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IEEE International Conference on Circuits and Systems 2006
Abstract— This paper presents a comparison of two
logarithmic codecs for low-medium resolution (6-12 bit)
adaptive beamforming in WCDMA terminals. The two
different logarithmic coding techniques are based on the
utilization of a look up table (LUT), where one method employs
a direct LUT conversion, and the other uses linear
interpolation requiring a multiplier and a comparatively small
LUT. The two logarithmic converters are analyzed with
Matlab simulation, where the address precision is varied to
generate observations on the adaptive filter misadjustment and
its effect on the probability of error. The results indicate that
both algorithms cause relatively small mean square error for
address precision above 9 bits for the direct LUT conversion
and 5 bits for the interpolative conversion. Both techniques
render low cost alternatives to a fixed-point implementation
while yielding an improved probability of error metric.
I. INTRODUCTION
Logarithmic coding is an important aspect in the application
of adaptive filters utilizing hybrid-linear-logarithmic
architecture. The Logarithmic Number System (LNS) has
been an active research topic spanning four decades and is
an interesting subject when considering multiplierless
realizations of digital filters [1]. Of particular note, the LNS
is compatible with the floating point number system offering
similar dynamic range and fairly straightforward
mathematical conversion between linear and logarithmic
domains requiring just two bits of overhead (a sign and zero
detection flag bit) and a codec for converting the mantissa
into a fractal number (of “m” bits). Generally, the LNS
actually executes logarithmic multiplication (utilizing lower
cost adders) with greater efficiency and tolerance to
multiplicative round-off errors [2]. The practical limitations
of logarithmic coding for equivalent floating point accuracy
involve either time consuming iterations or prohibitively
large look up tables even if interpolation (utilizing linear
regression or polynomial expansion) is considered to reduce
the memory requirements of the algorithm [3]. However for
low-medium precision, the use of the LNS becomes
considerably more practical. The hybrid linear-logarithmic
number system (HLNS) has been studied for 8-12 bit
matched filter applications in [4] and [5] for WCDMA
Downlink receivers, where the logarithmic codec employed
was direct LUT based. The size of the table grows
exponentially for every single bit increase in the numerical
accuracy prior to coding, where Lagrange interpolation
employing a low cost multiplier (and smaller LUT) offers a
viable alternative. Adaptive filters are generally quite
sensitive to low dynamic range, low precision number
formats particularly if used for signal processing
applications operating in the constraint of generating
sufficient statistics for the optimal detection of selectively
faded information in additive Gaussian noise channels. The
motivation of this paper is to evaluate an alternative
numerical technique (utilizing logarithmic coding) that
circumvents intensive usage of parallel array multipliers
while generating products in multiplicative chains with
comparatively low arithmetic round off error. The paper is
organized as follows: Section II presents the preliminaries
for the adaptive beamformer while Section III reviews the
logarithmic number system for the adaptive filter. Section
IV presents the results and discussion, and Section V yields
the summary and conclusion to this paper.
II. THE SINGLE USER LMMSE RECEIVER
Adaptive beamforming employing finite length, transversal
filters is one technique that can be used in the suppression of
multiple access interference in synchronous WCDMA
channels when transmitted through a frequency selective
multipath medium [6]. This is particularly so if long
scrambling PN sequences overlay the coded information
bearing signals transmitted by a basestation, which render
the more traditional and computationally exhaustive
Minimum Mean Square Error receiver [7] impractical to
implement in real time communication. The proposed
receiver utilizes LMMSE chip-level estimation obtained via
minimization of the quadratic functional
}{ )()(E [n][n][n][n]
minarg θrwθrw
w
−−
∂
∂ HHH
with w the unknown coefficient vector,
[n]
r the filter
