Chapter 7 IIR Filter Design

Chapter 7 IIR Filter Design
Content Preliminaries Characteristics of Prototype Analog Filters Analog-to-Digital Filter Transformations Frequency Transformations We now turn our attention to the inverse problem of designing systems from the given specifications. It is an important as well as a difficult problem. In digital signal processing there are two important types of systems. The first type of systems perform signal filtering in the time domain and hence are called digital filters. The second type of systems provide signal representation in the frequency domain and are called spectrum analyzers. Copyright © Shi Ping CUC

Copyright © 2005. Shi Ping CUC
Preliminaries How to design a digital filter The design of a digital filter is carried out in three steps: First: Specifications Before we can design a filter, we must have some specifications. These specifications are determined by the applications. Second: Approximations Once the specifications are defined, we use various concepts and mathematics to come up with a filter description that approximates the given set of specifications. This step is the topic of filter design. Copyright © Shi Ping CUC

Preliminaries Third: Implementation The product of the above step is a filter description in the form of either a difference equation, or a system function, or an impulse response. From this description we implement the filter in hardware or software on a computer. In this and the next chapter we will discuss in detail only the second step, which is the conversion of specification into a filter description. Copyright © Shi Ping CUC

Preliminaries The specifications In many applications, digital filters are used to implement frequency-selective operations; Therefore, specifications are required in the frequency-domain in terms of the desired magnitude and phase response of the filter; Generally a linear phase response in the passband is desirable; An FIR filter is possible to have an exact linear phase; An IIR filter is impossible to have linear phase in passband. Hence we will consider magnitude-only specifications. Copyright © Shi Ping CUC

Preliminaries There are two ways to give the magnitude specifications Absolute specifications Provide a set of requirements on the magnitude response function and generally used for FIR filters. Passband Stopband Transition band The tolerance (or ripple) in passband The tolerance (or ripple) in stopband The ending frequency of the passband. Bandwidth The beginning frequency of the stopband. Copyright © Shi Ping CUC

Preliminaries Relative specifications (dB) Provide requirements in decibels (dB). This approach is the most popular one in practice and used for both FIR and IIR filters W=0时的幅度被归一化为1。 The maximum tolerable passband ripple The minimum tolerable stopband attenuation Copyright © Shi Ping CUC

Preliminaries The basic technique of IIR filter design IIR filters have infinite-length impulse responses, hence they can be matched to analog filters. Analog filter design is a mature and well developed field. We can begin the design of a digital filter in the analog domain and then convert the design into the digital domain Copyright © Shi Ping CUC

Preliminaries There are two approaches to this basic technique
Design analog lowpass filter Apply freq. band transformation s → s Apply filter transformation s → z Designed IIR filter Approach 2 Design analog lowpass filter Apply filter transformation s → z Apply freq. band transformation z → z Designed IIR filter 本课程介绍第二种方法 return Copyright © Shi Ping CUC

Characteristics of Prototype Analog Filters
Magnitude-squared function Let be the frequency response of an analog filter 设计模拟滤波器是根据一组设计规范来设计模拟系统函数Ha(s)，使其逼近某个理想滤波器特性。可利用幅度平方函数、相位特性以及群延时特性来进行逼近。 is a passband ripple parameter is a stopband attenuation parameter is the passband cutoff frequency in rad/sec is the stopband cutoff frequency in rad/sec Copyright © Shi Ping CUC

The properties of is a real function Ha(s)是模拟滤波器的系统函数，应是稳定因果的，因此其极点一定在左半平面。 The poles and zeros of are distributed in a mirror-image symmetry with respect to the axis. For real filters, poles and zeros occur in complex conjugate pairs. Copyright © Shi Ping CUC

How to construct is the system function of the analog filter. It must be causal and stable. Then all poles of must lie within the left half-plane. All left-half poles of should be assigned to Zeros are not uniquely determined. They can be halved between and (Zeros in each half must occur in complex conjugate pairs) Ha(s)是模拟滤波器的系统函数，应是稳定因果的，因此其极点一定在左半平面。 If a minimum-phase filter is required, the left-half zeros should be assigned to Copyright © Shi Ping CUC

