1. Chapter 6 Design of FIR Digital Filter

2. IIR vs. FIR IIR FIR: Analog filter—Design simply.
Spectrum specification uneasy to control.
FIR:
Strictly linear spectrum characteristics.
h(n) is finite length, no problem of stability and
causality.

3. CONTENTS 6.1 Characteristics of linear phase FIR filter
6.2 Design of FIR filter by Windowing
6.3 Frequency sampling method
6.4 Comparison of IIR and FIR filter

4. Introduction IIR filter’s phase characteristics is hard to control.
FIR filter’s advantage：Exact liner phase.
(Arbitrary amplitude-frequency characteristics)
FIR filter’s impulse response h(n) is a finite length sequence
 z-transform convergent on whole z-plane (no stability problem )
 can delay some time( no causality problem)
FIR filter’s structure is non-recursive generally, there will
no problem of instability with finite precision computation.

5. 6.1 Characteristic of linear phase FIR filter
（1）Condition for linear phase
Phase
Amplitude
Linear phase:
在实际系统中，一个输入信号可以分解为多个正弦信号的叠加，为了使得输出信号不会产生相位失真，必须要求它所包含的这些正弦信号通过系统的时间是一样的。
在图像处理、雷达通信等领域，对各个频率成分之间相位关系要求非常严格，因此线性相位滤波器非常重要。

6. 6.1 Characteristic of linear phase FIR filter
（1）Condition for linear phase
If impulse response h(n) of FIR filter is real value, and meets
condition of even symmetric or odd symmetric, that is:
then filter’s phase characteristic is exact linear.
Phase
Amplitude
（2）Linear phase FIR filter’s amplitude and phase characteristic

7. even symmetric with =0,,2
Odd symmetric with =
Even symmetric impulse response
Phase response
N is odd
N is even

8. Odd symmetric impulse response
Odd symmetric with
=0,,2
Even symmetric
with =
Odd symmetric impulse response
N is odd
N is even
Phase response

9. （3）Zero location of linear phase FIR filter
System function:
If z=zi is zero of H(z), its reciprocal zi-1 is also zero of H(z).
Because h(n) is real sequence, H(z)’s zero are conjugate symmetric.
That is: zi* and (zi-1)* are also zero of H(z).
Characteristic of zero location:
Zeros are reciprocal conjugate
symmetric pair.
Exceptional:
Zeros are real value, then there are only two zeros.
Zeros are pure complex and lie on unit circle, then
there are only two zeros.
Zeros are real value and lie on unit circle,
then there are only one zeros.

10. Structure of linear phase FIR filter
Basic structure
Structure of linear phase FIR filter

11. Structure of linear phase FIR filter
N is even
Save half multiplier
N is odd

12. Not absolutely summable
6.2 Windowing
1. Basic idea of windowing method
Infinitely long
Not absolutely summable
Unrealizable!
Window function
Approximate

13. ？ 1. Basic idea of windowing method 单位脉冲响应 理想幅度特性 矩形窗 矩形窗幅度特性 h(n)偶对称
时域相乘
频域卷积

14. 时域相乘
频域卷积
理想幅度特性
矩形窗幅度特性

16. 截止频率

17. max

18. min

19. 矩形窗对理想
低通幅度特性
的影响

21. 讨论：增大N对窗函数的影响 N=10 N=20 增加窗函数的长度，能够减少窗函数频谱主瓣宽度；
不能改变主瓣和旁瓣的相对特性，该值取决于窗函数的形状。
Gibbs Phenomenon 吉布斯效应

22. 2. Effects of windowing on ideal filter specification
At disconnected points of cutoff frequency, Hd(w) becomes continuous curve
— transition region. It’s width is equal to window’s main lobe width.
Wider window’s main lobe  wider transition region 4/N
Effects of window’s side lobe: ripple appear in filter’s amplitude-frequency
characteristics, amplitude of ripple depends on side lobe’s relative amplitude.
Larger area of side lobe  larger ripple of passband and stopband.
Increase window’s length can only decrease main lobe width, not relative
characteristic between main lobe and side lobe, that value only depends on
window’s shape.
Increase N can only decrease width of transition region, not ripple
characteristics.

