小波包频段划分

1、MATLA小波包的频率分段我总觉得不是按照自然顺序排的，通过测得响应频谱 所得的结果能明显看出来。可苦于找不到相关的说明，哎，踏破铁鞋无觅处得来 全不费工夫，原来帮助里早都说了，只怪自己急功近利没好好学习，骑驴找驴， 反倒浪费不少时间。Buildi ng Wavelet PacketsThe computati on scheme for wavelet packets gen erati on is easy whe n using an orthogonal wavelet. We start with the two filters of length 2 N, where h( n)

2、and g(n), corresp onding to the wavelet.Now by in duct ion let us defi ne the follow ing seque nee of functions:(Wn(x), n = 0, 1,2,)by2JV-1叫芒)=2 y h(k)Wn(2x-k)i - 02N-1W加+ 1（尤）二血 X k）Wn（2x-kwhere W0(x) = $ (x) is the scaling function andWx) = (x) is the wavelet function.For example for the Haar wave

3、let we haveN =A(0) = A(l)=（0）=理The equati ons become用2斤(工)=Wn(2x)+ Wn(2x- 1)卬2斤 + (疋)=W科(2兀)-W”(2h 1)W0(x) = $ (x) is the Haar scaling function andW1(x) = (x) is the Haar wavelet, bothsupported in 0, 1. Then we can obta inW2n by addi ng two 1 /2-scaled versions ofWn withdist inct supports 0,1/2 and

4、1 /2,1 and obta in W2n+1 by subtract ing the same versions ofWn.For n = 0 to 7, we have the W-functions shown in Figure 6-370.50.5WOW1V/2W3W4W5W5W7Figure 6-37: Haar Wavelet PacketsThis can be obta ined using the follow ing comma nd:wfun ,xgrid = wpfu n( db1,7,5);which returns in wfun the approximate

5、 values ofWn for n = 0 to 7, computed on a 1/2 gridof the support xgrid .Starting from more regular original wavelets and using a similar construction, we obtainsmoothed versions of this system of W-functions, all with support in the in terval 0, 2N-1.Figure 6-38 presents the system ofW-functions fo

6、r the originaldb2 wavelet.Figure 6-38: db 2 Wavelet PacketsW7Wavelet Packet AtomsStarti ng from the functionsand followi ng the same line leadi ng toorthogo nal wavelets, we con sider the three-i ndexed family of an alyz ing fun cti ons (the waveforms):As in the wavelet framework, k can be in terpre

7、ted as a time-localizatio n parameter andjas a scale parameter. So what is the in terpretati on of n?The basic idea of the wavelet packets is that for fixed values ofj and k, Wj,n,k an alyzes the2Jand atn.and Figure 6-387 i-* 尺 ad the scalefluctuati ons of the sig nal roughly around the positi on va

8、rious freque ncies for the differe nt admissible values of the last parameterIn fact, exam ining carefully the wavelet packets displayed inFigure 6-37the n aturally orderedWn for n = 0, 1, ., 7, does not match exactly the order defi ned by thenumber of oscillations. More precisely, counting the numb

9、er of zero crossings(up-cross ings and dow n-cross in gs) for thedb1 wavelet packets, we have the follow ing.Natural order n01234567Number of zero crossi ngs for db 1 Wn 23549867So, to restore the property that the main freque ncy in creases monotoni cally with the order, it is convenient to defi ne

10、 thefreque ncy order obta ined from the n atural one recursively.Natural order n01234567Freque ncy order r(n) 013 2 6 7 5 4As can be see n in the previous figures,Wr(n)(x) oscillates approximately n times.To an alyze a sig nal (the chirp of Example 2 for in sta nee), it is better to plot the wavelet

11、 packet coefficients following the frequency order (on the right ofFigure 6-39 ) from the lowfreque ncies at the bottom to the high freque ncies at the top, rather tha n n aturally ordered coefficients (on the left ofFigure 6-39 ).Clnlli ir i 1 I m !lln:n :r I s l r H i r!r i n il NmlisiiSiy r 曰IIkriijlhi 51 ：n： nir (Zulnrk r/iic itn ik/rxFigure 6-39: Natural and Freque ncy Ordered Wavelet Packets Coefficie ntsColorsd CoTc ent ftr TsrmlnEil iNoda*When plotting the coefficients, the various o