ds-cdmareceiverbasedonafive-porttechnology · eurasip journal on applied signal processing 2005:11,...

EURASIP Journal on Applied Signal Processing 2005:11, 1628–1644c© 2005 Hindawi Publishing Corporation

DS-CDMA Receiver Based on a Five-Port Technology

IvoMaljevicThe Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON, Canada M5S 1A1Email: [email protected]

Elvino S. SousaThe Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON, Canada M5S 1A1Email: [email protected]

Received 29 February 2004; Revised 15 November 2004

High data rates, low-power consumption, and low complexity will be the most important parameters in the design of the next-generation mobile terminals. In this paper we are introducing a new paradigm in the design of direct sequence spread spectrumreceiver by combining analog and digital signal processing. Themain difference with respect to the conventional all-digital receiverdesign approach is that the proposed mixed analog/digital processing results in a symbol rate sampling rather than the high-ratesubchip sampling. Analog signal despreading is the key part of the proposed receiver solution, which is based on a five-port device,a passive RF square-law-type device. It is used to perform two important tasks at the same time, namely, the direct conversion andanalog despreading. To achieve lower complexity, the proposed receiver uses rectangular instead of pulse-matched despreading atthe cost of only a small performance degradation. Also, we propose a new noncoherent pseudonoise (PN) code tracking schemebased on error signal generated through the L1 norm. This results in comparable or even better PN code tracking performancethan L2 norm circuitry, using less complex hardware. Further, we explore how this technology can be applied in the design ofDS-CDMA RAKE receiver for mobile terminals. Depending on how the pilot signal is multiplexed, we propose two types of RAKEreceivers. It is shown that under Rayleigh fading channel such receiver structures offer robustness and high performance, whilemaintaining the low complexity achievable through the five-port device.

Keywords and phrases: five-port device, CDMA receiver, direct conversion, RAKE receiver.

1. INTRODUCTION

When it comes to designing wireless receivers, issues of com-plexity and low-power consumption have become as impor-tant as achieving high performance. Besides the all-digitalapproach that addresses the performance issue, the old, yetimproved, analog design approach is beginning to emerge.Hagenauer et al. [1] have demonstrated that it is possibleto implement data detection, equalization, and decoding inanalog VLSI circuits, resulting in a significant speed gain andlower power consumption. In an in-depth overview [2] of alow-power WCDMA system design, that also relies on ana-log processing, it has been shown that direct sequence is wellsuited for a multiple access system in terms of power con-sumption. It has been estimated [3] that more than 50% ofthe total processing power in DS-CDMA receivers is spenton despreading. Not surprisingly, a majority of work on low-power consumption is focused on a correlator design. For in-stance, analog implementation of a baseband correlator thatoperates at 128MHz with a power consumption of 75mWis presented in [4]. An alternative approach for power con-sumption reduction has been introduced in [5], where power

savings are achieved by reducing precision from 16-bit to a10-bit data path, and by reducing the sampling rate from twoto one sample per chip at the cost of a small performancedegradation.

Another important aspect of the radio receiver designthat affects its complexity is the problem of downconver-sion. Some relatively newwork has shown that five-port tech-nology can be used for direct conversion in SDR receivers[6, 7, 8]. This technology seems very promising because itcan operate with wideband signals and over wide range offrequencies, maintaining an accurate 90 degrees phase shift.This feature makes it particularly suitable for SDR architec-ture, as well as for the DS-CDMA receivers that are intendedfor multiband operation.

In this paper we propose a new, low-complexity DS-CDMA mobile terminal receiver that performs analog de-spreading and direct downconversion in a single stage. Thishas been made possible through the use of five-port device.Downconverted and despreaded signal can now be sampledat the symbol rate. In addition to lower sampling rate, digitalsignal processor now operates with narrowband rather thanwideband signal. Further, we propose a new noncoherent

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DS-CDMA Receiver Based on a Five-Port Technology 1629

g(·)

g(·)

g(·)

θ

θ

r(t)

l(t)

P0(t)

P1(t)

P2(t)

LPF

LPF

LPF

x0(t)

x1(t)

x2(t)

T

T

T

x0(kT)

x1(kT)

x2(kT)+

+

Figure 1: Functional block diagram of a five-port device.

analog PN code tracking scheme where the error signal thatdrives the tracking loop is generated by forming the absolutevalues (L1 norm) of the two tracking correlator outputs. Itis shown that it offers similar or better performance than thewell-known L2-norm-based noncoherent tracking loop.

Also, using the five-port device, we have designed a RAKEreceiver that works well under wide range of conditions. As-suming that the system uses pilot signal, we introduce two so-lutions, depending on how the pilot signal is multiplexed. Insystems that employ serially multiplexed pilot signal, we pro-pose a RAKE structure that has a combination of fractionallyand nonfractionally spaced fingers. Complexity of this re-ceiver is low because we use already existing PN code trackingcorrelators to form additional fingers. For systems with par-allel or continuous pilot transmission, a receiver with groupsof fingers is proposed. RAKE fingers that belong to the samegroup have fixed spacing and are tracked with a single track-ing circuitry, while the spacing between different groups de-pends on the multipath distribution. It is shown that in bothcases performance improvement and greater robustness ofthe RAKE receiver can be achieved.

The paper is organized as follows. In the next sectionwe explain the functionality of a five-port device, and showhow it can be used for simultaneous downconversion anddespreading in a DS-CDMA receiver. Moreover, we proposerectangular despreading that simplifies the receiver designat the cost of slight performance degradation. In Section 3,an L1-norm-based tracking loop is introduced and analyzed.Following that, two types of RAKE receiver solutions are pre-sented in Section 4. Analysis of the five-port-based receiversis carried out in Section 5, and the power savings estimate isgiven in Section 6. Numerical results are provided and dis-cussed in Section 7, followed by conclusions in Section 8.

2. FIVE-PORT-BASED DS-CDMA RECEIVER

The functional block diagram of a five-port device, similarto the one in [6], is shown in Figure 1. The nonlinear blockg(·) is considered in this paper to be an ideal squaring device,which can be described as g(x) = x2. The θ block performsphase shifting, and importantly, it maintains linearity over awide frequency range. This is important for receivers that areintended to operate in different frequency bands. Direct con-version and despreading of the input signal r(t) is performedby its addition to the local reference signal l(t) and squaring

the sum, followed by a subsequent filtering. In this paper weassume that technological problems associated with mono-lithic integration of a five-port device are not very difficult tosolve, as suggested in [6].

Before describing the despreading operation, we willbriefly describe how the DS-CDMA signal is generated. Weconsider a baseband system where ith source generates bi-nary or quaternary symbols di(n) such that |di(n)| = 1.Each symbol is spread by a PN sequence ci(k), such that|ci(k)| = 1, and convolved with a unit energy pulse shapep(t), resulting in a baseband signal for the ith user:

sbi(t) =∞∑

n=−∞

√Ecdi(n)

N−1∑k=0

ci(k)p(t − kTc − nT

), (1)

where Ec is the energy per chip, N is the number of chipsper symbol or the spreading factor, and Tc is the chip period.This model is applicable to any of the known CDMA modu-lation schemes. For example, if di(n) and ci(k) in (1) are realbinary symbols, the transmitted signal is BPSK-modulated;if only di(n) is real, then we have QPSK modulation identi-cal to the one used in IS-95, and if both di(n) and ci(k) arecomplex the result is a signal with complex spreading, usedinWCDMA. In this paper we assume that square-root-raisedcosine pulse-shaped chips p(t) are used.

Also, we will assume that the PN code is similar tothose used in IS-95 and WCDMA, that is, it consists of along PN code multiplied by short orthogonal codes (Walsh-Hadamard codes) that are used for different users. While thisrestriction is not necessary as the receiver is designed in thesame way for nonorthogonal codes, orthogonal codes are in-tended to eliminate the multiple access interference and canbe easily implemented in the downlink.

Typically, a matched filter is used at the receiver, that is,the impulse response of the receiving filter is p(t), and the re-sulting convolution function Rpp(τ) =

∫∞−∞ p(t)p(τ−t)dt is a

raised cosine pulse. We introduce a shorter symbolic notationfor the transmitted signal:

sbi(t) =√Ecdi(t)cpi(t), (2)

where di(t) and cpi(t) represent data and spreading codefor the user i, respectively. The information bearing signalsbi(t) is modulated with a carrier signal and then transmitted.

