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1、Copyright 2001, S. K. Mitra1Types of Transfer FunctionsThe time-domain classification of an LTI digital transfer function sequence is based on the length of its impulse response:- Finite impulse response (FIR) transfer function- Infinite impulse response (IIR) transfer functionCopyright 2001, S. K.
2、Mitra2Types of Transfer FunctionsIn the case of digital transfer functions with frequency-selective frequency responses, one classification is based on the shape of the magnitude functionthe form of the phase function q(w)Copyright 2001, S. K. MitraTransfer Function ClassificationTransfer function c
3、lassification based on Magnitude characteristicsIdeal FiltersBounded Real Transfer FunctionsAllpass Transfer FunctionTransfer function classification based on Phase characteristicsZero-Phase Transfer FunctionLinear-Phase Transfer FunctionMinimum-Phase and Maximum-Phase Transfer Functions3Copyright 2
4、001, S. K. Mitra4Ideal FiltersA 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 frequenciesCopyright 2001, S. K. Mitra5Ideal
5、 FiltersThe range of frequencies where the frequency response takes the value of one is called the passbandThe range of frequencies where the frequency response takes the value of zero is called the stopbandCopyright 2001, S. K. Mitra6Ideal FiltersFrequency responses of the four popular types of ide
6、al digital filters with real impulse response coefficients are shown below:LowpassHighpassBandpassBandstopCopyright 2001, S. K. Mitra7Linear-Phase Transfer FunctionsIn the case of a causal transfer function with a nonzero phase response, the phase distortion can be avoided by ensuring that the trans
7、fer function has a unity magnitude and a linear-phase characteristic in the frequency band of interestCopyright 2001, S. K. Mitra8Linear-Phase Transfer FunctionsThe most general type of a filter with a linear phase has a frequency response given bywhich has a linear phase, i.e., , from w = 0 to w =
8、2pNote alsoCopyright 2001, S. K. Mitra9Linear-Phase Transfer FunctionsThe output yn of this filter to an input is then given byIf D is an integer, then yn is identical to xn, but delayed by D samplesIf and represent the continuous-time signals whose sampled versions, sampled at t = nT, are xn and yn
9、 given above, then the delay between and is precisely DCopyright 2001, S. K. MitraLinear-Phase Transfer Functions10true signalNoise corrupted signalsignal obtained by Linear-phase filterCopyright 2001, S. K. Mitra11Linear-Phase Transfer FunctionsTherefore, if it is desired to pass input signal compo
10、nents 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 interestSince the signal components in the stopband are blocked, the phase response in the stopband can be of an
11、y shapeCopyright 2001, S. K. Mitra12Linear-Phase Transfer FunctionsFigure below shows the frequency response if a lowpass filter with a linear-phase characteristic in the passbandCopyright 2001, S. K. Mitra13Linear-Phase FIR Transfer FunctionsIt is nearly impossible to design a linear-phase IIR tran
12、sfer functionIt is always possible to design an FIR transfer function with an exact linear-phase responseConsider a causal FIR transfer function H(z) of length N+1, i.e., of order N:必考！必考！必考！必考！Copyright 2001, S. K. Mitra14Linear-Phase FIR Transfer FunctionsThe above transfer function has a linear p
13、hase, if its impulse response hn is either symmetric, i.e.,or is antisymmetric, i.e.,Copyright 2001, S. K. Mitra15Linear-Phase Transfer FunctionsExample - Determine the impulse response of an ideal lowpass filter with a linear phase response:Copyright 2001, S. K. Mitra16Linear-Phase Transfer Functio
14、nsApplying the time-shifting property of the DTFT to the impulse response of an ideal zero-phase lowpass filter we arrive atAs before, the above filter is noncausal and of doubly infinite length, and hence, unrealizableCopyright 2001, S. K. Mitra17Copyright 2001, S. K. Mitra18Copyright 2001, S. K. M
15、itra19CausalStableImplementableCopyright 2001, S. K. Mitra20Linear-Phase Transfer FunctionsBy truncating the impulse response to a finite number of terms, a realizable FIR approximation to the ideal lowpass filter can be developedThe truncated approximation may or may not exhibit linear phase, depen
16、ding on the value of chosenCopyright 2001, S. K. Mitra21Linear-Phase Transfer FunctionsIf we choose = N/2 with N a positive integer, the truncated and shifted approximationwill be a length N+1 causal linear-phase FIR filterCopyright 2001, S. K. Mitra22Linear-Phase Transfer FunctionsFigure below show
17、s the filter coefficients obtained using the function sinc for two different values of NCopyright 2001, S. K. Mitra23Linear-Phase FIR Transfer FunctionsSince the length of the impulse response can be either even or odd, we can define four types of linear-phase FIR transfer functionsFor an antisymmet
18、ric FIR filter of odd length, i.e., N even hN/2 = 0We examine next the each of the 4 casesCopyright 2001, S. K. Mitra24Linear-Phase FIR Transfer FunctionsType 1: N = 8Type 2: N = 7Type 3: N = 8Type 4: N = 7时域上时域上的特点的特点Copyright 2001, S. K. Mitra25Linear-Phase FIR Transfer FunctionsType 1: Symmetric
19、Impulse Response with Odd LengthIn this case, the degree N is evenAssume N = 8 for simplicityThe transfer function H(z) is given byCopyright 2001, S. K. Mitra26Linear-Phase FIR Transfer FunctionsBecause of symmetry, we have h0 = h8, h1 = h7, h2 = h6, and h3 = h5Thus, we can write Copyright 2001, S.
