測試信號英文版課件：Chapter4 LTI Discrete-Time Systems in the Transform-Domain_Lec2

1、1Types 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 function2Types of Transfer FunctionsIn the case of digita

2、l 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)Transfer Function ClassificationTransfer function classification based on Magnitude characteristicsIdeal FiltersBounded Real Transfe

3、r FunctionsAllpass Transfer FunctionTransfer function classification based on Phase characteristicsZero-Phase Transfer FunctionLinear-Phase Transfer FunctionMinimum-Phase and Maximum-Phase Transfer Functions34Ideal FiltersA digital filter designed to pass signal components of certain frequencies wit

4、hout distortion should have a frequency response equal to one at these frequencies, and should have a frequency response equal to zero at all other frequencies5Ideal FiltersThe range of frequencies where the frequency response takes the value of one is called the passbandThe range of frequencies whe

5、re the frequency response takes the value of zero is called the stopband6Ideal FiltersFrequency responses of the four popular types of ideal digital filters with real impulse response coefficients are shown below:LowpassHighpassBandpassBandstop7Linear-Phase Transfer FunctionsIn the case of a causal

6、transfer function with a nonzero phase response, the phase distortion can be avoided by ensuring that the transfer function has a unity magnitude and a linear-phase characteristic in the frequency band of interest8Linear-Phase Transfer FunctionsThe most general type of a filter with a linear phase h

7、as a frequency response given bywhich has a linear phase, i.e., , from w = 0 to w = 2pNote also9Linear-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

8、 whose sampled versions, sampled at t = nT, are xn and yn given above, then the delay between and is precisely DLinear-Phase Transfer Functions10true signalNoise corrupted signalsignal obtained by Linear-phase filter11Linear-Phase Transfer FunctionsTherefore, if it is desired to pass input signal co

9、mponents 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

10、 any shape12Linear-Phase Transfer FunctionsFigure below shows the frequency response if a lowpass filter with a linear-phase characteristic in the passband13Linear-Phase FIR Transfer FunctionsIt is nearly impossible to design a linear-phase IIR transfer functionIt is always possible to design an FIR

11、 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！14Linear-Phase FIR Transfer FunctionsThe above transfer function has a linear phase, if its impulse response hn is either symmetric, i.e.,or is antisymmetric, i.e.,15

12、Linear-Phase Transfer FunctionsExample - Determine the impulse response of an ideal lowpass filter with a linear phase response:16Linear-Phase Transfer FunctionsApplying the time-shifting property of the DTFT to the impulse response of an ideal zero-phase lowpass filter we arrive atAs before, the ab

13、ove filter is noncausal and of doubly infinite length, and hence, unrealizable171819CausalStableImplementable20Linear-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 appro

14、ximation may or may not exhibit linear phase, depending on the value of chosen21Linear-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 filter22Linear-Phase Transfer FunctionsFigure below shows t

15、he filter coefficients obtained using the function sinc for two different values of N23Linear-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 antisymmetric FIR filter of odd length,

16、i.e., N even hN/2 = 0We examine next the each of the 4 cases24Linear-Phase FIR Transfer FunctionsType 1: N = 8Type 2: N = 7Type 3: N = 8Type 4: N = 7時域上的特點25Linear-Phase FIR Transfer FunctionsType 1: Symmetric Impulse Response with Odd LengthIn this case, the degree N is evenAssume N = 8 for simplic

17、ityThe transfer function H(z) is given by26Linear-Phase FIR Transfer FunctionsBecause of symmetry, we have h0 = h8, h1 = h7, h2 = h6, and h3 = h5Thus, we can write 27Linear-Phase FIR Transfer FunctionsThe corresponding frequency response is then given byThe quantity inside the braces is a real funct

18、ion of w, and can assume positive or negative values in the range 28Linear-Phase FIR Transfer FunctionsThe phase function here is given bywhere b is either 0 or p, and hence, it is a linear function of w in the generalized sense29Linear-Phase FIR Transfer FunctionsIn the general case for Type 1 FIR

19、filters, the frequency response is of the formwhere the amplitude response , also called the zero-phase response, is of the form30Linear-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

20、 of the form31Linear-Phase FIR Transfer FunctionsMaking use of the symmetry of the impulse response coefficients, the transfer function can be written as32Linear-Phase FIR Transfer FunctionsThe corresponding frequency response is given byAs before, the quantity inside the braces is a real function o

21、f w, and can assume positive or negative values in the range33Linear-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 sense34Linear-Phase FIR Transfer FunctionsThe expression for the

22、frequency response in the general case for Type 2 FIR filters is of the formwhere the amplitude response is given by35Linear-Phase FIR Transfer FunctionsType 3: Anti-symmetric Impulse Response with Odd LengthIn this case, the degree N is evenAssume N = 8 for simplicityApplying the symmetry condition

23、 we get36Linear-Phase FIR Transfer FunctionsThe corresponding frequency response is given byIt also exhibits a generalized phase response given bywhere b is either 0 or p37Linear-Phase FIR Transfer FunctionsIn the general casewhere the amplitude response is of the form38Linear-Phase FIR Transfer Fun

24、ctionsType 4: Anti-symmetric Impulse Response with Even LengthIn this case, the degree N is oddAssume N = 7 for simplicityApplying the symmetry condition we get39Linear-Phase FIR Transfer FunctionsThe corresponding frequency response is given byIt again exhibits a generalized phase response given by

25、where b is either 0 or p40Linear-Phase FIR Transfer FunctionsIn the general case we havewhere now the amplitude response is of the form41Linear-Phase FIR Transfer FunctionsGeneral Form of Frequency ResponseIn each of the four types of linear-phase FIR filters, the frequency response is of the formTh

26、e amplitude response for each of the four types of linear-phase FIR filters can become negative over certain frequency ranges, typically in the stopband42Linear-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 3,443Li

27、near-Phase FIR Transfer 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 response44Zero Locations

28、of Linear-Phase FIR Transfer FunctionsConsider first an FIR filter with a symmetric impulse response: Its transfer function can be written asBy making a change of variable , we can write 45Zero Locations of Linear-Phase FIR Transfer FunctionsBut,Hence for an FIR filter with a symmetric impulse respo

29、nse of length N+1 we haveA real-coefficient polynomial H(z) satisfying the above condition is called a mirror-image polynomial (MIP)46Zero Locations of Linear-Phase FIR Transfer FunctionsNow consider an FIR filter with an anti-symmetric impulse response: Its transfer function can be written asBy mak

30、ing a change of variable , we get47Zero 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-coefficient polynomial H(z) satisfying the above condition is called a anti-mirror-image pol

31、ynomial (AIP) 48Zero 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, the zeros of H(z) occur in complex conjugate pairsHence, a zero at is associated with a zero at49Zero Locations

32、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 appear as a pairas its reciprocal is also its complex conjugate50The zeros of a linear-phase FIR filter exhibit a mirror-image symmetry with

33、 respect to the unit circle.四種線性相位FIR濾波器的零點所共有的結(jié)構(gòu)Zero Locations of Linear-Phase FIR Transfer 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

34、 it order N being even and the symmetry of hn, there must be an even no. of zeros at z=1 or zt z=-1 or at both locations. 5152Zero 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 53Zero Locations of Lin

35、ear-Phase FIR Transfer FunctionsLikewise, 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,54Zero Locations of Linear-Phase FIR Transfer FunctionsTypical zero locations shown below1Type 2Type 111Type 4Type 31四種線性相位FIR濾波器的零點所獨有的結(jié)構(gòu)Linear-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 particular zeros for the given type of