The Theory and Application of
Complex Orthogonal Designs
Sarah Spence Adams
Robert E. Gaines Professor of Mathematics, University of Richmond
Associate Professor of Mathematics, Franklin W. Olin College of Engineering
Research supported by NSA Grant H98230-07-1-0022
Motivation: Wireless Communications
• Transmit antenna sends
signal to receive antenna
• Signal subject to distortion,
interference, attenuation,
multipath fading, etc.
Time Diversity
Single Transmit Antenna Example
a1
Timestep t1
Timestep t2
t1
z1
t2
z1
Space Diversity
Two Transmit Antennas Example
a1
a2
t1 Z1 Z1
Power is split between two antennas
Combined Space and Time Diversity
Timestep t1
Timestep t2
Naïve Scheme for Two Transmit Antennas
a1
a2
Z1
Z2
t 2 Z2
Z1
t1
Practical Alamouti Scheme
„
Z1
Z2
-Z2*
Z1*
Orthogonality permits simple maximum-likelihood
decoding algorithm
(Alamouti, 1998)
Orthogonal Designs
„
Alamouti’s scheme implicitly depends on complex
orthogonal designs
„
Introduced in 1970’s by Seberry, Geramita, Geramita
Complex Orthogonal Designs (CODs)
„A
complex orthogonal design A is an n x n matrix
with entries
such that
The Alamouti COD
Existence Results
„ Based
on Hurwitz-Radon matrices
„ Number
„ Square
theoretic results from turn of century
CODs exist only for n = 1 or 2
Generalizing Alamouti’s Scheme
„
Tarokh, Jafarkhani, Calderbank (1999)
– Pioneered use of generalized complex orthogonal designs
for wireless communications
– Popularized term space-time block code
Generalized Complex Orthogonal Designs
„A
generalized complex orthogonal design A is
an r x n matrix with entries
such that
GCOD
STBC
# Rows
Decoding
Delay (8)
# of Columns
# Antennas (4)
Rate: A Measure of Efficiency
„ Ratio
of number of variables to number of
rows
„ Ratio
of number of information symbols to
number of required timesteps
Rate ¾ Example
# Rows
Decoding
Delay (8)
# of Columns
# Antennas (4)
Two Major Research Questions
„ What
„ What
is the maximum achievable rate?
is the minimum decoding delay for a
maximum rate STBC?
Maximum Rate (Liang, 2003)
„ For
a GCOD with 2m-1 or 2m columns, the
maximum achievable rate is
„ Liang
provides algorithm to construct max rate
designs for any number of columns
Maximum Rate
Examining Delay for Max Rate GCODs
Max rate ¾
Max rate ¾
Decoding Delay 8
Decoding Delay 4
Minimum Decoding Delay
„A
tight lower bound on the decoding delay
of a maximum rate GCOD with 2m-1 or 2m
columns is
„ Achievable
when the number of columns is
congruent to 0, 1, or 3 modulo 4
(Adams, Karst, Pollack, 2005)
Proof of Lower Bound
„ Involves
combinatorial properties of max rate
GCODs
„ Specialized
properties
software helped illuminate these
Equivalence Operations
„ Rearrange
order of rows or columns
„ Negating
rows, columns, or all instances of a
given variable
„ Conjugating
all instances of a given variable
An Important Submatrix
„
Within any max rate GCOD, the Bi submatrix is
achievable through equivalence operations for each
variable zi
(Liang, 2003)
B1
“Bi form”
„A
max rate GCOD A is in “Bi form” if
– the rows of Bi appear somewhere within A, up to
conjugations and negations
– the contiguous submatrix Bi is achievable through
only row rearrangements, conjugations, negations
B1 form
Column Rearrangements
„ Rearranging
columns within left half of A or
within right half of A preserves current Bi form
Importance of Single Rows
„ If
one row of Bi appears within A, then A is in
Bi form
Left/Right Column Swaps
„ If
A is in Bi form, then after swapping any
column from the left with any column from the
right, A is in some Bk form
Swapping 1st and 5th
columns moves from
B1 form to B6 form
Culminating Corollary
„ Any
arrangement of a max rate A is in Bi form
for some i
Zero Patterns
„ Every
zero pattern with m-1 (and m-2) zeros
must appear in some row of length 2m (2m-1)
„ Gives
lower bound of
on the number of required rows
Bound is Usually Achievable
„ When
the number of columns is equivalent to