1630 EURASIP Journal on Applied Signal Processing

Due to the propagation losses, only a portion of the transmit-ted energy, along with the noise and interference, is collectedby the antenna at the receiver. The received signal, with thecarrier frequency ωc and with an unknown phase φ, can beexpressed as

r(t) = √2�{[ Ku∑

i=0sbi(t) + nb(t)

]e j(ωct+φ)

}, (3)

where �[·] indicates the real part of a complex number, Ku

is the number of active users, index i = 0 corresponds to thepilot signal, and

nb(t) = nc(t) + jns(t) (4)

is the complex representation of the equivalent baseband ad-ditive noise.

2.1. Chip-matched despreading

It will be illustrated here, based on a single-user example,that five-port technology can be used for simultaneous di-rect conversion and analog despreading. Instead of using twophase-shifted sinusoids for the local reference signal (to pro-duce I andQ outputs), a single direct sequence BPSK/QPSK-modulated signal is used:

l(t) = √2�{c∗p (t)e− j(ωct−π/4)}. (5)

Using the simple relationship for the real part of a complexnumber�{z} = (z + z∗)/2, the result of squaring operationon the input signal becomes

P0(t) =[12r(t) +

12r∗(t)

]2= 1

2

[(√Ecd(t)cp(t) + nc(t) + jns(t)

)e j(ωct+φ)

+(√

Ecd∗(t)c∗p (t) + nc(t)− jns(t)

)e− j(ωct+φ)

]2= 1

2

[(√Ecd(t)cp(t) + nc(t) + jns(t)

)2e j2(ωct+φ)

+(√


)2e− j2(ωct+φ)

+ 2(√

Ecd(t)cp(t) + nc(t) + jns(t))

×(√


)].

(6)

After lowpass filtering, the output signal x0(t) becomes

x0(t) = Lp[(√


×(√


)],

(7)

where Lp[·] is the notation used for lowpass filtering opera-tion. Also, the squared sum of signal rθ(t) + l(t) (the θ indexnotation is used to indicate phase shift) is

P1(t) =[12

(rθ(t) + l(t) + r∗θ (t) + l∗(t)

)]2= 1

2

[2(√


×(√


)+ 2∣∣cp(t)∣∣2

+ 2c∗(t)(√

Ecd(t)cp(t) + nc(t) + jns(t))e j(φ+π/4−θ)

+ 2c(t)(√

Ecd∗(t)c∗p(t)+nc(t)−jns(t)

)e− j(φ+π/4−θ)

]+ double frequency terms.

(8)

The first term in (8) is identical to the baseband portion ofthe P0(t) output, and the second term is a known constant.After lowpass filtering, and setting θ = π/4, the output signalx1(t) becomes

x1(t) = Lp[�{√

Ecd(t)∣∣cp(t)∣∣2e jφ}

+�{c∗p (t)(nc(t) + jns(t))}]

+ x0(t) + const,

(9)

where const is a constant due to the average power of thelocal oscillator. Similarly, by repeating the same procedurefor x2(t), we get

x2(t) = Lp[−�

{√Ecd(t)

∣∣cp(t)∣∣2e jφ}−�{c∗p (t)(nc(t) + jns(t)

)}]+ x0(t) + const,

(10)

where �[·] indicates the imaginary part of a complex num-ber. The lowpass filter is used to correlate the received andlocal PN codes, and to remove the high-frequency compo-nents. We choose the filter constant to be 1/

√N . Then, af-

ter sampling at the symbol rate, and removing the x0(n) andconst parts, we obtain discrete variables,

y1(n) =√NEc�

{d(n)e jφ

}+�{ν(n)},

y2(n) =√NEc�

{d(n)e jφ

}+ �{ν(n)}, (11)

where ν(n) is the filtered version of additive Gaussian whitenoise. Since y1(n) and y2(n) are real and imaginary parts of acomplex value y(n) = y1(n)+ j y2(n), we can write the signalin a compact, complex form:

y(n) =√Esd(n)e jφ + ν(n), (12)


where Es = NEc is the energy per symbol. This illustratesthat the five-port device, through addition, squaring, and fil-tering acts as a despreader and direct converter, whose out-puts are I and Q signal components. If it is possible to matchthe five-port components properly to avoid phase and gainmismatch, subtraction of the signal x0(t) and const dc off-set from signals given in (9) and (10) can be performed be-fore sampling, reducing the number of required A/D con-verters.

2.2. Rectangular despreading

The derivation above demonstrates that five-port technologycan be used to simultaneously perform direct conversion anddespreading. The resulting output signal is narrowband com-pared to the input signal, which makes it possible to sampleit at the symbol rather faster than the chip rate. However,some important details in the receiver described above areoverlooked. The local reference signal that is used as one in-put in the five-port circuitry is a QPSK-modulated signal.The QPSK modulation with a given pulse shape is in itselfa computationally intensive operation. It involves generationof sampled version of chip pulses p(t), and D/A conversion atvery high speed followed by modulation with the carrier fre-quency. This becomes a problem, and the benefits of analogdespreading disappear in a receiver where a number of dif-ferent local reference signals is required to perform varioustasks, such as PN code tracking, channel estimation, espe-cially if RAKE structure is used. We now present a methodthat significantly reduces the complexity of local referencesignal generation at the cost of only a slight performanceloss. It is a simple approach, that involves using rectangu-lar pulses q(t) instead of pulses p(t). The rectangular pulsesq(t) with the duration of Tc are normalized to have unit en-ergy.

Our task here is to show that the performance loss dueto a nonmatched filtering approach is relatively small, whiletheQPSKmodulation for local reference generation is greatlysimplified because it can be implemented through a simpleswitching operation and does not require D/A converters.Following the same procedure as before, and this time in-cluding all the active users in the system, the sampled versionof the output signal for user i becomes

yi(n) = 1√N

∫ nT

(n−1)T

[ Ku∑i′=0

N−1∑k=0

di′(n)ci′(k)p(t−kTc−nT

)e jφ]

×N−1∑k=0

c∗i (k)q(t − τ − kTc − nT

)dt

+1√N

∫ nT

(n−1)Tnl(t)

N−1∑k=0


)dt,

(13)

where τ is a time delay between the received and local PNcode. If we denote the result of the filtering operation, as

Rpq(τ) =∫∞−∞ p(t)q(τ − t)dt and derotate the signal by the

angle φ, the output signal becomes

zi(n) =Ku∑i′=0

N−1∑k=0

N−1∑m=0

di′(n)ci′(k)c∗i (m)

×√

EcNRpq

((k −m)Tc + τ

)+ ν(n)

= di(n)√EsRpq(τ) + I(n, τ) +

Ku∑i′=0i′ �=i

Mi′(n, τ) + ν(n),

(14)

where |cn(k)|2 = 1, the self-noise term is

I(n, τ) = di(n)

√EcN

N−1∑k=0

∑m

m �=k

ci(k)c∗i (m)Rpq((k −m)Tc + τ

),

(15)

the multiple access interference (MAI) term from user i′ �= iis

Mi′(n, τ) = 1√N

∫ nT

(n−1)T

[ N−1∑k=0

ci′(k)p(t − kTc − nT

)e jφ]

×N−1∑k=0


)dt

= di′(n)

√EcN

N−1∑k=0

∑m

m �=k

ci′(k)c∗i (m)Rpq((k−m)Tc+τ

),

(16)

and ν(n) is the filtered output of the thermal noise. The termassociated with m = k in the MAI term is equal to zero dueto the assumed orthogonality of PN codes. The expressionsfor the self noise andMAI are also valid for the chip-matcheddespreading if Rpq(·) is replaced with Rpp(·).

3. ANALOG PN CODE TRACKING

In this section we will show how the five-port device can beused to perform the task of analog PN code tracking. We willassume that the PN code synchronization is achieved also byusing the five-port device without going into the details ofsynchronization due to the limited space. Once the coarsePN code synchronization is achieved, that is, the local PNcode is synchronized within half a chip period, the next stepis the fine PN code tracking. There is a number of differenttracking loops. Themost commonly used tracking loop is thecoherent delay-lock loop (C-DLL), which is optimal for track-ing the delay difference between the acquired and the localPN code in the presence of a white Gaussian noise. How-ever, this tracking loop does not work well in the presenceof phase error. A noncoherent squaring loop, based on L2norm (L2-DLL) is normally used to overcome this problem.


