数字信号处理英文版课件：Chap 7 LTI Discrete-Time Systems in the Transform Domain

1、 Chapter 7 LTI Discrete-Time Systems in the Transform Domain Transfer function classification based on magnitude characteristics Lowpass, Highpass,etc. transfer functions Allpass transfer functions Types of linear-phase FIR transfer functions Transfer function classification based on phase character

2、istics Minimum-phase transfer functions Linear-phase transfer functionsDSP group 2007 chap7-ed12 7.1 Transfer function classification based on magnitude characteristicsOr For stable systems:DSP group 2007 chap7-ed13 7.1.1 Digital Filters with Ideal Magnitude Response The range of frequencies where t

3、he frequency response takes the value of one is called the pass-band The range of frequencies where the frequency response takes the value of zero is called the stop-band Has a zero phase everywhere (in all frequencies). There are four popular types of ideal digital filters with real impulse respons

4、e coefficients :DSP group 2007 chap7-ed14 7.1.1 Digital Filters with Ideal Magnitude ResponseccHHP( e j )10ccHLP( e j )10HBP( e j )10c2c1c1c2HBS( e j )10c2c1c1c2LowpassBandstop HighpassBandpassDSP group 2007 chap7-ed15 7.1.1 Digital Filters with Ideal Magnitude ResponseccHLP( e j )10 Impulse respons

5、e Characteristics Doubly infinite length and not causal; Cannot be realized by an LTI filter with a transfer function of finite order .DSP group 2007 chap7-ed16 7.1.1 Digital Filters with Ideal Magnitude Response Other types of filters have the same characteristics as the ideal lowpass digital filte

6、r. Transition band was relaxed; The magnitude of response is allowed to vary by a small amount both in passband and stopband. Methods to develop stable and realizable digital filters: DSP group 2007 chap7-ed17 7.1.2 Bounded real transfer function A causal stable real-coefficient transfer function H(

7、z) is defined as a bounded real (BR) transfer function if|H (e j)| 1 for all values of (7.2)Example 7.1 Consider the causal stable IIR transfer function:where K, is a real constant.DSP group 2007 chap7-ed18 7.1.2 Bounded real transfer function Its square-magnitude function isSolution: Maximum value

8、of square-magnitude function isFor 0:DSP group 2007 chap7-ed19 7.1.2 Bounded real transfer functionLowpass filterHighpass filterDSP group 2007 chap7-ed110 7.1.2 Bounded real transfer function Property of BR transfer functions: Let xn and yn are the input and output of a digital filter characterized

9、by a BR transfer function H(z) respectively, their DTFTs are X(ej) and Y(ej), then Using Parsevals relation, BR system is a passive systemDSP group 2007 chap7-ed111 7.1.3 Allpass transfer function DefinitionAn llR transfer function A(z) with unity magnitude response for all frequencies , i . e . , A

10、n M-th order causal real-coefficient allpass transfer function is of the formDSP group 2007 chap7-ed112 7.1.3 Allpass transfer function denote:Then : Or in zero and pole form:mirror-image polynomial13 7.1.3 Allpass transfer functionExample All zeros of a causal stable allpass transfer function must

11、lie outside the unit circle in a mirror- image symmetry with its poles situated inside the unit circle.DSP group 2007 chap7-ed114 7.1.3 Allpass transfer functionproperties The phase of the allpass transfer function00.20.40.60.81-4-2024w / Phase, degreesPrincipal value of phase of A3(z) Note: the dis

12、continuity by the amount of 2 in the phase (). A causal stable allpass filterDSP group 2007 chap7-ed115 7.1.3 Allpass transfer functionpropertiesThe unwrapped phase function c() is a continuous function of : Note1: The unwrapped phase function c() is monotonically decreasing in the range 0 .DSP grou

13、p 2007 chap7-ed116 7.1.3 Allpass transfer functionproperties Note 2: The unwrapped phase function c() is nonpositive, i.e. c() 0 in the range 0 . Note 3: The group delay is always positive in the range 0 .Eg: 1-st order allpass filter P311DSP group 2007 chap7-ed117 7.1.3 Allpass transfer functionpro

14、perties(1) A causal stable real-coefficient allpass transfer function is a lossless bounded real (LBR) function or, equivalently , a causal stable allpass filter is a lossless structure(2) The magnitude function of a stable allpass function A(z) satisfies:DSP group 2007 chap7-ed118 7.1.3 Allpass tra

15、nsfer functionproperties(3) The group delay function g() of a causal stable real-coefficient allpass filter AM(z) is everywhere positive in the range 0 , and satisfies,DSP group 2007 chap7-ed119 7.1.3 Allpass transfer function- application delay equalizerG(z)A(z) transfer function of the total syste

16、m：H(z)= G(z) A(z) Frequency response of the overall system：H( e j)= G(e j) A(e j) h( )= g( ) + a( )Causal stable systemAllpassDSP group 2007 chap7-ed120 7.1.3 Allpass transfer function Example of application4th order filter with the specifications:P314 Fig 7.8DSP group 2007 chap7-ed121 7.2 Transfer

17、function classification based on Phase Characteristics7.2.1 zero-phase Transfer-functionAn ideal lowpass filter with frequency response:ccHLP( e j )10The phase function is:Linear-phaseDSP group 2007 chap7-ed122 7.2.2 Linear-phase Transfer-function Frequency response magnitude response: phase respons