0, 1, or 3 modulo 4, an algorithm by Lu, Fu,
Xia (2005) shows that the bound is achievable
Exceptional Case 2m
2 mod 4
„ The
best achievable delay for a max rate
GCOD with 2m 2 mod 4 columns is twice
the lower bound:
⎛ 2m ⎞
⎟⎟
2⎜⎜
m
−1
⎠
⎝
(Adams, Karst, Murugan, under review)
Multiplicity of Zero Patterns
one zero pattern appears κ times, then every zero
pattern appears κ times
„ If
„ If
delay r of A satisfies
then
„ Algorithms
exist that achieve
– Lu, Fu, Xia (2005), Liang (2003), Su, Xia, Lui (2004)
„ Remained
to determine why cannot achieve bound
New Proof Techniques
„ Connect
the problem to signed graphs
– edges labeled ±1
„ Signed
graph is balanced if the product of signs
of edges in every cycle is positive
„ Not
achieving lower bound on delay is equivalent
to a certain signed graph not being balanced
Introduce Standard Form
„ To
determine sign of (r,c) entry:
– Consider binary support of length 2m-1 row r
– Expand using basis for (2m-1)-dimensional binary
vector space, where basis definition depends on
column c
– Determine parity of the first c-1 coefficients of the
vector expansion
„ Resulting
signs effectively organize all 2x2
subdesigns
Equivalence Classes of GCODs
„ Standard
form achievable for all max rate, min
delay GCODs with 2m-1 columns
„ Hence
exactly one equivalence class of
designs of this order
Complete Minimum Delay Result
„ The
minimum achievable decoding delay for a
maximum rate GCOD with n = 2m-1 or 2m
columns is
when n is congruent to 0, 1, or 3 modulo 4, and
when n is congruent to 2 modulo 4.
Min Delay for Max Rate GCODs
Practical Implications
„ The
decoding delay grows quickly as the
number of antennas increases
„ Maximum
rate, minimum delay codes are
not practical for large numbers of antennas
„ Non-rate
optimal codes may be preferable
in practice
Motivation for Studying Rate ½ GCODs
„ Maximum
„ Trade-off:
rate approaches ½
Hopefully a savings in delay will be
worth some sacrifice in rate
Rate ½ Minimum Delay
„
Conjecture: The minimum decoding delay of a
rate ½ GCOD with n = 2m-1 or 2m columns is:
– 2m-1 if n is congruent to 0, 1, 2, or 7 modulo 8
– 2m if n is congruent to 3, 4, 5, 6 modulo 8
(Adams, Crawford, Davis, Greeley, Karst, Lee, Murugan)
Improved Delay for Rate ½
Max Rate vs. Rate ½
# of Antennas
Rate ½ /
Max Rate
Rate ½ Delay /
Max Rate Delay
6
.75
.267
8
.8
.143
10
.833
.0762
14
.875
.0213
20
.909
.00610
Best Balance?
„ Determine
minimum delay as a function of
arbitrary rate
„ Seek
optimal balance between rate and delay
Additional Consideration
„ Transceiver
„ Linearized
linearization
received signal expression allows for
– backwards compatibility with existing technologies
– the design of low complexity filters, equalizers
Connect Practical Consideration to
Combinatorial Property
„A
GCOD is conjugation-separated if the
nonzero entries in a given row are either all
conjugated or all non-conjugated
„ An
STBC can achieve transceiver linearization if
its underlying GCOD is conjugation-separated
(Su, Batalama, Pados, 2006)
Achieving Conjugation-Separation
„ Any
max rate, min delay GCOD with 2m-1
columns is equivalent to a conjugation-separated
GCOD that can achieve transceiver linearization
„ No
arrangement of a max rate GCOD with 2m
columns that achieves the lower bound on delay
is conjugation-separated
Delay vs. Linearization Trade-off
„ Max
rate GCODs with 2m columns that
achieve twice the lower bound can achieve
conjugation-separation
„ With
2m columns, cannot simultaneously
achieve the lower bound on delay and
conjugation-separation
Related Work
„ Quaternion
orthogonal designs for
space-time-polarization codes
„ Multidimensional
real and complex
orthogonal designs
„ Multilevel
Hadamard matrices
Summary
„ Classical
orthogonal designs utilized in
modern communications systems
„ Mathematical
properties of designs have
significance for implementation
„ Trade-offs
in system design are omnipresent