regressor that is of finite dimension and obtained from the
received sampled signal vector ηHθr += where
∑=
=
K
1i
iii bξSθ is the desired signal. H is the convolution
matrix of channel signatures (and is a tall matrix accounting
for more than one antenna), Si is the signature matrix of user
“i” and ∑=
K
1i
ii bi
ξS is the sum of unequal energy multiuser
information sequences with K the number of active users
LOGARITHMIC CODECS FOR ADAPTIVE BEAMFORMING IN WCDMA
DOWNLINK CHANNELS
C. Litchfield1
, R.J Langley2
, P. Lee1
1
University of Kent at Canterbury
2
University of Sheffield
Email: cl34@kent.ac.uk

and
i
ξ the ith
traffic channel amplitude. The receiver
coefficients are given by [8]
( ) ∂σ+=
−
θθ
H2
n
H
HHHw
11-
Σ (1)
where θθΣ and
2
nσ are the multichannel signal reference and
additive noise covariance’s respectively (the noise for a
oversampled system will be assumed to be a correlated
Gaussian random variable). ∂ is a sampled Dirac function
which is a vector of equal dimensionality to the number of
filter taps, where the unity scalar resides in the position
related to the delay operator for ensuring causality. The
implementation of the matrix solution in (1) does not
translate well for time varying channels (since it is a one-
shot optimization for a low rank filter). The coefficient
vector, w can alternatively be found from the solution to the
set of linear equations [9] Λw =rΣ with
(n)
P
1i
iiE rbξSΛ ⋅
=
∑= ˆ
i
and rΣ the signal covariance
matrix. P ≠ K is a quantity related to the number of
channels that are known/demodulated by the receiver (i.e. P
= 2 if only the Pilot and desired traffic channel signals are
despread). ibˆ is the decision directed training symbols. The
optimization problem can also be computed recursively
applying well – known steepest descent techniques, where
the adaptive solution applying the NLMS algorithm [5] is
given as:
1)-(n
*
1)-(n
1)-(n
H
1)-(n
1)-(n(n)
e
ρ
µ
r
rr
ww ⋅⋅
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
+
+= (2)
where ∑=
−=
P
1i
(n)
i
(n)
i
(n)
i(n)
H
(n)(n)e bξSrw ˆ , ρ a small constant
and µ the constant step size << 1. The Fig.1 shows a general
block diagram of the single user receiver utilizing chip
equalizers. The receiver consists of two antennas separated
by half a wavelength, where each antenna consists of a Root
Raised Cosine (RRC) matched filter and an adaptive
equalizer. The update algorithm applying (2) utilizes a
composite weight vector [ ]T
T
2
T
1 ,www = and regressor
[ ]T
T
2
T
1 ,rrr = where the error term (n)e is formulated at the
output of the dual antenna beamformer.
III. THE LOGARITHMIC CODEC
Any real quantized number, N for radix 2 coding can be
represented by
∏
−
−=
−−=
1
mi
S
i
i 2ε C
221)Z)((1N (3)
with m the number of fractal bits and ε=floor( log2|N| ) for
N ≠ 0. Let i
1
mi
iε2 2C
2
N
log ∑
−
−=
= bounded by
1
2
N
log0
ε2 ≤≤ represent the base 2 logarithmic
conversion where { } i01,Ci ∀∈ symbolizes the binary digit
at the ith
bit. { }01,S ∈ and { }01,Z ∈ are the sign and zero
flag bits conditioned where S = 1 for N < 0 and Z = 1 for
N = 0. The goal of the logarithmic codec is to approximate
the logarithm with sufficient accuracy such that the square
difference between (3) and the actual value of N is not so
large that numerical errors cause large misadjusment and
hence low efficiency.
Case 1: The direct LUT codec maps a quantized value of N,
q(N) to an address corresponding to a pre-assigned
logarithmic fraction where
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎯→⎯⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
≈
ε2εε2
2
N
qlog
2
N
q
2
N
log (4)
and 2ε
is a normalizing factor constraining the input for the
logarithmic conversion to be in the interval 2
2
N
1
ε
<≤ .
Case 2: The utilization of a direct LUT conversion for the
address range m
2q(N)1 <≤ renders accurate, but memory
limited solutions when m > 16. An alternative approach,
employing smaller look up tables at a cost of a small fixed-
point multiplier (a 12-bit multiplier is considered in this
paper), is Lagrange interpolation [10]. In this case, the
logarithm can be approximated by
⎭
⎬
⎫
⎩
⎨
⎧
⎟
⎠
⎞
⎜
⎝
⎛
+
⎟
⎠
⎞
⎜
⎝
⎛
−⋅⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+⎟
⎠
⎞
⎜
⎝
⎛
≈ −
ε
εεεεε2
2
N
qL
2
N
q
2
N
2
N
qL∆
2
N
qL∆
2
N
log 1
(5)
where ∆ is the LSB of the LUT addresses (assuming they
are uniformly spaced) and L(U) is the logarithmic
conversion function returned from the table.
