Digital Signal Processing Reference
In-Depth Information
Since tan(=4) = 1, we will replace the 1 in the numerator. Also in the denominator,
tan(
1
) = 1tan(
1
) = tan(=4)tan(
1
). This gives us:
tan(=4)tan(
1
)
1 + tan(=4)tan(
1
)
1
2
= tan
:
Now comes the point of all this manipulation. There is a trigonometric identity
2
that says:
tan() =
tan()tan()
1 + tan()tan()
:
It is no coincidence that the right side of the above identity looks a lot like part of
our expression for
2
. Once we use this identity, it simplies our equation:
1
(tan(=4
1
))
2
= tan
2
= =4
1
:
This says what we expect it to; that the Haar transform is a rotation by =4 radians,
or 45 degrees.
9.4
Daubechies Four-Coecient Wavelet
Let's also consider the case of four coecients in a conjugate quadrature lter.
Figure 9.6 shows a structure similar to Figure 9.5, except that its FIR lters have
four taps (use four coecients).
z[n]
d, −c, b, −a
−a, b, −c, d
Input
Output
w[n]
x[n]
a, b, c, d
d, c, b, a
y[n]
Figure 9.6: A two-channel lter bank with four coecients.
The expressions for w[n] and z[n] can be found in a similar manner as in section
9.1:
w[n] = ax[n] + bx[n1] + cx[n2] + dx[n3]
z[n] = dx[n]cx[n1] + bx[n2]ax[n3]:
2
Thanks to Mr. Srikanth Tirupathi for pointing this out.
