- sampled values of input signal
- prediction error, difference between actual and predicted value
- quantized prediction error
- predicted value
- reconstructed value of sampled signal
- value after DPCM coding (input value for DPCM decoding)
- predictor coefficients (weighting factors)

is current sample and is predicted value, predicted value is formed using prediction factors and previous samples, usually linear prediction is used, so predicted value can be given as a weighed linear combination of p  previous samples using , weighting factors:

Difference signal is then: 
We choose weighting factors in order to minimize some function of error between and (like mean-squared) this leads us to the minimization of quantization noise (better signal-to-noise ratio).

DPCM compression of images and video signals

DPCM conducted on signals with correlation between successive samples leads to good compression ratios.
Images and video signals are examples of signal which have above mentioned correlation. In images this means that there is a correlation between the neighboring pixels, in video signals correlation is between the same pixels in consecutive frames and inside frames (which is same as correlation inside image).
Formally written, DPCM compression method can be conducted for intra-frame coding and inter-frame coding. Intra-frame coding exploits spatial redundancy and inter-frame coding exploits temporal redundancy.
In the intra-frame coding the difference is formed between the neghboring pixels of the same frame, while in the inter-frame coding it is formed between the value of the same value in two consecutive frames. In both coding intra- and inter- frame the value of target pixel is predicted using the previously-coded neighboring pixels.

If we apply facts mentioned in DPCM description and Fig 1. and Fig 2. on image compression is the current pixel value and is formed using p  pixels prior to current pixel. is differential image formed as difference beteween actual pixel and previos pixels (as described above for any signal).

It is important to point out that in forming a prediction reciever i.e decoder has access only to reconstructed pixel values , since the process of quantization of differential image introduces error, reconstructed values, as expeceted diverges from the original values. Identical predictions of both receiver and transmitter are assured by transmitter configuration in which transmitter bases its prediction on the same values as receiver i.e predicted values. The facts that were mentioned in this paragraph are applicable to signals in general not just image and video signals.

Design of DPCM system means optimizing the predictor and quantizer components, because the quantizer is included in prediction loop there is complex dependancy between the prediction error and quantizaton error so joint optimization should be performed to assure optimal results. But, modeling such optimization is very complex so optimization of those two components are usually optimized separately. It has been shown that under the mean-squared error optimization criterion, apart constructions of quantizatior and predictor are good approximations of joint optimization. Same as in the previous paragraph, facts in this paragraph are also applicable to signals in general.

Delta modulation

Delta modulation (DM )is a subclass of differential pulse code modulation. It can be viewed as a simplified variant of DPCM, in which 1-bit quantizer is used with the fixed first order predictor, and was developed for voice telephony applications.
Principle of DM : DM output is 0 if waveform falls in value, 1 represents rise in value, each bit indicates direction in which signal is changing (not how much), i.e. DM codes the direction of differences in signal amplitude instead of the value of difference (DPCM).
Basic concept of delta modualation can be explained in the DM block diagram shown in Fig 3.

Fig 3. DM encoder

Input signal is compared to the integrated output and delta signal (difference between the input signal and the pulse signal) is brought to quantizer. Quantizer generates output according to difference signal if difference signal is positive quantizer generates positive impulse, and if the difference is negative quantizer generates negative signal. So, output signal contains bipolar pulses.
As it can be noticed in DM there is a feedback by which the output signal is brought to the integrator which integrates and the bipolar pulses forming a pulse signal which is being compared to the input value. Comparisson is conducted between signal value in n-1  time interval and input signal value in n  time interval, the result is a delta signal . Delta signal can be positive or negative and then (as described above) the output signal is formed. The output signal contains information about sign of signal change for one level comparing to previous time interval.
Important chacteristic of DM is that waveform that is delta modulated needs oversampling i.e. signal must be sampled faster than necessary, sampling rate for DM is much higher than Nyquist rate (twice bandwidth). But, at any sampling rate two types of distortion limits performance of DM encoder.
These distortions are: slope overload distortion and granular noise.
Slope overload distorsion - is caused by use of step size delta which is too small to follow portions of waveform that has a steep slope.
Can be reduced by increasing the step size.
Granular noise - is caused by too large step size in signal parts with small slope. It can be reduced by decreasing the step size.

An illustration of DPCM's advantages over PCM

A typical example of a signal good for DPCM is a line in a continuous-tone (photographic) image which mostly contains smooth tone transitions. Another example would be an audio signal with a low-biased frequency spectrum.

For illustration, we present two histograms made from the same picture which were coded in two ways. The histograms show the PCM and DPCM sample frequencies, respectively.

On the first histogram(Fig 4.), a large number of samples has a significant frequency and we cannot pick only a few of them which would be assigned shorter code words to achieve compression. On the second histogram(Fig 5.), practically all the samples are between -20 and +20, so we can assign short code words to them and achieve a solid compression rate.

Fig 4. Histogram of PCM sampled image

Fig 5. Histogram of DPCM sampled image

DPCM - practical uses

In practice, DPCM is usually used with lossy compression techniques, like coarser quantization of differences can be used, which leads to shorter code words. This is used in JPEG and in adaptive DPCM (ADPCM), a common audio compression method. ADPCM can be watched as a superset of DPCM.
In ADPCM quantization step size adapts to the current rate of change in the waveform which is being compressed.
Different ADPCM implementations have been studied. The more popular is IMA ADPCM, this ADPCM implementation is based on the algorithm proposed by Interactive Multimedia Association. IMA ADPCM standard specifies compression of PCM from 16 down to 4 bits per sample.
The good side of the ADPCM method is minimal CPU load, but it has significant quantization noise and only mediocore compression rates can be achieved(4:1).