1. Chapter 6 Digital Filter Structures
Content
Introduction
IIR Filter Structures
FIR Filter Structures
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Introduction
What is a digital filter
A filter is a system that is designed to remove some component or modify some characteristic of a signal
A digital filter is a discrete-time LTI system which can process the discrete-time signal.
There are various structures for the implementation of digital filters
The actual implementation of an LTI digital filter can be either in software or hardware form, depending on applications
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Introduction
Basic elements of digital filter structures
Adder has two inputs and one output.
Multiplier (gain) has single-input, single-output.
Delay element delays the signal passing through it by one sample. It is implemented by using a shift register. a
Block diagram
Signal-flow graph
z-1 a
z-1
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z-1 a2 b0 a1 a2 b0
z-1 1 2 5 3 4
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Introduction
The major factors that influence the choice of a specific structure
Computational complexity
refers to the number of arithmetic operations (multiplications, divisions, and additions) required to compute an output value y(n) for the system.
Memory requirements
refers to the number of memory locations required to store the system parameters, past inputs, past outputs, and any intermediate computed values.
Finite-word-length effects in the computations
refers to the quantization effects that are inherent in any digital implementation of the system, either in hardware or in software.
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6. IIR Filter Structures The characteristics of the IIR filter
IIR filters have Infinite-duration Impulse Responses
The system function H(z) has poles in
An IIR filter is a recursive system
可参考教材P75～76
The order of such an IIR filter is called N if aN≠0
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IIR Filter Structures
Direct form
In this form the difference equation is implemented directly as given. There are two parts to this filter, namely the moving average part and the recursive part (or the numerator and denominator parts). Therefore this implementation leads to two versions: direct form I and direct form II structures
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Direct form I b1 b2 b0
z-1
bM-1 bM a1 a2
z-1
aN-1 aN
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Direct form II
For an LTI cascade system, we can change the order of the systems without changing the overall system response b1 b2 b0
z-1
bM-1 bM a1 a2
aN-1 aN
z-1
直接II型比直接I型节省存储单元（软件实现），或节省寄存器（硬件实现）
但不论是直接I型还是直接II型，其共同的缺点是系数对滤波器的性能控制作用不明显，这是因为它们与系统函数的零极点关系不明确因而调整困难。
另外，这两种结构极点对系数的变化过于敏感，从而使系统频率响应对系数的变化过于灵敏，也就是对有限精度（有限字长）运算过于灵敏，容易出现不稳定或产生较大误差。
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IIR Filter Structures
Cascade form
In this form the system function H(z) is written as a product of second-order sections with real coefficients
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IIR Filter Structures
Each second-order section (called biquads) is implemented in a direct form ,and the entire system function is implemented as a cascade of biquads.
If N=M, there are totally biquads.
If N>M, some of the biquads have numerator coefficients that are zero, that is, either b2k = 0 or b1k = 0 or both b2k = b1k = 0 for some k.
为了简化级联形式，特别是在时分多路复用时，采用相同形式的子网络结构就更有意义，因而将实系数的两个一阶因子组合成二阶因子
若实系数的一阶因子有奇数个，则可认为某一个二阶因子的二阶系数为0
可以不考虑N<M的情况，因为这种情况转换为一个横向结构（只有分子，没有分母）和一个递归结构。
if N>M and N is odd, one of the biquads must have ak2 = 0, so that this biquad become a first-order section.
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b1k
b2k
z-1
a1k
a2k
biquad
First-order section
b1k
z-1
a1k
b1k
z-1
a1k
a2k
z-1
a1k
a2k
z-1
a1k
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Example
5.2644
z-1
-0.5
0.25 4 2
级联型的特点：
调整系数b1k和b2k就能单独调整滤波器的第k对零点，而不影响其它零点和极点。
同样，调整系数a1k和a2k就能单独调整滤波器 第k对极点，而不影响其它零点和极点。
这种结构便于准确实现滤波器零、极点，因而便于调整滤波器频率相应特性。
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IIR Filter Structures
Parallel form
In this form the system function H(z) is written as a sum of sections using partial fraction expansion. Each section is implemented in a direct form. The entire system function is implemented as a parallel of every section.
