JPL's Wireless Communication Reference WebsiteChapter:
Analog and Digital Transmission

The rake receiver is designed to optimally detected a DSCDMA 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., BPSKtype) of DSCDMA, we denote the transmit waveform as m_{0} for a digital "0" and m_{0} for a digital "1". The received signal is
r_{i}(t) = S h_{k} m_{i}(t  kT_{c}) + z(t)
where
p = E Sh_{k} h_{k}*
where E is the expectation operator, applied to find the localmean value, and * denotes a complex conjugate. We ignore intersymbol interference, that is, we assume that the bit duration T_{b} is much larger than the rms delay spread T_{rms}. However, typically T_{c} < T_{rms}.
In this section we apply the principle of a matched filter to a DSCDMA 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 r_{i}(t) with the expected received signal y(t) = S h_{k} m_{0}(t  kT_{c}). The decision variable v becomes
v = Re INT r_{i}(t) y_{i}*(t)
where INT is an integration of one bit time. Inserting y_{i}*(t)
v = Re INT r_{i}(t) S h_{k}* m_{0}*(t  kT_{c})
Interchanging integration and summing, we can rewrite this as
v = S Re h_{k}* INT r_{i}(t) m_{0}*(t  kT_{c})
Thus, we can implement K correlators, each correlating the incoming signal with the signal waveform m_{0}*(t  kT_{c}), timeshifted by kT_{c}. The outputs of these correlator are weighed by h_{k}*. Since the signal amplitude is also proportional to h_{k}, this correspons to maximum ratio combining diversity.
Note that thus far we have not assumed any specific properties of the spreading code.
v = S_{k1} Re h_{k1}* INT [S_{k2} h_{k2} m_{i}(t  k2 T_{c}) + z(t) ] m_{0}*(t  k1 T_{c})
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 = S_{k1} Re h_{k1}* INT S_{k2} h_{k2} m_{i}(t  k2 T_{c}) ) m_{0}*(t  k1 T_{c})
+ S_{k1} Re h_{k1}* INT z(t) m_{0}*(t  k1 T_{c})
We now use the autocorrelation properties R() of the spreading code, with
R(k1  k2) = INT m(t  k1 T_{c}) m(t  k2 T_{c}) 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 = S_{k1} Re h_{k1}* S_{k2} h_{k2} R(k1  k2)
+ S_{k1} Re h_{k1}* INT z(t) m_{0}*(t  k1 T_{c})
For speading codes with sufficiently good autocorrelation, one typically assumes that R(0) = T_{b} with T_{b} the duration of a bit and R(n) = 0 for n not equal to zero. So
v = T_{b} S_{k1} Re h_{k1}* h_{k1}
+ S_{k1} Re h_{k1}* INT z(t) m_{0}*(t  k1 T_{c})
So the signaltonoise ratio in the decision variable is
T_{b}^{2} [ S_{k1}h_{k1}*h_{k1} ]^{2} g =  = Sg_{k} N_{0}T_{b} S_{k1}h_{k1}*h_{k1}The BER is
P = (1/2) erfc[SQRT( S g_{k} )]
It is important to realize the limitations of this derivation.
The pdf of S g_{k} is found considering the Laplace images (or characteristic function) of the pdf of individual received powers. For an exponential signaltonoise ratio, the image is
U_{k}(s) = 1/(1+sG_{k}),
with G_{k} the localmean signaltonoise ratio.
So for all paths combined, the image of the postcombiner signaltonoise ratio is the product of individual images. Thus,
U(s) = P_{k} U_{k} = P_{k} 1/(1+sG_{k}),
Inverse transformation yields
f(g) = S C_{k} /G_{k} exp(g_{k}/G_{k})
where
C_{k} = P_{i, i not equal k} G_{k} /(G_{k}  G_{i})
We now average the instantaneous BER (for fixed {g_{k}}) over the pdf of the f(g) . The localmean BER is
P = (1/2) S_{k} C_{k} [ 1 SQRT(G_{k} / (1+G_{k} ) ) ]
This expression can be evaluated wity the SPEC, embedded below.