Dirty paper coding

In telecommunications, **dirty paper coding** (**DPC**) is a good way to send digital data  through a channel that is subject to some interference that is known to the sender. The sender does precoding of the data so as to cancel the effect of the interference.

Costa asked the following question:

When Costa asked his question, the Shannon–Hartley theorem (and the more general noisy-channel coding theorem) was well known.
The Shannon–Hartley theorem tells us that, all else being equal, a paper sent along a path that picks up less dirt can reliably deliver more information than another paper sent along a path that picks up more dirt.
People have also thought up many ways of dealing with such dirt added after the message is written—see error detection and correction for details.

Most people expected that the same thing would happen when dirt is added to the paper before the message was written—the more dirt, the less information can be reliably sent.

In 1983, Costa showed the surprising result that we can send just as much information on such a dirty piece of paper as we can when writing on a clean sheet of paper, and gave a way to get that capacity.

A **dirty paper code** is a way for the writer to adapt his message to the dirt already on the paper.
The writer and the reader agree ahead of time on which dirty paper code they will use for the messages.

## History

People have thought up several dirty paper codes, including Costa precoding (1983), Tomlinson-Harashima precoding (1971)  and the vector perturbation technique of Hochwald et al. (2005).

A similar problem called "writing on dirty tape (WDT)" is more complicated.
As of 2005, the capacity computation problem and the capacity-achieving
problem for writing on dirty tape are unsolved
.

"Writing on wet paper" is a related problem in steganography
.

### Wireless networks

Many wireless networks use dirty paper coding, especially MIMO systems.

In a wireless network, often a transmitter has many different messages, and each one needs to be sent to a different person.

The sum-rate capacity of a system that transmits all the messages at the same time—and uses dirty-paper codes to reduce the interference between messages—can be many times the sum-rate capacity of a similar system that only sends one message at a time (TDMA).

Any one receiver is only concerned with the messages for that receiver—all the other messages the transmitter is simultaneously sending to everyone else are—to that receiver—irrelevant noise that only interferes with the desired message.

The "dirty paper" story can be seen as a parable for wireless communication.
- In both cases, a person wants to get a message to some other person.
- The limited amount of power at the radio transmitter is analogous to the limited amount of ink.
- The "other messages the transmitter is simultaneously sending" are analogous to the dirt is already on the paper.
- The transmitter—which knows exactly all the other messages it is simultaneously sending—is analogous to the writer—who knows where the dirt is on the paper.
- Other channel noise that the transmitter does not know about until after the message is sent—environmental noise, other transmitters, etc. -- is analogous to dirt added after the message is written.

Recently, there has been interest in DPC as a possible solution to optimize the efficiency of wireless networks, in particular multiuser MIMO networks  and into an interference aware coding technique for dynamic wireless networks.

### Digital watermarking

People doing "informed digital watermarking" use dirty paper codes, using this analogy:
- The "cover work" to be watermarked is analogous to the dirty paper.
- The person adding the watermark already knows what the cover work looks like, analogous to the writer who knows where the dirt is on the paper.
- The person adding the watermark wants the watermarked work to look the same as the original "cover work", so he makes only small modifications, analogous to the writer using only a limited amount of ink.
- Changes that happen during normal processing or malicious tampering are analogous to dirt added after the message is written.
- The person who tries to detect the watermark in the watermarked work is analogous to the reader.
- Trying to detect the watermark in the watermarked work, without seeing the original -- "blind detection"—is analogous to "cannot distinguish between dirt and ink".

## Related pages

- Multiple-input multiple-output communications
- Advanced MIMO communications
- Cognitive radio