Wavelet basis functions in biomedical signal processing

10 J. Rafiee et al. / Expert Systems with Applications xxx (2010) xxx–xxx
Table 1A
Studied wavelet families in this research (Rafiee et al., 2010).
No. Family (short form) Order
1 Haar (db1) db 1
2–45 Daubechies(db) db 2–db 45
46–50 Coiflet (coif) coif 1–coif 5
51 Morlet (Morl) morl
52–147 Complex Morlet (cmor Fb-Fc) a Included Table 1B
148 Discrete Meyer (dmey) dmey
149 Meyer (meyr) meyr
150 Mexican Hat (mexh) mexh
151–200 Shannon (Shan Fb-Fc) a Included Table 1B
201–260 Frequency B-Spline (fbsp M-Fb- Fc) a Included Table 1B
261–267 Gaussian (gaus) gaus 1–gaus 7
268–275 Complex Gaussian (cgau) cgau 1–cgau 8
276–290 Biorthogonal (bior Nr.Nd) b Included Table 1B
291–305 Reverse Biorthogonal (rbio Nr.Nd) b Included Table 1B
306–324 Symlet (sym) sym 2–sym 20
a
Fb is a bandwidth parameter, Fc is a wavelet center frequency, and M is an
integer order parameter.
b Nr and Nd are orders: r for reconstruction/d for decomposition.
5. Conclusions
In summary, db44 appears to be the most similar function to
EMG, EEG, and VPA signals among 324 mother wavelet functions.
Symmetric mother wavelets have been shown more proper results
with biosignals in prior research. Daubechies are orthogonal compact
support functions and are not symmetric. However, db44 possesses
a near-symmetric attribute (see Fig. 9) that well-matched
the biosignals. Most biosignals possess sharp spikes that could be
appropriately analyzed using Daubechies’ wavelets to reflect sharp,
asymmetric spikes seen in these signals. These features of db44
suggest that it would be effective for many biosignal processing
methods based on the resemblance of the signal and mother wavelet
function.
The similarity between signal and mother wavelet function is
not always proper for signal processing based on wavelet transform
and it is just appropriate for those wavelet-based processing
methods based on the resemblance between signals and mother
functions.
Table 1B
Studied wavelet families in detail (Rafiee et al., 2010).
No Wave No Wave No Wave No Wave No Wave
52 1–1.5 100 3–1.1 148 dmey 196 2–0.6 244 2–0.2–0.4
53 1–1 101 3–1.2 149 meyr 197 2–0.7 245 2–0.2–0.5
54 1–0.5 102 3–1.3 150 mexh 198 2–0.8 246 2–0.2–0.6
55 1–0.3 103 3–1.4 151 0.1–0.1 199 2–0.9 247 2–0.2–0.7
56 1–0.2 104 3–1.5 152 0.1–0.2 200 1–1 248 2–0.2–0.8
57 1–0.1 105 3–1.6 153 0.1–0.3 201 1–0.1–0.1 249 2–0.2–0.9
