数字信号处理chapter4 Structure of Digital Filter

（1）Direct Form
M
aizi
System function: H ( z )
i0 N
1 bi z i
i1
Difference equation:
N
y( n) ai x( n i ) i0 N bi y( n i ) i1
Direct Form
2N delay units are required.
h(n) is a finite-length sequence, its system function is:
N1
H(z) h(n)zn n0
FIR doesn’t have feedback loop, it is not a recursive form filter,
so there is no problem of stability.
01
11
z
1
11
21
z 1
1M 2M
2020/4/13
0M z 1
1M z 1
（4）Parallel Form
y n
We can see: a) Pole can be adjusted
independently, zero cannot; b) If there are multi-order pole,
)
A
2 i 1
1
1i
z
1
1 1i z 1
2i z2 2i z2
[•]表示取整
x(n)
11 z-1 11 21 z-1 21
y(n)
1M z-1 1M 2M z-1 2M
H1( z )
We can see:
HM( z )
a) Cascade form’s structure is flexible, pole and zero can be
Flow graph
2020/4/13
12
3 Implementation of the Digital Filter
（1）Hardware (Special-purpose) Digital signal processor
（2）Software Digital Filter Algorithm
IIR filter’s structure is not unique.
2020/4/13
15
H1z10.8z110.15z2
Direct Form
H 2z101 .3z1101 .5z1 Cascade Form
H 3z1 0 1 ..3 5z112 0..5 5z1 Parallel Form
Such as: y ( n ) a 0 x ( n ) a 1 x ( n 1 ) b 1 y ( n 1 )
z-1
z-1
z-1
Flow graph of one-order digital filter Different structure will determine system’s accuracy, error, stability, cost efficiency and speed, etc al.
（2）Impulse response Such as: h(n)
（3）System function
Such as: H(z)Y X((zz))a10ba11zz 11
2020/4/13
10
2 Diagram
（1）Block-diagram
Such as: y ( n ) a 0 x ( n ) a 1 x ( n 1 ) b 1 y ( n 1 )
N1
H(z) h(k)zk k0
Infinite Impulse Response
M
aizk
H(z)
k0 N
1 bi z k
k 1
N1
y(n)h(k)x(nk) k0
y(n)h(k)x(nk) k0
h(n): Finite length Nonrecursively
h(n): infinite length Recursively
(Output Sequence)
2020/4/13
8
CONTENTS 4.1 Representation of Digital Filter 4.2 Structure of IIR Filter 4.3 Structure of FIR Filter 4.4 Review
2020/4/13
9
characteristics structures design approach.
2020/4/13
7
CHAPTER 4 BASIC STRUCTURE OF DIGITAL FILTER
h(k), k=0,1,…
x(n)
(Impulse Response)
y(n)
(Input Sequence)
H (z) A 0 i L 11 A p iiz 1 iM 11 r 0 1 iiz 1 r 1 iz 1 2 iz 2
where: N=L+2M，L, one-order network, M, two –order network.
2020/4/13
23
A0
A1
p1
z 1
x n
18
（2）Direct Form II
x(n)
a0
y(n) x(n)
a0
y(n)
b1
z-1 z-1
a1
b1
z-1 a1
b2
z-1 z-1
a2
b2
z-1 a2
bN
z-1 z-1
aN
H2( z )
H1( z )
Direct Form II
bN
z-1 aN
N Delay Unit.
2020/4/13
19
2020/4/13
14
4.2 Structure of IIR filter
1. Characteristics of IIR filter
System function:
M
aizi
H(z)
i0 N
1 bi ቤተ መጻሕፍቲ ባይዱ i
i1
At least one coefficient bi is not zero, its difference equation is:
2020/4/13
22
（4）Parallel Form
Transfer function can be expressed into partial fraction, and can be composed into parallel form:
H(z)A0iN 11A diizi
expanding into real boots and conjugate complex roots form:
2. Structure of FIR filter N1 （1） Direct-form y(n) h(i)x(ni)
4.1 Representation of digital filter 1 Expression of digital filter
（1）Difference equation Such as: one-order system
y ( n ) a 0 x ( n ) a 1 x ( n 1 ) b 1 y ( n 1 )
partial fraction expansion is difficult. c) High speed computation, lower error than cascade form. 24
4.3 Structure of FIR filter
1. Characteristic of FIR filter
Application of digital filter in engineering —— Preprocess of test data
Simple ? Complex ? Experiments or Theories ? Application and Textbook ?
2020/4/13
H(z)AiN 11
i1 N2
(1piz1) (1qiz1)(1qi*z1)
， ， i1
i1
g i p iare real roots, h i q iare complex roots，and N1+N2=N
2020/4/13
21
Two-order Cascade Form
N 1
H (
z
word-length Instability Errors.
2020/4/13
20
（3）Cascade Form
N-order transfer function can be expressed by its poles and zeros:
N
aizi
N
(1ciz1 )
H(z)
i0 N
A
i1 N
4 Classification of the Digital Filter
(1) Finite Impulse Response (FIR) FIR filter
(2) Infinite Impulse Response (IIR) IIR filter
2020/4/13
13
Finite Impulse Response
adjusted conveniently and do not influence each other.
b) Two order’s position can be selected arbitrarily, different
scheme has different error and can be optimized.
The same H(z) may have different implementation structure cost efficiency computation error stability finite word-length sensitivity
2020/4/13
16
2. Structure of IIR filter
N
N
y(n) aix(ni) biy(ni)
i0
i1
There are feedback loops in IIR structure, called recursively.
There are poles inside unit circle on z-plane, otherwise unstable.
Unit Delay
a0
z-1
z-1
a1 b1
Delay
z-1
a Constant multiplication
Diagram of one-order digital filter
Addition
First-order Nth-order
Block diagram
2020/4/13
11
（2）Flow-graph
2020/4/13
6
Digital filter
1.Filter is a particularly important class of linear time-invariant system in DSP.
2.Purpose: Filter, detection and prediction of signal. 3.Content:
We can find that:
a) Direct Form II requires N delay units. b) Common disadvantage of the two form:
Coefficient ai , bi can not be adjusted
conveniently. When N is large, too sensitive to finite
2020/4/13
17
（2）Direct Form II
w n HzH 1zH 2z
N
H1 z aizi i0
x n H 1 z w1 n H 2 z y n
H zH 1zH 2z
2020/4/13
H2 z
1
N
1 bi zi
i 1
x n H2 z w2 n H 1 z y n
H zH 2zH 1z
1
Original waveform
2020/4/13
2
Original Spectrum
2020/4/13
3
Frequency response of the selected filter
2020/4/13
4
Waveform after filtering
2020/4/13
5
Spectrum after filtering
1 bizi
(1diz1 )
i1
i1
Because H(z)’s parameter ai , bi is real value, zero ci and pole di
are real roots or conjugate complex roots, that is:
N1
N2
(1giz1) (1hiz1)(1hi*z1)