Raised-Cosine-Filter

The raised cosine filter, also known as a cosine roll-off filter is a digital signal in the processing, a portion of the communication equipment, Applied electronic filter that is used for forming signal pulses. It belongs to the group of the Nyquist filter.

The main feature of this filter satisfies the first Nyquist criterion. This means that temporally successive signal pulses which are formed by this filter, at a time interval of the sampling signal having zero points and thus do not interfere with preceding and succeeding pulses at the sampling instants. Therefore, this allows a time- discrete filter signal transmission in which no inter-symbol interference (ISI) occurs.

Representation

A raised cosine - filter always has the properties of a low-pass filter and is used in digital filter structures (such as a filter with a finite impulse response ( FIR) ) realized. Its transfer function (the amount of the frequency response ) up to a certain frequency, which depends on a roll- off factor, a constant and drops in addition to higher frequency cosine to the value zero. From this circumstance the name of this filter is derived.

The raised cosine filter is used for digital signal transmission to form the transmission pulses used on the transmission channel. The transmitted pulses, or symbols called, represent the discrete to transmitted information is supplied The transmission channel can be for example a radio channel or in a pipe. Therefore, applications are in digital broadcasting, such as mobile telecommunications.

The implementation of this filter is carried out in digital signal processors ( DSP), application specific integrated circuits (ASICs ), or programmable digital circuits, the so-called Field Programmable Gate Arrays (FPGA).

Transfer function

The transfer function of this filter is, except for the case of time- discrete transmission systems always substantially the symbol rate (1 / T) of a particular factor, the roll- off factor α depends. The term α for the roll-off factor is uniformly set in the literature. There are also terms like r or β.

With this factor, which can take values ​​between 0 and 1, the slope of the transfer characteristic is significantly affected: For the limit α = 0, an ideal, non-causal low-pass with a rectangular transfer function results. For α = 1 results in a maximally flat Kosinusflanke. For intermediate values ​​of the corresponding frequency response is approximately constant in a certain range, and then only for stays with a slightly steeper Kosinusflanke, as shown in the following pictures:

The larger the roll-off factor is, the more increases the bandwidth of the filter. The bandwidth may be increased only up to the symbol rate 1 / T (due to the discrete-time property). A smaller roll-off factor leads, because of the steeper flank of the filter to greater undesirable overshoot the ultimate result in real transmission systems, a larger phase noise and thus an inadequate pulse shaping. This can lead to errors in the demodulation.

In practice, however, a roll-off factor of less than 0.5 is generally used in the range of 0.2 to 0.5, since this valuable bandwidth can be saved. For example, using the UMTS mobile phone standard for the filter used is a pulse roll-off factor of α = 0.22.

The bandwidth of the filter at the symbol rate fs = 1 / T is calculated to be in the baseband:

Mathematical description of the transfer function

Normalized to 1, the transfer function H (f ) is given by

With the impulse response

The singularities at and have been fixed by continuous extension. The impulse response has on the course of the sinc function, which has at multiples of the symbol duration T of zeros and therefore is free of intersymbol interference.

Symbol rate

The transmitted symbol rate of a raised cosine filter with roll-off factor given α and B is a bandwidth of:

Since the bandpass range is twice the bandwidth available results:

In the baseband bandwidth efficiency in the limiting case of α = 0 is 2 symbols / s per hertz of bandwidth or with binary transfer 2 bits / s per hertz of bandwidth and halved when double bandwidth requirements in the bandpass region. In practical realizations of roll-off factors in the range α = 0.3 are (30%) selected, resulting in the case of binary transmission, a spectral efficiency of 1.5 bit / s per Hz bandwidth results.

Eye diagram

The following figures show eye diagrams of raised cosine filters with different roll-off factors. With this, the quality of the signals judge.

Root raised cosine filter

The root raised cosine filter, short RRC corresponds to the root (English root) from the raised cosine filter.