58 1–0.05 106 3–1.8 154 0.1–0.4 202 1–0.1–0.2 250 2–0.2–1
59 1–0.02 107 3–1.9 155 0.1–0.5 203 1–0.1–0.3 251 3–0.2–0.1
60 1–0.01 108 3–2 156 0.1–0.6 204 1–0.1–0.4 252 3–0.2–0.2
61 2–0.1 109 3–2.1 157 0.1–0.7 205 1–0.1–0.5 253 3–0.2–0.3
62 2–0.2 110 3–2.2 158 0.1–0.8 206 1–0.1–0.6 254 3–0.2–0.4
63 2–0.3 111 3–2.3 159 0.1–0.9 207 1–0.1–0.7 255 3–0.2–0.5
64 2–0.4 112 3–2.4 160 0.1–1 208 1–0.1–0.8 256 3–0.2–0.6
65 2–0.5 113 3–2.5 161 0.2–0.1 209 1–0.1–0.9 257 3–0.2–0.7
66 2–0.6 114 3–2.6 162 0.2–0.2 210 1–0.1–1 258 3–0.2–0.8
67 2–0.7 115 3–2.7 163 0.2–0.3 211 2–0.1–0.1 259 3–0.2–0.9
68 2–0.8 116 3–2.8 164 0.2–0.4 212 2–0.1–0.2 260 3–0.2–1
69 2–0.9 117 3–2.9 165 0.2–0.5 213 2–0.1–0.3 276 1.1
70 2–1 118 3–3 166 0.2–0.6 214 2–0.1–0.4 277 1.3
71 2–1.1 119 4–0.1 167 0.2–0.7 215 2–0.1–0.5 278 1.5
72 2–1.2 120 4–0.2 168 0.2–0.8 216 2–0.1–0.6 279 2.2
73 2–1.3 121 4–0.3 169 0.2–0.9 217 2–0.1–0.7 280 2.4
74 2–1.4 122 4–0.4 170 0.2–1 218 2–0.1–0.8 281 2.6
75 2–1.5 123 4–0.5 171 0.5–0.1 219 2–0.1–0.9 282 2.8
76 2–1.6 124 4–0.6 172 0.5–0.2 220 2–0.1–1 283 3.1
77 2–1.8 125 4–0.7 173 0.5–0.3 221 3–0.1–0.1 284 3.3
78 2–1.9 126 4–0.8 174 0.5–0.4 222 3–0.1–0.2 285 3.5
79 2–2 127 4–0.9 175 0.5–0.5 223 3–0.1–0.3 286 3.7
80 2–2.1 128 4–1 176 0.5–0.6 224 3–0.1–0.4 287 3.9
81 2–2.2 129 4–1.1 177 0.5–0.7 225 3–0.1–0.5 288 4.4
82 2–2.3 130 4–1.2 178 0.5–0.8 226 3–0.1–0.6 289 5.5
83 2–2.4 131 4–1.3 179 0.5–0.9 227 3–0.1–0.7 290 6.8
84 2–2.5 132 4–1.4 180 0.5–1 228 3–0.1–0.8 291 1.1
85 2–2.6 133 4–1.5 181 1–0.1 229 3–0.1–0.9 292 1.3
86 2–2.7 134 4–1.6 182 1–0.2 230 3–0.1–1 293 1.5
87 2–2.8 135 4–1.8 183 1–0.3 231 1–0.2–0.1 294 2.2
88 2–2.9 136 4–1.9 184 1–0.4 232 1–0.2–0.2 295 2.4
89 2–3 137 4–2 185 1–0.5 233 1–0.2–0.3 296 2.6
90 3–0.1 138 4–2.1 186 1–0.6 234 1–0.2–0.4 297 2.8
91 3–0.2 139 4–2.2 187 1–0.7 235 1–0.2–0.5 298 3.1
92 3–0.3 140 4–2.3 188 1–0.8 236 1–0.2–0.6 299 3.3
93 3–0.4 141 4–2.4 189 1–0.9 237 1–0.2–0.7 300 3.5
94 3–0.5 142 4–2.5 190 1–1 238 1–0.2–0.8 301 3.7
95 3–0.6 143 4–2.6 191 2–0.1 239 1–0.2–0.9 302 3.9
96 3–0.7 144 4–2.7 192 2–0.2 240 1–0.2–1 303 4.4
97 3–0.8 145 4–2.8 193 2–0.3 241 2–0.2–0.1 304 5.5
98 3–0.9 146 4–2.9 194 2–0.4 242 2–0.2–0.2 305 6.8
99 3–1 147 4–3 195 2–0.5 243 2–0.2–0.3
Please cite this article in press as: Rafiee, J., et al. Wavelet basis functions in biomedical signal processing. Expert Systems with Applications (2010),
doi:10.1016/j.eswa.2010.11.050