
2 EURASIP Journal on Advances in Signal Processing
the average transmit power P,whereP ≤−12 dBm. Some
papers indicate that 53.3 kbps is the expected bit rate but
they do not give details on how this bound was obtained
[1, 3].
A common inaccuracy made when computing the capac-
ity of digital channels is in making the assumption that the
inputs and outputs of the channel are analog Gaussian ran-
dom variables and then using the Shannon capacity bound
for an AWGN channel (refer to Section 2,(4)) [1, 4]. Since
the DSP hardware used in digital modems utilize a finite sig-
nal set with finite precision, it is clear that the inputs of the
channel are not Gaussian and Shannon bound is not exact.
The question that naturally arises is in what region and for
what parameters of the A/D, D/A converters we can rely upon
the analog channel approximation? Our purpose in this pa-
per is to propose these conditions given the following con-
straints.
(a) First we consider a channel whose inputs x
∈ X and
outputs y
∈ Y are chosen from a finite set of possi-
bilities. Next we consider a special case of this channel,
one with a finite set of inputs and an infinite set of out-
puts.
(2) There exists an average power constraint P on the in-
put signals (see Section 2,(3)).
(3) The channel is an ISI channel represented by the cir-
culant matrix H, whose rows are circular shifts of the
ISI channel fading coefficients. The channel is assumed
known at the transmitter.
Our conclusions are that the performance of the quan-
tized block transmission channel approaches that of the ana-
log channel when we constrain the quantized channel to ap-
proximate the analog channel, by increasing peak-to-average
power ratio. We will apply the theoretical framework devel-
oped in this paper, to a practical numerical example which is
the downlink dial-up connection. Using this example we aim
to test how accurate is the bound of 53.3 kbps for this chan-
nel, under a reasonable scenario for the twisted pair connec-
tion. The results show that the bound of 53.3 kbps can be
improved upon.
Note that the block transmission systems we have de-
scribed can be modelled as MIMO systems where one
user communicates with an NSP. As the size of the block
goes to
∞, the throughput of the block transmission tech-
nique will give the capacity of the channel. In a gen-
eralized MIMO system (involving multiple users and the
NSP), by adding a cyclic prefix to each user’s block, the
matrix H would be block circulant. In this paper we
have sometimes used the terminolog y “MIMO” in place of
“block transmission” especially where we want to conserve
space.
The problem of obtaining the capacity of a quantized
MIMO channel has been preceded by such work as [4], in
which Shannon obtained the capacity of an AWGN chan-
nel and showed that this capacity is achievable by a Gaus-
sian input distribution. Arimoto [5]andBlahut[6], derived
a numerical method for computing the capacity of discrete
memoryless channels. In their work, Kavcic [7] and Varnica
et al. [8] presented an equivalent expectation-maximization
version of the Blahut-Arimoto algorithm. In [8] further-
more, the Blahut-Arimoto algorithm is modified to incor-
porate an average power constraint. In [9], Honary et al. in-
vestigated the capacity of a scalar, quantized, AWGN chan-
nel numerically. Ungerboeck [10] showed numerical results
that the performance of a memory less, quantized, AWGN
channel approached the performance of a memoryless, un-
quantized, AWGN channel, with a certain number of in-
put levels and the work of Ozarow and Wyner [11], pro-
vided analytically bounds that support the numerical results
of [10]. In [12], Shamai et al. obtained bounds on the aver-
age mutual information rates of a discrete-time, peak power
limited ISI channel with additive white Gaussian noise. In
Varnica et al. [13], Varnica [14]considerMarkovsources
transmitted over memoryless and ISI channels with an av-
erage power constraint and a peak-to-average power ratio
constraint. They obtained lower bounds on the capacity of
the ISI channel. In [15], Bellor ado et al. obtain the capac-
ity of a Rayleigh flat-fading MIMO channel with QAM con-
stellations independent across antennas and dimensions. In
our work, we seek to obtain the exact numerical capacity of
the quantized MIMO system with average power constraint.
This system is obtained by the inclusion of a cyclic prefix
to blocks of data symbols in order to supress edge effects.
Therefore the capacity of the quantized MIMO system ob-
tained is a lower bound on the capacity of the ISI channel.
We compare this capacity to the capacity of the unquan-
tized MIMO system and propose, as a result of our com-
parisons, conditions under which we can come arbitrarily
close to the Shannon bound of (4) at low SNR operating re-
gions.
To achieve our pur pose, we use the constrained Blahut-
Arimoto algorithm presented in [11], which incorporates the
average power constraint P on the channel inputs. How-
ever, we replace the interval-halving procedure in [8]by
a Newton-Raphson method. We derive this constrained
Blahut-Arimoto algorithm in Section 2.InSection 3 we
present and discuss results considering the SISO channel.
Section 3 provides some useful insights for the block trans-
mission channel, whose results we present in Section 4.We
implement a practical example and give the results ob-
tained, in Section 5. Finally we draw our conclusions in
Section 6.
In the notations used in this paper, boldface font (e.g ., x)
is used to denote vectors and matrices (and the correspond-
ing random variables). Calligraphic font (e.g., X) is used to
denote the alphabet of the channel inputs or outputs. Sum-
mations such as
x
refer to summations taken over all the
elements in a set under consideration, in this case x
∈ X.
Unless otherwise stated, natural logarithms are used, thus the
unit of capacity is in nats per channel use. We consider the
real-valued ISI channel, however the results we obtain ap-
ply (mainly with changes in notation) to the complex-valued
ISI channel representative of passband systems, where the
inputs, outputs, and ISI channel coefficients are complex-
valued.