1、1Chapter7 LTI Discrete-Time Systems in the Transform Domain Transfer Function ClassificationTypes of Linear-Phase Transfer FunctionsSimple Digital Filters 2Types of Transfer FunctionsThe time-domain classification of a digital transfer function based on the length of its impulse response sequence:-

2、Finite impulse response (FIR) transfer function.- Infinite impulse response (IIR) transfer function.3Types of Transfer FunctionsIn the case of digital transfer functions with frequency-selective frequency responses, there are two types of classifications:(1) Classification based on the shape of the

3、magnitude function |H(ei)|. (2) Classification based on the form of the phase function ().47.1 Transfer Function Classification Based on Magnitude CharacteristicsDigital Filters with Ideal Magnitude ResponsesBounded Real Transfer FunctionAllpass Transfer Function57.1.1 Digital Filters with Ideal Mag

4、nitude ResponsesA digital filter designed to pass signal components of certain frequencies without distortion should have a frequency response equal to 1 at these frequencies, and should have a frequency response equal to 0 at all other frequencies.6Digital Filters with Ideal Magnitude ResponsesThe

5、range of frequencies where the frequency response takes the value of 1 is called the passband.The range of frequencies where the frequency response takes the value of 0 is called the stopband.7Digital Filters with Ideal Magnitude ResponsesMagnitude responses of the four popular types of ideal digita

6、l filters with real impulse response coefficients are shown below:8Digital Filters with Ideal Magnitude ResponsesThe frequencies c, c1, and c2 are called the cutoff frequencies.An ideal filter has a magnitude response equal to 1 in the passband and 0 in the stopband, and has a 0 phase everywhere.9Di

7、gital Filters with Ideal Magnitude ResponsesEarlier in the course we derived the inverse DTFT of the frequency response HLP(ej)of the ideal lowpass filter: hLPn=sincn/n, - n0, ( |H(ej)|2 )max = K2/(1- )2 | =0 ( |H(ej)|2 ) min = K2/(1+ )2 | =On the other hand, for 0, (2cos )max = -2 | = (2cos )min =

8、2 | =0Here, ( |H(ej)|2 )max = K2/(1+ )2 | = ( |H(ej)|2 )min = K2/(1- )2 | = 019Bounded Real Transfer FunctionsHence,is a BR function for K(1-),Plots of the magnitude function for =0.5 with values of K chosen to make H(z) a BR function are shown on the next page.20Bounded Real Transfer FunctionsLowpa

9、ss filterHighpass filter217.1.3 Allpass Transfer FunctionThe magnitude response of allpass system satisfies: |A(ej)|2=1,for all .The H(z) of a simple 1th-order allpass system is:Where a is real, and .Or a is complex ,the H(z) should be:22Allpass Transfer Functionone real pole one complex pole 23Allp

10、ass Transfer FunctionTwo order allpass transfer function ploes：zeros：24Allpass Transfer FunctionGeneralize, the Mth-order allpass system is:If we denote polynomial:So:25Allpass Transfer FunctionThe numerator of a real-coefficient allpass transfer function is said to be the mirror-image polynomial of

11、 the denominator, and vice versa. We shall use the notation to denote the mirror-image polynomial of a degree-M polynomial DM(z) , i.e.,26Allpass Transfer FunctionThe expressionimplies that the poles and zeros of a real-coefficient allpass function exhibit mirror-image symmetry in the z-plane.27Allp

12、ass Transfer FunctionTo show that |AM(ej)|=1 we observe that:Therefore:Hence:28Allpass Transfer FunctionProperties: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.The magnitu

13、de function of a stable allpass function A(z) satisfies:29Allpass Transfer Function(3) Let g() denote the group delay function of an allpass filter A(z), i.e.,The unwrapped phase function c() of a stable allpass function is a monotonically decreasing function of w so that g() is everywhere positive

14、in the range 0 w p.30Application of allpass systemAny causal stable system can be denoted as: H(z)=Hmin(z)A(z)Where Hmin(z) is a minimum phase-delay system.Use allpass system to help design stable filters.Use allpass system to help design linear phase system.A simple example.(P361,Fig7.7)317.2 Trans

15、fer Function Classification Based on Phase Characteristic1、The phase delay will cause the change of signal waveform時(shí)間 tAmp原始信號(hào)時(shí)間 t幅度相移90o時(shí)間 t幅度相移 180o322、The nonlinearity of system phase delay will cause the signal distortionf1 f2f時(shí)延f1 f2f時(shí)延f1 f2f()f1 f2f()Time delay of signal is depended on systemp

