1、1,Chapter7 LTI Discrete-Time Systems in the Transform Domain,Transfer Function Classification Types of Linear-Phase Transfer Functions Simple Digital Filters,2,Types of Transfer Functions,The time-domain classification of a digital transfer function based on the length of its impulse response sequen
2、ce:- Finite impulse response (FIR) transfer function.- Infinite impulse response (IIR) transfer function.,3,Types of Transfer Functions,In the case of digital transfer functions with frequency-selective frequency responses, there are two types of classifications: (1) Classification based on the shap
3、e of the magnitude function |H(ei)|. (2) Classification based on the form of the phase function ().,4,7.1 Transfer Function Classification Based on Magnitude Characteristics,Digital Filters with Ideal Magnitude ResponsesBounded Real Transfer FunctionAllpass Transfer Function,5,7.1.1 Digital Filters
4、with Ideal Magnitude Responses,A 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.,6,Digital Filters with Ideal Magnit
5、ude Responses,The 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.,7,Digital Filters with Ideal Magnitude Responses,Magnitude responses of the four popular
6、 types of ideal digital filters with real impulse response coefficients are shown below:,8,Digital Filters with Ideal Magnitude Responses,The 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 ha
7、s a 0 phase everywhere.,9,Digital Filters with Ideal Magnitude Responses,Earlier in the course we derived the inverse DTFT of the frequency response HLP(ej)of the ideal lowpass filter:hLPn=sincn/n, - n We have also shown that the above impulse response is not absolutely summable, and hence, the corr
8、esponding transfer function is not BIBO stable.,10,Digital Filters with Ideal Magnitude Responses,Also, hLPn is not causal and is of doubly infinite length. The remaining three ideal filters are also characterized by doubly infinite, noncausal impulse responses and are not absolutely summable. Thus,
9、 the ideal filters with the ideal “brick wall” frequency responses cannot be realized with finite dimensional LTI filter.,11,Digital Filters with Ideal Magnitude Responses,To develop stable and realizable transfer functions, the ideal frequency response specifications are relaxed by including a tran
10、sition band between the passband and the stopband. This permits the magnitude response to decay slowly from its maximum value in the passband to the 0 value in the stopband.,12,Digital Filters with Ideal Magnitude Responses,Moreover, the magnitude response is allowed to vary by a small amount both i
11、n the passband and the stopband.,Typical magnitude response specifications of a lowpass filterare shown as:,13,7.1.2 Bounded Real Transfer Functions,A causal stable real-coefficient transfer function H(z) is defined as a bounded real (BR) transfer function if:,for all values of w,Let xn and yn denot
12、e, respectively, the input and output of a digital filter characterized by a BR transfer function H(z) with X(ej) and Y(ej) denoting their DTFTs.,14,Bounded Real Transfer Functions,Integrating the above from- to , and applying Parsevals relation we get:,Then the condition implies that:,15,Bounded Re
13、al Transfer Functions,Thus, for all finite-energy inputs, the output energy is less than or equal to the input energy. It implies that a digital filter characterized by a BR transfer function can be viewed as a passive structure.,If , then the output energy is equal to the input energy, and such a d
14、igital filter is therefore a lossless system.,16,Bounded Real Transfer Functions,The BR and LBR transfer functions are the keys to the realization of digital filters with low coefficient sensitivity.,A causal stable real-coefficient transfer function H(z) with is thus called a lossless bounded real
15、(LBR) transfer function.,17,Bounded Real Transfer Functions,Example: Consider the causal stable IIR transfer function:,where K is a real constant. Its square-magnitude function is given by:,18,Bounded Real Transfer Functions,Thus, for 0, ( |H(ej)|2 )max = K2/(1- )2 | =0 ( |H(ej)|2 ) min = K2/(1+ )2
16、| = On the other hand, for 0, (2cos )max = -2 | = (2cos )min = 2 | =0 Here, ( |H(ej)|2 )max = K2/(1+ )2 | = ( |H(ej)|2 )min = K2/(1- )2 | = 0,19,Bounded Real Transfer Functions,Hence,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 functio
17、n are shown on the next page.,20,Bounded Real Transfer Functions,Lowpass filter,Highpass filter,21,7.1.3 Allpass Transfer Function,The 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(
