Re: Describing a filter in a wavelet basis

Corrected a bad typo below...

Rune


On 23 May, 21:13, Rune Allnor <all...@tele.ntnu.no> wrote:
> On 23 May, 09:23, Palle Uldenborg <palle.uldenb...@gmail.com> wrote:
>
> > Suppose that I have used the same orthogonal (Dauberchies) wavelet
> > basis to describe a filter I(t) and a set of signales x1(t), x2(t),
> > x3(t). Now I want to calculate the convolutions (I*x1)(t), (I*x2)(t),
> > (I*x3)(t).. and express them in the same wavelet basis as the signals
> > and the filter.
>
> > Is there a smart way to do this within the wavelet formalism?
>
> Before discussing wavelets, let's see how time-domain
> convolution and the DTFT are related (view with fixed-width
> font):
>
> [All sums from -infinite to infinite. sum means "sum over k"]
>                                        k
>
> y[n] = x[n] (x) h[n] = sum x[k] h[n - k]            [1]
>                         k
>
> Y(w) = sum y[n] exp(jwn)
>         n
>
>      = sum sum x[k] h[n-k] exp(jwn)                 [2]
>         n   k
>
> set p = n-k, find n = p+k and substitute:
>
> Y(w) = sum sum x[k] h[p] exp(jw(k+p))               [3]
>         p   k
>
>      = sum sum x[k] h[p] exp(jwn)exp(jwp)           [4]
>         p   k
>
>      = sum h[p] exp(jwp) sum x[k] exp(jwp)          [5]
>         p                 k
>
>      = X(w)H(w)                                     [6]
>
> Equation [6] is the key why the DTFT is so useful when
> computing convolutions, since the numercal work of
> computing [6] often is far less than computing [1] directly.
>
> But do take a closer look into how we arrived at [6], and you
> will find one utterly essential step:
>
> Transforming from equation [3] to equation [4].
>
> Here, one uses the property of the exponential function that
>
>    F(a+b) = F(a)F(b).                  [7]
>
> I don't know of any other function than the exponential
> function that has this property. Not that this is the

NotE that this is the...

> reason why one can set up equation [5] without any (p,k)
> cross terms, which in turn is essential for the whole result.
>
> > Is it better to do inverse wavelet transform, FFT, and then wavelet
> > transform the result?
>
> If the reason is to save FLOPs, then yes. Unless your
> wavelet function satisfies [7] above.
>
> Rune