Operation of the Discrete Wavelet Transform: basic overview with examples

Transcription

1 Operation o the Dicrete Wavelet Tranorm: baic overview with example Surname, name Antonino Daviu, Joe Alono (joanda@die.upv.e) Department Centre Ecuela Técnica Superior de Ingeniero Indutriale Departamento de Ingeniería Eléctrica Univeritat Politècnica de València

2 Summary The aim o thi paper i to preent a didactic overview o the operation o the Dicrete Wavelet Tranorm (DWT). Unlike the Fat Fourier Tranorm (FFT), extenively exploited to analyze tationary quantitie, the DWT i a ignal proceing tool which i epecially uited or the analyi o non-tationary ignal. Thee ignal are preent in a countle number o everyday application and procee. Hence, the ue o uitable tool or their proceing, uch a the DWT, i a topic o increaing relevance. In thi particular work, the bae o the DWT are reviewed. The operation o the tool i briely decribed under an engineering perpective, without deepening in complex mathematical detail, which are eaily available in well-known reerence. Some illutrative example o the practical operation o the tool are included in the inal part o the document. 2 Introduction Stationary ignal are ignal whoe pectral characteritic do not change with time. Nonethele, mot ignal in the nature do not have thi characteritic. Intead, thee ignal have a time-varying pectral content. A very obviou one i the human peech, in which requencie change a we peak. In act, the tranmitted meage relie upon the requency change and on the time equencing o the requencie [5]. Thee ignal are known a non-tationary ignal, due to the act that their baic pectral eature do not remain contant but change with time. In order to analyze non-tationary ignal, the FFT i not longer uitable. Note that in the cae, ince the requencie change with time, the analyi tool mut be capable o extracting the time evolution o the requency component preent in the analyzed ignal. FFT analyi doe not enable thi, ince it implie a lo o time inormation. In order word, FFT only extract the requency content o the analyzed ignal, but it doe not inorm on when each requency occur. A an example, conider the ollowing unction, which ha been built a an addition o our inuoidal ignal with imilar amplitude and with requencie 5, 5, 3 and 5 Hz: ( t) = co(2 π 5 t) + co(2 π 5 t) + co(2 π 3 t) + co(2 π 5 t) Equation. Function baed on the addition o our inuoidal ignal. Figure (a) how the repreentation o the unction (t), given by Equation. It can be oberved that all requencie are preent at every time. Hence, the ignal ha a tationary nature, conidered a the invariability o it baic pectral eature with time. Figure (b) how it FFT analyi: it reveal our requency peak at the aorementioned requencie. On the other hand, conider now the unction depicted in Figure 2(a); in that unction, the ame requency component appear but, in thi cae, they occur at dierent time intant. The ignal in thi cae i no longer tationary. Figure 2(b)

3 how the FFT analyi o thi ignal: it alo reveal the preence o our peak at the correponding requencie. Amplitud Tiempo (a) (b) Figure. (a) Repreentation o (t), (b)fft analyi o (t) Frecuencia Frequency (Hz) (Hz) Frequency(Hz) Amplitud Tiempo () Frecuencia (Hz) Frequency(Hz) (a) (b) Figure 2. (a) Dierent requencie at dierent time, (b)fft analyi Thee example illutrate one o the drawback o the FFT analyi: ince the tranorm implie a lo o time inormation, imply extracting the requency component, two rather dierent ignal (uch a thoe plotted in Figure (a) and Figure 2(a)) can have imilar repreentation in term o their FFT pectra. In other word, the FFT only extract the requency content o a ignal, which may be enough or tationary ignal, but not or non-tationary, in which the knowledge o the time at which each requency occur i undamental or the comprehenion o the ignal tructure. In thi context i where novel time-requency decompoition (TFD) tool, uited or the analyi o non-tationary ignal rie. Thee tool enable to extract, not only the requency content preent in a certain ignal, but alo the time inormation (i.e. when the requencie occur). The TFD tool lead to a time-requency repreentation o the analyzed ignal.

