njit-etd2003-081 - New Jersey Institute of Technology

66 [-Ω,Ω
], then its STFT will be localized in the region [-T,T]x [– Q, n] of the timefrequency
plane. Of course, the uncertainty principle prevents the possibility of having
arbitrary high resolution in both time and frequency domains, since it lower-bounds the
time bandwidth product of any basis function by ΔTΔΩ>=
1
where (ΔT)2 and
4,r
(A0) 2 are the variances of the time function and its Fourier transform respectively
[43][44].
An important parameter of a window function is its size (or scale). The selection
of an appropriate window size poses a fundamental problem in signal analysis. Thus, by
varying the window function used, one can trade the resolution in time for the resolution
in frequency. An intuitive way to achieve this is to have short time duration high
frequency basis functions, and long time duration low frequency ones. Fortunately, the
wavelet transform provides for this desired feature and is defined as,
where a E R+ ,b E R . Here
a, and b are the scale and shift variables respectively, and
they are continuous variables. Depending on the scaling parameter a, the wavelet
function ψ(t) dilates or contracts in time causing the corresponding contraction or
dilation in the frequency domain. Therefore a flexible time-frequency resolution is
achievable with the wavelet transform. Another significant difference of these
transforms is that the STFT is never a real function on the time-frequency plane
regardless of the choice of co(t), but the wavelet transform is real if the basic wavelet
ψ(t) is chosen to be real.