16、hase characteristic333、If we ignore the phase information, then輸入波形DFT變換忽略相位信息IDFT變換輸出波形34linear phase requirement：4、The linear phase FIR filter design group delay357.2 Transfer Function Classification Based on Phase Characteristic Zero-Phase Transfer FunctionLinear-Phase Transfer FunctionMinimum-Ph

17、ase and Maximum-Phase Transfer Functions367.2.1 Zero-Phase Transfer FunctionOne way to avoid any phase distortions is to make the frequency response of the filter real and nonnegative,to design the filter with a zero phase characteristic. But for a causal digital filter it is impossible.37Zero-Phase

18、 Transfer FunctionOnly for non-real-time processing of real-valued input signals of finite length, the zero phase condition can be met.Let H(z) be a real-coefficient rational z-transform with no poles on the unit cycle, then F(z)=H(z)H(z-1) has a zero phase on the unit cycle.38Zero-Phase Transfer Fu

19、nctionPlease look at book P362.xnvnunwnH(z)H(z)un=v-n, yn=w-nThe function filtfilt implements the above zero-phase filtering scheme. Please look at book P412P7.5.7.2.2 Linear-Phase Transfer FunctionLinear-phase transfer function: Where k is a real constant in the frequency band of interest .Example

20、7.3 Linear-Phase LPF7.2.2 Linear-Phase Transfer Function The above filter is noncausal and of doubly infinite length and, hence, is unrealizable. By truncating it, we get (N+1) points :7.2.2 Linear-Phase Transfer FunctionWhere , called the zero-phase response or amplitude response, is a real functio

21、n of .Obviously :7.2.3 Minimum-Phase and Maximum-Phase Transfer FunctionsConsider the two 1st-order transfer functions: Both transfer functions have a pole inside the unit circle at the same location z=-a and are stable7.2.3 Minimum-Phase and Maximum-Phase Transfer Functions But the zero of H1(z) is

22、 inside the unit circle at z=-b, whereas, the zero of H2(z) is at z=-1/b situated in a mirror-image symmetryH1(z)H2(z)Fig 7.127.2.3 Minimum-Phase and Maximum-Phase Transfer FunctionsHowever, both transfer functions have an identical magnitude function as7.2.3 Minimum-Phase and Maximum-Phase Transfer

23、 FunctionsThe corresponding phase functions are The unwrapped phase responses of the two transfer functions for a = 0.8 and b = -0.5Fig 7.137.2.3 Minimum-Phase and Maximum-Phase Transfer FunctionsFrom this figure it follows that H2(z) has an excess phase lag with respect to H1(z)The excess phase lag

24、 property of H2(z) with respect to H1(z) can also be explained by observing that we can rewrite7.2.3 Minimum-Phase and Maximum-Phase Transfer FunctionsWhere A(z)=(bz+1)/(z+b) is a stable allpass functionThe phase functions of H1(z) and H2(z) are thus related through argH2(ej)= argH1(ej)+ argA(ej)As

25、the unwrapped phase function of a stable first-order allpass function is a negative function of , it follows from the above that H2(z) has indeed an excess phase lag with respect to H1(z) 7.2.3 Minimum-Phase and Maximum-Phase Transfer FunctionsGeneralizing the above result, let Hm(z) be a causal sta

26、ble transfer function with all zeros inside the unit circle and let H(z) be another causal stable transfer function satisfying |H(ej)|= |Hm(ej)|These two transfer functions are then related through H(z)=Hm(z)A(z) where A(z) is a causal stable allpass function7.2.3 Minimum-Phase and Maximum-Phase Tra

27、nsfer FunctionsThe unwrapped phase functions of Hm(z) and H(z) are thus related through argH (ej)= argHm(ej)+ argA(ej)H(z) has an excess 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 function7.2.3 Minimum-Pha

28、se and Maximum-Phase Transfer FunctionsA causal stable transfer function with all zeros outside the unit circle is called a maximum-phase transfer functionA causal stable transfer function with zeros inside and outside the unit circle is called a mixed-phase transfer function7.2.3 Minimum-Phase and

29、Maximum-Phase Transfer FunctionsExample_7.4: consider the mixed-phase transfer function We can rewrite H(z) as7.2.3 Minimum-Phase and Maximum-Phase Transfer FunctionsProperties: Since the group delay functions are always nonnegative, so ,it follows that1. For the causal stable transfer functions7.2.