23. Trade-off 3. Principle of window selection Transition width increase 
Small side lobe amplitude, especially the first side lobe amplitude.
High speed decreasing of side lobe amplitude, in order to increase
stopband attenuation.
Narrow main lobe width, in order to get narrow transition width.
Trade-off
Small side lobe amplitude: Smoothing amplitude-frequency characteristics 
Transition width increase 
Narrow main lobe width: Narrower transition width 
Ripple of passband and stopband is large 

24. 常用窗函数
幅频特性
矩形窗
Rectangle
三角窗
Triangle
哈明窗
Hamming

25. 常用窗函数
幅频特性
汉宁窗
Hann
布莱克曼窗
Blackman
凯塞窗
Kaiser

26. 4. Process of FIR filter design using window method
(1) Determine filter’s unit impulse response:
If filter’s frequency response is Hd(ejw), then:
If edge frequency, attenuation of passband, stopband is known,
we can choose ideal filter as approximation function:
Hd(ejw)  IDFT  hd(n)

27. (2) Determine window function and its length based on requirement
Suppose width of transition is , it is an approximation of window’s main lobe.
 is reverse ratio to window’s length N: NA/ , A  window form;
For example: Rectangular: A=4; Hamming: A=8;
Window
Peak attenuation of side lobe (dB)
Transition width 
Minimum attenuation of stopband (dB)
Rectangular
-13
4/N
-21
Hann
-31
8/N
-44
Hamming
-41
-53
Blackman
-57
12/N
-74
Kaiser
10/N
-80
Principle: Select window function with narrower main lobe when stopband attenuation meets the requirement.

28. (3) Compute filter’s unit impulse response h(n)
——Selected window.
（4）Verification
Compute designed filter’s frequency response:
FFT Algorithm

29. 5 KAISER （1）Basic property A most useful and optimum window.
Zeroth-order modified Bessel function
Parameter  can control shape of the window.
Characteristic: Different width of transition band under same N.
（1）If  =5.658，width of transition band 7.8/N,
least attenuation of stop band 60dB,
（2）If  =4.538， width of transition band 5.8/N,
least attenuation of stop band 50dB.

30. Kaiser window for different 
=1
=5.44
=8.5
Kaiser window for different 

31. p, s and As is known, acquire the following parameter:
（2）Design approach
Kaiser obtain a pair of formulas that permit designer to predict in advance
parameters needed to meet a given frequency selective filter specification.
p, s and As is known, acquire the following parameter:
Width of transition band
Order of filter

32. Example: Design a digital FIR lowpass filter with Kaiser window, specification is as follows:
p=0.2，s =0.3 ， As =50dB
Determine impulse response and filter’s frequency response diagram.
MATLAB demo: KAISERFIR_design.m

33. 6.3 Frequency sampling method
Suppose system function of designed filter doesn’t have closed form equation.
N points regular interval sampling
between =0 to2
Condition for linear phase

34. 6.4 Comparison of IIR and FIR filter
（1）Performance
IIR filter:
Require less memory and calculation to achieve a given
filter response characteristic, high cost efficiency.
Better selectivity, phase nonlinearity.
FIR filter:
Require more memory and calculation to achieve a given
filter response characteristic. Delay time is large.
Exact linear phase
Under the same requirement, IIR’s complexity will increase dramatically, so FIR outweigh IIR in performance and cost.

35. （2）Structure IIR filter:
Recursive structure, pole must be inside unit circle,
or system instability will appear.
Finite-precision arithmetic can cause significant problems
due to the use of feedback.
FIR filter
Non-recursive structure, no instability problem in theory and practice. Low computation error.
Fast Fourier Transform (FFT), high speed with same order.

36. （3）Design approach IIR filter:
A variety of frequency-selective filters can be designed using
closed form design formulas, so it can save computation time.
Limited to frequency-selective filters, inflexible.
FIR filter:
Closed form design equation does not exit for FIR filter. Although
window method is straightforward to apply, some iteration is
necessary to meet a prescribed specification.
Flexible, arbitrary frequency response with exact linear phase.

37. （4）Application IIR filter:
Application with no requirement for linear phase.
FIR filter:
Application with high requirement for linear phase.
In practical application, requirement, tools and cost must be considered.

38. RIVIEW Comparison of FIR filter and IIR filter
Specification, structure, phase characteristics, design approach, application
Condition for FIR linear phase (proof)
Window method
（1）Idea
（2）Conclusion
Width of transition band is determined by width of windows’ main lobe;
Peak ripples of bandstop and bandpass region is determined by side
lobe characteristic.
Length of windows: N  Influence main lobe, not side lobe.
（3）Selective of windows
Width of main lobe and side lobe characteristics (restrict each other )
4. Commonly used windows.