Five-portdevice

LPF

LPF +

+

dc offset removal

y−(t)

Norm

∑−

Norm

y+(t)

+

+

Loopfilter

VCC

LPF

LPF

QPSKmod

QPSKmod

−∆

∆

Five-portdevice

l(t)r(t)

c = ci + jcq

dc offset removal

Figure 2: Baseband analog I-Q, early-late PN code tracking loop based on a five-port device.

Here we propose a simple to implement, noncoherent track-ing scheme based on L1 instead of L2 norm, as shown inFigure 2. The idea of using L1 norm has been used in [2] forchannel estimation, but not for PN code tracking. The pro-posed tracking loop consists of two arms, where each armis implemented by using a five-port device, followed by thenorm finding circuitry and tracking module.

The received DS-CDMA signal, assuming a single-userscenario for simplicity, has the following form:

r(t) = �{√

2Ecd(t − τ)cp(t − τ)e( jωct+φ(t))}

+�{[nc(t) + jns(t)]e jωct

},

(17)

where τ ≤ Tc/2 is the unknown time delay at the receiver.If analog implementation is not possible because of five-portimperfections (which we did not include in this analysis), orbecause the pilot is serially multiplexed with the data signal,a hybrid analog/digital implementation would be required,where the outputs of early and late correlators are sampled,and the error signal forming and loop filter are implementedin digital domain. The output of a loop filter is then passedthrough a D/A converter to drive the voltage-controlled PNcode generator (VCC). We focus our attention here on theanalog implementation.

Local oscillator, QPSK-modulated with advanced and de-layed versions of the PN sequence, results in local early andlate reference signals:

le(t) = �{√

2c∗q(t − τ − ∆Tc

)e−( jωct−π/4)

},

ll(t) = �{√

2c∗q(t − τ + ∆Tc

)e−( jωct−π/4)

},

(18)

where τ is the estimated time delay of the received signal. Theanalog complex outputs of early and late arms of the tracking

circuit shown in Figure 2 are

y±(t) =√EsRpq(ε± ∆)e jφ + ν±(t, ε,∆), (19)

where ε = (τ − τ)/Tc is the PN code timing error,and ν±(t, ε,∆) are the noise components in early and latebranches. The error signal for L1-DLL is formed as

eA(t) =∣∣�[y−(t)]∣∣ + ∣∣�[y−(t)]∣∣− ∣∣�[y+(t)]∣∣− ∣∣�[y+(t)]∣∣

=√EsSA(ε,∆;φ) +wA(t),

(20)

where

SA(ε,∆;φ) = 1√EsE[eA(t)

∣∣φ]= 1√

Es

[M−

I +M−Q −M+

I −M+Q

] (21)

is the average tracking curve conditioned under φ, with

M±I = σn

√2πe−(m

±I )

2/2σ2n +∣∣m±

I

∣∣ erf (∣∣m±I

∣∣2σ2n

),

M±Q = σn

√2πe−(m

±Q)

2/2σ2n +∣∣m±

Q

∣∣ erf (∣∣m±Q

∣∣2σ2n

),

(22)

where σ2n = N0/2 and

m±I =

√EsRpq(ε ± ∆) cosφ,

m±Q =

√EsRpq(ε ± ∆) sinφ.

(23)


The noise term is

wA(t) = eA(t)− E[eA(t)

]. (24)

The standard approach of DLL circuitry analysis thatconsists of finding the noise autocorrelation function first,and then its power spectral density, is difficult to carry out inthis case because the absolute values are involved. To avoidthis difficulty, an easier approach has been derived that pro-duces results that are in relatively good agreement with sim-ulations. As pointed out in [9], performance of analog DLLis similar to the performance of digital DLL. For digital DLLit is much easier to find the noise power spectral density atf = 0, and then use its scaled version to approximate powerspectral density of the analog noise process. The scaling fac-tor is the symbol period T . Therefore, we write the noisepower spectral density of the equivalent digital L1-DLL as

SWA( f = 0, ε = 0) = V−I +V−

Q +V+I +V+

Q , (25)

where

V±I = σ2n +

(m±

I

)2 − (M±I

)2,

V±Q = σ2n +

(m±

Q

)2 − (M±Q

)2.

(26)

If the signal-to-noise ratio is not too low, DLL circuits canbe analyzed by the approximate linear model. If we denotethe “S” curve slope as

ηA = ∂SA(ε)∂ε

∣∣∣∣ε=0

, (27)

then, following a procedure outlined in [10], the variance ofthe timing error is

σ2εA =SWA(0, 0)T

Esη2A

∫∞−∞

∣∣H(2π f j)∣∣2df = 2SWA(0, 0)BLT

Esη2A

,

(28)where

BL = 12

∫∞−∞

∣∣H(2π f j)∣∣2df (29)

is the loop filter bandwidth, and the closed loop transferfunction is defined as

H(s) =√EsηAKF(s)

s +√EsηAKF(s)

, (30)

where F(s) is the transfer function of the loop filter, and Kis the gain of the VCC. The tracking error variance for theL1-DLL in (28) is conditioned under φ, and the average jittercan be determined as

σ2εA =∫φσ2εA|φ p(φ)dφ. (31)

If we define signal-to-noise ratio of the noise loop asγD = Es/(N0BLT), we can compare the RMS jitter of the L1-DLL with the RMS jitter of coherent and L2-DLL (noncoher-ent) tracking circuitries. The tracking error variance for thecoherent DLL is [11]

σ2εc =2[Rpq(0)− Rpq(2∆)

]γDη

2C

, (32)

and for the L2-DLL the tracking error variance is [10]

σ2εN =2[Rpq(0)− Rpq(2∆)

]γDη

2CρN

, (33)

where ρN represents the noncoherent loss for L2-DLL givenas

ρN = η2Nη2C

γDBLT

(4/3)[Rpq(0) + Rpq(2∆)

]+ 4γDBLTR

2pq(∆)

.

(34)

4. FIVE-PORT-DEVICE-BASED RAKE RECEIVERS

So far we have seen that it is possible to design a DS-CDMAreceiver with analog despreading and direct conversion us-ing five-port device. The same approach can be used in adesign of RAKE receivers. RAKE receivers consist of a num-ber of correlators, where the local reference for each correla-tor is generated with a different PN code offset. Different PNcode delays are locked to different times of arrival of multi-path components (time-delayed replicas of the same signalsdue to reflection) of the received signal. The weighted sum ofthe correlator outputs is formed to produce a decision vari-able, where the combining coefficients are selected such thatthe decision variable has maximal SNR. Different multipathcomponents can fade independently, and their combinationat the output of a RAKE receiver results in a smaller varia-tion of the SNR, which in turn improves the performance ofthe receiver. The desirable property of the propagation chan-nel is that the delay spread extends over several chip periodintervals, and that each correlator output is independent ofthe other. This would make the task of tracking of multi-path components that fade independently much easier. How-ever, the independence between two relatively closely spacedcorrelators (in the case of RAKE receiver the correlators arecalled fingers) does not always exist. The correlation betweenthe multipath components with a relatively close spacing canbemodelled with a discrete set of multipath components thatare separated by less than one chip period in time. This sce-nario is known as nonresolvable1 multipath, and it can havedetrimental effect on the receiver performance [12, 13]. Thisis particularly true in the case of fast fading, where channelchanges faster than the averaging time of the loop filter inthe tracking module. In this case, the loop filter averages out

1Resolvable multipath refers to multipath components that are separatedby one or more than one chip period apart.


the fading signal. Then, the average instead of instantaneousfading signal determines the tracking point. As a result, atracking bias occurs, which leads to a significant performancedegradation of the RAKE receiver.

