Digital Signal Processing Reference
In-Depth Information
rT
Propertiesof
pseudocirculantmatrices
D.1Introduction
Pseudocirculantmatricesariseinfilterbankprecodertheory(Sec.3.9.2).We
alsofindthemarisinginthecontextofaliascancellationandinthedescrip-
tionofblockedversionsoftransferfunctions[VaidyanathanandMitra,1988],
[Vaidyanathan,1993].Inthissectionwementionafewmathematicalproperties
ofpseudocirculantswhichhavebeenfoundtobeusefulinresearch,e.g.,inthe
useoffilterbanksindigitalcommunications.
D.2Circulantmatrices
Webeginbyreviewingcirculantmatrices,andthenwemoveontopseudocircu-
lants.Todefineacirculantmatrixfirstconsideranexample:
⎡
⎤
c
(0)
c
(4)
c
(3)
c
(2)
c
(1)
c
(1)
c
(0)
c
(4)
c
(3)
c
(2)
c
(2)
c
(1)
c
(0)
c
(4)
c
(3)
c
(3)
c
(2)
c
(1)
c
(0)
c
(4)
c
(4)
c
(3)
c
(2)
c
(1)
c
(0)
⎣
⎦
C
=
.
(D
.
1)
Notethattheleftmostcolumnisarbitrary,buttheothercolumnsareobtained
byshifting
down
thepreviouscolumnandrecirculatingtheelementthatspills
over.Thismatrixisthereforecalledacirculant,ormorespecificallya
down
circulant
(todistinguishfromanupcirculant,whichisdefinedinanobvious
way).Inthistexttheterm
circulant
alwaysreferstodowncirculants.Notice
thatinadown-circulantmatrix,anyrowisobtainedfromtheprecedingrow
771
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