30、3 Minimum-Phase and Maximum-Phase Transfer Functions 2. Let hmn and hn denote the impulse responses of Hm(z) and H(z), thenand547.3 Types of Linear-Phase FIR Transfer FunctionsIt is nearly impossible to design a linear-phase IIR transfer function.It is always possible to design an FIR transfer funct

31、ion with an exact linear-phase response.We now develop the forms of the linear-phase FIR transfer function H(z) with real impulse response hn.55Linear-Phase FIR Transfer FunctionsIf H(z) is to have a linear-phase, its frequency response must be of the formWhere c and are constants, and , called the

32、amplitude response, also called the zero-phase response, is a real function of .Consider a causal FIR transfer function H(z) of length N+1, i.e., of order N:56Linear-Phase FIR Transfer FunctionsFor a real impulse response, the magnitude response |H(ej)| is an even function of , i.e., |H(ej)| = |H(e-

33、j)| Since , the amplitude response is then either an even function or an odd function of , i.e.57Linear-Phase FIR Transfer FunctionsThe frequency response satisfies the relation H(ej)=H*(e-j), or equivalently, the relationIf is an even function, then the above relation leads to ej=e-j implying that

34、either =0 or =58Linear-Phase FIR Transfer FunctionsFrom We haveSubstituting the value of in the above we get59Linear-Phase FIR Transfer FunctionsReplacing with in the previous equation we getMaking a change of variable l=N-n, we rewrite the above equation as60Linear-Phase FIR Transfer FunctionsAs ,

35、we have hne-j(c+n)= hN-nej(c+N-n)The above leads to the condition when c=-N/2 hn=hN-n, 0nNThus, the FIR filter with an even amplitude response will have a linear phase if it has a symmetric impulse response.61Linear-Phase FIR Transfer FunctionsIf is an odd function of , then fromWe get ej= -e-j as T

36、he above is satisfied if =/2 or =- /2 ，ThenReduces to62Linear-Phase FIR Transfer FunctionsThe last equation can be rewritten as:As , from the above we get63Linear-Phase FIR Transfer FunctionsMaking a change of variable l=N-n, we rewrite the last equation as:Equating the above withWe arrive at the co

37、ndition for linear phase as:64Linear-Phase FIR Transfer Functions hn=-hN-n, 0nNwith c=-N/2Therefore a FIR filter with an odd amplitude response will have linear-phase response if it has an antisymmetric impulse response.65Linear-Phase FIR Transfer FunctionsSince the length of the impulse response ca

38、n be either even or odd, we can define four types of linear-phase FIR transfer functionsFor an antisymmetric FIR filter of odd length, i.e., N even hN/2 = 0We examine next the each of the 4 cases66Linear-Phase FIR Transfer FunctionsType 1: N = 8Type 2: N = 7Type 3: N = 8Type 4: N = 767Linear-Phase F

39、IR Transfer FunctionsType 1: Symmetric Impulse Response with Odd LengthIn this case, the degree N is evenAssume N = 8 for simplicityThe transfer function H(z) is given by68Linear-Phase FIR Transfer FunctionsBecause of symmetry, we have h0=h8, h1 = h7, h2 = h6, and h3 = h5Thus, we can write69Linear-P

40、hase FIR Transfer FunctionsThe corresponding frequency response is then given by The quantity inside the braces is a real function of w, and can assume positive or negative values in the range 0|70Linear-Phase FIR Transfer Functionswhere b is either 0 or p, and hence, it is a linear function of w in

41、 the generalized senseThe group delay is given by indicating a constant group delay of 4 samples The phase function here is given by71Linear-Phase FIR Transfer FunctionsIn the general case for Type 1 FIR filters, the frequency response is of the formwhere the amplitude response , also called the zer

42、o-phase response, is of the form72Linear-Phase FIR Transfer FunctionsType 2: Symmetric Impulse Response with even LengthType 3: Antisymmetric Impulse Response with odd LengthType 4: Antisymmetric Impulse Response with even LengthP371-372 about these FIR transfer functions.73Four Types of Linear Phas

43、e Filter 74Four Types of Linear Phase Filter 75Linear-Phase FIR Transfer Functions which is seen to be a slightly modified version of a length-7 moving-average FIR filterThe above transfer function has a symmetric impulse response and therefore a linear phase responseExample - Consider76Linear-Phase FIR Transfer Function