Biomedical Signal Processing - Computational Intelligence in Biomedical Engineering - page 32

2.3.7 The Hadamard Transform
The Walsh-Hadamard transform is similar to the Walsh transform but has
rows of the transform matrix ordered differently. The Hadamard-ordering
sequence is derived from the Walsh-ordered sequence by first expressing the
Walsh function in binary, bit reversing the values and converting the binary
values to Gray code, and finally to real values. It is considered as a generalized
class of Fourier transforms.

2.4 Spectral Analysis and Estimation
Random signals are best characterized statistically by averages (Proakis and
Dimitris, 1996), for example, fluctuations in the weather of temperature and
pressure could be viewed as random events. Information in these kind of sig-
nals can be obtained from their power spectrum, where the most relevant mea-
sure is the autocorrelation function used to characterize the signal in the time
domain. The Fourier transform of this autocorrelation function gives the power
density spectrum, which is a transformation to the frequency domain. There
are several methods of obtaining estimates of the power spectrum (Proakis
and Dimitris, 1996; Oppenheim et al., 1999; Clarkson, 1993).

2.4.1 Autocorrelation and Power Density Spectrum
Consider a digital signal x ( n ) obtained by sampling some analog signal x ( t ), if
the digital signal is of finite length the power spectrum can only be estimated
because complete information of the signal is not available. Given a longer
sequence of digital samples, our estimates are more accurate provided that the
signal is stationary. If the signal is nonstationary it may be more worthwhile
to obtain power spectrum estimates from shorter digital sequences and treat
them as estimates of signal segments instead of the full signal. Too short a
sequence, however, severely distorts our estimate and the goal is to determine
the minimum sequence length providing acceptable estimates.
Recall from Equation 2.3 the energy of an analog signal x ( t )as

∞

2 d t<

E = | x ( t ) | ∞ (2.64)

−∞

If the signal has finite energy, then its Fourier transform exists and is given as

∞

x ( t )e −j 2 πF t d t X ( F )= (2.65)

−∞