1. Cascade and Parallel Realizations for IIR filters
EMU-E&E-ENG

13. Examples:
Assume we have a discrete-time system with the following difference equation
𝑦 𝑛 =−0.1𝑦 𝑛−1 +0.2𝑦 𝑛−2 +3𝑥 𝑛 +3.6𝑥 𝑛−1 +0.6𝑥 𝑛−2
Represent this filter using the cascade and parallel form realizations.
𝑌 𝑧 +0.1 𝑧 −1 𝑌 𝑧 −0.2 𝑧 −2 𝑌 𝑧 =3X z +3.6 𝑧 −1 𝑋 𝑧 +0.6 𝑧 −2 𝑋 𝑧
𝐻 𝑧 = 𝑌(𝑧) 𝑋(𝑧) = 𝑧 − 𝑧 − 𝑧 −1 −0.2 𝑧 −2
CASCADE FORM:
𝐻 𝑧 = 𝑌(𝑧) 𝑋(𝑧) = 𝑧 − 𝑧 − 𝑧 −1 −0.2 𝑧 −2 = 𝑧 − 𝑧 − 𝑧 −1 1−0.4 𝑧 −1

14. 𝐻 1 𝑧 = 3+0.6 𝑧 −1 1+0.5 𝑧 −1 and 𝐻 2 𝑧 = 1+ 𝑧 −1 1−0.4 𝑧 −1
𝐻 𝑧 = 𝑌(𝑧) 𝑋(𝑧) = 𝑧 − 𝑧 − 𝑧 −1 −0.2 𝑧 −2 = 𝑧 − 𝑧 − 𝑧 −1 1−0.4 𝑧 −1
𝐻 1 𝑧 = 𝑧 − 𝑧 − and 𝐻 2 𝑧 = 1+ 𝑧 −1 1−0.4 𝑧 −1
x[n]
z--1 3
0.6
-0.5
z--1 1
0.4
y[n]

15. PARALLEL FORM
𝐻 𝑧 = 𝑌(𝑧) 𝑋(𝑧) = 𝑧 − 𝑧 − 𝑧 −1 −0.2 𝑧 −2
Perform a long division and partial fractions to get H(z) = C + H1(z) + H2(z)
𝐻 𝑧 =− 𝑧 − 𝑧 −1 −0.2 𝑧 −2
−3+ 𝐴 1−0.4 𝑧 −1 + 𝐵 𝑧 −1 -3
−0.2 𝑧 − 𝑧 −1 +1
0.6 𝑧 − 𝑧 −1 +3
0.6 𝑧 −2 −0.3 𝑧 −1 −3
0+3.9 𝑧 −1 +6

16. 3.9 𝑧 −1 +6=A 𝑧 −1 +B 1−0.4 𝑧 −1
0.5A B = (1)
A+B = (2)
From (1) and (2) we have A = 7 and B = -1
𝐻 𝑧 =−3+ 7 1−0.4 𝑧 −1 + − 𝑧 −1

17. z-1 1
0.4 -3 7
0.5 -1
y[n]
x[n]

18. Ex: Determine the cascade realization for the filter below:
𝐻 𝑧 = 𝑧 − 𝑧 − 𝑧 −3 1− 𝑧 − 𝑧 −2 − 𝑧 −3
𝐻 𝑧 = (1+3+3 𝑧 −2 + 𝑧 −3 ) 1− 𝑧 − 𝑧 −2 − 𝑧 −3
b=[ ]
a=[ ]
SOS = tf2sos(b,a,’up’)
If flag is ‘up’ then first row will contain poles closest to the origin
and
Last row will contain poles closest to the unit circle

19. SOS
Numerator
Denominator a0 a1 a2 b0 b1 b2
First Row 1
Second Row 2
0.3355
H z = 1+ 𝑧 −1 1− 𝑧 −1 ∗ 1+ 2𝑧 −1 + 𝑧 −2 1− 𝑧 − 𝑧 −2