Butterworth lowpass filters This filter is characterized by the property that its magnitude response is flat in both passband and stopband. The magnitude-squared function of an Nth-order lowpass filter is given by Copyright © Shi Ping CUC

The properties of Butterworth lowpass filters At , for all N At , for all N, which implies a 3dB attenuation at is a monotonically decreasing function of 不管N为多大，所有曲线都经过－3dB点，或说衰减3dB，这就是3dB不变性。 approaches an ideal lowpass filter as is maximally flat at since derivatives of all orders exist and are equal to zero Copyright © Shi Ping CUC

There are 2N poles of , which are equally distributed on a circle of radius with angular spacing of radians. If the N is odd, there are poles on real axis. If the N is even, there are not poles on real axis. The poles are symmetrically located with respect to the imaginary axis. A pole never falls on the imaginary axis, and falls on the real axis only if N is odd. Copyright © Shi Ping CUC

In general, we consider and this results in a normalized Butterworth analog prototype filter When designing an actual filter with , we can simply do a replacement for s, that is 这里分子系数可由低频特性决定，带入Ha(0)=1可得。 Copyright © Shi Ping CUC

Since the actual N chosen is larger than required, specifications can be either met or exceeded at or To satisfy the specifications exactly at N is the smallest integer larger than N’ 用通带方程求出的OmegaC，带到阻带方程求衰减时，衰减值比指标大，即超过了指标的要求。用阻带方程求出的OmegaC，带到通带方程求波纹时，波纹值比指标小，即超过了指标的要求。 We can choose any OmegaC between the above two numbers. To satisfy the specifications exactly at Copyright © Shi Ping CUC

Chebyshev lowpass filters There are two types of Chebyshev filters Chebyshev-I: equiripple in the passband and monotonic in the stopband. Chebyshev-II: monotonic in the passband and equiripple in the stopband. Chebyshev filters can provide lower order than Butterworth filters for the same specifications. Chebyshev-I 滤波器也是个全极点滤波器，在通带内等波纹，阻带内单调下降 Chebyshev-I Copyright © Shi Ping CUC

Designing equations Given , two parameters are required to determine a Chebyshev-I filter: Delta2增加，则N也会随之增加。即若要求阻带衰减越大（过渡带越陡），则滤波器阶数N就越高。给出omigaS的公式是为了给出3dB带宽的公式，前提是3dB带宽必须大于omigaC Note: this is only for Copyright © Shi Ping CUC

Determine system function To determine a causal and stable , we must find the poles of and select the left half-plane poles for The poles are obtained by finding the roots of It can be shown that if are the (left half-plane) roots of the above polynomial, then Copyright © Shi Ping CUC

Determine poles by geometric method The poles of fall on an ellipse with major axis and minor axis 2N个等角间隔将大圆和小圆各自均分为2N份，在大圆和小圆上各自出现了六个交点，他们共同决定六个极点的位置。大圆上的交点决定极点的纵坐标，小圆上的交点决定极点的横坐标。 Copyright © Shi Ping CUC

Design a lowpass Chebyshev-I filter to satisfy: Passband cutoff: Passband ripple: Stopsband cutoff: Stopband attenuation: Solution: 注：上述求解N的公式与下列MATLAB程序中用的公式虽然形式不同，但结果相同。 (digital signal processing\chap6\chebyshev_filter_design\afd_cheb1.m) 中求N的公式 Copyright © Shi Ping CUC

Analog-to-Digital Filter Transformations
Impulse invariance transformation Definition To design an IIR filter having a unit sample response h(n) that is the sampled version of the impulse response of the analog filter. That is T : Sampling interval Since this is a sampling operation, the analog and digital frequencies are related by Copyright © Shi Ping CUC

Properties Using All semi-infinite left strips of width map into Thus this mapping is not unique but a many-to-one mapping Since the entire left half of the s-plane maps into the unit circle, a causal and stable analog filter maps into a causal and stable digital filter. Copyright © Shi Ping CUC

Aliasing occurs if the filter is not exactly band-limited Frequency response If then There will be no aliasing. 教材P240 若增加抽样频率，即减小抽样周期T，则系统频率响应各周期延拓分量之间相距更远，因而可减小频率响应的混叠效应。当给定数字频率时，无法通过减小T降低混叠效应。因为w=ΩT，当T减小时，为保持w不变，Ω要同比例地增加。 To minimize the effects of aliasing, the T should be selected sufficiently small. If the filter specifications are given in digital frequency domain, we cannot reduce aliasing by selecting T. Copyright © Shi Ping CUC