Fig. 1 The Single User Receiver employing a 2-element adaptive
antenna array

While multiplications in the logarithmic domain translate to
additions, the case is not as straightforward for linear
addition in the logarithmic domain where a large LUT will
be required to calculate the non-linear function
( ))L(N)L(N 1221log2
−
+ . To overcome this, a mixed linear-
logarithmic (hybrid logarithmic) approach is employed
where L(U)
2 (for the input range 1L(U)0 <≤ ) is evaluated
with a LUT containing a maximum 212
address elements
(that is dependent on the numeric precision employed by the
Lin – Log converter). This is sufficient for the purposes of
this paper for converting between logarithmic and linear
domain to perform binary addition.
IV. DISCUSSION
The receiver executes linear domain multiplication as an
addition in the logarithmic domain utilizing one of the Lin –
Log codecs proposed in section III. Prior to beamformer
combining or product accumulation, the RRC and adaptive
filter inner products (computed in the logarithmic domain)
must be converted back to the linear domain with an
antilogarithmic codec. The received vector r is modeled as
a continuous sum of multipath signal components where the
original signal consisted of a multiplex of individually
coded orthogonal users with variable spreading factor
yielding identical chip rates of 3.84Mcps. The spreading
sequences used in simulation were orthogonal Walsh-
Hadamard codes overlaid with Gold sequence PN
scrambling codes of far greater duty cycle. All multiplexed
user signals exhibit chip pulse shaping applying a Root
Raised Cosine filter with excess bandwidth of 0.22. The
following parameters for the Matlab simulations were
observed: The multiuser QPSK symbols were modulated on
a carrier of 2.1GHz, where the number of users
simultaneously accessing the channel (synchronously) was
set to K = 10 with random distribution of signaling powers.
A 4-path Rayleigh fading channel was implemented with
maximum normalized Doppler shift of 5.185×10-3
and delay
spread of 7.8µs.
Table.1 The filter misadjustment for various ranges of address precision
Fig. 2 The BER as a function of the mean SNR for the dual antenna Rake
Receiver and the LMMSE Beamformer employing 6-bit and floating –
point arithmetic.
The symbol rate for the desired user was set to 240Kbps
utilizing a spreading factor, G = 32. The first set of
observations (Table. 1) concurs with what was stated in
Section I- that the arithmetic imprecision is less for
logarithmic coding than with equivalent fixed-point binary
solutions. Also evident, is that the interpolation technique
outperforms the direct LUT conversion – particularly for
larger logarithmic approximations (i.e. for smaller LUT
sizes). The arithmetic imprecision metric was the filter
misadjustment in steady state- which is a quantity
representing the mean square error (MSE) formed from the
expectation of the difference between the ideal filter output
(with infinitely small quantization step) and the finite-
quantized filter output. Increasing the precision/accuracy of
the logarithmic codec decreases the misadjustment, where
the MSE is small for a 10-bit address for direct LUT
conversion and 8-bit address for the interpolative
conversion.
The Fig. 2 shows the results for the probability error
(Pe) as a function of reference signal to noise ratio obtained
for the Rake and LMMSE beamformer with 6-bit and
floating-point precision. The 2D Rake receiver and single
user bound are included in the plot for interest. It is clear
from the result that for 6-bit precision/LUT range, the
receiver performance is significantly degraded compared to
the floating-point version. It is also of some relevance that
the performance of both logarithmic codecs exceeds that of
the equivalent precision fixed point version. The
performance gain (~3dB for high SNR) offered by the
linear-Lagrange codec over the direct LUT conversion is
considerable (albeit, at accentuated complexity).
The final simulation results obtained (Fig. 3)
considered varying the address precision for both
logarithmic codecs and comparing the Pe (at SNR = 12dB)
with an equivalent 2s complement fixed-point model. An
implementation with logarithmic domain addition (i.e. the
classical LNS approach utilizing a LUT for evaluation of

this non-linear function) is included for completeness. The
first major point of interest is that at the boundary of
approximately 9 bits of precision, the Pe results are very
close for all three cases (the interpolative approach requires
approximately 6-bit address LUT for similar performance),
where increasing the precision beyond 9-bits yields very
minor decreases in Pe since the floating point bound is
approached asymptotically (provided m < 24). For lower
number of LUT addresses, the interpolative technique
outperforms all other conversions. Note also that the direct
LUT conversion outperforms the case for 2’s complement
coding. The main useful observation presented in this case
study is that use of either logarithmic codec yields results
that are not degraded compared to the 2’s complement
coding (in fact, offer superior performance) for bit lengths <
10 bits. However the LUT complexity dependence of
logarithmic coding is very limited in that only applications
with low-medium fixed-point precision is practical. In the
upper limit of floating point calculations, obtaining
equivalent accuracy with logarithmic numbers using LUT’s
is currently a bottleneck. Numerous algorithms have been
proposed and patented; although their practical application
for low cost receivers is limited- see [11] and the references
therein.
