m+1) + p(m)x(k-m)

then the corresponding predictor vector for toolbox functions is

predictor = [0, p(1), p(2), p(3),..., p(m-1), p(m)]

**Note**

The initial zero in the predictor vector makes sense if you view the vector as the polynomial transfer function of a finite impulse response (FIR) filter.

### Example: DPCM Encoding and Decoding

A simple special case of DPCM quantizes the difference between the signal's
current value and its value at the previous step. Thus the predictor is just
`y(k) = x (k - 1)`

. The code below
implements this scheme. It encodes a sawtooth signal, decodes it, and plots both the
original and decoded signals. The solid line is the original signal, while the
dashed line is the recovered signals. The example also computes the mean square
error between the original and decoded signals.

predictor = [0 1]; % y(k)=x(k-1) partition = [-1:.1:.9]; codebook = [-1:.1:1]; t = [0:pi/50:2*pi]; x = sawtooth(3*t); % Original signal % Quantize x using DPCM. encodedx = dpcmenco(x,codebook,partition,predictor); % Try to recover x from the modulated signal. decodedx = dpcmdeco(encodedx,codebook,predictor); plot(t,x,t,decodedx,'--') legend('Original signal','Decoded signal','Location','NorthOutside'); distor = sum((x-decodedx).^2)/length(x) % Mean square error

The output is

distor = 0.0327

### Optimize DPCM Parameters

#### Section Overview

The section Optimize Quantization Parameters describes how to use training
data with the `lloyds`

function to help find quantization
parameters that will minimize signal distortion.

This section describes similar procedures for using the
`dpcmopt`

function in conjunction with the two functions
`dpcmenco`

and `dpcmdeco`

, which first
appear in the previous section.

**Note**

The training data you use with `dpcmopt`

should be
typical of the kinds of signals you will actually be quantizing with
`dpcmenco`

.

#### Example: Comparing Optimized and Nonoptimized DPCM Parameters

This example is similar to the one in the last section. However, where the
last example created `predictor`

, `partition`

,
and `codebook`

in a straightforward but haphazard way, this
example uses the same codebook (now called `initcodebook`

) as
an initial guess for a new *optimized* codebook parameter.
This example also uses the predictive order, 1, as the desired order of the new
optimized predictor. The `dpcmopt`

function creates these
optimized parameters, using the sawtooth signal `x`

as training
data. The example goes on to quantize the training data itself; in theory, the
optimized parameters are suitable for quantizing other data that is similar to
`x`

. Notice that the mean square distortion here is much
less than the distortion in the previous example.

t = [0:pi/50:2*pi]; x = sawtooth(3*t); % Original signal initcodebook = [-1:.1:1]; % Initial guess at codebook % Optimize parameters, using initial codebook and order 1. [predictor,codebook,partition] = dpcmopt(x,1,initcodebook); % Quantize x using DPCM. encodedx = dpcmenco(x,codebook,partition,predictor); % Try to recover x from the modulated signal. decodedx = dpcmdeco(encodedx,codebook,predictor); distor = sum((x-decodedx).^2)/length(x) % Mean square error

The output is

distor = 0.0063