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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
EELE 4310: Digital Signal Processing (DSP) - Ch.9 Dr. Musbah Shaat 2 / 19
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.
EELE 4310: Digital Signal Processing (DSP) - Ch.9 Dr. Musbah Shaat 3 / 19
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.
EELE 4310: Digital Signal Processing (DSP) - Ch.9 Dr. Musbah Shaat 4 / 19
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.
EELE 4310: Digital Signal Processing (DSP) - Ch.9 Dr. Musbah Shaat 5 / 19
Structures for FIR SystemsDirect-form Structure ... 2
When the FIR system has linear phase (it will described later),i.e.
h(n) = ±h(M − 1− n).
EELE 4310: Digital Signal Processing (DSP) - Ch.9 Dr. Musbah Shaat 6 / 19
Structures for FIR SystemsDirect-form Structure ... 3
Ex. Determine a direct realization for the following linear phasefiltersh(n) = {1
↑, 2, 3, 4, 3, 2, 1}
EELE 4310: Digital Signal Processing (DSP) - Ch.9 Dr. Musbah Shaat 7 / 19
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
EELE 4310: Digital Signal Processing (DSP) - Ch.9 Dr. Musbah Shaat 8 / 19
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
EELE 4310: Digital Signal Processing (DSP) - Ch.9 Dr. Musbah Shaat 9 / 19
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.
EELE 4310: Digital Signal Processing (DSP) - Ch.9 Dr. Musbah Shaat 10 / 19
Structures for IIR SystemsDirect-Form Structures ... 2
Direct form I realization
EELE 4310: Digital Signal Processing (DSP) - Ch.9 Dr. Musbah Shaat 11 / 19
Structures for IIR SystemsDirect-Form Structures ... 3
Direct form II realization
EELE 4310: Digital Signal Processing (DSP) - Ch.9 Dr. Musbah Shaat 12 / 19
Structures for IIR SystemsSignal Flow Graphs and Transposed Structures ... 1
A signal flow provides an alternative but equivalent graphicalrepresentation to a block diagram structure.
EELE 4310: Digital Signal Processing (DSP) - Ch.9 Dr. Musbah Shaat 13 / 19
Structures for IIR SystemsSignal Flow Graphs and Transposed Structures ... 2
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).
EELE 4310: Digital Signal Processing (DSP) - Ch.9 Dr. Musbah Shaat 14 / 19
Structures for IIR SystemsSignal Flow Graphs and Transposed Structures ... 3
To get a transposed direct form II system
EELE 4310: Digital Signal Processing (DSP) - Ch.9 Dr. Musbah Shaat 15 / 19
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
EELE 4310: Digital Signal Processing (DSP) - Ch.9 Dr. Musbah Shaat 16 / 19
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.
EELE 4310: Digital Signal Processing (DSP) - Ch.9 Dr. Musbah Shaat 17 / 19
Structures for IIR SystemsParallel-form Structure ... 2
EELE 4310: Digital Signal Processing (DSP) - Ch.9 Dr. Musbah Shaat 18 / 19
End of Chapter # 9