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Problems 289
z −1
x(n)
K
2
2 0.5
K
z −1 z −1
y(n) x(n)
0.5
−0.9 1.8
z −1 z −1 z −1
y(n)
(i)
FIGURE P6.1 Block diagrams for Problem 6.2
(ii)
P6.2 Two block diagrams are shown in Figure P6.1. Answer the following for each structure.
1. Determine the system function H(z) =Y (z)/X(z).
2. Is the structure canonical (i.e., with the least number of delays)? If not, draw a
canonical structure.
3. Determine the value of K so that H(e j 0 )=1.
P6.3 Consider the LTI system described by
y(n) =ay(n − 1) + bx(n) (6.82)
1. Draw a block diagram of this system with input node x(n) and output node y(n).
2. Now perform the following two operations on the structure drawn in part 1: (i) reverse
all arrow directions and (ii) interchange the input node with the output node. Notice
that the branch node becomes the adder node and vice versa. Redraw the block diagram
so that input node is on the left side and the output node is on the right side. This is the
transposed block diagram.
3. Determine the difference equation representation of your transposed structure in part 2,
and verify that it is the same equation as (6.82).
P6.4 Consider the LTI system given by
H(z) =
1 − 2.818z −1 +3.97z −2 − 2.8180z −3 + z −4
1 − 2.536z −1 +3.215z −2 − 2.054z −3 +0.6560z −4 (6.83)
1. Draw the normal direct form I structure block diagram.
2. Draw the transposed direct form I structure block diagram.
3. Draw the normal direct form II structure block diagram. Observe that it looks very
similar to that in part 2.
4. Draw the transposed direct form II structure block diagram. Observe that it looks very
similar to that in part 1.
P6.5 Consider the LTI system given in Problem P6.4.
1. Draw a cascade structure containing 2nd-order normal direct-form-II sections.
2. Draw a cascade structure containing 2nd-order transposed direct-form-II sections.
3. Draw a parallel structure containing 2nd-order normal direct-form-II sections.
4. Draw a parallel structure containing 2nd-order transposed direct-form-II sections.
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