1、11)Transfer function classificationlTransfer function classification Based on Magnitude CharacteristicslTransfer function classification based on Phase CharacteristicslTypes of Linear-Phase Transfer FunctionChat 7 LTI Discrete-Time System in the Transform Domain27.1 Transfer function Classification
2、Based on Magnitude CharacteristicsIn the case of digital transfer functions with frequency-selective frequency responses there are two types of classifications1)Classification based on the shape of the magnitude function)(jeH2)Classification based on the form of the phase function)(31)Ideal magnitud
3、e response A digital filter designed to pass signal components of certain frequencies without distortion should have a frequency response equal to one at these frequencies,and should have a frequency response equal to zero at all other frequenciesDefinition7.1.1 Digital Filters with Ideal Magnitude
4、Response4 The range of frequencies where the 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-bandExplanation Has a zero phase everywhere(in all frequencies)5Diagrammatical Representatio
5、nFrequency responses of the four popular types of ideal digital filters with real impulse response coefficients are shown below:Lowpass HighpassBandpass Bandstop6 Ideal lowpass filter a)Analytical Expressionb)Characteristics Not absolutely summable,hence,the corresponding transfer function is not BI
6、BO stableEarlier in the course we derived the inverse DTFT of the frequency response of the ideal lowpass filter)(jLPeH(7.1)nnnnhcLP,)sin(Not causal and is of doubly infinite length7The reason for its infinite length response is that have“brick wall”frequency responsesResolve method To develop stabl
7、e and realizable transfer functions,the ideal frequency response specifications are relaxed by including a transition band between the passband and the stopband This permits the magnitude response to decay slowly from its maximum value in the passband to the zero value in the stopband Ideal lowpass
8、filter 8 Moreover,the magnitude of response is allowed to vary by a small amount both in passband and stopbandLowpass filter 7.1.2 Bounded real transfer function91)DefinitionA causal stable real-coefficient transfer function H(z)is defined as a bounded real(BR)transfer function if 2)Characteristics(
9、7.2)H(e j)1 for all values of Let xn and yn denote,respectively,theinput and output of a digital filter characterized by a BR transfer function H(z)with X(ej)and Y(ej)denoting their DTFTs10|H(ej)|1lThen the condition implies that(7.5)22()()jjY eX e Integrating the above from to ,and applying Parseva
10、ls relation we get(7.6)nnnxny227.1.2 Bounded real transfer function11Example Consider the causal Stable IIR transfer fuction(7.3)10 ,1)(1zKzHwhere K is a real constant Its square-magnitude function is given by(7.4)cos2)1(|)()()(2212KzHzHeHjezj12222()(1)2cosjKHe2 The value of|()|is obtain when 2 cos
11、in the denominator is a and the value is obtamaximin whummaximumminen 2 cosimumminimum is a.jH eFor 0,value of 2 cos is equal tmaximumminimumo 2 at 0,and value is-2 at.22222 Thus,for 0,the value of|()|is equal to/(maximumminimu1-)at=0 and the value is equal to/(1m)at=jH eKK 13 On the other hand,for
12、,the maximum value of is equal to at and the minimum value is equal to at 0cos22200 Here,the maximum value of is equal to at and the minimum value is equal to at22(1)K2)(jeH22(1)K Hence,the maximum value can be made equal to 1 by choosing ,in which case the minimum value becomes(1|)K 22(1|)(1|)222()
13、(1)2cosjKHe14 Hence,is a BR function for10 ,1)(1zKzH(1|)K lPlots of the magnitude function for5.0Example 7.1.3 Allpass transfer function151)DefinitionAn llR transfer function A(z)with unity magnitude response for all frequencies,i.e.,is called an all pass transfer function2)Analytical descriptionAn
