3. Structures for Discrete-Time Systems

3.1. Signal Flow Graphs (6.2)

3.2. Basic Structures for IIR Systems (6.3)

3.3. Basic Structures for FIR Systems (6.5)

3.4. Transposition Theorem (6.4)

The input-output relation of a linear time-invariant discrete-time

system can be characterized by an impulse response, a frequency

response, a system function or a linear constant-coefficient difference

equation. When the input-output relation is given, the system can be

implemented in different structures. These structures are different in

accuracy, speed, cost, and others. We discuss causal systems only.

3.1. Signal Flow Graphs

The structure of a linear time-invariant discrete-time system can be

represented by a signal flow graph. Basically, a signal flow graph is a

network of nodes and directed branches (figure 3.1).

A node carries out addition. Each output of a node equals the sum

of all its inputs. Usually the number of the inputs of a node is limited

to no more than 2.

A directed branch carries out multiplication or delay. Its output is

equal to its input multiplied by a constant (usually omitted if equal to

X1(z)

X2(z)

Y1(z)

Y2(z)

Figure 3.1. Elements of Signal Flow Graphs.

(a) Y1(z)=Y2(z)=X1(z)+X2(z)

X(z) Y(z)

(b) Y(z)=aX(z)

a

z1

X(z) Y(z)

(c) Y(z)=z1X(z)

1) or delayed by 1.

Example. Determine the system function of the signal flow graph

given below.

X(z)

z1

Y(z)

1 W1(z) W2(z)

W3(z) W4(z)

Figure 3.2. A Signal Flow Graph.

3.2. Basic Structures for IIR Systems

The basic structures for IIR systems include the direct form I, the

direct form II, the cascade form and the parallel form. These

structures, as well as other structures for IIR systems, have feedback

loops.

Consider an IIR system with system function

,

za1

zb

)z(HN

1k

k

k

M

0k

k

k

(3.1)

where ak and bk are assumed to be real numbers. Let us use different

structures to implement this system.

3.2.1. Direct Form I

From (3.1), we obtain

,)z(Xzb)z(Yza)z(YM

0k

k

k

N

1k

k

k

(3.2)

where X(z) and Y(z) are the z-transforms of the input and the output,

respectively. (3.2) can be used to construct the direct form I structure

X(z) Y(z)

z1

b0

b1

b2

bM1

bM

z1

z1

…

z1

z1

z1

a1

a2

aN1

aN

…

Figure 3.3. Direct Form I.

3.2.2. Direct Form II

From (3.1), we obtain

of the system (figure 3.3).

X(z) Y(z)b0

b1

b2

bM1

bM

…

z1

z1

z1

a1

a2

aM1

aM

…

Figure 3.4. Direct Form II.

W(z)

aN1

aN

…

z1

).z(X

za1

zb

)z(YN

1k

k

k

M

0k

k

k

(3.3)

(3.3) is also written as

),z(Wzb)z(YM

0k

k

k

(3.4)

).z(X

za1

1)z(W

N

1k

k

k

(3.5)

From (3.4) and (3.5), we obtain

,)z(Wzb)z(YM

0k

k

k

(3.6)

).z(X)z(Wza)z(WN

1k

k

k

(3.7)

(3.6)-(3.7) can be used to construct the direct form II structure of the

system (figure 3.4).

The direct form II structure can also be derived from the direct

form I structure by changing the order of the forward network and

the feedback network and combining the same nodes and directed

branches.

The direct form II structure may need less delay elements than the

direct form I structure. The direct form I structure needs M+N delay

elements, but the direct form II structure only needs max(M,N) delay

elements.

3.2.3. Cascade Form

(3.1) can be expressed as

,)z(H)z(HL

1k

k

(3.8)

where Hk(z) is a real-coefficient rational fraction of z1. (3.8) can be

used to construct a cascade form structure of the system (figure 3.5).

The system is a cascade of multiple subsystems, and each subsystem

is implemented in a direct form structure.

The numerator and the denominator of Hk(z) have an order lower

than or equal to 2 generally. In a lot of cases, we try to make Hk(z) a

ratio of two second-order polynomials.

It is easy to adjust a zero or pole in a cascade form structure.

X(z) Y(z)H1(z) H2(z) … HL(z)

Figure 3.5. Cascade Form.

3.2.4. Parallel Form

(3.1) can be expressed as

,)z(H)z(H)z(HL

1k

k0

(3.9)

where H0(z) is a real-coefficient polynomial of z1, and Hk(z) is a

real-coefficient partial fraction of z1. (3.9) can be used to build a

parallel form structure of the system (figure 3.6). The system is a

parallel of multiple subsystems, and each subsystem is implemented

in a direct form structure.

We can also pair single real poles so that the resulting Hk(z)’s are

ratios of 1st-order polynomials to 2nd-order polynomials.

