To generate orthogonal wavelet bases, we will need the set B φ t n n to be an orthogonal set. This set consists of the scaling function φ t and its integer-translations. The orthogonality condition is written as t φ t φ t n δ n But not just any φ t satisfies this orthogonality condition. How can we be sure that a set of coefficients h n will generate a scaling function φ t that satisfies ?

Projections Define the set B to be B φ t n n B is called an orthonormal set if t φ n t φ m t δ n m where ≔ φ n t φ t n . Define the set 𝒱 to be 𝒱 Span φ t n n The set of functions of the form n a n φ t n Given a signal x t , how can we find the signal x ^ t in 𝒱 that is closest to x t ? If B is an orthonormal set, then it is simple to find the signal x ^ t in 𝒱 that minimizes the square error ε t x t x ^ t 2 where x ^ t n a n φ n t We just need to find the coefficients a n , which can be done by differentiating ε with respect to a k and setting it to zero. a n ε a n t x t k a k φ k t 2 t a n x t k a k φ k t 2 t 2 x t k a k φ k t φ n t 2 t x t φ n t 2 k a k t φ k t φ n t 2 t x t φ n t 2 k a k δ k n 2 t x t φ n t 2 a n Setting a n ε to zero gives ≔ a n t x t φ n t x t φ n t Therefore, if φ t is orthogonal to its integer-translates, then the coefficients a n that give the best square-error signal in 𝒱 are given by integrating x t with φ n t . We get x ^ t n x t φ n t φ n t This is called the projection of x t onto 𝒱.

The Normalization induced by Orthogonality It turns out that if a scaling function φ t satisfies the orthogonality condition, t φ t φ t n δ n then t φ t ± 1 To derive this normalization on φ t induced by , begin with . t φ t φ t n δ n Sum each side over n, n t φ t φ t n n δ n Then we have t φ t n φ t n 1 Recall from this equation that n φ t n t φ t (provided φ t is continuous), so that we can write t φ t α φ α 1 or t φ t α φ α 1 t φ t 2 1 or t φ t ± 1 t φ t φ t n δ n t φ t ± 1

Orthogonality conditions in h(n) If we assume that the scaling function φ t does satisfy , what conditions can we derive for h n ? Assume is true. Then use the dilation equation to expand . Substitute φ t 2 k h k φ 2 t k and φ t n 2 m h m φ 2 t n m into . t 2 k h k φ 2 t k 2 m h m φ 2 t 2 n m δ n Then we have 2 k h k m h m t φ 2 t k φ 2 t 2 n m δ n Use the change of variables α 2 t k to get k h k m h m t φ α φ α k 2 n m δ n Using , k h k m h m δ k 2 n m δ n Because δ k 2 n m is zero except when m k 2 n , we have k h k h k 2 n δ n This is the condition on h n that corresponds to condition . t φ t φ t n δ n k h k h k 2 n δ n

Constraint induced by the doubleshift orthogonality condition Let's write out explicitly when h n is of length 4.

We get h 0 2 h 1 2 h 2 2 h 3 2 1 h 0 h 2 h 1 h 3 0 k h k h k 2 n δ n h n is of even length

Autocorrelation form of the doubleshift orthogonality condition Let's define the autocorrelation of h n to be the sequence ≔ r n h n h n k h k h k n Autocorrelation sequences have several properties. r n r n R z 𝒵 r n H z H 1 z . Note that R z 𝒵 r n 𝒵 h n h n 𝒵 h n 𝒵 h n H z H 1 z R f ω DTFT r n H f ω 2 . This comes from the following equations. R f ω R ω H ω H ω H f ω H f ω H f ω 2 We can write the orthogonality condition k h k h k 2 n δ n in terms of the autocorrelation very compactly as r 2 n δ n . ⇔ k h k h k 2 n δ n r 2 n δ n In other words, if the scaling function φ t is orthogonal to its integer-translates, then the autocorrelation r n must be a halfband filter. That is, r n 0 for even n, except for n 0 , r 0 1 .

Fourier form of the doubleshift orthogonality condition The Fourier form of the orthogonality condition is based on the following property. ⇔ r 2 n δ n r n 1 n r n 2 δ n Let's take the 𝒵-transform of both sides of this equation. 𝒵 r n 1 n r n 𝒵 2 δ n Note that 𝒵 1 n r n R z Therefore, we have the equivalence ⇔ r 2 n δ n R z R z 2 Now let's write in terms of the Fourier transform. DTFT r n 1 n r n DTFT 2 δ n Note that DTFT 1 n r n R ω f Therefore, we have the equivalence ⇔ r 2 n δ n R ω f R ω f 2 summarizes the different forms of the orthogonality condition. ⇔ k h k h k 2 n δ n r 2 n δ n ⇔ R z R z 2 ⇔ R ω f R ω f 2 Using the relation R z H z H 1 z , or equivalently R ω f H ω f H ω f , we get . ⇔ k h k h k 2 n δ n H z H 1 z H z H 1 z 2 ⇔ H ω f 2 H ω f 2 2 gives the Fourier transform and 𝒵-transform forms of the orthogonality condition using the autocorrelation sequence r n h n h n . gives the Fourier transform and 𝒵-transform forms of the orthogonality condition using the scaling filter h n itself.