UNIT - III PART A. 2. Mention any two techniques for digitizing the transfer function of an analog filter?
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1 UNIT - III PART A. Mention the important features of the IIR filters? i) The physically realizable IIR filters does not have linear phase. ii) The IIR filter specification includes the desired characteristics of the magnitude response only.. Mention any two techniques for digitizing the transfer function of an analog filter? The bilinear transformation and the impulse invariant transformation are the two techniques available for digitizing analog filter transfer function. 3. What is frequency warping? [AU Dec 04] The non-linear relation between analog and digital frequency introduces distortion in the frequency axis is called frequency warping. 4. What happens to the magnitude response as the order of the filter (N) increases? The magnitude response approaches the ideal response as the value of N increases. 5. What is the relation between digital and analog frequency in impulse invariant method? W = T where w digital frequency. T Sampling Time period - analog frequency. 6. What is the transposition of a flow graph? The transposition of a flow graph can be obtained by reversing the directions of all signals and by interchanging mates and summing junctions. 7. Classify the structure for realizing IIR systems? a. Direct form I structure. b. Direct form II structure. c. Cascade form structure.
2 d. Parallel form structure. 8. Why pre warping is employed? In IIR filter design using bilinear transformation, the conversion of the specified digital frequencies to analog frequencies is called pre warping and this is necessary to eliminate the effect of warping on amplitude response. 9. What is the advantage in direct form II structure when compared to direct form structure? In direct form II structure the number of delay elements required its exactly half that of direct form I structure when the number of poles and zero are equal. Hence it requires less amount of memory. 0. Mention the advantage and disadvantage and bilinear transformation?. One to one mapping.. Can be mapped into realizable, stable digital system. 3. No aliasing. Disadvantages:. Neither impulse response nor the phase response of the analog filter is preserved in a digital filter.. Mapping is highly non-linear.. What are the methods to converts the analog filter transfer function into a digital filter transfer function? The techniques are. Impulse Invariance transformation.. Bilinear transformation method. 3. Approximation of derivatives. 4. Matched transform.. What is the relation between digital and analog frequency in Bilinear transformation? w tan T where w = Digital frequency = analog frequency T = Sampling Time period.
3 3. How bilinear transformation is performed? The bilinear transformation is performed by letting S in the analog filter transfer function. T i.e. H(z) = H a (s) S 4. What is aliasing? S T The phenomena of high frequency sinusoidal components acquiring the identity of low frequency sinusoidal components after sampling is called aliasing. 5. List out any two differences between analog and digital filter? Analog filter Digital filter.. Operates on analog Signals Operates on digital Samples. It consist of adders, multipliers and delays implemented in digital logic It consist of electrical component like resistors, capacitors and inductor. 6. Define Ripple in a filter. The limits of the tolerate in the magnitude of pass band and stop band are called ripples 7. Define IIR filter The filter designed by considering all the infinite samples of impulse response are called IIR filter. 8. Mention any two comparison between FIR & IIR filter. FIR filter. Only N samples of impulse response are considered. Linear phase filters can be easily designed IIR Filter All the infinite samples of impulse response are considered Linear phase characteristics cannot be achieved 9. Classify the filters based on frequency response. Based on frequency response, the filters can be classified into low pass, high pass, band pass and band stop filters. 0. What is the importance of poles in filter design? The stability of a filter is related to the location of the poles. For a stable analog filter the poles should lie on the left half of s-plane. For a stable digital filter the poles should
4 lie inside the unit circle in the -plane. What are the advantages of FIR filters?. Linear phase FIR filters can be easily designed. Efficient realization of FIR filter exist as both recursive and non recursive structures. 3. FIR filters realized non recursively art always stable 4. The round off noise can be made small in non recessive realization of FIR filter.. List the different types of structures for realizing FIR systems. a) Direct form Realization b) Cascade Realization c) Linear phase realization 3. What is the advantage in linear phase realization of FIR system The advantage in linear phase realization is that the system can be realized with minimum number of multipliers. 4. Write the properties of frequency response of LTI system?. The frequency response is periodic function of with a period. The frequency response is a continuous function of w. 5. What is the relation between Fourier transform and transform? X ( w) X ( z) e x( n) e n jwn jw 6. Write the properties of chebyshev type-. filters. the magnitude response is equiripple in the passband and monotonic in the stop band. the chebbyshev type- filters are all pole designs 3. The normalized magnitude function has a value of frequency c 4. The magnitude response as the value of N increases. / at the cut-off 7. How the poles of chebyshev transfer function are exated in s-plane? The poles of the chebyshev transfer function symmetrically lies on an cllipse in s- plane
5 8. How will you determine the order N of chebyshev filter? Calculate a parameter TV, using the following equation and correct it to nearest integer / A TV Cosh Choose N such NN 9. Write the transfer function of unnormalized chebyshev low pass filter When N is even H a (S) = N Bkc k S bk cs Ckc N B When TV is odd, H (S)= S+C k Bkc 0 c a 0 C S bks CKC Ck where bk y ~ stn ; N (k ) Ck yn COS ; Co yn N / N / YN 30. How the order of the filter affects the frequency response of the cheby shev filter. From the magnitude response of type- chebyshev filter it can be observed make the magnitude response as the order of the filter is increased write the magnitude function of type one cbebyshev low pass filter. Ha( ) c CN 3. What is type chebyshev approximation?
