# Symbol crosstalk

The symbol crosstalk , also known as intersymbol interference (ISI), describes interference between temporally successive transmission symbols in digitally coded transmission techniques . In contrast to crosstalk , which describes mutual crosstalk between spatially adjacent transmission paths and can also affect analog transmissions, symbol crosstalk relates to the time sequence of the symbols used for digital transmission on one and the same transmission channel.

## causes

In the case of digital transmission, the discrete-value information is transmitted in successive symbols over a channel . This channel can, for example, be a radio link or a wired transmission. Symbol crosstalk is triggered by the following causes:

1. By limiting the bandwidth of the transmission channel. As a result, the shape of the transmission symbols can be spectrally deformed and time-extended by different group delays of the individual frequency components. As a result, the individual symbols “flow” into one another over time.
2. Multipath propagation , especially for free space connections with echoes. In this case, the different lengths of transit times of the individual signal paths at the receiver result in a time-shifted superposition of the symbols transmitted. In the extreme case, the disturbance can completely erase symbol sequences, at least temporarily, through destructive interference .
3. With some pulse shaping filters , such as the root-raised cosine filter , intersymbol interference is deliberately allowed on the transmission channel with the aim of minimizing the required bandwidth. A filter adapted to the pulse shaping of the transmitter is used on the receiver , so that in total, both pulse shaping filters and transmission path, ISI-free data transmission is possible.

## Band-limited channels

Since each transmission channel has a band limitation and, on the other hand, a transmission channel should be used as efficiently as possible, certain boundary conditions arise under which ISI-free transmission is still possible with a given bandwidth. These are summarized in the two Nyquist conditions which must be met in order to avoid symbol crosstalk as a result of a band limitation.

### First Nyquist condition

A pulse train of five RC pulses that meet the first Nyquist condition

The first Nyquist criterion states that the impulse response h (t) of the total transmission system with the sampling rate T at the sampling times n * T ( n satisfy an integer), the following condition must:

${\ displaystyle h [n \ cdot T] = {\ begin {cases} 1; & n = 0 \\ 0; & n \ neq 0 \ end {cases}}}$

This means that a specific transmission symbol which is sent at the time n = 0 must be zero at all other sampling times. So-called pulse shaping filters are used to meet this requirement. An example is the raised cosine filter (RC pulse), the impulse response of which is shown on the right for a sequence of five transmission pulses. Each RC pulse is only exactly at its sampling time 1 and at all other sampling times it is equal to 0, which means that no symbol crosstalk occurs. As a further example, the ideal low-pass filter also fulfills the first Nyquist condition, but can not be implemented due to the lack of causality .

### Second Nyquist condition

Eye diagram

The second Nyquist condition represents a tightening of the first Nyquist condition and additionally requires that the impulse response h (t) of the filter must have the value 0 exactly between two sampling times. This fact can be illustrated graphically in the eye diagram : The first Nyquist condition requires the maximum opening of the eye at sampling time t = 0 in the vertical direction. The second Nyquist criterion calls for the maximum opening of the eye in the horizontal direction of the symbol duration T .

In the eye diagram opposite, the second Nyquist condition is barely fulfilled. The RC pulse of a raised cosine filter only fulfills the second Nyquist condition for the so-called roll-off factor of β = 1. In this case, the signal edge of the RC pulse represents a Nyquist edge .

In many practical transmission systems, the second Nyquist condition is not exactly met. The less the second Nyquist condition is fulfilled or the further the eye is closed in the horizontal direction, the more precise the symbol clock must be at the receiver.