Suppose M=N
当M<N，但M不等于N时，见教材P207（式5－7）
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IIR Filter Structures
if N is odd, the system has one first-order section and second-order sections.
if N is even, the system has second-order sections.
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Example
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-12.9
z-1
7/8
-3/32
-14.75
26.82 1
-1/2
24. 5
并联型可以用调整a1k和a2k的方法来单独调整一对极点的位置，但是不能单独调整零点的位置。
此外，并联型结构中，各并联基本节的误差相互没有影响，所以比级联型的误差一般来说要稍小一些，因此在要求准确的传输零点的场合下，宜采用级联型结构。
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IIR Filter Structures
Transposition theorem
If we reverse the directions of all branch transmittances and interchange the input and output in the flow graph, the system function remains unchanged.
The resulting structure is called a transposed structure or a transposed form.
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19. FIR Filter Structures The characteristics of the FIR filter
FIR filters have Finite-duration Impulse Responses, thus they can be realized by means of DFT
The system function H(z) has the ROC of , thus it is a causal system
An FIR filter is a nonrecursive system
FIR filters can be designed to have a linear-phase response
It has N-1 order poles at
and N-1 zeros in
The order of such an FIR filter is N-1
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FIR Filter Structures
Direct form
In this form the difference equation is implemented directly as given.
z-1
h(1)
h(2)
h(0)
h(N-2)
h(N-1)
It requires N multiplications
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FIR Filter Structures
Cascade form
In this form the system function H(z) is converted into products of second-order sections with real coefficients
It requires (3N/2) multiplications
z-1
b11
b21
b01
b12
b22
b02
b1x
b2x
b0x
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FIR Filter Structures
Linear-phase form
Linear phase:
The phase response is a linear function of frequency
Linear-phase condition
Symmetric impulse response
Antisymmetric impulse response
When an FIR filter has a linear phase response, its impulse response exhibits the above symmetry conditions. In this form we exploit these symmetry relations to reduce multiplications by about half.
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If N is odd
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If N is odd
z-1 ±1
1 for symmetric
-1 for antisymmetric
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If N is even
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If N is even
z-1 ±1
1 for symmetric
-1 for antisymmetric
The linear-phase filter structure requires 50% fewer multiplications than the direct form.
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FIR Filter Structures
Frequency sampling form
This structure is based on the DFT of the impulse response h(n) and leads to a parallel structure. It is also suitable for a design technique based on the sampling of frequency response H(z)
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It has N equally spaced zeros on the unit circle
All-zero filter
or comb filter 2
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resonant filter
It has N equally spaced poles on the unit circle
The pole locations are identical to the zero locations and that both occur at , which are the frequencies at which the designed frequency response is specified. The gains of the filter at these frequencies are simply the complex-valued parameters
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z-1
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FIR Filter Structures
Problems
It requires a complex arithmetic implementation
It is possible to obtain an alternate realization in which only real arithmetic is used. This realization is derived using the symmetric properties of the DFT and the factor.
It has poles on the unit circle, which makes this filter critically unstable
We can avoid this problem by sampling H(z) on a circle |z|=r where the radius r is very close to one but is less than one.
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jIm[z]
Re[z]
unit circle
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By using the symmetric properties of the DFT and the factor, a pair of single-pole filters can be combined to form a single two-pole filter with real-valued parameters.
For poles
h(n) is a real-valued sequence, so
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So the kth and (N-k)th resonant filters can be combined to form a second-order section with real-valued coefficients
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jIm[z]
Re[z]
N is even
|z|=r
jIm[z]
Re[z]
N is odd
|z|=r
If N is even, the filter has a pair of real-valued poles
If N is odd, the filter has a single real-valued pole
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second-order section
b1k
z-1
b0k
first-order sections
z-1 r
z-1 r
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N is even
N is odd
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FIR Filter Structures
Fast convolution form
x(n): N1-point sequence
h(n): N2-point sequence
L ≥ N1 + N2 - 1
L-point
DFT
IDFT
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(digital signal processing\chap5\linear_phase.m)
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