18、e: group delay: relation between the output yn and input xn :DSP group 2007 chap7-ed123 7.2.3 Minimum-Phase and Maximum -Phase Transfer Functions DefinitionsConsider the two 1st-order transfer functions :H1(z)H2(z)DSP group 2007 chap7-ed124 7.2.3 Minimum-Phase and Maximum -Phase Transfer Functions s

19、quare-magnitude response phase responseDSP group 2007 chap7-ed125 7.2.3 Minimum-Phase and Maximum -Phase Transfer Functions unwrapped phase response curvesa = 0.8 and b 0.5/Phase in radiansH2(z) has an excess phase lag with respect to H1(z) relation between H1(z) and H2(z) Stable allpass systemDSP g

20、roup 2007 chap7-ed126 7.2.3 Minimum-Phase and Maximum -Phase Transfer Functions Conclusion: Let Hmin(z) be a causal stable transfer function with all zeros inside the unit circle and let H(z) be another causal stable transfer function with the same magnitude response as Hmin(z) , i.eThen Where A(z)

21、is a stable allpass transfer function.DSP group 2007 chap7-ed127 7.2.3 Minimum-Phase and Maximum -Phase Transfer Functions Conclusion:H(z) has an excess phase lag with respect to Hmin(z). Minimum-phase transfer function A causal stable transfer function with all zeros inside the unit circle is calle

22、d a minimum-phase transfer function. Maximum-phase transfer function A causal stable transfer function with all zeros outside the unit circle is called a maximum-phase transfer function.DSP group 2007 chap7-ed128 7.2.3 Minimum-Phase and Maximum -Phase Transfer Functions Mixed-phase transfer function

23、 A causal stable transfer function with zeros inside and outside the unit circle is called a mixed-phase transfer function.Example 7.4 Consider the mixed-phase transfer function:DSP group 2007 chap7-ed129 7.2.3 Minimum-Phase and Maximum -Phase Transfer Functions exampleRewrite it as: Properties of m

24、inimum-phase system (compared with the systems of the same magnitude response): the smallest group delay; the smallest phase lag; the smallest energy lag.DSP group 2007 chap7-ed130 lossless systemG(z)Gc(z)Causal stable system the overall system ：A(z)= G(z) Gc(z) Frequency response of the overall sys

25、tem：A( e j)= G(e j) Gc (e j)A(z)；allpass|A( e j)|=1 magnitude response of the overall system ： 7.2.3 Minimum-Phase and Maximum -Phase Transfer Functions example31 7.1.3 Allpass transfer function Example of applicationEg.A3: Assume Let： causal and stable32 7.1.3 Allpass transfer function Example of a

26、pplicationAllpassMinimum-phase systemInverse systemDSP group 2007 chap7-ed133 7.3 Types of Linear-Phase Transfer Function7.3.1 Frequency Response for a FIR Filter with a Linear-PhaseWhere: c and are constants, A() amplitude response/zero-phase response; it is a real function of .DSP group 2007 chap7

27、-ed134 7.3.1 Frequency Response for a FIR Filter with a Linear-Phase For real impulse response hnOr Or P323 DSP group 2007 chap7-ed135 7.3.1 Frequency Response for a FIR Filter with a Linear-PhaseThe FIR filter with an even amplitude response will have a linear phase if it has a symmetric impulse re

28、sponseA FIR filter with an odd amplitude response will have linear phase if it has an antisymmetric impulse responseDSP group 2007 chap7-ed136 7.3.1 Frequency Response for a FIR Filter with a Linear-Phase Four Types of Linear-Phase FIR FilterN evenN oddType 1Type 2Type 3Type 4Where : 0 n N DSP group

29、 2007 chap7-ed137 7.3.1 Frequency Response for a FIR Filter with a Linear-Phase Type 1 Linear-Phase FIR Filter0 n N and N is even.Type 1: N = 8DSP group 2007 chap7-ed138 7.3.1 Frequency Response for a FIR Filter with a Linear-Phase Type 1 phase function group delay magnitude function:DSP group 2007

30、chap7-ed139 7.3.1 Frequency Response for a FIR Filter with a Linear-Phase Type 2 Linear-Phase FIR Filter0 n N and N is odd.Type 2: N = 7DSP group 2007 chap7-ed140 7.3.1 Frequency Response for a FIR Filter with a Linear-Phase Type 2 phase function group delay magnitude functionDSP group 2007 chap7-ed

31、141 7.3.1 Frequency Response for a FIR Filter with a Linear-Phase Type 3 Linear-Phase FIR Filter0 n N and N is even.Type 3: N = 8DSP group 2007 chap7-ed142 7.3.1 Frequency Response for a FIR Filter with a Linear-Phase Type 3 phase function group delay magnitude function:DSP group 2007 chap7-ed143 7.

32、3.1 Frequency Response for a FIR Filter with a Linear-Phase Type 4 Linear-Phase FIR Filter0 n N and N is odd.Type 2: N = 7DSP group 2007 chap7-ed144 7.3.1 Frequency Response for a FIR Filter with a Linear-Phase Type 4 phase function group delay magnitude function45 7.3.1 Frequency Response for a FIR

33、 Filter with a Linear-Phase ExampleExample A: modified moving-average system:00.20.40.60.8100.20.40.60.81w/pMagnitudemodified filtermoving-averageconventionalmodified46 7.3.2 Zero Locations of Linear-Phase FIR Transfer FunctionsFor type 1 and 2 0 n N and N is even/ odd.DSP group 2007 chap7-ed147 7.3.2 Zero Locations of Linear-Phase FIR