6 In type- chebyshev approximation, the error function is selected such that the magnitude response is monotonic is pass band and equiripple in stop band. The type- magnitude response is called inverse chebyshev response. 3. What is type- chebyshev approximation? In type-chebyshev approximation, the error function is selected such that, the magnitude response is equiripple in the pass band and monotic in the stopband. 33. What is chebyshe approximation? In chebyshev approximation, mc approximation function is selected such that the error is minimized over a prescribed band of frequencies. 34. Write the properties of Butterworth filter. The butter worth filters are all pole decisions. it the cutoff frequency, the magnitude of normalized butterworth filter is 3. The filter order N, completely specifies the filter and as approaches the ideal response 4. The magnitude is maximally flat at the orgin and mononically decreasing with increasing 35. How will you choose the order N for a butter worth filter calculate a parameter N, using the following equation and correct it to nearest integer A log A N / log Choose the order N of the filter such that N N. 36. Write the transfer function of unnormalized butter worth low pass filter? When N is even, Transfer function of analog low pass butter worth filter
7 N / c Ha() s K S bkcs When N is odd Transfer function of analog Low pass Butterworth filter N C Ha( S) S S bk CS C k C C Where b k = Sin k n N N =order of the filter C = Analog cutoff frequency 37. Write the magnitude function of lowpass butterworth filter The magnitude function of low pass butterworth filter is given by ha () l N where C is the cutoff frequency and N is the order of the filter. 38. How the poles of butterworth transfer function are located in s-plane? The poles of the normalized butterworth transfer function symmetrically lies on the unit circle in s-plane with angular spacing of /N. 39. What is butter worth approximation? In butter worth approximation, the error function is selected such that the magnitude is maximally flat in the orgin (i-e at=0) and monotonically decreasing with increasing. 40. What is impulse invariant transformation? The transformation of analog filter to digital filter without modifying the impulse response of the filter is called impulse invariant transformation (ie-in this transformation the impulse response of the digital filter will be the sampled version of the impulse response of the analog filter)
8 4. How will you choose the order TV for a butter worth filter Calculate a parameter TV using the following equation and correct it to nearest integer N log / Log Choose the order N of me filter such that N N 4. Mention the disadvantage in digital filter ) The bandwidth of the discrete signal is limited by sampling frequency ) The performance of digital filter depends on the hardware used to implement the filter 43. Define FIR filter Filter designed by considering only finite samples of impulse response is known as FIR filter. 45. Compare the digital and analog filter. Digital Filter. Operates on digital samples (or sampled version) of the signal.. It is governed (or defined) by linear difference equation. 3. It consists of adders, multipliers and delays implemented in digital logic (either in hardware or software or both). 4. In digital filters the filter coefficients are designed to satisfy the desired frequency response. Analog Filter. Operates on analog signals (or actual signals). It is governed (or defined) by linear differential equation. 3. It consists of electrical components like resistors, capacitors and inductors. 4. In analog filters the approximation problem is solved to satisfy the desired frequency response. 46. What are the advantages and disadvantages of digital filters?
9 Advantages of digital filters i. High thermal stability due to absence of resistors, inductors and capacitors. ii. iii. iv. The performance characteristics like accuracy, dynamic range, stability and tolerance can be enhanced by increasing the length of the registers. The digital filters are programmable. Multiplexing and adaptive filtering are possible. Disadvantages of digital filters i. The bandwidth of the discrete signal is limited by the sampling frequency. ii. The performance of the digital filter depends on the hardware used to implement the filter. 47. What are the requirements for a digital filter to b e stable and causal? i. The digital filter transfer function H(z) should be a rational function of z and the coefficients of z should be real. ii. The poles should lie inside the unit circle in z-plane. iii. The number of zeros should be less than or equal to number of poles. 