Having this in mind, our goal is to design a low com-plexity, low-power consumption RAKE receiver that workswell under a wide range of circumstances. Assuming that DS-CDMA systems use a pilot signal in the forward link, we in-troduce two solutions, depending on how the pilot signal ismultiplexed. In systems that employ serially multiplexed pi-lot signal, we propose a RAKE structure that has a combi-nation of fractionally (less than one chip period) and non-fractionally (more or equal to the one chip period) spacedfingers. Complexity of this receiver is low because, as we willsee, we can use already existing PN code tracking correla-tors to form additional fingers. For systems with parallel orcontinuous pilot transmission, a receiver with groups of fin-gers is proposed. RAKE fingers that belong to the same grouphave fixed spacing and are tracked with a single tracking loop,while the spacing between different groups depends on themultipath distribution. It is shown that in both cases per-formance improvement and greater robustness of the RAKEreceiver can be achieved.

The two proposed solutions do not have to be necessar-ily tied to the two previously mentioned types of pilot signaltransmission. Both of these solutions can be applied to thereceiver that operates under any type of pilot transmission,and even to receivers used in a system without a pilot signal.However, from the complexity point of view, it seems reason-able to classify them in the aforementioned way.

4.1. RAKE receiver with fractional finger spacing

If the pilot is serially multiplexed with the data channel, thePN code tracking of the five-port-based receiver can be im-plemented digitally to achieve better performance. In thiscase, we need more A/D converters than in the case of analogPN code tracking, but the same correlators can be used forRAKE combining during the data transmission.Wewill showthat it is possible to improve the performance of the RAKEreceiver with a small number of fingers, by converting eachconventional RAKE finger into three fractionally spaced fin-gers without changing the hardware structure of the receiver.The idea of using fractionally spaced fingers has already beensuggested in [14], where it is shown that the performance ofthe RAKE receiver in the presence of signal correlation canbe improved by using fractional chip finger spacing. Insteadof using optimal fractional finger spacing, that may be hardto calculate at the receiver because it depends on the chan-nel, our solution uses fixed, half-chip finger spacing. Thoughnot optimal, this solution leads to an improved receiver per-formance. The so-called early and late correlators that areused in the proposed receiver already exist in the receiverand are part of the PN code tracking circuitry. We will alsoinvestigate a very simple receiver structure [15], where onlyearly and late correlator outputs are used in the combiningprocess, and we will call it an early-late combiner (ELC). Itwill be shown that even without using the on-time correla-tor, there is a performance gain under fast fading, but this

Five-port

Five-port

Five-port

LPF

LPF

LPF

ADC

ADC

ADC

f symbolr(t)

le(t)

ll(t)

ld(t)

Analog despreadingand downconversion

Figure 3: One finger of a RAKE receiver.

improvement disappears under slow fading. The proposedreceiver with fractionally spaced fingers is identified as early-on-late combiner (EOLC), because it uses early and late cor-relators like the previous one and on-time correlator in theprocess of combining. Conventional RAKE receiver consistsof on-time correlator only for each finger.

One finger of the conventional RAKE receiver based ona five-port device, that will be transformed either in two(ELC receiver), or three (EOLC receiver) fingers, is shownin Figure 3. Local reference signal le(t) for early correlation,ld(t) and ll(t) for late correlation are generated by mul-tiplying the carrier frequency with rectangular pulses q(t)through the switching operation.

4.2. RAKE receiver with groups of fingers

When the pilot is continuously transmitted, the five-port-based receiver can perform PN code tracking in analog do-main. However, even with such low complexity, if we try toimplement a conventional RAKE receiver, each finger wouldrequire a separate tracking circuitry. Therefore, such imple-mentation would result in a replication of too many com-ponents. A new solution is introduced here, where a num-ber of uniformly spaced fingers are added, but the numberof tracking circuitries is kept low. The PN codes for thesenew correlators are delayed or advanced by one chip periodwith respect to the existing on-time correlator, resulting in agroup of uniformly spaced fingers. Importantly, each groupof fingers formed in this way shares one tracking circuitry,that is, the tracking is the same as for the conventional RAKEreceiver. Normally, dominant multipath components are notisolated, and it is possible to pick up some energy from thesurrounding components even without a need to track them.The proposed receiver consists of a relatively small number ofgroups of RAKE fingers, as shown in Figure 4.

A block diagram of one group of the proposed receiverthat consists of three correlators is shown in Figure 5.

5. PERFORMANCE ANALYSIS

In the analysis of the proposed RAKE receivers we will usea discrete Rayleigh fading channel model. Multipath compo-nents are grouped together in resolvable clusters, where eachcluster consists of unresolvable components. In a systemwith


RAKE group 1at τ1

RAKE group 2at τ2

RAKE groupMat τM

Optimal combining

· · ·

r(t)

Decision

Figure 4: RAKE receiver with groups of equally spaced fingers.

Ku active users, we will denote by ri(t) the signal intended foruser i. Then, the composite signal at the input of the receivercan be written as

r(t) =L∑l=1

Ul∑ul=1

Ku∑i=0

aul (t)ri(t − τul

), (35)

where L is the number of resolvable multipath clusters, Ul isthe number of unresolvable components within the lth mul-tipath cluster, aul (t) are independent complex Gaussian ran-dom processes used to represent Rayleigh fading, and indexi = 0 corresponds to the pilot signal. Since we are consider-ing downlink reception, it should be noted that pilot and alldata signals are subjected to the same channel.

The result of despreading of the lth cluster and samplingevery T seconds at the ith user receiver can be written as

yil(n, τl

) = 1√N

∫∞−∞

r(t)c∗qi(t − τl

)e− jωctdt

= sl(n, τl

)+ I(τl)+ P

(τl)+∑i′i′ �=i

Mi′l(n, τl

)+ ν(n),

(36)

where cqi(t) is the local PN code reference, N is the numberof chips per symbol, and

sl(n, τl

) = Ul∑ul=1

√Esaul (t)di(n)Rpq

(τl − τul

)(37)

is the desired signal component. The noise components aredecomposed in the self-noise part

I(τl) = 1√

N

Ul∑ul=1

N∑r=1r �=s

N∑s=1

√Ecaul (t)di(n)ci(r)c

∗i (s)

×Rpq((r − s)Tc + τl − τul

),

(38)

interpath interference from the other multipath clusters:

P(τl) = 1√

N

∫∞−∞

L∑l′=1l′ �=l

Ul′∑u′l=1

√Ecaul′ (t)di(t)cpi

(t−τul′

)c∗qi(t−τl

)dt,

(39)

Five-port

Five-port

Five-port

LPF

LPF

LPF

ADC

ADC

ADC

f symbolr(t)

ld(t − Tc)

ld(t + Tc)

ld(t)

Analog despreadingand downconversion

Figure 5: Themth group for the proposed LORC RAKE receiver.

contribution from an unwanted user i′, or the MAI term

Mi′l(n, τl

) = 1√N

∫∞−∞

L∑l=1

Ul∑ul=1

√Ecaul (t)di′(t)cpi′

(t − τul

)×c∗qi

(t − τl

)dt,

(40)

and ν(n) is the filtered version of additive Gaussian noise.To account for the effect of each of these components on

the decision variable, we derive their second-order statistics.Each of the noise components has zero mean.

5.1. Self-noise

If the self-noise is treated as a zero mean random process,then the correlation function of its real part is defined as

RI(τl,∆τ

) = E[�[I(τl)]�[I(τl + ∆τ

)]]. (41)

After a quite involved random variable exercise, the correla-tion function conditioned on the multipath is found to be

RI(τl,∆τ

) =Es2N�{ Ul∑

ul=1

Ul∑vl=1

[aula

∗vl

∑m �=0

Rpq(mTc + τl − τul

)×Rpq

(mTc + τl + ∆τ − τvl

)+ aulavl

∑m �=0

Rpq(mTc+τl−τul

)

×Rpq(τl+∆τ−τvl−mTc

)]},

(42)

where Es = NEc is energy per symbol, and Ec is the energyper transmitted chip.

The self-noise contribution to the noise correlation ma-trix for fractionally spaced fingers can be directly calculatedfrom the above correlation function. Also, for ∆τ = 0 thevariance of the self-noise at the finger position τl can be


found. Similarly, the noise variance for the simple one-fingerreceiver and without fading can be calculated as

VI = RI(τ, 0) = 12EsNιpq(τ), (43)

where

ιpq(τ) =∑m �=0

[R2pq

(mTc + τ

)+ Rpq

(mTc + τ

)Rpq

(mTc − τ

)].

(44)

Clearly, if q(t) = p(t), the self-noise component disappearsfor τ = 0, that is, ιpq(0) = 0, because Rpq(mTc) = 0 form �= 0.

