JPL's Wireless Communication Reference Website

Chapter: Analog and Digital Transmission
Section: CDMA, Rake Receiver

Analysis of Rake Receiver

The rake receiver is designed to optimally detected a DS-CDMA signal transmitted over a dispersive multipath channel.

 

 
Different reflected waves arrive with different delays. A rake receiver can detect these different signals separately. These signals are then combined, using the diversity technique called maximum ratio combining.  

 

Considering antipodal (i.e., BPSK-type) of DS-CDMA, we denote the transmit waveform as m0 for a digital "0" and -m0 for a digital "1". The received signal is

   ri(t) = S hk mi(t - kTc) + z(t)

where

The power in the k-th resolvable bin is hk hk* The local-mean power is

     p = E Shk hk*

where E is the expectation operator, applied to find the local-mean value, and * denotes a complex conjugate. We ignore intersymbol interference, that is, we assume that the bit duration Tb is much larger than the rms delay spread Trms. However, typically Tc < Trms.

Deriving the Rake Algorithm

In this section we apply the principle of a matched filter to a DS-CDMA signal, received over a Linear Time Invariant (LTI) dispersive channel, with additive white Gaussian noise (AWGN). Using the matched filter theory, we derive the optimum receive algorithm, which is also called a "rake receiver".

An optimal matched filter receiver exploits information of the transmit waveform and the channel characteristics. It correlates the incoming signal ri(t) with the expected received signal y(t) = S hk m0(t - kTc). The decision variable v becomes

    v = Re INT ri(t) yi*(t)

where INT is an integration of one bit time. Inserting yi*(t)

   v = Re INT ri(t) S hk* m0*(t - kTc)

Interchanging integration and summing, we can rewrite this as

   v = S Re hk* INT ri(t) m0*(t - kTc)

Thus, we can implement K correlators, each correlating the incoming signal with the signal waveform m0*(t - kTc), time-shifted by kTc. The outputs of these correlator are weighed by hk*. Since the signal amplitude is also proportional to hk, this correspons to maximum ratio combining diversity.

Note that thus far we have not assumed any specific properties of the spreading code.

Bit Error Rate in Fixed, Known Channel

We assume that the receiver perfectly knows and applies estimates of the channel hk. We insert the received signal r(t) into the decision variable. That allows us to derive the received signal-to-noise ratio and the BER.

   v = Sk1 Re hk1* INT [Sk2 hk2 mi(t - k2 Tc) + z(t) ] m0*(t - k1 Tc)

Here, we simplified the effect of partial correlations. See a more detailed discussion.
Splitting the expression for the devision variable for signal and noise,

   v = Sk1 Re hk1* INT Sk2 hk2 mi(t - k2 Tc) ) m0*(t - k1 Tc)

      + Sk1 Re hk1* INT z(t) m0*(t - k1 Tc)

We now use the autocorrelation properties R() of the spreading code, with

   R(k1 - k2) = INT m(t - k1 Tc) m(t - k2 Tc) dt

where the integration is over one bit duration, which also equals one period of the code sequence. This gives the following decision variable:

   v = Sk1 Re hk1* Sk2 hk2 R(k1 - k2)

      + Sk1 Re hk1* INT z(t) m0*(t - k1 Tc)

For speading codes with sufficiently good autocorrelation, one typically assumes that R(0) = Tb with Tb the duration of a bit and R(n) = 0 for n not equal to zero. So

   v = Tb Sk1 Re hk1* hk1

      + Sk1 Re hk1* INT z(t) m0*(t - k1 Tc)

So the signal-to-noise ratio in the decision variable is

         Tb2 [ Sk1hk1*hk1  ]2
    g = ------------------------- = Sgk 
         N0Tb   Sk1hk1*hk1  

The BER is

   P = (1/2) erfc[SQRT( S gk )]

It is important to realize the limitations of this derivation.

Bit Error Rate in Rayleigh Channel

According to Rayleigh's model, the channel coefficients hk are independent complex Gaussian random variables. The power received or the signal-to-noise ratio in the k-th bin of the delay spread is gk, which is exponentially distributed. Powers in various paths are independent, but usually not identically distributed. The delay profile describes the expected power in each path.

The pdf of S gk is found considering the Laplace images (or characteristic function) of the pdf of individual received powers. For an exponential signal-to-noise ratio, the image is

Uk(s) = 1/(1+sGk),

with Gk the local-mean signal-to-noise ratio.

So for all paths combined, the image of the post-combiner signal-to-noise ratio is the product of individual images. Thus,

U(s) = Pk Uk = Pk 1/(1+sGk),

Inverse transformation yields

   f(g) = S Ck /Gk exp(-gk/Gk)

where

   Ck = Pi, i not equal k Gk /(Gk - Gi)

We now average the instantaneous BER (for fixed {gk}) over the pdf of the f(g) . The local-mean BER is

   P = (1/2) Sk Ck [ 1- SQRT(Gk / (1+Gk ) ) ]

This expression can be evaluated wity the SPEC, embedded below.  

BER SPEC

This Special Purpose Embedded Calculator calculates the BER for a DS-CDMA signal over a dispersive channel with known delay profile.

BER for DS-CDMA

Pre-programmed default parameters:

Exponential delay spread.
RMS Delay spread / Chip duration
Local-mean SNR dB
Number of
fingers in Rake
dB
 
 

Resolvable path parameters

Path 1: Local-mean SNR dB
Path 2: Local-mean SNR dB
Path 3: Local-mean SNR dB
Path 4: Local-mean SNR dB
Path 5: Local-mean SNR dB
Path 6: Local-mean SNR dB
 
 

Result:

BER for BPSK  



JPL's Wireless Communication Reference Website 1999.