It is easy to adjust a pole in a parallel form structure. Besides, in a

parallel form structure, an error cannot propagate from one section to

another.

X(z) Y(z)

Figure 3.6. Parallel Form.

…

H1(z)

HL(z)

Example. Consider a system with system function

.z125.0z75.01

zz21)z(H

21

21

(3.10)

Implement this system in the following structures:

H0(z)

H2(z)

(1) The direct form I.

(2) The direct form II.

(3) The cascade form.

(4) The parallel form.

3.3. Basic Structures for FIR Systems

The basic structures for FIR systems include the direct form and

the cascade form. These structures have no feedback loops. Most of

other structures for FIR systems have no feedback loops either.

Let h(n), 0nN1, be the impulse response of an FIR system.

Then, the system function of this system is expressed as

.z)n(h)z(H1N

0n

n

(3.11)

Assume that h(n) is a real sequence. Let us use different structures to

implement this system.

3.3.1. Direct Form

From (3.11), we obtain

.)z(Xz)n(h)z(Y1N

0n

n

(3.12)

The direct form structure of the system can be constructed according

to (3.12) (figure 3.7).

X(z)

Y(z)

z1

h(0) …

Figure 3.7. Direct Form.

h(1) h(2)

z1

h(N1)

The number of multiplications can be halved if the system belongs

to the four types of FIR generalized linear-phase systems.

X(z)

Y(z)

z1

h(0)

…

Figure 3.8. Direct Form for a Type-I

FIR Generalized Linear-Phase System.

h(1) h(2)

z1 z1

h[(N1)/2]

z1z1z1

h[(N3)/2]

For a type-I FIR generalized linear-phase system, (3.12) becomes

(3.13) .)z(Xz)z(Xz)n(h

)z(Xz]2/)1N[(h)z(Y

2/)3N(

0n

)n1N(n

2/)1N(

The corresponding structure is shown in figure 3.8.

For a type-II FIR generalized linear-phase system, (3.12) becomes

(3.14) .)z(Xz)z(Xz)n(h)z(Y12/N

0n

)n1N(n

X(z)

Y(z)

z1

h(0)

…

h(1) h(2)

z1

z1

h(N/21)

z1 z1

Figure 3.9. Direct Form for a Type-II

FIR Generalized Linear-Phase System.

The corresponding structure is shown in figure 3.9.

For a type-III FIR generalized linear-phase system, (3.12)

becomes

(3.15) .)z(Xz)z(Xz)n(h)z(Y2/)3N(

0n

)n1N(n

X(z)

Y(z)

z1

h(0)

…

h(1) h(2)

z1

h[(N3)/2]

z1

z1z1z1

Figure 3.10. Direct Form for a Type-III

FIR Generalized Linear-Phase System.

The corresponding structure is shown in figure 3.10.

For a type-IV FIR generalized linear-phase system, (3.12)

becomes

(3.16) .)z(Xz)z(Xz)n(h)z(Y12/N

0n

)n1N(n

X(z)

Y(z)

z1

h(0)

…

h(1) h(2)

z1

z1

h(N/21)

z1 z1

Figure 3.11. Direct Form for a Type-IV

FIR Generalized Linear-Phase System.

The corresponding structure is shown in figure 3.11.

3.3.2. Cascade Form

(3.11) can be expressed as

.)z(H)z(HL

1k

k

(3.17)

Here Hk(z) is a real-coefficient polynomial of z1. (3.17) can be used

to construct a cascade form structure of the system (figure 3.12). The

entire system is a cascade of multiple subsystems. Each subsystem is

implemented in the direct form structure.

Figure 3.12. Cascade Form.

X(z) Y(z)H1(z) H2(z) … HL(z)

Hk(z) is a 1st- or 2nd-order polynomial generally. In many cases,

we try to make Hk(z) a 2nd-order polynomial.

It is easy to adjust a zero in a cascade form structure.

Let the system belong to the four types of FIR generalized linear-

phase systems. Usually, we make Hk(z) correspond to such a group

of zeros: a, a*, 1/a and 1/a*. Since it also belongs to the four types of

FIR generalized linear-phase systems, Hk(z) can be implemented in a

special direct form structure given above.

3.4. Transposition Theorem

Signal flow graphs can be used to derive new system structures.

According to certain rules, we can convert a signal flow graph into

another form with the same input-output relation and thus obtain a

new structure.

We study one of these rules, the transposition theorem. In a signal

X(z) Y(z)

z1

b1

z1

b2

b0

a1

a2

Figure 3.13. A Signal Flow Graph.

flow graph, if the directions of all the directed branches are reversed

and the roles of the input and the output are interchanged, then the

resulting signal flow graph and the original signal flow graph express

the same input-output relation.

Example. Transpose the following signal flow graph and show that

the input-output relation keeps unchanged.