Signal Processing - January 2017 - 51

when SNR is large enough. Zheng and
a positive constant a, # e - at dt does
a
not have a simple closed form while
Tse [3] obtained the following well3 - at 2
known DMT:
e
dt
does. Another example is
#- 3
the diversity and multiplexing tradeoff
(DMT) obtained by Zheng and Tse
d (r) = (m - r) (n - r),
[3] for multiple-input, multiple-output
(MIMO) antenna systems in wireless
where m and n are the numbers of transcommunications, which becomes a necmit and receive antennas, respectively.
essary parameter in
One can see that both
designing a MIMO
r and d (r) are sort
i consider that small data
wireless communiof
indices, and they
corresponds to algebra,
cation system. Let
are only meaningful
mid data corresponds to
when SNR approachR be the transmisanalysis, and big data
sion rate in bits/seces infinity, i.e., in a
corresponds to topology
ond/hertz. Let r be
massive transmission
in mathematics.
rate case or big data
the normalized rate
case. This is the case
r = R/ log (SNR),
when it is impossible to count one elewhere SNR stands for signal-to-noise
ment by one element for a massive data,
ratio and is the channel SNR. When
and one needs to sort out its index, such as
SNR is huge, one may expect that R is
exponentials and/or genus, in some way
huge as well by Shannon's channel capacto describe and/or extract features from
ity formula that is about log (SNR), i.e.,
the massive/big data. I think this belongs
massive data (or big data) can be transto topology in mathematics. Thus, in my
mitted through the channel. In this case,
opinion, topology in mathematics correcounting R may be not possible, while
sponds to big data, where it is impossible
counting r becomes more reasonable,
or not necessary to count one element by
where r is called the multiplexing gain.
one element.
Let Pe be the error probability at the
receiver of a MIMO modulation scheme
with transmission rate R. Let
Summary and discussion
In summary, I consider that small data
log (Pe)
corresponds to algebra, mid data corre(5)
d (r) = - lim
.
SNR " 3 log (SNR)
sponds to analysis, and big data corresponds to topology in mathematics. Was
Then, d (r) is the index of the negative
big data started when it was named?
exponential of the error probability Pe
Of course not. Big data has existed for
and called the diversity gain.
a long time, as massive groups of fish
move in the ocean, massive groups of
Pe . SNR - d (r),
(6)
b

2

birds fly in the sky, and/or a massive
number of people on the ground travel
around the world. Today, massive bits
are transmitted through both wired and
wireless channels called the Internet.
The key is how to get some indices,
trends, or patterns from these massive
data and/or how to find a needle in the
ocean. What will big data lead to tomorrow? Or, how deep can we go toward
infinity tomorrow? Or, how fast will a
computer be tomorrow?

Author
Xiang-Gen Xia (xianggen@gmail.com)
is the Charles Black Evans Professor in
the Department of Electrical and
Computer Engineering, University of
Delaware, Newark. His main research
interests include wireless communications and radar signal processing. He is a
Fellow of the IEEE.

References

[1] D. B. Shapiro, Compositions of Quadratic
Forms. New York: De Gruyter, 2000.
[2] K. Lu, S. Fu, and X.-G. Xia, "Closed-form
designs of complex orthogonal space-time block
codes of rate (k+1)/(2k) for 2k-1 or 2k transmit antennas," IEEE Trans. Inform. Theory, vol. 51, pp. 4340-
4347, Oct. 2005.
[3] L. Zheng and D. N. C. Tse, "Diversity and multiplexing: A fundamental tradeoff in multiple antenna
channels," IEEE Trans. Inform. Theory, vol. 49,
pp. 1073-1096, May 2003.
[4] Wikipedia. Genus. (2016). [Online]. Available:
https://en.wikipedia.org/wiki/Genus_(mathematics)
[5] A. Cichocki. (2014). Era of big data processing: A
new approach via tensor networks and tensor decompositions. arXiv: 1403.2048v4 (cs.ET) [Online].
Available: http://arxiv.org/abs/1403.2048v4

sp

errata

R

eference [8] in the "SP Education"
column of the November 2016 issue
of IEEE Signal Processing Magazine [1] was published missing a URL.

We apologize for any confusion this may
have caused. The corrected reference is
shown in [2].

Signal Process. Mag., vol. 33, no. 6, pp. 123-127,
Nov. 2016.
[2] BCI hand control download [Online]. Available:
http://classes.engineering.wustl.edu/ese497/BigFiles/
IpsiHand_v5.wmv

References

Digital Object Identifier 10.1109/MSP.2016.2636258
Date of publication: 11 January 2017

[1] E. Richter and A. Nehorai, "Enriching the undergraduate program with research projects," IEEE

IEEE Signal Processing Magazine

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January 2017

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https://en.wikipedia.org/wiki/genus_(mathematics http://www.cs.ET http://www.arxiv.org/abs/1403.2048v4 http://classes.engineering.wustl.edu/ese497/BigFiles/
Table of Contents for the Digital Edition of Signal Processing - January 2017
