dditional delay. In the earlier section on ADSL and VDSL we highlighted that higher per user data rates are achieved by adding an additional process into the transmit and receive path by moving from the frequency domain to the time domain and back again. The combination of time-domain and frequency-domain processing at the system level is already widely implemented - SO for example in GSM and third-generation networks gain is achieved both in the frequency and time domain. A time domain to frequency domain transform in an OFDM system, however, has the express purpose of increasing per user data rates. To a lesser extent it should also increase average data rates for all users, but this depends on the efficiency and effectiveness of contention protocols and queuing and allocation algorithms (treated in Chapter 3). Transforms are the process of changing something into something else. Transforms can be unidirectional and irreversible. Setting fire to something is an example of a unidirectional and irreversible transform. More usefully, transforms can be bi-directional and reversible. AC/DC and DC conversion in electrical engineering is one of the most ubiquitous examples of a day- to-day use of transforms (hence the use of the word transformer). Our prime interest though is in the use and application of transforms in communications engineering. Transforms change the representation of a signal without changing or destroying its infor- mation content. They change the way the signal is viewed. The purpose of the transform is to make a signal easier to process. Typically, this means making the signal easier to compress and easier to send. 'Easier' implies an ability to process the signal at a lower power and/or to process the signal using less storage or (radio and network) transmission bandwidth. 2.13.1 Transforms in Historical Context Transforms can be described and implemented mathematically and exploit techniques that we take for granted but are the result of several thousand years of mathematical progress. Maths (a