Harmonic transients in time signals - Signal Processing in MRS with Biomedical Applications - page 158

known to be very important. Of course, theoretical developments always favor
the analysis of the general delayed Hankel matrix H n (c s ), since the needed
associated counterparts H n (c 0 ) and H n (c 1 ) appear as two special cases taken
at s = 0 and s = 1, respectively. Naturally, a general approach using H n (c s )
has a chance to yield certain advantages over a particular procedure given
by the evolution matrix H n (c 1 ) , particularly in the domain of generating
more fruitful algorithms. This will be shown to be the case in the present
chapter. Specifically, it will be demonstrated that the sole introduction of a
nonzero initial time t s

= 0 has farreaching consequences for spectral analysis,
even when one actually intends to keep the whole signal{c n }(0≤n≤

N−1) of total length N. For instance, processing the evolution matrix H n (c 1 )
usually proceeds through matrix diagonalizations, or equivalently, rooting the
corresponding characteristic equation to find the spectral parameters{u k ,d k }

of the signal (4.9). By contrast, both of these customary procedures can be
alleviated altogether by using H n (c s ) for a fixed integer s > 0 rather than
setting s = 1, as in the evolution matrix H n (c 1 ). Then the sought spectral
parameters{u k ,d k }could be obtained nonconventionally from convergence
of the coe cients of the corresponding 'delayed' continued fractions as the
values of number s are systematically augmented [5].

In contradistinction to the FFT, the space methods, such as the FPT, from
the Schrodinger picture of quantum mechanics can spectrally analyze Hankel
matrices with an arbitrary initial time t s . These signal processors employ the
evolution operator U which generates the state Φ s of the considered system
at the delayed moment t s by the transformation. Φ s = U s Φ 0 . In this ex
pression, the time delay is described by starting the analysis from Φ s rather
than from Φ 0 which refers to s = 0. Of course, evolution of the system in
the time interval [t 0 ,t s ] of nonzero length has to be taken into account. The
needed correction is easily made in the state space methods through multi
plication of the vector Φ 0 by the operator U −s = exp (is ˆ τ). This cancels
out the evolution effect accumulated in the state vector in the time interval
t∈[0,t s ]. Thus, when the first s signal points are skipped, all the matrix
elements in the statespace version of the FPT must be altered by the coun
teracting operator exp (is ˆ τ). As such, in these signal processors, taking the
instant t s = 0 in place of t 0 = 0 for the initial time engenders no di culty
whatsoever. Moreover, the statespace methods accomplish spectral analysis
by diagonalizing the Hankel matrix H n (c s ) for a fixed integer s in the same
fashion as for H n (c 0 ) and H n (c 1 ). The same is true for the FPT which re
places diagonalization by rooting the characteristic polynomials. Importantly,
relative to H n (c 1 ), it is advantageous to diagonalize the delayed evolution ma
trix H n (c s ) = U (s)

={U (s)
i,j

}, where the general element U (s)
i,j

is due to the

n

i,j = (Φ j |U s |Φ i ). The advantage is in the key possi
bility of identifying spurious roots. To accomplish this task in practice, one
usually diagonalizes the matrix U (s n for s = 1 and s = 2 in order to compare
the obtained eigenfrequencies. The same frequencies for different values of s

U from U (s)

s th power of