18、z) should be:,22,Allpass Transfer Function,Generalize, the Mth-order allpass system is:,If we denote polynomial:,So:,23,Allpass Transfer Function,The numerator of a real-coefficient allpass transfer function is said to be the mirror-image polynomial of the denominator, and vice versa.,We shall use t
19、he notation to denote the mirror-image polynomial of a degree-M polynomial DM(z) , i.e.,24,Allpass Transfer Function,The expression,implies that the poles and zeros of a real-coefficient allpass function exhibit mirror-image symmetry in the z-plane.,25,Allpass Transfer Function,To show that |AM(ej)|
20、=1 we observe that:,Therefore:,Hence:,26,Allpass Transfer Function,Properties: 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 magnitude function of a stable allpass fun
21、ction A(z) satisfies:,27,Allpass 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 in the range 0 w p.,28,Applicati
22、on of allpass system,Any 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),29,7.2 Transfer Function Classifica
23、tion Based on Phase Characteristic,Zero-Phase Transfer FunctionLinear-Phase Transfer FunctionMinimum-Phase and Maximum-Phase Transfer Functions,30,7.2.1 Zero-Phase Transfer Function,One way to avoid any phase distortions is to make the frequency response of the filter real and nonnegative,to design
24、the filter with a zero phase characteristic.,But for a causal digital filter it is impossible.,31,Zero-Phase Transfer Function,Only 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 wit
25、h no poles on the unit cycle, then F(z)=H(z)H(z-1) has a zero phase on the unit cycle.,32,Zero-Phase Transfer Function,Please look at book P362.,The function filtfilt implements the above zero-phase filtering scheme.Please look at book P412P7.5.,33,7.2.2 Linear-Phase Transfer Function,The phase dist
26、ortion can be avoided by ensuring that the transfer function has a unity magnitude and a linear-phase characteristic, that is: H(ej)=e-jD How to perform the linear-phase filter? yn=xn-DY(ej)= e-jDX(ej) H(ej)= Y(ej)/ X(ej)= e-jD,34,Linear-Phase Transfer Function,Example - Determine the impulse respon
27、se of an ideal lowpass filter with a linear phase response:,Applying the frequency-shifting property of the DTFT to the impulse response of an ideal zero-phase lowpass filter we arrive at:,35,Linear-Phase Transfer Function,By truncating the impulse response to a finite number of terms, a realizable
28、FIR approximation to the ideal lowpass filter can be developed. The truncated approximation may or may not exhibit linear phase, depending on the value of n0 chosen. If we choose n0= N/2 with N a positive integer, the truncated and shifted approximation:,36,Linear-Phase Transfer Function,Figure belo
29、w shows the filter coefficients obtained using the function sinc for two different values of N.,37,7.2.3 Minimum-Phase and Maximum-Phase Transfer Function,Based on the expression:,The phase function is:,Where, k andk are zeros and poles, respectively.,38,Minimum-Phase and Maximum-Phase Transfer Func
30、tion,We define the expression (ej- k )and (ej - k ) as zero vectors and pole vectors. When k and k are inside the unit circle, and change from 0 to 2, the change of phase of the zero (pole) vectors are 2. When k and k are outside the unit circle, and change from 0 to 2, the change of phase of the ze
31、ro (pole) vectors are 0.,39,Minimum-Phase and Maximum-Phase Transfer Function,So, only zeros or poles inside the unit circle can affect phase function of H(ej). For causal stable system, we can deduce:,So, its the phase delay (lag) system. When all zeros are all inside the unit circle, we get:,Its t
32、he minimum phase delay system.,40,Minimum-Phase and Maximum-Phase Transfer Function,When all zeros are all outside the unit circle, we get:,Where M is the number of zeros. And its the maximum phase delay system. A transfer function with zeros inside and outside the unit circle is called a mixed-phas
33、e transfer function. Example7.4(P367).,41,7.3 Types of Linear-Phase FIR Transfer Functions,It is nearly impossible to design a linear-phase IIR transfer function. It is always possible to design an FIR transfer function with an exact linear-phase response. We now develop the forms of the linear-phas
34、e FIR transfer function H(z) with real impulse response hn.,42,Linear-Phase FIR Transfer Functions,If H(z) is to have a linear-phase, its frequency response must be of the form,Where c and are constants, and , called the amplitude response, also called the zero-phase response, is a real function of