5.2. Interpath interference

For the PN code offset larger than one chip period Tc, theinterpath interference (IPI) from the desired user, as well asfrom the other users, acts like the multiple access interferencefrom the other users in a nonorthogonal CDMA system (e.g.,reverse link). The interpath interference, given above, can bewritten as

Pp(n, τl

) ≈ 1√N

L∑l′=1l′ �=l

Ul′∑u′l=1

√Ecaul′di(n)

N∑r=1

ci′(r)c∗i (r)

×Rpq(τl − τul′

)dt,

(45)

where the approximation comes from the fact that the crossterms that come with r �= s are much smaller than the termsthat come with r = s. By following the same procedure as forthe self-noise, the correlation function is

RP(τl,∆τ

) = Es2N�

L∑l′=1l′ �=l

Ul∑ul′=1

Ul∑vl′=1

aul′ a∗vl′Rpq

(τl − τul′

)

×Rpq(τl + ∆τ − τvl′

).(46)

The IPI variance, obtained from the expression above aftersetting∆τ = 0, changes together with the change in the signalpower, which makes the analysis of our receiver very compli-cated. To avoid this, we can use the average IPI power. Also,it is possible to remove the effect of the IPI if the multipathcomponent delays are known [16].

After removing the conditioning on the multipath, andsetting ∆τ = 0, and Es = 2Eb, the average IPI variance is

VP = EbN

L∑l′=1l′ �=l

Ul∑ul′=1

∣∣aul′∣∣2R2pq

(τl − τul′

). (47)

This expression is similar to the one [17] for the MAI termsin nonorthogonal CDMA systems. The IPI term from the pi-lot signal exists for parallel multiplexed pilot.

5.3. MAI from the samemultipath clusterTo analyze the effects of multiple access interference, it is suf-ficient to analyze the effects of a single MAI user, because theinterference from each user will be mutually independent,and thus can be added together. The expression for the MAIcontribution of an unwanted i′th user in the expanded formis

Mi′l(n, τl

) =1√N

Ul∑ul=1

N∑r=1r �=s

N∑s=1

√Ecaul (t)di(n)ci′(r)c

∗i (s)

×Rpq((r − s)Tc + τl − τul

)+ Ppl′

(τl),

(48)

where Ppl′(τ′l ) is the interpath interference from the MAI,and we note that the sum of terms in (48) for r = s is equal 0,which is a consequence of downlink PN code orthogonalitybetween different users. The MAI correlation function is

RM(τl,∆τ

) =Es2N�{ Ul∑

ul=1

Ul∑vl=1

aula∗vl

∑m �=0

Rpq(mTc + τl − τul

)×Rpq

(mTc + τl + ∆τ − τvl

)}(49)

which is an expression similar to the one obtained for theself-noise.

For the single-finger receiver andwith no fading, theMAIvariance is

VM = RM(τ, 0) = 12EsNµpq(τ), (50)

where

µpq(τ) = 1N

∑m �=0

∑q

q �=p

R2pq

((p − q)Tc + τ

) = ∑m �=0

R2pq

(mTc + τ

).

(51)

Again, if q(t) = p(t), the MAI components vanish for τ = 0,that is, µpq(0) = 0, because Rpq(mTc) = 0 form �= 0, which isa direct consequence of orthogonality between the PN codesused in the forward link.

5.4. Additive noise correlationThe real and imaginary parts of the additive white noise termhave the correlation function

Rν(τl,∆τ

) = N0

2Rqq(∆τ). (52)


5.5. Total noise correlation

The total noise correlation function can now be obtained byadding all of the previously derived correlations:

Rw(τl,∆τ

) = RI(τl,∆τ

)+RP

(τl,∆τ

)+RM

(τl,∆τ

)+Rν

(τl,∆τ

)= N0

2Rqq(∆τ) + KuVPRpq(∆τ) + εδ(∆τ)

≈ I02Rqq(∆τ) +VIMδ(∆τ),

(53)

where Ku is the number of users that are occupying the chan-nel, I0 is the effective noise power spectral density, andVIM =VI +VM , where VI and VM are self-noise and MAI variances,approximately takes into account the effects of self-noise andMAI from the same cluster. For typical SNR values and forhigh processing gain, its value will be well below the I0 level.

5.6. Single-finger uncoded bit error rate

The single-finger receiver signal output without fading isgiven in (14). The decision variable is based on the real orboth real and imaginary parts, depending on the modula-tion scheme. Since the imaginary part has the same statisticsas the real part, we will focus on the real part only. Statis-tics of the decision variable are determined by its mean andvariance. The mean and the variance of the real part of thedecision variable are

E[�{zi(n)}] = ±√EbRpq(τ),

Var[�{z(n)}] = VI +VM +VN ,

(54)

where VI and VM = ∑i′=0i′ �=i

VMi′ are previously determined

variances for the self-noise and MAI, and

VN = 12σ2ν (n) =

120.903N0 (55)

is the variance of the real part of the thermal noise, as deter-mined in Appendix A.

The noise variance due to the self-noise and MAI de-pends on the modulation scheme. For instance, Es = Eb forIS-95 type of modulation, and Es = 2Eb for WCDMA. Also,noise statistics for BPSK-modulated direct sequence signalwill be identical to ones for WCDMA. For example, the self-noise variance for BPSK and WCDMA is two times largerthan the self-noise variance for the IS-95. The same resulthas been obtained through a different derivation process in[17]. This also applies for the MAI variance.

Finally, the SNR per bit [18] of the sampled signal at theoutput of the five-port device becomes

γ = 12

[E[�{zi(n)}]]2

Var[�{zi(n)}] = Eb

N0ξpq(τ), (56)

where

ξpq(τ) =R2pq(τ)

0.903+(2/N)(Eb/N0

)ιpq(τ)+(2/N)

(Eb/N0

)Kuµpq(τ)

(57)

is the SNR loss due to the self-noise,MAI, and PN code track-ing error τ. The uncoded bit error rate for a given trackingerror τ can be calculated as [17]

Pe(τ) = 12erfc

(√γ) = 1

2erfc

(√EbN0

ξ(τ)

). (58)

5.7. Uncoded bit error rate for RAKE receivers

We will determine uncoded bit error rates for the proposedRAKE receivers and compare them with the conventionalRAKE receiver. We start with introducing the signal andnoise vectors for the receivers in question. The RAKE receiversignal, before combining, can be written in a complex vectorform as

yx = sx +wx, (59)

where yx is the vector whose elements are correlator outputsyl(τl) given in (36), the subscript x will be replaced with thecorresponding initials for each receiver, and wx is the noisevector that includes the effects of additive Gaussian noise,self-noise, IPI, andMAI. For the conventional RAKE receiverwith L fingers, the signal vector is

sC =[s1(τ1)s2(τ2) · · · sL(τL)]T , (60)

where sl(τl) are given in (37). Similarly, the noise vector is

wC =[w1(τ1)w2(τ2) · · ·wL

(τL)]T

. (61)

The ELC combining RAKE receiver has the signal vectorgiven by

sELC

= [s1E(τ1−∆Tc)s1L(τ1+∆Tc

)· · ·sLE(τL−∆Tc)sLL(τL+∆Tc

)]T,

(62)

and noise vector is formed in the similar way. The EOLCcombiner has the signal vector

sEOLC =[s1L(τ1 − ∆Tc

)s1O(τ1)s1R(τ1 + ∆Tc

)· · · sLL

(τL − ∆Tc

)sLO(τL)sLR(τL + ∆Tc

)]T,(63)

where ∆ = 0.5. The RAKE receiver with groups of fingers,that we will call LORC, has the same vector form as the EOLCreceiver, with the difference that ∆ = 1.