V. CONCLUSION
Logarithmic coding is proposed for adaptive beamforming
and shown to yield equivalent or better results than 2’s
complement fixed-point arithmetic without necessitating use
of large hardware array multipliers. The results generated
indicate that both logarithmic codecs yield lower
misadjustment and hence superior resistance to arithmetic
round-off errors as well as decreased probability of error.
The case of a hybrid-logarithmic implementation of similar
performance offered by a 12-bit fixed-point solution yields a
12.2kbit LUT for a direct conversion and a 768bit LUT
(with fixed 12-bit multiplier) for the interpolative
conversion. The merit of logarithmic coding with
interpolation is superior performance to direct LUT
conversion with equal sized address range (albeit with
increased complexity). For a codec with small address
range, interpolation will create a more accurate logarithmic
conversion. However, for LUTs with 8-12 address bits, then
a direct conversion without interpolation will be simpler, if
not quite as accurate. The work presented in this paper
shows that high precision calculations are not necessary
when considering the probability of error metric, therefore
yielding an easy translation from a 10-bit fixed-point
solution to a logarithmic architecture built with look up
tables requiring only 210
addresses. Work is currently
approaching completion on modeling the effect of arithmetic
precision on adaptive filters.
Fig. 3 The BER (for SNR = 12dB) as a function of the multiplier size (bits)
for fixed point 2’s complement binary, and/or number of address bits
for the logarithmic conversion LUT.
REFERENCES
[1] Vainio, O.; Neuvo, Y.; "Logarithmic arithmetic in FIR filters", IEEE
Transactions on Circuits and Systems, Vol. 33, No. 8, pp. 826 - 828,
1986
[2] J.L. Barlow, E.H. Bareiss; “On Roundoff Error Distributions in
Floating Point and Logarithmic Arithmetic’, Computing, Vol. 34,
1985, pp. 325-347.
[3] D. M. Lewis; "Interleaved memory function interpolators with
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Compututers, Vol. 43, pp. 974-982, 1994
[4] Litchfield, C.; Langley, R.J.; Lee, P.; Batchelor, J.; "The use of hybrid
logarithmic arithmetic for root raised cosine matched filters in
WCDMA downlink receivers", IEEE Wireless Communications and
Networking Conference, 2005, Vol. 1, 13-17 March 2005, pp. 596 -
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[5] Litchfield, C.; Langley, R.J.; Lee, P.; Batchelor, J.; "Least Squares
Adaptive Algorithms Suitable for Multiplierles LMMSE Detection in
3rd
Generation Mobile Systems”, IEEE International Symposium on
Personal, Indoor, and Mobile Radio Communications, 2005.
[6] Hooli, K.; Latva-aho, M.; Juntti, M.; "Multiple access interference
suppression with linear chip equalizers in WCDMA downlink
receivers", IEEE Global Telecommunications Conference, 1999, Vol.
1A, 1999 PP. 467 – 471.
[7] U. Madhow and ML Honig, “MMSE interference suppression for
direct-sequence spread-spectrum CDMA”, IEEE Transactions on
Communications, vol. 42, pages. 3178-3188, Dec 1994.
[8] Krauss, Hillery, Zoltowski.; "Downlink Specific Equalization for
Frequency Selective CDMA Cellular Systems", Journal of VLSI
Signal Processing, Vol. 30, pp. 143 – 161, 2002.
[9] Proakis, J.; “Digital Communications”, Prentic Hall, 2001.
[10] M. Arnold; “Design of a Faithful LNS Interpolator”, Euremicro
Digital System Design, Warsaw, pp. 336 – 344, 2001.
[11] M. Arnold, T. Bailey, J. Cowles; "Error Analysis of the
Kmetz/Maenner Algorithm," Journal of VLSI Signal Processing, vol
30, pp. 37-53, 2002.

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