Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 154

DO 22 I=1,M
DO 22 J=1,I
HOLD=VH(I,J)
VH(I,J)=DCONJG(VH(J,I))
VH(J,I)=DCONJG(HOLD)
22 CONTINUE
C Sort singular values into descending order, exchange columns of U
C and rows of VH.
N=1
23 CONTINUE
IMAX=N
DO 24 I=N,M
IF(S(I).GE.S(IMAX)) IMAX=I
24 CONTINUE
C Exchange singular values.
HOLD=S(N)
S(N)=S(IMAX)
S(IMAX)=HOLD
C Exchange columns of U and rows of VH.
DO 25 I=1,M
HOLD=U(I,N)
U(I,N)=U(I,IMAX)
U(I,IMAX)=HOLD
HOLD=VH(N,I)
VH(N,I)=VH(IMAX,I)
VH(IMAX,I)=HOLD

25 CONTINUE

N=N+1
IF(N.LE.M) GO TO 23
RETURN
END

2.4 Fourier series and transforms
In the discrete, equally spaced case, the discrete Fourier transform (DFT) pair,
(2.147) and (2.148), provide a one-to-one mapping between the time and frequency
domains. If we hold the record length fixed at T but increase the sampling rate
indefinitely, N →∞ and Δ t → 0, making the sequence g j a continuous function of

time, g( t ). Thus,

T /2
− T /2 g( t ) e − i 2π f k t dt ,

G k → (2.244)

and

=Δ f ∞

G k e i 2π f k t

g j → g( t ) . (2.245)

k =−∞

This is the infinite Fourier series representation of the continuous function of time,
g( t ), defined for − T /2 < t < T /2,

∞

1
T

G k e i 2π T t g( t ) = , (2.246)

k =−∞