The z-transform of is Compared with Conclusions: The pole in s-plane is mapped to the pole in z-plane The partial fraction expansion coefficient of is the same as that of The zeros in the two domains do not satisfy the same relationship Copyright © Shi Ping CUC

Advantages and disadvantages The digital filter impulse response is similar to that of a analog filter. This means we can get a good approximations in time domain. It is a stable design and that the frequencies and are linearly related. So a linear phase analog filter can be mapped to a linear phase digital filter. Due to the presence of aliasing, this method is useful only when the analog filter is essentially band-limited to a lowpass or bandpass filter in which there are no oscillations in the stopband. Copyright © Shi Ping CUC

Design procedure Given the digital lowpass filter specifications Choose T and determine the analog frequencies Design an analog filter using the specifications Using partial fraction expansion, expand into Given the digital lowpass filter specifications Wp, Ws, Rp, As, we want to determine H(z) by first designing an equivalent analog filter and then mapping it into the desired digital filter. Transform analog poles into digital poles to obtain the digital filter Copyright © Shi Ping CUC

Design a lowpass digital filter using a Butterworth prototype to satisfy Solution Let T=1, and then Design an analog filter using the specifications Using partial fraction expansion, expand into Copyright © Shi Ping CUC

Transform analog poles into digital poles to obtain the digital filter 部分分式展开：[r,p,k] = residue(b,a), r是系数Ak，p是极点sk k is empty if length(b) < length(a), that is k=[ ]. b为分子多项式系数（降幂），a为分母多项式系数（降幂）。 r = i i i i i i p = i i i i i i Copyright © Shi Ping CUC

Bilinear transformation Definition This is a conformal mapping that transforms the axis into the unit circle in the z-plane only once, thus avoiding aliasing of frequency components. This mapping is the best transformation method. Conformal：保角映射。将一平面映射或显示到另一平面上，其内切曲线的角保持不变 Copyright © Shi Ping CUC

Parameter c Keeping a good corresponding relationship between the analog filter and the digital filter in low frequencies. i.e in low frequencies Keeping a good corresponding relationship between the analog filter and the digital filter in a specific frequency (for example, in the cutoff frequency, ) Copyright © Shi Ping CUC

Properties Using , we obtain So Using , we obtain The imaginary axis maps onto the unit circle in a one-to-one fashion. Hence there is no aliasing in the frequency domain. The entire left half-plane maps into the inside of the unit circle. Hence this is a stable transformation. Copyright © Shi Ping CUC

Advantages and disadvantages It is a stable design; There is no aliasing; There is no restriction on the type of filter that can be transformed;. The frequencies and are not linearly related. So a linear phase analog filter cannot be mapped to a linear phase digital filter. Copyright © Shi Ping CUC

Design procedure Given the digital lowpass filter specifications Choose a value for T. We may set T=1 Prewarp the cutoff frequencies and ; that is Design an analog filter to meet the specifications Finally, set and simplify to obtain as a rational function in Copyright © Shi Ping CUC

Comparison of three filters Given the digital filter specifications: Using different prototype analog filters will give out different N and the minimum stopband attenuations. prototype Order N Stopband Att. Butterworth 6 15 dB Chebyshev-I 4 25 dB Elliptic 3 27 dB Clearly, the elliptic prototype gives the best design. However, if we compare their phase responses, then the elliptic design has the most nonlinear phase response in the passband. Bilinear transformation is used in this comparison. return Copyright © Shi Ping CUC

Frequency Transformations
Introduction The treatment in the preceding section is focused primarily on the design of digital lowpass IIR filters. If we wish to design a highpass or a bandpass or a bandstop filter, it is a simple matter to take a lowpass prototype filter and perform a frequency transformation. There are two approaches to perform the frequency transformation The treatment in the preceding section is focused primarily on the design of digital lowpass IIR filters. If we wish to design a highpass or a bandpass or a bandstop filter, it is a simple matter to take a lowpass prototype filter and perform a frequency transformation. There are two methods to perform the frequency transformation Frequency transformations in the analog domain To perform the frequency transformation in the analog domain and then to convert the analog filter into a corresponding digital filter by mapping of the s-plane into the z-plane. Frequency transformations in the digital domain To convert the analog lowpass filter into a digital lowpass filter and then to transform the digital lowpass filter into the desired digital filter by a digital transformation. Frequency transformations in the analog domain Frequency transformations in the digital domain Copyright © Shi Ping CUC