4 3 Objetive The preent work ha three main goal: To explain the operation o a particular TFD tool, the Dicrete Wavelet Tranorm, under an engineering perpective, without getting into complex mathematical detail that are available in well-known reerence. To provide everal illutrative example acilitating the comprehenion o the DWT operation. To review the advantage and drawback o the tranorm, in comparion with claical FFT analyi and with other TFD tool. The preent work can be epecially ueul or tudent or reearcher involved in the tudy o application implying the analyi o non-tationary ignal. In thi context, the DWT ha recently revealed itel a a very powerul tool, providing important advantage veru other technique. 4 Development 4. Foundation When the Dicrete Wavelet Tranorm (DWT) i applied to a certain ampled unction (t), thi unction i decompoed a the addition o a et o ignal, named wavelet ignal: an approximation ignal at a certain decompoition level n (a n ) plu n detail ignal (d j with j varying rom to n). The mathematical expreion characterizing thi n j proce i given by Equation 2, where α, β are the caling and wavelet coeicient, φ n (t), ψ j (t) are the caling unction at level n and wavelet unction at level j, repectively, and n i the decompoition level [-3]. i i n n n j j i ϕ i (t) + βi ψi (t) = an + dn j= i = ( t) α d i Equation 2. Decompoition o the ignal (t) in term o wavelet ignal. Each one o the wavelet ignal (approximation and detail) ha an aociated requency band, the limit o which are well-etablihed, once the ampling rate () o the original analyzed ignal i known, in accordance with an algorithm enunciated by S. Mallat (Subband Coding Algorithm) [2]. The expreion ued to calculate the limit o the requency band aociated with each wavelet ignal, according to the Mallat algorithm, are peciied in Figure 3 [4]. It i oberved how the limit o the requency band or each wavelet ignal depend on the ampling rate () a well a on the level o the correponding wavelet ignal (j). A an example, i the ampling rate ued or capturing (t) i = ample/econd, and we perorm the DWT decompoition in n=8 level, the requency band aociated with each wavelet ignal are thoe peciied in Table I.

5 a n [, 2 -(n+) ] Hz (t) DWT d n [2 -(n+), 2 -n ] Hz d j [2 -(j+), 2 -j ] Hz d [2-2, 2 - ] Hz Figure 3. DWT decompoition in wavelet ignal and aociated requency band Wavelet ignal Frequency band a8 [-9 5] Hz d8 [9 5-39] Hz d7 [39-78 ] Hz d6 [ ] Hz d5 [ ] Hz d4 [ ] Hz d3 [625-25] Hz d2 [25-25] Hz d [25-5] Hz Table. Frequency band aociated with wavelet ignal or = khz and n=8 The intuitive idea underlying the application o the DWT relie on the ollowing act: each one o the wavelet ignal act a a ilter, extracting the temporal evolution o the component o the original ignal contained within the requency band aociated with that wavelet ignal. For intance, in the previou example, the wavelet ignal d7 (detail ignal 7) will relect the time evolution o every harmonic component o the original ignal when it requency all in the band [39-78 ] Hz. For intance, i the ignal i a pure 5 Hz inuoidal waveorm, the whole ignal evolution would be relected in that ignal d7. In concluion, the DWT perorm a dyadic band-pa iltering proce in requency band whoe limit depend on and on n. Thi iltering i illutrated in Figure 4. an d3 d2 d 2 n Figure 4. Dyadic iltering proce carried out by the DWT

6 4.2 Example In thi ection, everal didactic example o the operation o the DWT are explained. They are ueul to undertand how the tranorm work in a very imple way. In all three example, the DWT decompoition i carried out in n=9 level and DB-44 i ued a mother wavelet or the analye. The correponding requency band aociated with each wavelet ignal are peciied beide each igure Example : DWT analyi o a pure inuoidal ignal Figure 5 how the DWT decompoition or the cae o a 5 Hz pure inuoidal ignal (ignal, plotted at the top o the igure). It i oberved how, in accordance with the iltering proce carried out by the tranorm, the whole ignal i iltered into the detail ignal d7. Thi i due to the act that thi ignal relect the evolution o every component evolving within the range [39-78,]Hz. Since there i a ingle 5 Hz component in the original ignal, d7 exactly relect the evolution o the whole component and, hence, o the ignal. The ret o wavelet ignal are approximately zero, ince no other requency component exit in the original ignal. - a 9 - d 9 - d 8 - d 7 - d 6 - d 5 - d 4 - d 3 - d 2 - d - 5 Hz -9.7 Hz Hz Hz Hz Hz Hz Hz Hz Hz 25-5 Hz Figure 5. DWT analyi o a 5 Hz pure inuoidal ignal Example 2: Superpoition o inuoidal ignal Figure 6 how the DWT analyi o a ignal (plotted at the top o that igure) which ha been built by adding our inuoidal ignal with repective requencie 5 Hz, 5 Hz, 3 Hz and 5 Hz. The reult i a tationary ignal in which all our requencie are preent at every time. The iltering nature o the DWT enable to extract each requency component in a eparate wavelet ignal, in agreement with the value o their repective band limit. A it i oberved, the 5Hz component i iltered in a9, the 5 Hz component in d9, the 3 Hz component in d8 and the 5 Hz component in d7, remaining almot zero the