14、M-th order causal real-coefficient all pass transfer function is of the formallfor,1|)(|2jeA(7.7)(7.8)MMMMMMMMMzdzdzdzzdzddzA111111111.)(16l If we denote the denominator polynomials of AM(z)as DM(z)then it follows that A(z)can be written as:l Note from the above that if is a pole of a real coefficie
15、nt all pass transfer function,then it has a zero atjrez 1/jzr e3)Zero and pole Characteristics(7.9)MMMMMzdzdzdzD11111.)(7.10)()()(zDzDzMMMMzA17.1.3 Allpass transfer function17l It implies that the poles and zeros of a real-coefficient all pass function exhibit image-symmetry in the z-planel The nume
16、rator of a real-coefficient all pass transfer function is said to be the mirror image polynomial of the denominator,and vice versa,then we have(7.10)()()(zDzDzMMMMzA132132132.018.04.014.018.02.0)(zzzzzzzA7.1.3 Allpass transfer function184)Why is the AM(z)is the allpass transfer functionl Now,the pol
17、es of a causal stable transfer function must lie inside the unit circle in the z-plane.Hence,all zeros of a causal stable allpass transfer function must lie outside the unit circle in a mirror-image symmetry with its poles situated inside the unit circle)()(11)(zDzDzMMMMzA Therefore)()()()(111)()(zD
18、zDzzDzDzMMMMMMMMzAzA Hence1)()(|)(|12jezMMjMzAzAeA7.1.3 Allpass transfer function195)The phase of the allpass transfer functionFigure below shows the principal value of the phase of the 3rd-order allpass function(7.11)32132132.018.04.014.018.02.0)(zzzzzzzA00.20.40.60.81-4-2024/Phase,degreesPrincipal
19、 value of phase Note the discontinuity by the amount of in the phase 2)(7.1.3 Allpass transfer function20 Note:The unwrapped phase function is a continuous function of 7.1.3 Allpass transfer function216)Properties(1)A causal stable real-coefficient allpass transfer function is a lossless bounded rea
20、l(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:(7.20)1zfor ,11zfor ,11zfor ,1)(zA7.1.3 Allpass transfer function22(3)Let denote the group delay functionof an allpass filter A(z),i.e.,)()()(cdd
21、The unwrapped phase function of a stable allpass function is a monotonically decreasing function of so that is everywhere positive in the range)(c)(0 The group delay of an M-th order stable real-coefficient allpass transfer function satisfies:(7.21)0()dM 7.1.3 Allpass transfer function237)Simple App
22、licationA simple but often used application of an allpass filter is as a delay equalizerDelay equalizer(均衡均衡)Implementationl Let G(z)be the transfer function of a digital filter designed to meet a prescribed magnitude responsel The nonlinear phase response of G(z)can be corrected by cascading it wit
23、h an allpass filter A(z)so that the overall cascade has a constant group delay in the band of interest7.1.3 Allpass transfer function24Overall group delay is the given by the sum of the group delays of G(z)and A(z)G(z)A(z)Since ,we have1)(jeA|)(|)()(|jjjeGeAeG7.1.3 Allpass transfer function25Left fi
24、gures shows the group delay of a 4th order filter with the specifications,35,1 ,3.0dBdBsppRight figure shows the group delay of the original filter cascaded with an 8th order allpass designed to equalize the group delay in the passband7.1.3 Allpass transfer function7.2 Transfer function classificati
25、on based on Phase Characteristics267.2.1 zero-phase Transfer-function1)Introduction l In many applications,it is necessary that the digital filter designed does not distort the phase of the input signal components with frequencies in the passbandl One way to avoid any phase distortion is to make the