The elements of the noise covariance matrix, Rwx =E[wxwH

x ], are formed from the correlation function definedin Section 2. Due to the fractional finger spacing for theEOLC receiver, the noise covariance matrix is not an iden-tity matrix. Thus, to ensure that noise is not correlated, com-bining process is preceded with the noise decorrelation orwhitening. Since noise covariance matrix is positive definite,a positive definite square root matrix R1/2

wxexists [19], such

that Rwx = R1/2wxR1/2wx, where

R1/2wx= UwxΛ

1/2wxUH

wx(64)

for the unitary decomposition Rwx = UwxΛwxUHwx. Based on

this, we can choose a noise decorrelation matrix to be

D = R−1/2wx= UwxΛ

−1/2wx

UHwx, (65)

because

E[Dw(Dw)H

] = I. (66)

The received signal, after noise decorrelation and beforecombining has the following form:

Dyx = Dsx +Dwx, (67)

and the optimal RAKE combining coefficients are then α =(Dsx)H . Then, the SNR per bit at the output of a combiner isgiven by the quadratic form

γx = EbI0sxHR−1wx

sx, (68)

which can, through a series of matrix transformations, bebrought into another quadratic form, as shown in the ap-pendix:

γx = EbI0

∑i

∣∣ux,i∣∣2λx,i =∑i

∣∣ux,i∣∣2γx,i, (69)

where |ux,i|2γx,i are χ2 distributed independent random vari-ables. While the process of arriving at the expression (69)for fractionally and nonfractionally spaced fingers is slightlydifferent, the final expression is the same. Clearly, after thequadratic form transformation, the SNR has become a well-known linear combination of independent central χ2 ran-dom variables. Thus, the pdf for γx can be found by us-ing the characteristic function approach. It is convenient touse Laplace transform, since λx,i are real positive constants.The pdf can be found directly by finding the inverse Laplacetransform of the characteristic function for the case when theeigenvalues λx,i are distinct:

px(γx) = L−1

[ Px∏i=1

11 + sγx,i

]=

Px∑i=1

Ai

γx,ie−γ/γx,i , (70)

where

Ai =Px∏j=1j �=i

λx,iλx,i − λx, j

, (71)

and Px = L for the conventional RAKE receiver, Px = 2L forthe ELC receiver, and Px = 3L for EOLC and LORC receivers.The cdf function can be interpreted as the outage probabilityfor some threshold γ. The uncoded bit error rate can be easilyfound as [18]

BERx = 12

[ Px∑i=1

Ai

(1−

√γx,i

1 + γx,i

)]. (72)

Similarly, if some of the eigenvalues in (69) are nondistinct,the pdf, cdf, and uncoded bit error rate expressions can easilybe found by using partial fraction expansion.

6. POWER CONSUMPTION SAVINGS ESTIMATE

In this section we will try to give an estimate of the powersaving due to the lower sampling rate and reduced process-ing load on the DSP achieved with the receiver that usesfive-port device with respect to the power consumption ofa conventional all-digital DS-CDMA receiver. The exact to-tal amount of the power saving obtained with the proposedreceiver could be made only after the receiver has been im-plemented. We will assume that it is not difficult to integratea number of five-port devices in a single chip, so that theirpower consumption should not be significant.

As a “proof of the concept,” we will use a DS-CDMA re-ceiver with the following parameters: QPSK-modulated sig-nal with the spreading factor N = 64, oversampling ratio ofO = 2 samples per chip (for the conventional receiver), anda symbol rate of Rs = 1Mbps.

6.1. A/D conversion savings

An all-digital DS-CDMA receiver requires two ADCs, one forI and the other for Q signal components, with a samplingrate of fs1 = Rs × N × O = 128Msps each. On the otherhand, a five-port-based RAKE receiver with three fingers re-quires at least 6 five-port devices for data detection and chan-nel estimation (PN code tracking, if implemented by ana-log circuitry, does not require A/D conversion even though itrequires five-port circuits). Each five-port device is followedby at least two ADCs, which means that twelve ADCs withthe sampling rate of 1Msps are required for the five-port-device-based receiver.2 The approximate way of estimatingthe power consumption saving by using a lower sampling

2The number of symbol rate sampling ADCs can be smaller becausemany converters can sample more than one input signal. For example, 12single-channel ADCs with the sampling rate of 1Msps could be replacedwith 3 four-channel ADCs with the sampling rate of 4 Msps.


rate is to use the figure of merit for ADCs [20]:

F = 2SNRbits fsPdis

, (73)

where SNRbits represents the effective number of bits, fs isthe sampling frequency, and Pdis is the dissipation power. As-suming the same figure of merit for two different samplingfrequencies, we form the ratio

fs1fs2= Pdis 1

Pdis 2. (74)

In our example, fs1/ fs2 = Pdis 1/Pdis 2 = 128, and after takinginto account the number of A/D converters, we have a powerdissipation savings ratio of 21.3 in favour of analog despread-ing receiver. This number may be too optimistic, but signifi-cant power saving is to be expected nonetheless.

For the purpose of illustration, we will use an off-the-shelf single-channel 10-bit AD9411 ADC by Analog Deviceswith the sampling rate of up to 170Msps [21], that couldbe used for the all-digital DS-CDMA receiver. Two suchconverters sampling at 128Msps consume about 2W. Onthe other hand, a 10-bit AD7440 single-channel converterwith a total power dissipation of 9mW at 1Msps can beused with each five-port device. twelve ADCs, required fora three-finger RAKE receiver, would consume about 0.108Wof power. This result is similar to the one obtained by using afigure of merit.

6.2. DSP savings

Once the signal is digitized with ADCs, the radio interfaceis implemented in software by means of digital signal pro-cessing, or in hardware by using ASIC or FPGAs. The mostcomputationally intensive arithmetic operation that has tobe done in the WCDMA receiver is the correlation with thespreading code. The complexity of correlation increases withthe increase in spreading factor and with the number ofRAKE fingers. We will use the same example to illustrate thereduction in the number of digital operations by moving thedespreading in the analog domain. By following the simi-lar methodology to the one outlined in [5], we find that thematched filter requires 2NO multiplications and 2NO addi-tions per symbol. The output of the matched filter is down-sampled to one sample per chip. Therefore, 4N multiplica-tions and 4N additions are required to despread a complexsignal. Thus, if we multiply this number by 4 to account forpilot, data, and two tracking correlators, we have 16N +8NOfloating point operations (FLOPS) per one data symbol perRAKE finger. The number of required FLOPS per symbol is6.144 GFLOPS for a three-finger RAKE receiver. The esti-mate in [3], which is a result of a DARPA small unit oper-ations project, shows that despreading/correlation accountsfor 50% of the total number of operations at the DS-CDMAtransceiver. Dynamic power consumption is proportional tothe node switching activity [5], which means that about halfof the power required for the DSP/ASIC can be saved by per-forming the analog instead of digital despreading.

10−1

10−2

10−30 1 2 3 4 5 6 7

Eb/N0

Pe

MF-analyticalMF-sim

Rect-analyticalRect-sim

Figure 6: Uncoded BER with MF and rectangular despreading, α =0.3, N = 64.

We have shown that there is potential power saving whena DS-CDMA receiver is designed by using a five-port device.On the other hand, five-port-based receiver would have someother losses. For example, replication of five-port devices andthe need for additional PN code tracking circuitry would re-claim some portion of the power consumption savings dis-cussed above. However, in addition to power consumptionreduction, the five-port approach with analog despreadingmakes very high data rates possible that would otherwise beimpossible due to technological limitations associated withsampling rates of ADCs and processing power of DSPs.

7. NUMERICAL RESULTS

The uncoded bit error rate of a single-finger receiver with-out fading when rectangular despreading is used is comparedwith the uncoded bit error rate of the receiver that uses chip-matched filter in Figure 6. Here we can see that the perfor-mance loss due to the rectangular despreading is minimal.Also, we see that the simulation results are in good agreementwith the analytical expressions derived above. A 17-bit LFSRshift register is used for PN code generation in our simula-tion, and Hadamard code is used for orthogonalization. Thechip oversampling ratio is 10, and the number of chips persymbol is N = 64.

The RMS jitter of the proposed PN code tracking loopis compared with the RMS jitter of the conventional coher-ent and noncoherent L2-norm-based loop in Figure 7. Fromhere we can see that the performance of L1-DLL obtainedthrough simulation is better than predicted analytically dueto the overestimation of the noise power spectral density inthe adopted analytical model, and slightly better than theperformance of the L2-DLL. The effects of different valuesfor φ have also been analyzed, and it has been observed thatin spite of decrease of SNR for case when φ = 0, the filter


10−1

2 4 6 8 10 12 14

Eb/N0

σε

CoherentL1L2

Figure 7: Normalized RMS jitter σεX , for ∆ = Tc/2.

bandwidth BL is also smaller, and vice versa, resulting innegligible difference between worst- and best-case RMS jit-ter.