Approach 1 Analog lowpass filter Frequency transformation s → s Filter transformation s → z Designed IIR filter Approach 2 Analog lowpass filter Filter transformation s → z Frequency transformation z → z Designed IIR filter Approach1 的公式见教材P291 Approach2 的公式见教材P301 Copyright © Shi Ping CUC

Frequency transformations in the digital domain the given prototype lowpass digital filter the desired frequency-selective digital filter Define a mapping of the form Such that To do this, we simply replace everywhere in by the function Copyright © Shi Ping CUC

Given that is a stable and causal filter, we also want to be stable and causal. This imposes the following requirements: The unit circle of the z-plane must map onto the unit circle of the Z-plane The inside of the unit circle of the z-plane must also map onto the inside of the unit circle of the Z-plane. must be a rational function in so that is implementable. Copyright © Shi Ping CUC

Let and be the frequency variables of and , respectively. That is Then Hence the is an all-pass function By choosing an appropriate order N and the coefficients , we can obtain a variety of mappings Copyright © Shi Ping CUC

Design procedure Determine the specifications of the digital prototype lowpass filter; Determine the specifications of the analog prototype lowpass filter; Design the analog prototype lowpass filter; Transform the analog prototype lowpass filter into a digital prototype lowpass filter using bilinear transformation; Perform the frequency transformation in digital domain to obtain the desired frequency-selective filters. Copyright © Shi Ping CUC

Examples Given the specifications of Chebyshev-I lowpass filter and its system function 该例题的程序在 (digital signal processing\chap6\frequency_transformation\example1.m) Design a highpass filter with the above tolerances but with passband beginning at Copyright © Shi Ping CUC

Using the Chebyshev-I prototype to design a highpass digital filter to satisfy Solution Determine the specifications of the digital prototype lowpass filter Choose the passband frequency with a reasonable value: 该例题的程序在 (digital signal processing\chap6\frequency_transformation\highpass_example.m) Determine the stopband frequency by Copyright © Shi Ping CUC

Design an analog Chebyshev-I prototype lowpass filter to satisfy the specification: Transform the analog prototype lowpass filter into a digital prototype lowpass filter using bilinear transformation Copyright © Shi Ping CUC

Using the Chebyshev-I prototype to design a bandpass digital filter to satisfy Solution Determine the specifications of the digital prototype lowpass filter 该例题的程序在 (digital signal processing\chap6\frequency_transformation\bandpass_example.m) Choose the passband frequency with a reasonable value: Copyright © Shi Ping CUC

Determine the specifications of the analog prototype lowpass filter Set T = 1 and prewarp the cutoff frequencies Design an analog Chebyshev-I prototype lowpass filter to satisfy the specification: Copyright © Shi Ping CUC

Transform the analog prototype lowpass filter into a digital prototype lowpass filter using bilinear transformation Perform the frequency transformation in digital domain to obtain the desired digital bandpass filter Copyright © Shi Ping CUC

Using the Chebyshev-I prototype to design a bandstop digital filter to satisfy Solution Determine the specifications of the digital prototype lowpass filter 该例题的程序在 (digital signal processing\chap6\frequency_transformation\bandstop_example.m) Choose the passband frequency with a reasonable value: Copyright © Shi Ping CUC

Determine the specifications of the analog prototype lowpass filter Set T = 1 and prewarp the cutoff frequencies Design an analog Chebyshev-I prototype lowpass filter to satisfy the specification: Copyright © Shi Ping CUC

Transform the analog prototype lowpass filter into a digital prototype lowpass filter using bilinear transformation Perform the frequency transformation in digital domain to obtain the desired digital bandstop filter return Copyright © Shi Ping CUC

Chapter 7 IIR Filter Design

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Chapter 7 IIR Filter Design

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