26、 frequency response of the filter real and with a zero phase characteristic272)Zero-phase Transfer functionl However,it is not possible to design a causal digital filter with a zero phasel zero-phase filtering can be very simply implemented by relaxing the causality requirementl One zero-phase filte
27、ring scheme is sketched belowxnvnunwnH(z)H(z)un=v-n,yn=w-n7.2.1 zero-phase Transfer-function28l It is easy to verify the above system with zero phase response in the frequency domainCombining the above equations we get Let ,and denote the DTFTs of ,and ,respectively.We have)(jeX)(jeW)(jeV)(jeU)(jeYn
28、xnvnunynw )()()()()()(jjjjjjeUeHeWeXeHeV*()()()()jjjjU eV eY eW e*2*()()()()()()()()()()()jjjjjjjjjjjY eWeHeUeHeV eHeH eX eH eX e7.2.1 zero-phase Transfer-function297.2.2 Linear-phaseTransfer-function In the case of a causal transfer function with a nonzero phase response,the phase distortion can be
29、 avoided by ensuring that the transfer function has a unity magnitude and a linear-phase characteristic in the frequency band of interest 1)Importance of the Linear-phase filter2)Description of the Linear-phase filter The most general type of a filter with a linear phase has a frequency response byD
30、jjeeH)(303)The Effect of the Linear-phase Filter on the Input Signal If it is desired to pass input signal components in a certain frequency range undistorted in both magnitude and phase,then the transfer function should exhibit a unity magnitude response and a linear-phase response in the band of i
31、nterestNote also1)(jeHD)(The output yn of this filter to an inputis then given bynjAenx)(DnjnjDjAeeAeny7.2.2 Linear-phaseTransfer-function314)Diagrammatic description of the Linear-phase filterl Figure right shows the frequency response if a lowpass filter with a linear-phase characteristic in the p
32、assbandl Since the signal components in the stop band are blocked,the phase response in the stopband can be of any shape7.2.2 Linear-phaseTransfer-function325)Examplel Determine the impulse response of an ideal lowpass filter with a linear phase response:l Applying the frequency-shifting property of
33、 the DTFT to the impulse response of an ideal zero-phase lowpass filter we arrive at)(jLPeHccnjoe,00,nnnnnnhoocLP,)()(sin7.2.2 Linear-phaseTransfer-function33l Noncausality:As before,the above filter is noncausal and of doubly infinite length,and hence,unrealizablel Resolve Method:By truncating the
34、impulse response to a finite number of terms,a realizable FIR approximation to the ideal lowpass filter can be developedl Key Point:The truncated approximation may or may not exhibit linear phase,depending on the value of no chosen(to ensuring the symmetry)7.2.2 Linear-phaseTransfer-function34l Exam
35、ple:If we choose no N/2 with N a positive integer,the truncated and shifted approximationwill be a length N+1 causal linear-phase FIR filterNnNnNnnhcLP0,)2/()2/(sin7.2.2 Linear-phaseTransfer-function35l Diagrammatic Description:Figure below shows the filter coefficients obtained using the function s
36、inc for two diffrent values of NN=12N=137.2.2 Linear-phaseTransfer-function36l Frequency Response)()(2/0LPNjNnnjLPjLPHeenheH Because of the symmetry of the impulse response coefficients as indicated in the two figures,the frequency response of the truncated approximation can be expressed as:where ,c
37、alled the zero-phase response or amplitude response,is a real function of)(LPH7.2.2 Linear-phaseTransfer-function377.2.3 Minimum-Phase and Maximum-Phase Transfer Functions1)DefinitionConsider the two 1st-order transfer functions:l Pole Location:Both transfer functions have a pole inside the unit cir
38、cle at the same location z=-a and are stablel Zero location:But the zero of H1(z)is inside the unit circle at z=-b,whereas,the zero of H2(z)is at z=-1/b,Situated in a mirror image symmetry11121bazHzHazbzazbz,)(,)(38l Zero and pole figurel Magnitude Characteristics:Both transfer functions have an ide