In the analysis of RAKE receiver structures we use fastand slow fading channels. The terms slow and fast Rayleighfading channels are related to the rate of channel change withrespect to the averaging time of the tracking circuitry. Typi-cally, the averaging time of the tracking circuitry is in the or-der of 100 symbols. Thus, if the coherence time of the chan-nel is smaller than the duration of 100 symbols, we considerthe fading to be fast. Otherwise, the fading is slow. Analyti-cal results obtained in Section 5 are directly applicable to thecase of fast fading, because for the fixed sampling instanceτl, which could be treated as a result of tracking of the aver-age fading, the signal component sl(n, τl) is a complex Gaus-sian random variable since it is a linear combination of com-plex Gaussian random variables. However, for a slow varyingchannel, τl changes with the channel (because τl now cor-responds to the position of the instantaneous peak correla-tion), and sl(n, τl) is not a complex Gaussian random vari-able any more, since coefficients Rpqh(τl − τul ) are no longerconstants. Therefore, the analysis based on complex Gaus-sian random variables is no longer applicable, and simula-tion is used to measure the performance of the receiver un-der the slow fading. For a slow fading scenario, simulation isperformed with the narrowband (despreaded) signals, whichsignificantly reduces the running time but does not incorpo-rate the effects of self-noise and MAI. However, this does notaffect the obtained results much, because as the simulation ofthe fast fading shows, these effects are small for a large pro-cessing gain.

The first set of results applies to the RAKE receiver that isdesigned for serially multiplexed pilot signal. In Figure 8 we

100

10−1

10−2

10−30 2 4 6 8 10 12 14

E [γC]

On timeELCEOLC

On time (sim.)ELC (sim.)EOLC (sim.)

Pe

Figure 8: Bit error rates for conventional, ELC, and EOLC RAKEreceivers under fast fading with L = 1, U1 = 2, τ2 − τ1 = (1/2)Tc,and E[|a1|2]/E[|a2|2] = 1, Ku = 4dB.

see the comparison of uncoded bit error rates for three dif-ferent receivers, namely, conventional, ELC, and EOLC com-biners, under fast fading. Two unresolvable components areconsidered, spaced 1/2Tc apart and their average power ratiois SMR = 1dB. The average SNR in the plot corresponds toγC for the best tau. Evidently, even the very simple receiverbased on early and late correlators, or ELC combiner, out-performs the receiver based on the on-time correlation. Thiscan be explained if we remember that there is a tracking biasdue to the fast fading and presence of unresolvable multi-path components. Thus, under fast fading scenario, it wouldbe possible to use only two correlators that are also used fortracking, without a need for the on-time correlator. Also, wesee a good agreement between the theoretical result and sim-ulation. The self-noise andMAI do not have a significant im-pact on the receiver performance for a small number of users.For a given unresolvable multipath, the SNR gain achievedwith the EOLC receiver, for the uncoded bit error rate of 0.01,is 5 dB. This performance gain is reduced to 1 dB for the caseof slow fading, whereas the ELC receiver fares worse than theconventional receiver, as shown in Figure 9. The results dis-cussed above are obtained for ideal, noiseless tracking andideal channel estimation.

It is instructive to compare these receivers for differentsampling instances, or tracking delay errors. Bit error rates ofthe three receivers with serially multiplexed pilot, for the fastfading case are shown in Figure 10. The average SNR at theon-time correlator is γC = 10dB. Results for τ = 0.2 trackingerror are in agreement with the results shown in Figure 8.Similarly, for the slow fading, Figure 11 shows that thoughthe performance gain is not as significant as in the case offast fading, the EOLC receiver is much less sensitive to thetracking error.


100

10−1

10−2

10−30 2 4 6 8 10 12 14

SNR

On timeELCEOLC

Pe

Figure 9: Bit error rates for conventional, ELC, and EOLC RAKEreceivers under slow fading with L = 1, U1 = 2, τ2 − τ1 = 1/2Tc,and E[|a1|2]/E[|a2|2] = 1dB.

100

10−1

10−2

10−3−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2

τ

On timeELCEOLC

On time (sim)ELC (sim)EOLC (sim)

Pe

Figure 10: Comparison of analytical and simulation results for con-ventional, ELC, and EOLC RAKE receivers under fast Rayleigh fad-ing with L = 1, U1 = 2, τ2 − τ1 = 0.5Tc, and E[|a1|2]/E[|a2|2] =1dB; γC = 10dB, Ku = 4, and N = 64.

The rest of the results is for the receiver with groups offingers under the fast fading scenario. In Figure 12 we com-pare uncoded bit error rates versus SNR for the conventionaland proposed LORC receiver, and in Figure 13 we compareBERs as functions of tracking errors. Obtained results aresimilar to the ones for the fractionally spaced fingers. Sim-ilarly, Figures 14 and 15 show results for the slow fading.

100

10−1

10−2

10−3−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

τ

On timeELCEOLC

Pe

Figure 11: BER for conventional, ELC, and EOLC receivers un-der slow fading with L = 1, U1 = 2, τ2 − τ1 = 1/2Tc, andE[|a1|2]/E[|a2|2] = 1dB.

10−1

10−2

10−3

10−40 2 4 6 8 10 12

SNR

On-time combiningLeft-on-right combining

Pe

Figure 12: The BER comparison for conventional and LORC RAKEreceivers under fast fading with L = 2 resolvable multipath clusters,Ul = 2 unresolvable components per cluster, τi+1 − τi = 0.5Tc, andE[|ai|2]/E[|ai+1|2] = 3dB, for i = 0 and i = 2.

8. CONCLUSIONS AND FURTHERWORK

In this paper we propose a new low-complexity, low-powerconsumption DS-CDMA receiver that can be used for highdata rate applications. The receiver is based on a passivefive-port device. By proposing simultaneous direct conver-sion and analog despreading, we have potentially reducedthe sampling rate by two orders of magnitude. The perfor-mance of the proposed receiver that uses rectangular instead


100

10−1

10−2

10−3−1.5 −1 −0.5 0 0.5 1 1.5 2

τ


Pe

Figure 13: The BER comparison for conventional and proposedRAKE receivers under fast fading with L = 2 resolvable clusters,Ul = 2 unresolvable components per cluster, τi+1 − τi = 0.5Tc, andE[|ai|2]/E[|ai+1|2] = 3dB, for i = 0 and i = 2; SNR = 8 dB.

10−1

10−2

10−3

10−4

0 2 4 6 8 10 12

SNR


Pe

Figure 14: The BER comparison for conventional and proposedRAKE receivers under slow fading with L = 2 resolvable multi-path clusters, and Ul = 2 unresolvable components per cluster,τi+1 − τi = 0.5Tc, and E[|ai|2]/E[|ai+1|2] = 3dB, for i = 0 andi = 2.

of pulse-shaped despreading has been analyzed. Also, we pro-pose a new noncoherent tracking scheme based on L1 normthat offers similar or even better performance than L2 normtracking circuitry, at lower hardware complexity. Followingthat, we introduce two five-port-based RAKE receiver struc-tures that result in better performance in terms of uncoded

100

10−1

10−2

10−3−1.5 −1 −0.5 0 0.5 1 1.5

τ


Pe

Figure 15: The BER comparison for conventional and proposedRAKE receivers under slow fading with L = 2 resolvable clusters,Ul = 2 unresolvable components per cluster, τi+1 − τi = 0.5Tc, andE[|ai|2]/E[|ai+1|2] = 3dB, for i = 0 and i = 2; SNR = 8dB.

bit error rate and tracking error sensitivity. Both fast andslow fading scenarios are considered. It has been shown thatsignificant performance improvement with the proposed re-ceiver structures is achieved in a fast fading scenario. Theperformance improvement for a slow fading is not as large.However, the advantage of the proposed receiver structuresbecomes evident when the tracking error is taken into ac-count.

Many problems still remain open for further research.One of them is the problem of a limited dynamic range asso-ciated with the squaring operation performed in the five-portdevice. Possible solutionmay involve placing an analog adap-tive gain control circuitry in front of five-port devices. Thiscould be combined with some type of digital control thatcompensates for dc offset and other effects (e.g., caused bytemperature variations) that may cause the nonlinear blockto work outside the squaring region.

