Root raised cosine filter

The root raised cosine filter , abbreviated RRC filter , is an electronic filter used in digital signal processing , which is used to form signal pulses for transmission over a channel , such as a radio channel .

General

The root-raised-cosine filter corresponds to the root (engl. Root ) from the raised cosine filter , and serves to distribute the characteristics of the raised cosine to the transmitter and receiver evenly. It then represents a so-called matched filter and maximizes the signal-to-noise ratio in the receiver . A special feature is that a root raised cosine filter by itself has intersymbol interference (ISI), which means that the individual transmission pulses “flow” into one another over time on the transmission channel. Only the combination of the two RRC filters at the transmitter and receiver results in an ISI-free transmission path over the entire route, which allows the individual pulses to be differentiated over time. The individual RRC pulses are orthogonal to one another.

Along with the Gaussian filter, the root-raised cosine filter is one of the most frequently used filters for pulse shaping in digital transmission systems.

Transfer function

Impulse response h (t) of an RRC for various roll-off factors β

The magnitude of the transfer function H _rrc (jω) of an RRC filter is given by:

${\displaystyle |H_{\mathrm {RRC} }(j\omega )|={\sqrt {|H_{\mathrm {RC} }(j\omega )|}}}$

where H _rc (jω) represents the transfer function of the raised cosine filter .

The impulse response h (t) of an RRC filter is characterized by the roll-off factor β, which determines the bandwidth , and the duration of a transmission symbol T _s and has the following form:

${\displaystyle h(t)={\begin{cases}{\dfrac {1}{\sqrt {T_{s}}}}\left(1-\beta +4{\dfrac {\beta }{\pi }}\right),&t=0\\{\dfrac {\beta }{\sqrt {2T_{s}}}}\left[\left(1+{\dfrac {2}{\pi }}\right)\sin \left({\dfrac {\pi }{4\beta }}\right)+\left(1-{\dfrac {2}{\pi }}\right)\cos \left({\dfrac {\pi }{4\beta }}\right)\right],&t=\pm {\dfrac {T_{s}}{4\beta }}\\{\dfrac {1}{\sqrt {T_{s}}}}{\dfrac {\sin \left[\pi {\dfrac {t}{T_{s}}}\left(1-\beta \right)\right]+4\beta {\dfrac {t}{T_{s}}}\cos \left[\pi {\dfrac {t}{T_{s}}}\left(1+\beta \right)\right]}{\pi {\dfrac {t}{T_{s}}}\left[1-\left(4\beta {\dfrac {t}{T_{s}}}\right)^{2}\right]}},&{\mbox{andernfalls}}\end{cases}}}$

literature

• John G. Proakis, Masoud Salehi: Communication Systems Engineering . 2nd Edition. Prentice Hall, Upper Saddle River NJ 2002, ISBN 0-13-095007-6 .
• John B. Anderson: Digital Transmission Engineering . 2nd Edition. Wiley-Interscience, 2005, ISBN 0-471-69464-9 , pp. 26-30 .