Fig. 1. Basis functions and time frequency resolution of the short-time Fourier
transform (STFT). (a) Basis functions. (b) Coverage of time-frequency plane.
Of course, the uncertainty principle excludes the possibility of having arbitrarily
high resolution in both time and frequency, since it lower bounds the time-bandwidth
product of possible basis functions by (1/ 4 )T
π
Δ
⋅ΔΩ≥ , where
2
()TΔ and
2
()ΔΩ are
the variances of the absolute values of the function and its Fourier transform,
respectively. However, by varying he window used, one can trade resolution in time
for resolution in frequency. The can be achieved with the wavelet transform, where
the basis functions are obtained from a single prototype wavelet by translation and
dilation/contraction:
,
1
() ( )
ab
ta
ht h
b
a
−
= (2)
where
aR
+
∈ , bR∈ . For large a , the basis function becomes a stretched version of the
prototype wavelet, that is a low frequency function, while for small
a , the basis
function becomes a contracted wavelet, that is a short high frequency function (see
Fig. 2(a)).
The wavelet transform (WT) is defined as
*
1
(,) () ( )
ta
X
ab xth dt
b
a
∞
−∞
−
=
∫
(3)
The time-frequency resolution of the WT involves a different tradeoff to the one used
by the STFT : at high frequencies the WT is sharper in time, while at low frequencies,
the WT is sharper in frequency (see Fig. 2(b)).
(a)
(b)
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