Part IV MULTIPLE ACCESS AND ADVANCED TRANSCEIVER SCHEMES 363
18.2 Code Division Multiple Access
18.2.4 Effects of Multipath Propagation on Code Division
The above, strongly simplified, description of a CDMA system assumed a flat-fading channel. This assumption is violated under all practical circumstances. The basic nature of a CDMA system is to spread the signal over a large bandwidth; thus, it can be anticipated that the transfer function of the channel exhibits variations over this bandwidth.
The effect of frequency selectivity (delay dispersion) on a CDMA system can be understood by looking at the impulse response of the concatenation spreader–channel–despreader. If the channel is slowly time variant, the effective impulse response can be written as5
heff(ti, τ )= ˜p(τ )∗h(ti, τ ) (18.8) where the effective system impulse responsep(τ )˜ is the convolution of the transmit and receive spreading sequence:
˜
p(τ )=pTX(τ )∗pRX(τ )=ACF(τ ) (18.9) In the following, we assume an ideal spreading sequence (Eq. 18.1). The despreader output then exhibits multiple peaks: more precisely, one for each Multi Path Component (MPC) that can be resolved by the receiver – i.e., spaced at leastTC apart. Each of the peaks contains information about the transmit signal. Thus, all peaks should be used in the detection process: just using the largest correlation peak would mean that we discard a lot of the arriving signal. A receiver that can use multiple correlation peaks is the so-calledRake receiver, which collects (“rakes up”) the energy from different MPCs. As shown in Figure 18.6, a Rake receiver consists of abank of correlators.
Each correlator is sampled at a different time (with delayτ), and thus collects energy from the MPC with delayτ. The sample values from the correlators are then weighted and combined.
Alternatively, we can interpret the Rake receiver as a tapped delay line, whose outputs are weighted and added up. The tap delays, as well as the tap weights, are adjustable, and matched
5Note that this expression is identical to that in correlative channel sounders (see Chapter 8). Correlative channel sounders and CDMA systems have the same structure, it is just the goals that are different: the channel sounder tries to find the channel impulse response from the received signal and knowledge of the transmit data, while the CDMA receiver tries to find the transmit data from knowledge of the received signal and the channel impulse response.
Spread Spectrum Systems 395
r(t)
hL
r
tLr– 1 tLr– 2 t1
*
hL
r – 1
h1
*
*
p(t) Code reference
Spread spectrum correlator with hard decision
dt Z s∧
Multipath interference suppression Channel
estimation
Figure 18.6 Rake receiver.
Reproduced with permission from Molisch [2000]©Prentice Hall.
to the channel. Note that the taps are usually spaced at least one chip duration apart, but there is no requirement for the taps to be spaced at regular intervals. The combination of the receiver filter and the Rake receiver constitutes a filter that is matched to the receive signal. The receive filter is matched to the transmit signal, while the Rake receiver is matched to the channel.
Independent of this interpretation, the receiver adds up the (weighted) signal from the different Rake fingers in a coherent way. As these signals correspond to different MPCs, their fading is (approximately) statistically independent – in other words, they provide delay diversity (frequency diversity). A Rake receiver is thus a diversity receiver, and all mathematical methods for the treatment of diversity remain valid. As for the performance of Rake receiver systems, we can now simply refer to the equations of Sections 13.4–13.5.
Example 18.1 Performance of a Rake receiver: Compute the Bit Error Rate (BER) of BPSK in (i) a narrowband system and (ii) with a CDMA system that can resolve all multipaths, using a six-finger Rake receiver at a 15-dB SNR, in an International Telecommunications Union (ITU) Pedestrian-A channel.
The tapped delay line model from an ITU Pedestrian-A channel is
|h(n)|dB=
0 −9.7 −19.2 −22.8
(18.10)
|h(n)| =1 0.3273 0.1096 0.0724 (18.11) The average channel gain of the flat-fading channel is
|h(n)|2=1+0.332+0.112+0.072=1.1 (18.12)
and the transmit SNR has to be
¯
γTX=101.5
1.1 =28.75 (18.13)
so that a receive SNR of 15 dB is achieved.