APPENDICES

A. THERMAL NOISE

We begin by writing the noise portion of the signal:

ν(n) = 1√NTc

N−1∑k=0

c∗n (k)∫ (n−1)T−Tc/2+(k+1)Tc

(n−1)T−Tc/2+kTc

nl(t)dt. (A.1)

It is easy to show that the noise variance can be expressed asa power spectral density integral:

σ2ν = Tc

∫∞−∞

sinc2(f Tc

)Sn( f )df , (A.2)


where Sn( f ) is the power spectral density of the noise pro-cess, and sinc(x) = sin(πx)/(πx). If the input noise processhas an infinitely wide power spectral density of Sn( f ) = N0,the noise variance becomes

σ2ν = N0Tc

∫∞−∞

sinc2(f Tc

)df = N0. (A.3)

However, an infinite noise power spectral density is not arealistic scenario, because receivers normally have an im-age rejecting prefilter that limits the noise and interferencewithin the band of interest. Therefore, after the prefiltering,the noise process is band-limited within the bandwidth ofthe signal of interest. The bandwidth of the desired widebandsignal for square-root-raised cosine pulse waveforms, beforedespreading, is given by the expression

BW = 1 + α

2Tc, (A.4)

where α is the excess bandwidth. If we can set the noise band-width to be BN = 1/Tc, thus allowing for some margin, thenoise power of the filtered version of the thermal noise is

σ2ν = N0Tc

∫ BN

−BN

sinc2(f Tc

)df = 0.903N0 (A.5)

which means that the noise effect has been somewhat re-duced by the rectangular spreading operation. This is onlyvalid if the noise is band-limited.

B. RAKE RECEIVER SNR

The SNR in the expression (68) is given in a quadratic formwhich can easily be transformed into the quadratic form ofuncorrelated complex Gaussian vectors [19]:

γx = EbI0sxHR−1/2sx R1/2

sx R−1wxR1/2sx R−1/2sx sx

= EbI0

(R−1/2sx sx

)HR1/2sx R−1wx

R1/2sx

(R−1/2sx sx

)= Eb

I0

(Ux

HR−1/2sx sx)H

Λx(Ux

HR−1/2sx sx)

= EbI0

∑i

∣∣ux,i∣∣2λx,i =∑i

∣∣ux,i∣∣2γx,i,(B.1)

where Λx and Ux are the diagonal and unitary decomposi-tions of the matrix R1/2

sx R−1wxR1/2sx , respectively, ux,i are the unit

variance complex Gaussian random variables, and λx,i are theelements of Λx. Here we assumed ideal channel estimation.For receivers with nonfractional finger spacing, the noise cor-relation matrix Rwx is an identity matrix, and so is its inverseR−1wx

.

ACKNOWLEDGMENT

This work was funded in part by SOMA Networks, Inc.

REFERENCES

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[2] S. Sheng and R. Brodersen, Low Power CMOS Wireless Com-munications: a Wideband CDMA System Design, Kluwer Aca-demic, Boston, Mass, USA, 1998.

[3] J. E. Gunn, K. S. Barron, andW. Ruczczyk, “A low-power DSPcore-based software radio architecture,” IEEE J. Select. AreasCommun., vol. 17, no. 4, pp. 574–590, 1999.

[4] K. Onodera and P. R. Gray, “A 75 mW 128 MHz DS-CDMAbaseband correlator for high-speed wireless applications,” inProc. Symposium on VLSI Circuits Digest of Technical Papers,pp. 117–118, June 1997.

[5] V. Chandrasekhar, F. Livingston, and J. Cavallaro, “Reducingdynamic power consumption in next generation DSCDMAmobile communication receivers,” in Proc. IEEE 14th Inter-national Conference on Application-specific Systems, Architec-tures and Processors (ASAP ’03), pp. 251–261, the Hague, theNetherlands, June 2003.

[6] M. Ratni, D. Krupezevic, Z. Wang, and J.-U. Jurgensen,“Broadband digital direct down conversion receiver suitablefor software defined radio,” in Proc. IEEE 13th InternationalSymposium on Personal, Indoor, and Mobile Radio Communi-cations (PIMRC ’02), vol. 1, pp. 100–104, Lisbon, Portugal,September 2002.

[7] X. Xu, K. Wu, and R. G. Bosisio, “Software defined radio re-ceiver based on six-port technology,” in Proc. IEEE MTT-SInternational Microwave Symposium Digest, vol. 2, pp. 1059–1062, Philadelphia, Pa, USA, June 2003.

[8] X. Huang, M. Caron, and D. Hindson, “Adaptive I/Q regen-eration in 5-port junction based direct receivers,” in Proc. 5thAsia-Pacific Conference on Communications and 4th Optoelec-tronics and Communications Conference (APCC/OECC ’99),vol. 1, pp. 717–720, Beijing, China, October 1999.

[9] H. Meyr, M. Moeneclaye, and S. A. Fechtel, Digital Commu-nication Receivers: Synchronization, Channel Estimation, andSignal Processing, John Wiley & Sons, New York, NY, USA,1998.

[10] J. K. Holmes, Coherent Spread-Spectrum Systems, Wiley Inter-science, New York, NY, USA, 1982.

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[12] B. W. Hart, R. D. J. van Nee, and R. Prasad, “Performancedegradation due to code tracking errors in spread-spectrumcode-division multiple-access systems,” IEEE J. Select. AreasCommun., vol. 14, no. 8, pp. 1669–1679, 1996.

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[14] K. J. Kim, S. Y. Kwon, E. K. Hong, and K. C. Whang, “Effectof tap spacing on the performance of direct-sequence spread-spectrum RAKE receiver,” IEEE Trans. on Commun., vol. 48,no. 6, pp. 1029–1936, 2000.

[15] I. Maljevic and E. S. Sousa, “Performance degradation due tocode tracking errors on a RAKE spread-spectrum receiver inthe presence of unresolvable multipath,” in Proc. IEEE 13th In-ternational Symposium on Personal, Indoor, and Mobile RadioCommunications (PIMRC ’02), vol. 4, pp. 1815–1819, Septem-ber 2002.


[16] M. Guenach and L. Vandendorpe, “Downlink performanceanalysis of a BPSK-based WCDMA using conventional rakereceivers with channel estimation,” IEEE J. Select. Areas Com-mun., vol. 19, no. 11, pp. 2165–2176, 2001.

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[19] A. M. Mathai and S. B. Provost, Quadratic Forms in RandomVariables: Theory and Applications, Marcel Dekker, New York,NY, USA, 1992.

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[21] Analog device data sheets, http://www.analog.com.

Ivo Maljevic was born in Bar, Serbia andMontenegro, on November 22, 1966. Hereceived the B.S. degree from the Univer-sity of Podgorica in 1991, the M.S. degreefrom the University of Belgrade in 1995,and the Ph.D. degree from the Universityof Toronto, Canada, in 2004, all in electricalengineering. From 1993 to 1997 he workedas a Research Engineer at Mihajlo PupinInstitute, Belgrade. In 1997 he joined Mo-torola as a DSP software engineer, working on voice compressionalgorithms. Since 2004 he has been with Soma Networks, Toronto.His current research interests include CDMA systems, software-defined radio, signal processing, and digital communications the-ory.

Elvino S. Sousa holds the rank of Professorin the Department of Electrical and Com-puter Engineering, University of Toronto,Canada, and is the Bell University Lab Chairin computer engineering with a mandate inwireless communications. He is the Direc-tor of the Wireless Lab, which has under-taken research in spread spectrum systemsand CDMA wireless systems for the past 16years. His current research interests are inthe areas of high-speed CDMA systems, smart antenna systems,software radio, ad hoc networks, and wireless system concepts for4th-generation networks. Professor Sousa has been a Consultantto industry and governments in the area of wireless systems inter-nationally. He was the Technical Program Chair for PIMRC ’95,Vice-Technical Program Chair for Globecom ’99, Chair of the IEEETechnical Committee on Personal Communications, and has beeninvolved in the technical program committee of numerous inter-national conferences. He has spent sabbatical leaves at Qualcommand Sony CSL/ATL, where he was the holder of the Sony SabbaticalChair.

http://www.analog.com

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