0.12 Appendix a  (Page 2/2)

If we look at [link] as being the inverse Fourier transform of [link] and sample $a\left(t\right)$ at $t=k$ , we have

but this integral is the form of an inverse discrete-time Fourier transform (DTFT) which means

If the integer translates of $\phi \left(t\right)$ are orthogonal, $a\left(k\right)=\delta \left(k\right)$ and we have our result

If the scaling function is not normalized

which is similar to Parseval's theorem relating the energy in the frequency domain to the energy in the time domain.

Proof 6 Equation [link] states a very interesting property of the frequency response of an FIR filter with the scaling coefficients as filtercoefficients. This result can be derived in the frequency or time domain. We will show the frequency domain argument. The scaling equation [link] becomes [link] in the frequency domain. Taking the squared magnitude of both sides of a scaled version of

gives

Add $k\pi$ to $\omega$ and sum over $k$ to give for the left side of [link]

which is unity from [link] . Summing the right side of [link] gives

Break this sum into a sum of the even and odd indexed terms.

which after using [link] gives

which gives [link] . This requires both the scaling and orthogonal relations but no specific normalization of $\phi \left(t\right)$ . If viewed as an FIR filter, $h\left(n\right)$ is called a quadrature mirror filter (QMF) because of the symmetry of its frequencyresponse about $\pi$ .

Proof 10 The multiresolution assumptions in [link] require the scaling function and wavelet satisfy [link] and [link]

and orthonormality requires

and

for all $k\in \mathbf{Z}$ . Substituting [link] into [link] gives

Rearranging and making a change of variables gives

Using [link] gives

for all $k\in \mathbf{Z}$ . Summing over $\ell$ gives

Separating [link] into even and odd indices gives

which must be true for all integer $k$ . Defining $h_e\left(n\right)=h\left(2n\right)$ , $h_o\left(n\right)=h(2n+1)$ and $\tilde{g}\left(n\right)=g(-n)$ for any sequence $g$ , this becomes

From the orthonormality of the translates of $\phi$ and $\psi$ one can similarly obtain the following:

This can be compactly represented as

Assuming the sequences are finite length [link] can be used to show that

where ${\delta }_k\left(n\right)=\delta (n-k)$ . Indeed, taking the Z-transform of [link] we get using the notation of  Chapter: Filter Banks and Transmultiplexers $H_p\left(z\right)H_p^T\left(z^{-1}\right)=I$ . Because, the filters are FIR $H_p\left(z\right)$ is a (Laurent) polynomial matrix with a polynomial matrix inverse. Therefore the determinant of $H_p\left(z\right)$ is of the form $\pm z^k$ for some integer $k$ . This is equivalent to [link] . Now, convolving both sides of [link] by ${\tilde{h}}_e$ we get

Similarly by convolving both sides of [link] by ${\tilde{h}}_o$ we get

Combining [link] and [link] gives the result

Proof 11 We show the integral of the wavelet is zero by integrating both sides of ( [link] b) gives

But the integral on the right hand side is $A_0$ , usually normalized to one and from [link] or [link] and [link] we know that

and, therefore, from [link] , the integral of the wavelet is zero.

The fact that multiplying in the time domain by ${(-1)}^n$ is equivalent to shifting in the frequency domain by $\pi$ gives $H_1\left(\omega \right)=H(\omega +\pi )$ .