Rake receiver

As a rake receiver , even rake receiver , referred to receivers for digital signals that the multipath are designed. A rake receiver consists of several sub-receivers that receive and evaluate the received signal one after the other. By stringing together the individual sub-recipient results in a structure shown in the schematic representation as a rake (Engl. Rake ) looks like and this reception structure gives its name.

The rake receiver is used in particular for mobile radio transmissions which are based on code division multiple access (CDMA) or wideband CDMA . These are, for example, receivers for cellular radio standards such as UMTS , CDMA2000 or the wireless local area network (WLAN).

Working principle

By means of correlation , the rake receiver is able to selectively pick out signal components from a signal that spreads over several paths because of the multipath reception and is superimposed in time, and to receive them constructively, i.e. with an improvement in the signal / noise ratio.

In order to be able to equalize the multipath propagation with a rake receiver, the data signal to be transmitted must be multiplied with an in most cases antipodal (bipolar) spreading sequence before the modulation . The bandwidth is spread through the Direct Sequence Spread Spectrum (DSSS) used . The exact knowledge of the spreading sequence and its start time at the receiver end then enables the components of the transmission signal that have propagated over paths with different transit times to be recovered . The properties of the autocorrelation and the cross-correlation of the spreading sequence used are used here.

Principle of the spread spectrum

The data signal is a bipolar data signal, which consists of square-wave pulses ${\displaystyle d(t)}$

${\displaystyle d(t)=\sum \limits _{\nu =-\infty }^{\infty }c_{\nu }\cdot g_{r}(t-\nu T_{d})}$with and ${\displaystyle c_{\nu }\in \{-1,1\}}$ ${\displaystyle g_{r}(t)={\begin{cases}1&t\in [{\frac {-T_{d}}{2}};{\frac {T_{d}}{2}}]\\0&{\text{sonst.}}\end{cases}}}$

The symbol duration of the data signal is thus . This transmission signal takes up a certain spectral bandwidth . This results from the Fourier transformation of the data signal. Because of the qualitatively opposing relationship between symbol duration and spectral area bandwidth, which occurs at least with equally probable transmission bits, it immediately appears plausible that the bandwidth occupancy is linked to the symbol duration by directly opposing. The multiplication with the spreading sequence, which is also bipolar ${\displaystyle d(t)}$ ${\displaystyle T_{d}}$ ${\displaystyle \Delta f_{d}}$ ${\displaystyle d(t)}$

${\displaystyle p(t)=\sum \limits _{\mu =-\infty }^{\infty }c_{\mu }\cdot g_{p}(t-\mu T_{c})}$with and and on ${\displaystyle c_{\mu }\in \{-1,1\}}$ ${\displaystyle g_{p}(t)={\begin{cases}1&t\in [{\frac {-T_{c}}{2}};{\frac {T_{c}}{2}}]\\0&{\text{sonst.}}\end{cases}}:}$ ${\displaystyle {\frac {T_{d}}{T_{c}}}=:N{\text{ mit }}N\in \mathbb {N} }$

leads to the signal

${\displaystyle s(t)=\left(\sum \limits _{\mu =-\infty }^{\infty }c_{\mu }\cdot g_{p}(t-\mu T_{c})\right)\cdot \left(\sum \limits _{\nu =-\infty }^{\infty }c_{\nu }\cdot g_{r}(t-\nu T_{d})\right)}$

Because the symbol duration of the data signal is an integral multiple of the symbol duration of the spreading sequence, it is immediately clear that higher frequencies occur in than in . The symbol duration of the spreading sequence is called the chip duration. A chip describes a state, i.e. a symbol, of the spreading sequence. A more detailed analysis reveals that the frequency band occupied by has become a factor wider. For this reason, the size that corresponds to the length of the spreading code is called the spreading factor. A more detailed description of the principle of spectral spreading can be found in the article code division multiplexing . ${\displaystyle s(t)}$ ${\displaystyle d(t)}$ ${\displaystyle d(t)}$ ${\displaystyle N}$ ${\displaystyle N}$

Code sequence requirements

Rake receivers are particularly suitable for equalizing so-called macro paths. A macro path is characterized by the fact that the runtime of the signal on the respective path is much longer than the chip duration. An essential property of the spreading sequence used in single user transmission is its autocorrelation property. The autocorrelation describes the linear similarity of a signal to itself if the signal is shifted by a certain time. See the article on autocorrelation .

A suitable spreading sequence ideally has a sharp and periodic autocorrelation function. In this context, sharp means that the periodic autocorrelation function of the spreading sequence assumes its maximum value for shifts that correspond to an integral multiple of , but the linear similarity of the spreading sequence to itself has already disappeared for shifts of . ${\displaystyle N\cdot T_{c}}$ ${\displaystyle \pm T_{c}}$

Spreading sequences that theoretically perfectly meet this requirement are signals, the so-called M sequences , which can be systematically obtained with the help of shift registers. One says: two different M-sequences delayed differently by at least one chip duration are exactly orthogonal to one another. This property, together with the fact that the multiplication of a (bipolar) spreading sequence with itself, i.e. with an equal delay, results in exactly 1 at any point in time, is the basis of the functioning of the rake receiver.