7 ret o ignal, ince no other component exit within their band. Thi example illutrate the iltering proce carried out by the tranorm and it ability to eparate the dierent component o the ignal, provided that they all in dierent requency band covered by the wavelet ignal. 4-4 a 9 - d 9 - d 8 - d 7 - d 6 - d 5 - d 4 - d 3 - d 2 - d - 5 Hz + 5 Hz + 3 Hz + 5 Hz -9.7 Hz Hz Hz Hz Hz Hz Hz Hz Hz 25-5 Hz Figure 6. DWT analyi o a ignal baed on the uperpoition o our inuoidal ignal with requencie 5 Hz, 5 Hz, 3 Hz and 5 Hz Example 3: Concatenation o inuoidal ignal Figure 7 repreent the DWT analyi o a ignal (plotted at the top o the igure) which ha been built by concatenating our inuoidal ignal with repective requencie 5 Hz, 5 Hz, 3 Hz and 5 Hz. The reult i a nontationary ignal, in which each requency component i preent only during it correponding time interval. The application o the DWT lead to ilter each component in the wavelet ignal covering the requency band in which it i included. Hence, the 5Hz component i iltered in a9, the 5 Hz component in d9, the 3 Hz component in d8 and the 5 Hz component in d7, remaining almot zero the ret o ignal ince no component exit within their band. Moreover, the tranorm indicate when each component tart and end in the analyzed ignal; or intance, a9 how how the 5 Hz component i preent during the initial,25 econd, d9 how that the 5 Hz component i preent between,25 and,5, d8 reveal that the 3 Hz component occur between,5 and,75 econd and, inally, d7 how that the 5 Hz component i preent between,75 and econd. Thi example illutrate a clear advantage o the DWT veru the claical FFT approach. Wherea with the FFT, the time inormation wa lot and two rather dierent ignal (uch a thoe analyzed in Example 2 and 3) could be repreented by imilar FFT pectra (ee Figure and 2), the DWT preerve the time inormation, enabling to identiy not only which requencie are preent

8 but alo when they occur. Thereore, DWT lead to a three-dimenional repreentation o the analyzed ignal: requency (becaue each wavelet ignal cover a requency band), time (ince each wavelet ignal i repreented veru time) and amplitude (the amplitude o the wavelet ignal inorm on the correponding amplitude o it iltered component in the analyzed ignal). Thi i why DWT i known a a time-requency decompoition tool. - a 9 - d 9 - d 8 - d 7 - d 6 - d 5 - d 4 - d 3 - d 2 - d - 5 Hz 5 Hz 3 Hz 5 Hz -9.7 Hz Hz Hz Hz Hz Hz Hz Hz Hz 25-5 Hz Figure 7. DWT analyi o a ignal baed on the concatenation o our inuoidal ignal with requencie 5 Hz, 5 Hz, 3 Hz and 5 Hz. 5 Cloing Thi work ha preented a baic overview o the operation o the Dicrete Wavelet Tranorm (DWT). Thi i a time-requency decompoition tool which ha been ued with ucce or the analyi o non-tationary ignal, overcoming ome drawback o the FFT when analyzing uch ignal, uch a the lo o time inormation. The DWT enable a band pa iltering o the analyzed ignal in well-etablihed requency band. Moreover, it preerve the time inormation, ince each wavelet ignal in repreented veru time. The intention o the work ha been to introduce the operation o the tranorm under a imple engineering perpective, without deepening in it mathematical background, which i eaily acceible in well-known textbook. In thi regard, the work emphaize the iltering proce carried out by the tranorm, detailing the expreion to calculate the limit o the band aociated with the dierent wavelet ignal. Moreover, ome example illutrating the operation o the tranorm are included: the analyi o a pure inuoidal ignal, the analyi o a ignal baed on the uperpoition o inuoidal ignal and the analyi o a ignal baed on the concatenation o inuoidal ignal. All three didactic example are ueul to how how the wavelet ignal operate.

9 A a concluion o the idea expoed in thi work we can ummarize ome o the the advantage o the DWT in the ollowing point: - Simplicity - General availability o the DWT algorithm in conventional otware package. - Eay interpretation o the reult - Low computational burden With regard to it drawback we can remark, among other, the ollowing one: - Lower lexibility (limit o the band are ixed, once the ampling rate i known) - Reduced requency reolution or the high requencie - Poible diicult dicrimination o component when they all within the ame band. 6 Reerence 6. Textbook: [] Burru, C.S.; Gopinath, R.A.; Guo, H.: Introduction to Wavelet and Wavelet Tranorm. A Primer. Prentice Hall, 997. [2] Mallat, S.: A Wavelet Tour o Signal Proceing, Third Edition. Academic Pre, dec. 28. [3] Chui, C.K.: "Wavelet: A Mathematical Tool or Signal Analyi", SIAM, Journal paper: [4] Antonino-Daviu, J.; Riera-Guap, M.; Roger-Folch, J.; Molina, M.P.: Validation o a New Method or the Diagnoi o Rotor bar Failure via Wavelet Tranormation in Indutrial Induction Machine, IEEE Tranaction on Indutry Application, Vol. 42, No. 4, July/Augut 26, pp Electronic ource reerence: [5] Sala Mayato, R.F.; Trujillo González, R. (24). Análii de eñale Tiempo- Frecuencia. Curo organizado por el Departamento de Fíica Fundamental. Diponible en: