082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

Eliminating the unwanted values inthe first column, wefind
⎛
R 1
R 2
→R 2
+R 1
⎝ 2 −1 1 2
⎞
0 0 2 6 ⎠
R 3
→R 3
−3R 1 0 0 −5 −15
8.10 Gaussian elimination 291
Entriesunderthesteparenowzero.Toreducethematrixtoechelonformwemustensure
each successive row has more leading zeros than the row before.We continue:
⎛
R 1
R 2
⎝ 2−112
⎞
0 026⎠
R 3
→ 2R 3
+5R 2 0 0 0 0
which isnow inechelon form.
In this form there is a row which starts off with a non-zeroxvalue, that is the first
row, there is a row which starts off with a non-zerozvalue, but no row which starts off
withanon-zeroyvalue.Therefore,wechooseytobethefreevariable,y = λsay.From
thesecondrowwehavez = 3andfromthefirst2x −y+z = 2,sothat2x = λ−1,that
isx = (λ −1)/2.
Engineeringapplication8.2
TheVandermondeMatrix
Many data storage devices, for example the Blu-ray disc , and data transmission
standards, for example WiMAX ® , have built-in techniques to reduce the effects of
errors which occur during normal use. These errors originate from noise and interference
and they can result in the loss of data. One such technique is the use of an
error-correctingcodetoprovideameasureofprotectionagainstdataerrors.Errorcorrecting
codes are used to encode the data when it is stored or transmitted. The
data is then decoded when it is read or received. In the case of a transmission line,
theerror-correctingcodemakesitpossibletodetecterrorsinthereceivedsignaland
tomake corrections,so thatthe errors arenotsubsequently retransmitted.
Oneimportantclassoferror-correctingcodesarebasedonReed-Solomoncodes,
details ofwhicharebeyond the scopeofthistext.However animportantmathematical
concept usedinReed-Solomoncodes is thatofaVandermondematrix.
The Vandermonde matrix can be illustrated by considering the problem of representing
a signal by a polynomial. For example, to approximate a signal f (t) by a
second-degree polynomialwewrite
f(t)≈a 0
+a 1
t+a 2
t 2
wherea 0
,a 1
anda 2
are coefficients of the polynomial which must be found. These
are found by forcing the original signal f (t) and its polynomial approximation to
agree at three different values oft, sayt 0
,t 1
andt 2
. This gives rise to the following
systemofequations:
⎛
1t 0
t 2 ⎞ ⎛ ⎞ ⎛ ⎞
0 a
⎜
⎝1t 1
t1
2 ⎟ 0
f(t 0
)
⎠ ⎝a 1
⎠ = ⎝f(t 1
) ⎠
1t 2
t2
2 a 2
f(t 2
)
➔