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2 Fundamentals of Information, Signal and System Theory

decreases (∆f is large) and vice versa. For example, using a time resolution of 1 ms,

the achievable frequency resolution would be 1000 Hz.

An alternative to the STFT is the wavelet transform. The word wavelet describes

the form of the integral kernel function ψ(t). The transformation rule for the wavelet

transformation is

S (a, τ) =

1

√a ^{∫s(t)ψ ( t −τ}

a

) dt .

(2.74)

The parameter τ in the argument of the wavelet causes a time shift, the parameter a a

compression or stretching of the wavelet. Similar to the Fourier transform, the wavelet

transform can be interpreted as a correlation of the signal s(t) with the wavelet ψ(t),

where the correlation value S(a, τ) depends on the stretching a and the time shift τ.

The factor ¹/√a before the integral is necessary for the normalization of the transform

to the wavelet width. The functional description of ψ(t) is not fixed. Rather, ψ(t) can

be largely freely designed and adapted to the signal s(t), which is an advantage of the

wavelet function over transforms with a given integral kernel. For the wavelet syn-

thesis only two conditions must be fulfilled. First, the area fractions of the function

above and below the zero line must be equal. Thus applies

∫ψ(t)dt = 0 .

(2.75)

Equation 2.75 thus describes the wave character of the wavelet. The second require-

ment for ψ(t) is formulated as an admissibility condition:

∫^{Ψ2 (ω)}

ω

dω < ∞,

(2.76)

with Ψ(ω) as the Fourier transform of ψ(t). A consequence of Equation 2.76 is

lim

ω→0 ^{Ψ (ω) = 0 .}

(2.77)

In signal processing wavelets based on the Gaussian function

e^−t2

2

(2.78)

are often used. Equation 2.78 forms the envelope when multiplied by a second func-

tion, which corresponds to the effect of the Gaussian window in STFT. Examples are

the Morlet¹⁷ wavelet and the Mexican-Hat wavelet. The Morlet wavelet has the general

structure

ψ(t) = e^−t2

2σ e^−jct .

(2.79)

Here, σ is a scaling parameter used to determine the width of the wavelet, and c is a

modulation parameter for determining the frequency of the oscillation described by

17 Jean Morlet (1931–2007): French geophysicist and one of the founders of the wavelet transform.