39、ntical magnitude function H1(z)H2(z)()()()(122111zHzHzHzH7.2.3 Minimum-Phase and Maximum-Phase Transfer Functions39l phase functionsFigure right shows the unwrapped phase responses of the two transfer functions for a=0.8 and b-0.5H2(z)has an excess phase lag with respect to H1(z)cossin1cossin11tanta
40、n)(argabjeHcossin1cos1sin12tantan)(argabbjeH7.2.3 Minimum-Phase and Maximum-Phase Transfer Functions40l The Explanation of the Phase The excess phase lag property of with respect to can also be explained by observing that we can write)(1zH)(2zH )()(211)(1zAzHbzbzazbzazbzzHWhere is a stableallpass fu
41、nction)()1()(bzbzzA7.2.3 Minimum-Phase and Maximum-Phase Transfer Functions41l Phase Relationship The phase functions of and are thus related through)(1zH)(2zH)(arg)(arg)(arg12jjjeAeHeH As the unwrapped phase function of a stable first-order allpass function is a negative function of ,it follows fro
42、m the above that has indeed an excess phase lag with respect to)(2zH)(1zH7.2.3 Minimum-Phase and Maximum-Phase Transfer Functions42 Decomposition of the Transfer Function into Minimum-Phase and allpass Generalizing the above result,let be a causal stable transfer function with all zeros inside the u
43、nit circle and let H(z)be another causal stable transfer function satisfying)(zHm|()|()|jjmH eH e These two transfer functions are then related through where A(z)is a causal stable allpass function)()()(zAzHzHm7.2.3 Minimum-Phase and Maximum-Phase Transfer Functions43Phase Relationship:H(z)has an ex
44、cess phase lag with respect to Hm(z)A causal stable transfer function with all zeros inside the unit circle is called a minimum-phase transfer functionA causal stable transfer function with zeros inside and outside the unit circle is called a mixed-phase transfer functionl Minimum-phase transfer fun
45、ction)(arg)(arg)(argjjmjeAeHeH7.2.3 Minimum-Phase and Maximum-Phase Transfer Functions44l EXampleConsider the mixed-phase transfer function We can rewrite H(z)as)5.01)(2.01()4.0)(3.01(2)(1111zzzzzHfunction Allpass11function phase-Minimum 11114.014.0)5.01)(2.01()4.01)(3.01(2)(zzzzzzzH7.2.3 Minimum-Ph
46、ase and Maximum-Phase Transfer Functions457.3 Types of Linear-Phase Transfer Function7.3.1 Frequency Response for a FIR Filter with a Linear-Phase1)Linear Phase Requirements for Impulse Response l Even Symmetry:(7.39)()()()jj cH eeHIf H(z)is to have a linear-phase,its frequency response must be of t
47、he formLet 0()NnnH zh n z46l Discussion of the j|()|and H(e)jH eWhere c and are constants,and ,called the amplitude response,also called the zero-phase response,is a real function of .)(H)()(HHSince ,the amplitude response is then either an even function or an odd function of ,i.e.|()|()|jH eH For a
48、 real e response,the magnitude response is an even function of|()|jH e7.3.1 Frequency Response for a FIR Filter with a Linear-phase47()We have ()(j cjHeH e*The frequency response satisfies the relation ()()jjH eHe()()or,equivalently,the relation ()()(7.40)j cjceHeH If()is an even function,then the a
49、boverelation leads to jjjH eee()implying that either 0 or From ()()jj cH eeH 7.3.1 Frequency Response for a FIR Filter with a Linear-phase48Making a change of variable,werewrite the above equation asNnReplacing in the previous equation we get-with(7.43)(7.42)(0)(cjNehH(7.41)(0)()(get weabove thei of
50、 value thengSubstitutincjNnjjcenheHeHn(0(-)Nj c N-n)nHh N-n e 7.3.1 Frequency Response for a FIR Filter with a Linear-phase49Thus,the FIR filter with an even amplitude response will have a linear phase if it has a symmetric impulse responseImportant conclusion)()(nNcjncjenNhenh(7.44)Nn0 ,nNhnh)()(HH
