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Problems 295
1. Direct form
2. Linear-phase form
3. Cascade form
4. Frequency sampling form
P6.18 A linear time-invariant system is given by the system function
H(z) =2+3z −1 +5z −2 − 3z −3 +4z −5 +8z −7 − 7z −8 +4z −9
Determine and draw the block diagrams of the following structures.
1. Direct form
2. Cascade form
3. Lattice form
4. Frequency sampling form
P6.19 Using the conjugate symmetry property of the DFT
H (k) =
{
H (0) , k =0
H ∗ (M − k) , k =1,...,M − 1
and the conjugate symmetry property of the W −k
M
factor, show that (6.12) can be put in
the form (6.13) and (6.14) for real FIR filters.
P6.20 To avoid poles on the unit circle in the frequency sampling structure, one samples H(z) at
z k = re j2πk/M ,k=0,...,M − 1 where r ≈ 1(but < 1), as discussed in Section 6.3.
1. Using
H ( re j2πk/M) ≈ H (k) ,
show that the frequency-sampling structure is given by
H (z) = 1 − (rz)−M
M
{ L
∑
k=1
}
2 |H (k)| H k (z)+ H (0) H (M/2)
+
1 − rz−1 1+rz −1
where
H k (z) = cos [̸ H (k)] − rz −1 cos [̸ ]
H (k) − 2πk
M
1 − 2rz −1 cos ( ) , k =1,...,L
2πk
M + r2 z −2
and M is even.
2. Modify the MATLAB function dir2fs (which was developed in Section 6.3) to
implement this frequency-sampling form. The format of this function should be
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