Spectral Transformations - Biomedical Signal Analysis: Contemporary Methods and Applications - page 53

φ (t)

ψ

(t)

φ (2t)

1

0 1 1/2

0 1/2 0

(a) (b) (c)

Ψ(ω)

Φ(ω)

ω

ω

(d) (e)

Figure 2.9
(a) and (b) Haar basis functions; (c) Haar wavelet; (d) Fourier transform of the
scaling function; (e) Haar wavelet function.

Haar mother wavelet function is given by

⎧
⎨

1 , 0 ≤ t< 1 / 2

ψ ( t )= ϕ (2 t ) − ϕ (2 t − 1) = − 1 , 1 / 2 ≤ t< 1 (2.63)

⎩

0 , else

The Haar wavelet function is an up-down square wave, and can be
described by a half-box minus a shifted half-box. We also can see that the
wavelet function can be computed directly from the scaling functions. In
the Fourier domain it describes a bandpass, as can be easily seen from
figure 2.9e. This is given by

sin 2 ω/ 4
ω/ 4

Ψ( ω )= je −j 2

.

(2.64)