Harmonic transients in time signals - Signal Processing in MRS with Biomedical Applications - page 152

signals. Both physics and mathematics support this fact. Physics, because
time signals in MRS originate from attenuated harmonic oscillators that are
described by quantummechanical autocorrelation functions. In fact, for such
time signals, the FPT is the exact filter in the sense of being capable of filtering
out all the harmonics without invoking any approximation [5]. Mathematics,
because the exact spectrum for these time signals is given by a polynomial
quotient. Moreover, if a function to be described is itself a ratio of two poly
nomials, as is actually the case with the exact Green function, the fast Pade
transform represents, by construction, the exact theory. Hence optimality of
the FPT for signal processing.
We have presented two equivalent Pade variants of the Green functions,
G (+) and G (−) , corresponding to the incoming and the outgoing boundary
conditions, inside and outside the unit circle in the complex plane of the
harmonic variable z. Recall that these variants of the FPT are denoted by
FPT (+) and FPT (−) , respectively. The FPT (+) and FPT (−) are identical to
the causal and anticausal Padez transforms studied extensively in mathe
matical statistics and the engineering literature on signal processing [5]. In
chapter 3, we applied the FPT (+) and FPT (−) to time signals from MRS to
explicitly demonstrate that both versions of the fast Pade transform can solve
exactly a standard quantification problem from MRS.
It was seen that the FPT is a very e cient signal processor, since it belongs
to the category of fast algorithms as does the fast Fourier transform. This
becomes possible with the FPT by employing the Euclid algorithm imple
mented with continued fractions [86, 87]. In this latter algorithm, the FPT
necessitates N(log 2 N) 2 multiplications, that are comparable to the conven
tional Nlog 2 N multiplications within the FFT.

4.3 General time series

In examinations of time evolution of a given system, one usually begins from
an initial state Φ 0 prepared at the instant t 0 = 0, which is associated with
n = 0. Integer n counts the discrete time t≡t n = nτ (n = 0, 1, 2,...) where
τ is the sampling time or dwell time. One of the important properties of any
time signal is its total duration T = Nτ which is in encoding called the total
acquisition time. In the Schrodinger picture of quantum mechanics, for a given
state Φ 0 , the vector Φ n is deduced from Φ n = UΦ 0 . Here, U = U(τ) is the evo
lution operator U = exp (−i ˆ )τ where ˆ represents the system's dynamical

nonHermitean operator, which is the Hamiltonian H in quantum mechanics.
Moreover, the time signal{c n }(0≤n≤N−1) of length N is equivalent
to the quantummechanical autocorrelation function c n = (Φ 0 |U n |Φ 0 ). As
customary with dissipative dynamics, the symmetric scalar product is em