20、K. Mitra27Linear-Phase FIR Transfer FunctionsThe corresponding frequency response is then given byThe quantity inside the braces is a real function of w, and can assume positive or negative values in the range Copyright 2001, S. K. Mitra28Linear-Phase FIR Transfer FunctionsThe phase function here is
21、 given bywhere b is either 0 or p, and hence, it is a linear function of w in the generalized senseCopyright 2001, S. K. Mitra29Linear-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 zero-phas
22、e response, is of the formCopyright 2001, S. K. Mitra30Linear-Phase FIR Transfer FunctionsType 2: Symmetric Impulse Response with Even LengthIn this case, the degree N is oddAssume N = 7 for simplicityThe transfer function is of the formCopyright 2001, S. K. Mitra31Linear-Phase FIR Transfer Function
23、sMaking use of the symmetry of the impulse response coefficients, the transfer function can be written asCopyright 2001, S. K. Mitra32Linear-Phase FIR Transfer FunctionsThe corresponding frequency response is given byAs before, the quantity inside the braces is a real function of w, and can assume p
24、ositive or negative values in the rangeCopyright 2001, S. K. Mitra33Linear-Phase FIR Transfer FunctionsHere the phase function is given bywhere again b is either 0 or pAs a result, the phase is also a linear function of w in the generalized senseCopyright 2001, S. K. Mitra34Linear-Phase FIR Transfer
25、 FunctionsThe expression for the frequency response in the general case for Type 2 FIR filters is of the formwhere the amplitude response is given byCopyright 2001, S. K. Mitra35Linear-Phase FIR Transfer FunctionsType 3: Anti-symmetric Impulse Response with Odd LengthIn this case, the degree N is ev
26、enAssume N = 8 for simplicityApplying the symmetry condition we getCopyright 2001, S. K. Mitra36Linear-Phase FIR Transfer FunctionsThe corresponding frequency response is given byIt also exhibits a generalized phase response given bywhere b is either 0 or pCopyright 2001, S. K. Mitra37Linear-Phase F
27、IR Transfer FunctionsIn the general casewhere the amplitude response is of the formCopyright 2001, S. K. Mitra38Linear-Phase FIR Transfer FunctionsType 4: Anti-symmetric Impulse Response with Even LengthIn this case, the degree N is oddAssume N = 7 for simplicityApplying the symmetry condition we ge
28、tCopyright 2001, S. K. Mitra39Linear-Phase FIR Transfer FunctionsThe corresponding frequency response is given byIt again exhibits a generalized phase response given bywhere b is either 0 or pCopyright 2001, S. K. Mitra40Linear-Phase FIR Transfer FunctionsIn the general case we havewhere now the amp
29、litude response is of the formCopyright 2001, S. K. Mitra41Linear-Phase FIR Transfer FunctionsGeneral Form of Frequency ResponseIn each of the four types of linear-phase FIR filters, the frequency response is of the formThe amplitude response for each of the four types of linear-phase FIR filters ca
30、n become negative over certain frequency ranges, typically in the stopbandCopyright 2001, S. K. Mitra42Linear-Phase FIR Transfer FunctionsThe magnitude and phase responses of the linear-phase FIR are given by=0 or for Type 1, 2 = /2 for type 3,4Copyright 2001, S. K. Mitra43Linear-Phase FIR Transfer
31、Functionssince in general is not a constant, the output waveform is not a replica of the input waveformAn FIR filter with a frequency response that is a real function of w is often called a zero-phase filterSuch a filter must have a noncausal impulse responseCopyright 2001, S. K. Mitra44Zero Locatio
32、ns of Linear-Phase FIR Transfer FunctionsConsider first an FIR filter with a symmetric impulse response: Its transfer function can be written as By making a change of variable , we can write Copyright 2001, S. K. Mitra45Zero Locations of Linear-Phase FIR Transfer FunctionsBut,Hence for an FIR filter