As can be found from Chapter 12, the BER for the flat-fading channel is BER=E|PBER(γFlat)]= π/2
0
1 πMγFlat
− 1 sin2θ
dθ (18.14)
and
BER= π/2
0
1 π
sin2θ
sin2θ+ ¯γTX|h(n)|2dθ=7.724×10−3 (18.15) When combining the signals as done in the Rake receiver we have
γRake=γ1+ ã ã ã +γ6 (18.16) Since only four MPCs carry energy, only four Rake fingers are effectively used. If theγ1, . . . , γ4 are independent, the joint pdf offγ1...γ4(γ1, . . . , γ4)=fγ1(γ1)ã. . .ãfγ4(γ4)(see also Eq. 13.39):
BER=
dγ1pdfγ1(γ1)
dγ2pdfγ2(γ2)ã ã ã
dγ4pdfγ4(γ4) π/2
0
dθf1(θ )
Nr
k=1
exp(−γkf2(θ ))
= π/2
0
1 π
4 k=1
γk
fγk(γk)e
−sin2γk θ
dγkdθ
= π/2
0
1 π
4 k=1
Mγk
− 1 sin2θ
dθ (18.17)
Thus,
SER= π/2
0
1 π
4 k=1
sin2(θ ) sin2(θ )+ ¯γk
dθ (18.18)
For the same transmit SNRγ¯TX=28.75, we then get:
BER= π/2
0
1 π
sin2θ sin2θ+ ¯γTX
sin2θ sin2θ+0.332γ¯TX
sin2θ sin2θ+0.12γ¯TX
sin2θ sin2θ+0.072γ¯TX
dθ
=9.9×10−4 (18.19)
Another consequence of the delay diversity interpretation is the determination of the weights for the combination of Rake finger outputs. The optimum weights are the weights for maximum-ratio combining – i.e., the complex conjugates of the amplitudes of the MPC corresponding to each Rake finger. However, this is only possible if we can assign one Rake finger to each resolvable MPC (the termall Rakehas been largely used in the literature for such a receiver). Up toLr=τmax/TCtaps, whereτmax is the maximum excess delay of the channel (see Chapter 6), are required in this case.
Especially for outdoor environments, this number can easily exceed 20 taps. However, the number
Spread Spectrum Systems 397
of taps that can be implemented in a practical Rake combiner is limited by power consumption, design complexity, and channel estimation. A Rake receiver that processes only a subset of the available Lr resolved MPCs achieves lower complexity, while still providing a performance that is better than that of a single-path receiver. The Selective Rake (SRake) receiver selects the Lb best paths (a subset of the Lr available resolved MPCs) and then combines the selected subset using maximum-ratio combining. This combining method is “hybrid selection: maximum ratio combining” (as discussed in Chapter 13); however, note that the average power in the different diversity branches is different. It is also noteworthy that the SRake still requires knowledge of the instantaneous values ofall MPCs so that it can perform appropriate selection. Another possibility is thePartial Rake (PRake), which uses the firstLf MPCs. Although the performance it provides is not as good, it only needs to estimateLf MPCs.
Another generally important problem for Rake receivers is interpath interference. Paths that have delay τi compared with the delay the Rake finger is tuned to are suppressed by a factor ACF(τi)/ACF(0), which is infinite only when the spreading sequence has ideal ACF properties.
Rake receivers with nonideal spreading sequences thus suffer from interpath interference.
Finally, we note that in order for the Rake receiver to be optimal there must be no ISI – i.e., the maximum excess delay of the channel must be much smaller thanTS, though it can be larger than TC. If there is ISI, then the receiver must have an equalizer (working on the Rake output – i.e., a signal sampled at intervalsTS) in addition to the Rake receiver. An alternative to this combination of Rake receiver and symbol-spaced equalizer is the chip-based equalizer, where an equalizer works directly on the output of the despreader sampled at the chip rate. This method is optimum, but very complex. As we showed in Chapter 16, the computational effort for equalizers increases quickly as the product of sampling frequency and channel delay spread increases.