This document contains biographical information about Tad Matsumoto and describes his work on single carrier signaling and MIMO equalization. It discusses how single carrier signaling is impacted by multipath propagation and how the number of signal points increases exponentially with more multipath components. It then presents Matsumoto's 2002 equation for a block-wise time domain MIMO equalizer, noting its computational complexity is dominated by matrix inversion. The document concludes by describing Matsumoto's 2005 equation for a frequency domain equalizer, which has complexity at a logarithmic order of the frame length and is independent of the path number.

1. 1-1 Asahidai, Nomi, Ishikawa 923-1292, Japan
http://www.jaist.ac.jp/is/labs/matsumoto-lab
Tad Matsumoto, Fellow, IEEE
Visiting Fellow, Royal Academy of Engineering in UK
Japan Advanced Institute of Science and Technology (JAIST)
matumoto@jaist.ac.jp
Paradigm Shift in Turbo Processing:
From P2P to Networks
Centre for Wireless Communications, University of Oulu
tadashi.matsumoto@ee.oulu.fi
P.O.Box 4500, University of Oulu FIN-90014, FINLAND

2. School of Information Science
Symbol Duration T
･･･
･･･
τ
1G
･･･
･･･
τ
First path
Last Path
2G
τ
･･
･･
3G
τ<<T No inter-symbol interference (ISI)
τ≒T only minor ISI τ>>T Severe ISI
Single Carrier Signaling：Time Domain Behavior
Tx Rx
Delay Spread τ
“0” “1” “1” “0”
“0” “1” “1” “0”
“0” “1” “1” “0”
“0” “1” “1” “0”
Characterized by Multipath Propagation

3. School of Information Science
Single carrier signaling is Green!:
The number of the signal points increases exponentially!
What happens if there are a lot
more multipath components?
Single Carrier Signaling：Time Domain Behavior
2 x 2 MIMO setup QPSK signal
transmitted over 20 path
propagation channel!

4. Principle of MIMO SC-MMSE Turbo
equalization
Soft symbols constructed
from decoder feedback
sˆ
[ ])(2 kbλ
[ ])(1 kbλ

5. Block-wise Time Domain MIMO
SC-MMSE Equalizer: My 2002 Equation
where
diag denotes a diagonal matrix constructed
by its argument elements
Computational complexity is dominated
by the matrix inversion Σ−1.
Tx antenna
Soft symbols constructed
from decoder feedback
Covariance of the soft cancellation output
[ ] [ ]{ }22
ˆ)ˆ)(ˆ( ssEdiagssssE H
−=−−=Λ
Complexity: O[N3(L+K)3]
Problematic!!!
Soft Cancellation Output
Frame Length
Number of Receive Antennas
Path Length

6. Finally, the equalizer output vector is given by:
Diagonal Block
By Exploiting Beneficial Properties of
Frequency Domain Processing, it Ends up with
My 2005 Equation (Green!!!)
( )[ ]FΞIΔΞΞΞFΓ f
m
fHffH
m
H
m diag
12 −
+≈ σ
(2x2 MIMO)
Now, analysis, including EXIT analysis, is EASY!

7. Number of Users = U, Each user uses 2 Tx antennas and Rx uses 4 antenns.
(Multi-user MIMO) Frame length = 512
Complexity is at a logarithm order of the
frame length K and is Independent of the
Path Number L!
2002
2005