Water-filling algorithm: Difference between revisions

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The '''water-filling algorithm''' is a technique used in [[digital communications]] systems for allocating power among different channels in multicarrier schemes. It was described by R. C. Gallager in 1968<ref name="gallager">{{cite book |last=Gallager |first=R. C. |publisher=Wiley |year=1968 |title=Information Theory and Reliable Communications}}</ref> along with the '''water-filling theorem''' which proves its optimality for channels having [[Additive White Gaussian Noise]] (AWGN) and [[intersymbol interference]] (ISI).

{{merge from|Water-pouring algorithm|discuss=Talk:Water filling algorithm#Merge proposal: Water-pouring algorithm|date=June 2025}}

For this reason, it is a standard baseline algorithm for various digital communications systems, such as [[MIMO|MIMO wireless systems]].<ref>{{ cite patent

| country = USA

| number = 6973122

| title = Power allocation scheme for DMT-based modems employing simplex transmission

| pubdate = Dec 6 2005

| fdate = Jan 26 2001

| inventor = Miller II et al

| quote = The optimum approach for Additive White Gaussian Noise (AWGN), has been proved to be a `water pouring` algorithm of power distribution, where the g.sub.k.multidot.N.sub.k.sup.1 profile is considered to be equivalent to the `terrain` and the available power budget is likened to `water that is poured` on the terrain. In this analogy, the water depth at position k is equivalent to the power allocated to the frequency bin k.

}}</ref>

The intuition that gives the algorithm its name is to think of the communication medium as if it was some kind of water container with an uneven bottom. Each of the available channels is then a section of the container having its own depth, given by the reciprocal of the frequency-dependent [[Signal-to-noise ratio|SNR]] for the channel.<ref name="gallager"/><ref name="horrible">{{cite journal |last=Biglieri |first=Ezio |title=Coding and modulation for a horrible channel |journal=IEEE Communications Magazine |volume=41 |issue=5 |date=May 2003 |pages=92–98 |doi=10.1109/MCOM.2003.1200107 }}</ref>

'''Water filling algorithm''' is a general name given to the ideas in [[communication systems]] design and practice for [[adaptive equalizer|equalization]] strategies on [[channel (communications)|communications channels]]. As the name suggests, just as water finds its level even when filled in one part of a vessel with multiple openings, as a consequence of [[Pascal's law]], the amplifier systems in communications network repeaters, or receivers amplify each channel up to the required power level compensating for the channel impairments. See, for example, channel power allocation in [[MIMO]] systems.

To allocate power, imagine pouring water into this container (the amount depends on the desired maximum average transmit power). After the water level settles, the largest amount of water is in the deepest sections of the container. This implies allocating more power to the channels with the most favourable SNR. Note, however, that the ratio allocation to each channel is not a fixed proportion but varies nonlinearly with the maximum average transmit power.

==Single channel systems==

In a single-channel communication system the deamplification and loss present on them can be simplistically taken as attenuation by a percentage ''g'', then amplifiers restore the signal power level to the same value at transmission setup by operating at a gain of 1/ (1&nbsp;−&nbsp;''g''). E.g. if we experience 6&nbsp;dB attenuation in transmission, i.e. 75% loss, then we have to amplify the signal by a factor of 4''x'' to restore the signal to the transmitter levels.

==Multichannel systems==

Same ideas can be carried out in presence impairments and a multiple channel system. Amplifier nonlinearity, crosstalk and power budgets prevent the use of these waterfilling algorithms to restore all channels, and only a subset can benefit from them.

==See also==

* [[Water-pouring algorithm]]

* [[Zero-forcing equalizer]]

* [[Robert Lucky]]

* [[Amplifier system]]

* [[EDFA]]

==References==

* Proakis, Digital Communication Systems, 4th Ed., McGraw Hill, (2001).

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