35、.,Consider a causal FIR transfer function H(z) of length N+1, i.e., of order N:,43,Linear-Phase FIR Transfer Functions,For a real impulse response, the magnitude response |H(ej)| is an even function of , i.e., |H(ej)| = |H(e-j)|,Since , the amplitude response is then either an even function or an od
36、d function of , i.e.,44,Linear-Phase FIR Transfer Functions,The frequency response satisfies the relation H(ej)=H*(e-j), or equivalently, the relation,If is an even function, then the above relation leads to ej=e-j implying that either =0 or =,45,Linear-Phase FIR Transfer Functions,From,We have,Subs
37、tituting the value of in the above we get,46,Linear-Phase FIR Transfer Functions,Replacing with in the previous equation we get,Making a change of variable l=N-n, we rewrite the above equation as,47,Linear-Phase FIR Transfer Functions,As , we havehne-j(c+n)= hN-nej(c+N-n),The above leads to the cond
38、ition when c=-N/2 hn=hN-n, 0nN,Thus, the FIR filter with an even amplitude response will have a linear phase if it has a symmetric impulse response.,48,Linear-Phase FIR Transfer Functions,If is an odd function of , then from,We get ej= -e-j as,The above is satisfied if =/2 or =- /2 ，Then,Reduces to,
39、49,Linear-Phase FIR Transfer Functions,The last equation can be rewritten as:,As , from the above we get,50,Linear-Phase FIR Transfer Functions,Making a change of variable l=N-n, we rewrite the last equation as:,Equating the above with,We arrive at the condition for linear phase as:,51,Linear-Phase
40、FIR Transfer Functions,hn=-hN-n, 0nN with c=-N/2 Therefore a FIR filter with an odd amplitude response will have linear-phase response if it has an antisymmetric impulse response.,52,Linear-Phase FIR Transfer Functions,Since the length of the impulse response can be either even or odd, we can define
41、 four types of linear-phase FIR transfer functions For an antisymmetric FIR filter of odd length, i.e., N evenhN/2 = 0 We examine next the each of the 4 cases,53,Linear-Phase FIR Transfer Functions,54,Linear-Phase FIR Transfer Functions,Type 1: Symmetric Impulse Response with Odd Length In this case
42、, the degree N is even Assume N = 8 for simplicity The transfer function H(z) is given by,55,Linear-Phase FIR Transfer Functions,Because of symmetry, we have h0=h8, h1 = h7, h2 = h6, and h3 = h5 Thus, we can write,56,Linear-Phase FIR Transfer Functions,The corresponding frequency response is then gi
43、ven by,The quantity inside the braces is a real function of w, and can assume positive or negative values in the range 0|,57,Linear-Phase FIR Transfer Functions,where b is either 0 or p, and hence, it is a linear function of w in the generalized sense The group delay is given by,indicating a constan
44、t group delay of 4 samples,The phase function here is given by,58,Linear-Phase FIR Transfer Functions,In the general case for Type 1 FIR filters, the frequency response is of the form,59,Linear-Phase FIR Transfer Functions,Type 2: Symmetric Impulse Response with even Length Type 3: Antisymmetric Imp
45、ulse Response with odd Length Type 4: Antisymmetric Impulse Response with even Length P371-372 about these FIR transfer functions.,60,Linear-Phase FIR Transfer Functions,which is seen to be a slightly modified version of a length-7 moving-average FIR filter The above transfer function has a symmetri
46、c impulse response and therefore a linear phase response,Example - Consider,61,Linear-Phase FIR Transfer Functions,A plot of the magnitude response of along with that of the 7-point moving-average filter is shown below,62,Linear-Phase FIR Transfer Functions,Note the improved magnitude response obtai
47、ned by simply changing the first and the last impulse response coefficients of a moving-average (MA) filter It can be shown that we can express,which is seen to be a cascade of a 2-point MA filter with a 6-point MA filterThus, H0(z) has a double zero at z=-1, i.e., (w = p),63,7.3.1 Zero Locations of
48、 Linear-Phase FIR Transfer Functions,The zeros of the real-coefficient hn is in mirror-image pairs. Moreover, for a FIR filter with a real impulse response, the zeros occur in complex conjugate pairs. Particularly, zeros position for different linear-phase types:,64,Zero Locations of Linear-Phase FI
49、R Transfer Functions,(1)Type 1 :Either an even number or no zeros at z=1& z=-1. (2)Type 2 :Either an even number or no zeros at z=1,and an odd number of zeros at z=-1. (3)Type 3 :An odd number of zeros at z=1& z=-1. (4)Type 4 :An odd number of zeros at z=1, and either an even number or no zeros at z=-1.,
