eele 4310: digital signal processing...

EELE 4310: Digital Signal Processing (DSP)

Chapter # 9 : Implementation of Discrete-Time Systems

Spring, 2012/2013

EELE 4310: Digital Signal Processing (DSP) - Ch.9 Dr. Musbah Shaat 1 / 19

Outline

1 Structures for the Realization of Discrete-Time Systems

2 Structures for FIR Systems

3 Structures for IIR Systems


Structures for the Realization of Discrete-Time Systems

Consider an LTI system characterized by the following differenceequation:

y(n) = −∑N

k=1 aky(n − k) +∑M

k=0 bkx(n − k)

The system can be characterized by the system function:

H(z) =∑M

k=0 bkz−k

1+∑N

k=1 akz−k

The previous equations can be implemented in a variety of waysdepending on the form in which the equations are arranged.

The equations can be decomposed into several difference equations.

The equations can be represented by a block diagram which called”realization” of the system or ”structure” for realizing the system.

The factors that influence the choice of a specific realization arecomputational complexity, memory requirements, and other differentfactors.


Structures for FIR Systems

Generally, an FIR system is described by the difference equation

y(n) =∑M−1

k=0 bkx(n − k)

or equivalently by the system function

H(z) =∑M−1

k=0 bkz−k

The impulse response of the FIR system is

h(n) =

{bn, 0 ≤ n ≤ M − 1

0, otherwise

The variable M denotes the length of the FIR filter.


Structures for FIR SystemsDirect-form Structure ... 1

The direct-form realization follows directly form the nonrecursivedifference equation of the FIR filter

y(n) =∑M−1

k=0 bkx(n − k).

The structure resembles a tapped delay line an is calledtapped-delay-line filter.



When the FIR system has linear phase (it will described later),i.e.

h(n) = ±h(M − 1− n).



Ex. Determine a direct realization for the following linear phasefiltersh(n) = {1

↑, 2, 3, 4, 3, 2, 1}


Structures for FIR SystemsCascade-Form Structures ... 1

The system function H(z) can be factorized into second orderFIR-systems so that

H(z) =∏K

k=1Hk(z)

where

Hk(z) = bk0 + bk1z−1 + bk2z

−2, k = 1, 2, · · · ,K


Structures for FIR SystemsCascade-Form Structures ... 2

In case of linear-phase FIR filters, a fourth order sections of FIRsystem can be used as follows

Hk(z) = ck0 + ck1z−1 + ck2z

−2 + ck1z−3 + ck0z

−4


Structures for IIR SystemsDirect-Form Structures ... 1

The IIR system function can be veiwed as two systems in cascade,that is,

H(z) = H1(z)H2(z)

where H1(z) consists of the zeros of H(z), and H2(z) consists of thepoles of H(z),

H1(z) =∑M

k=0 bkz−k

and

H2(z) = 11+

∑Nk=1 akz

−k

Two different direct-form realization can be characterized bywhether H1(z) precedes H2(z) or vice versa.



Direct form I realization



Direct form II realization


Structures for IIR SystemsSignal Flow Graphs and Transposed Structures ... 1

A signal flow provides an alternative but equivalent graphicalrepresentation to a block diagram structure.



Flow graph reversal theorem: if we reverse the directions of all branchtransmittances and interchange the input and output in the flow graph,the system remains unchanged (we get transposed structure).



To get a transposed direct form II system


Structures for IIR SystemsCascade-form Structure

The system function can be factorized as H(z) =∏K

k=1Hk(z)

where Hk(z) = bk0+bk1z−1+bk2z

−2

1+ak1z−1+ak2z−2


Structures for IIR SystemsParallel-form Structure ... 1

The parallel form can be obtained by performing a partial-factionexpansion of H(z).

If N ≥ M, then H(z) = C +∑N

k=1Ak

1−pkz−1 .

To avoid multiplications by complex numbers, pairs ofcomplex-conjugates poles can be combined to form two-polesubsystems.

Hk(z) = bk0+bk1z−1

1+ak1z−1+ak2z−2

See example 9.3.1 page 554.


Structures for IIR SystemsParallel-form Structure ... 2


End of Chapter # 9

eele 4310: digital signal processing...

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