Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

The Fourier integral
- +- -io t T -iw t
n
z(un) = I Z(t)e
dt = J Z(t)e
n
dt
-@2 0
will be evaluated r.umerica1l.y according to th,e trapezoidal formula
of approximation. Setting
and observing tha-t
zo = $[zct 0 ) + 2(tN)-j
"n?n
= 2a fmtn = -- 2 limn
N'
the discrete Fourier transform of Z(t) is
2
L *
A linear trend of the record within the chosen interval may be
wri.tte11 as
with d = Z(tN) - Z(to).
A cor3rection for this trend i.n the frequency domain implies that
the imaginary Fourier transform of Z1(t),
-
is substracted from Z .
m
A second correction arises from the fact that Z(t) does not v?.nish
necessarily outside of the chosen intervall. In that case the
original time function can he multiplied in the rime domain wirh n
weight function W(t) which is zero for t < to and t > tN. The
Fourier transform of the product
W(t) . Z(t)
-
- - - -
is given by7a convolution of Z with the Fourier transform of W:
- 1
W(o) 35 ZCo) - Ff C Wm-& Z; .
A
m
A frequent.1~
used weight function is
' 1 2li T -
WCt) = -11 + cos --(t-t .-
2 T 0
2