Example of two-way propagation

Block diagram of a rake receiver with two correlators

In this section, the transmission chain of the data signal d (t), which is spread on the transmission side with the spreading sequence p (t), is illustrated using the example of two-way propagation. The signal p (t) at least approximately fulfills the requirement for a sharp periodic autocorrelation function. The scenario is shown in the adjacent picture.

The term is greater than the term . The rake receiver with two correlators (rake fingers) is arranged vertically in the right part of the graphic, while the transmission channel can be seen as a horizontally arranged runtime system. The two rake fingers and the operations that take place in them are now considered: ${\displaystyle t_{1}}$ ${\displaystyle t_{0}}$

In the first finger, the received signal is weighted with the delayed spreading sequence and then integrated (averaged) over a symbol duration of the data signal. Then it is multiplied by the channel-specific gain . The product of and is a real number different from zero, which depends on the selection and deselection as well as the meaningful weighting of the rake fingers. Then the following applies to the output signal of the first rake finger: ${\displaystyle t_{1}}$ ${\displaystyle c_{1}^{*}}$ ${\displaystyle c_{1}^{*}}$ ${\displaystyle c_{1}}$

${\displaystyle {\begin{aligned}y_{1}(t)&={\frac {c_{1}^{*}}{T_{\mathrm {d} }}}\int \limits _{0}^{T_{\mathrm {d} }}[c_{0}\cdot d(t-t_{0})\cdot p(t-t_{0})+c_{1}\cdot d(t-t_{1})\cdot p(t-t_{1})]\cdot p(t-t_{1})\,\mathrm {d} t\\&={\frac {c_{1}^{*}}{T_{\mathrm {d} }}}\int \limits _{0}^{T_{\mathrm {d} }}[c_{0}\cdot d(t-t_{0})\cdot p(t-t_{0})]\cdot p(t-t_{1})\,\mathrm {d} t+{\frac {c_{1}^{*}}{T_{\mathrm {d} }}}\int \limits _{0}^{T_{\mathrm {d} }}[c_{1}\cdot d(t-t_{1})\cdot p(t-t_{1})]\cdot p(t-t_{1})\,\mathrm {d} t\end{aligned}}}$

However, because of the correlation properties of the spreading sequence required above

${\displaystyle p(t-t_{0})\cdot p(t-t_{1})=0\forall t}$and . ${\displaystyle p(t-t_{1})\cdot p(t-t_{1})=1\forall t}$

The following applies:

${\displaystyle y_{1}(t)=c_{1}\cdot c_{1}^{*}\cdot {\bar {d}}(t-t_{1})}$

While in the second rake finger, the received signal is initially delayed by the transit time difference on both paths, which immediately makes sense when you consider that the two signal components that propagate over the two channel paths with the delays and are added in the correct phase for a constructive superposition Need to become. The following then applies: ${\displaystyle t_{1}}$ ${\displaystyle t_{0}}$

${\displaystyle {\begin{aligned}y_{2}(t)&={\frac {c_{0}^{*}}{T_{\mathrm {d} }}}\int \limits _{0}^{T_{\mathrm {d} }}[c_{0}\cdot d(t-t_{0}-t_{1}+t_{0})\cdot p(t-t_{0}-t_{1}+t_{0})+c_{1}\cdot d(t-2t_{1}+t_{0})\cdot p(t-2t_{1}+t_{0})]\cdot p(t-t_{1})\,\mathrm {d} t\\&={\frac {c_{0}^{*}}{T_{\mathrm {d} }}}\int \limits _{0}^{T_{\mathrm {d} }}c_{0}\cdot d(t-t_{1})\cdot p(t-t_{1})\cdot p(t-t_{1})\,\mathrm {d} t+{\frac {c_{0}^{*}}{T_{\mathrm {d} }}}\int \limits _{0}^{T_{\mathrm {d} }}c_{1}\cdot d(t-2t_{1}+t_{0})\cdot p(t-2t_{1}+t_{0})\cdot p(t-t_{1})\,\mathrm {d} t\end{aligned}}}$

Because of the required correlation properties of the spreading sequence, the following applies analogously to the first rake finger:

${\displaystyle p(t-2t_{1}+t_{0})\cdot p(t-t_{1})=0\forall t}$and . ${\displaystyle p(t-t_{1})\cdot p(t-t_{1})=1\forall t}$

and thus for the output signal of the second rake finger:

${\displaystyle y_{2}(t)=c_{0}\cdot c_{0}^{*}\cdot {\bar {d}}(t-t_{1})}$

The data signal can thus be recovered from the receiving end. ${\displaystyle d(t)}$ ${\displaystyle y(t)=y_{1}(t)+y_{2}(t)}$

literature

• John G. Proakis, Masoud Salehi: Communication Systems Engineering . 2nd Edition. Prentice-Hall, 2002, ISBN 0-13-095007-6 .
• C. Lüders: Mobile radio systems . Vogel Buchverlag, Würzburg 1996.