33、 with a symmetric impulse response of length N+1 we haveA real-coefficient polynomial H(z) satisfying the above condition is called a mirror-image polynomial (MIP)Copyright 2001, S. K. Mitra46Zero Locations of Linear-Phase FIR Transfer FunctionsNow consider an FIR filter with an anti-symmetric impul
34、se response: Its transfer function can be written asBy making a change of variable , we getCopyright 2001, S. K. Mitra47Zero Locations of Linear-Phase FIR Transfer FunctionsHence, the transfer function H(z) of an FIR filter with an antisymmetric impulse response satisfies the conditionA real-coeffic
35、ient polynomial H(z) satisfying the above condition is called a anti-mirror-image polynomial (AIP) Copyright 2001, S. K. Mitra48Zero Locations of Linear-Phase FIR Transfer FunctionsIt follows from the relation that if is a zero of H(z), so isMoreover, for an FIR filter with a real impulse response,
36、the zeros of H(z) occur in complex conjugate pairsHence, a zero at is associated with a zero atCopyright 2001, S. K. Mitra49Zero Locations of Linear-Phase FIR Transfer FunctionsThus, a complex zero that is not on the unit circle is associated with a set of 4 zeros given byA zero on the unit circle a
37、ppear as a pairas its reciprocal is also its complex conjugateCopyright 2001, S. K. Mitra50The zeros of a linear-phase FIR filter exhibit a mirror-image symmetry with respect to the unit circle.四种线性相位四种线性相位FIR滤波器的滤波器的零点零点所共有所共有的的结构结构Copyright 2001, S. K. MitraZero Locations of Linear-Phase FIR Trans
38、fer FunctionsThe principle difference among the 4 types of linear-phase FIR filters is with regards to the no. of zeros at z=1 and z=-1.Consider a Type 1 FIR filter If it has a zero at z=1 or at z=-1, because of it order N being even and the symmetry of hn, there must be an even no. of zeros at z=1
39、or zt z=-1 or at both locations. 51Copyright 2001, S. K. Mitra52Zero Locations of Linear-Phase FIR Transfer FunctionsNow a Type 2 FIR filter satisfieswith degree N odd Henceimplying , i.e., H(z) must have a zero at Copyright 2001, S. K. Mitra53Zero Locations of Linear-Phase FIR Transfer FunctionsLik
40、ewise, a Type 3 or 4 FIR filter satisfiesThusimplying that H(z) must have a zero at z = 1On the other hand, only the Type 3 FIR filter is restricted to have a zero at since here the degree N is even and hence,Copyright 2001, S. K. Mitra54Zero Locations of Linear-Phase FIR Transfer FunctionsTypical z
41、ero locations shown below1Type 2Type 111Type 4Type 31四种线性相位四种线性相位FIR滤波器的零点滤波器的零点所独所独有有的结构的结构Copyright 2001, S. K. MitraLinear-Phase FIR Transfer Functions with Zeros N-Filter Order, number of zeros, (filter length-1) Step 1: find the mirror-image pairs of given zeros;Step 2: write down the particula
42、r zeros for the given type of linear-phase FIR filters;Step 3:represent the transfer function in the factored form.Copyright 2001, S. K. Mitra56Zero Locations of Linear-Phase FIR Transfer FunctionsSummarizingType 1 FIR filter: Either an even number or no zeros at z = 1 andType 2 FIR filter: Either a
43、n even number or no zeros at z = 1, and an odd number of zeros atType 3 FIR filter: An odd number of zeros at z = 1 and Copyright 2001, S. K. Mitra57Zero Locations of Linear-Phase FIR Transfer FunctionsType 4 FIR filter: An odd number of zeros at z = 1, and either an even number or no zeros atThe pr
44、esence of zeros at leads to the following limitations on the use of these linear-phase transfer functions for designing frequency-selective filters Copyright 2001, S. K. Mitra58Zero Locations of Linear-Phase FIR Transfer FunctionsA Type 2 FIR filter cannot be used to design a highpass filter since i
45、t always has a zeroA Type 3 FIR filter has zeros at both z = 1 and , and hence cannot be used to design either a lowpass or a highpass or a bandstop filterCopyright 2001, S. K. Mitra59Zero Locations of Linear-Phase FIR Transfer FunctionsA Type 4 FIR filter is not appropriate to design a lowpass filter due to the presence of a zero at z = 1A Type 1 FIR filter has no such restrictions and can be used to design almost any type of filter