48. Compare IIR and FIR filters. IIR Filter i. All the infinite samples of impulse response are considered. ii. The impulse response cannot be directly converted to digital filter transfer function. iii. The design involves design of analog filter and then transforming analog filter to digital filter. iv. The specifications include the desired characteristics for magnitude response only. v. Linear phase characteristics cannot be achieved. FIR Filter i. Only N samples of impulse response are considered. ii. The impulse response can be directly converted to digital filter transfer function. iii. The digital filter can be directly designed to achieve the desired specifications. iv. The specifications include the desired characteristics for both magnitude and phase response. v. Linear phase filters can be easily designed. 49. Classify the structure for realizing IIR systems. a) Direct form I b) Direct form II c) Cascade structure
10 d) Parallel structure Lattice structure. PART B. Apply the BT to Ha(S) Ha ( S). ( S)( S0) ( S )( S ) with T = find H (z) given that ( ) T ( ) Put S= inh ( S) H ( ) H( ) ( ( ) T T ( ) a z T T z T T T ( ) T ( ) T ( ) ( T T )( T T ) PutT sec (+-) ( ) H()= ( )( z ) S 4 ( ) ( ) 0.5( z ) (3 z ) ( 3) ( 3). Obtain H() from H a (s) when T=sec and
11 Ha ( S) S S 0.5 Put ( ) s = T ( ) H( ) ( ) T ( z ) ( ) ( ) 0. T T ( ) 4 T 4( z ) 0.4( z ) [ T ( )] [ T( )] Put T =Sec 4( ) (-z )( ) H() = 4( ) 0.4( )( ) ( ) 4( ) 0.87( ) // 3. Explain the IIR filter design by PBilinear Transformation Let the system function of the analog filter he H(S)=b/ s+a () The differential equation describing the analog filter can be obtained from () ys ( ) H ( s) b / s a xs () Sy( S) ay( s) b( S) () Take Inverse Laplace the dy(t) ay( t) bx( t) () dt (L) is integrated b/w the (nt-t) and nt
12 nt nt n dy() t dt a y ( t ) dt b x ( t ) dt (3) dt nt T nt T nt T The trapezoidal rule for numeric integration is given nt nt T T a( t) dt [ a( nt ) a( nt T (4) Apply (4) in (3) at at y(nt)-y(nt-t)+ y nt y nt T T bt b xnt xnt T (5) Take transform the system function of digital filter is y z H z x z z z 0.5 z Determine H(z) using the impulse invariant technique for the along system function. Hs () ( s 0.5) s 0.5s using partial function A Bs L Hs () s 0.5 S 0.55 A S 0.5S BS C S 0.5 comparing the co eff of S,S constant on either A+ B =0 0.5 A+ 0.5 B + C =0 A C = Solving we get A= 0.5 B =-0.5 C=0
13 s H( S) s s s s s s s s s s 0.5 s s 0.5 s S 0.5 we know s a e at / cosbt z s a b e cos bt z e b s a b at at - e at / sin bt z e cos bt z e at at - m m () () d (3) m m s S m!ds st i s s e i using the equation () & (3) 0.5 -e (cos.399t)z H ( ) -0.5 e -e (cos.399t)z -0.5T - 0.5T -0.5T - +e -0.5T T e sin.399t e cos.3997 e Let T= sec -0.5T 0.5T H( ) T e Sin(.399 T ) e 0.5T 0.5T - e ( Cos.399T)
14 on simplifying we get H( ) on simplying we get - - H( z) z z z z z Convert analog filter with system function HS ( ) S 0. s 0. using BT the digital filter should have resonant frequency of From H(s) we note that c=3 T can be determined as into digital IIR filter w r 4 w tan r T Using BT T w r tan tan 0.76s c 3 3 H ( ) H ( S) S= T 0. T H( z) 0. 9 T T / T sub T=0.76S H()=
15 6. A digital filter with a 3 db band with of 0.5 is t be designed from analog filter C whose system response is HS ( ) use BT and obtain H(z) S w r 0.88 tan tan 0.5 T T T H ( ) H ( S) S T 0.88 c T 0.88 c T T T 0.88 ( ) 0.88( ) on simplying H( ) Using BT obtain H() it H( s) and T=0.8 S for the BT H ( z) H ( s) S T T Sub T= 0.s H( z) 0 Further simplifying z z 9
16 H( z) z z 8. Obtain direct from I realization for the given function. H SOL: ( ) X H z where X x()=o/p and x()= i/p Y ( z) Y Y Y 3 X Y Y Y Y Obtain direct form I realization for the system described by y( n) x( n) 0.5 x( n ) 0.4 x( n ) 0.6 y( n ) -0.7y(n-) taking z transform
17 Y Y( ) X ( ) Y ( z) 0.6 Y ( z) 0.7 Y ( z) X ( z) X Y Y X X 0. Find the direct form I realization for the system function rcos w z H( z) r cos w0z r z 0 y z rcos w z H( z) x z rcos w z r z 0 0 y( z) r cos w0 z y z r z y z x z -rcos w 0z x z y( z) x z az x z ay z r z y z. Find direct form Ii realization for the system described by the transfer function.
18 3 8 4 H( z) 4 Y W Y X X W Y X Y W X W z W W W X z W z X z W W W Y W z W z W W z. Obtain direct form II realization for the difference equation yn- y n x n x n x n y n sol: y z w x y z. x z x z w z w z y z 0.6z 0.7z y z w z 0.5z 0.4z W z W W X W z X z W W Y W z W z W
19 3. Obtain cascade form realisation for the system described by the transfer function. H( ) 4 4. H H 4 4 where H H ( ) 4 Y H( ) X 4 Y ( ) Y Y ( ) X( ) 4 Y ( ) X( ) Y Y 4 Y H( ) Y Y W Y. Y Y W W ( ) Y ( ) Y ( ) W ( )
20 W z W W Y W z Y z W W Y W z W z 4. Obtain the parallel form realisation for given function. H( ) H( ). 4 Y( z) H( z) X( z) 35 8 Y( z) X( z) Y ( z) X 3 X ( z) X ( z) 8 4 Y ( z) Y ( z) Y ( z) Y ( z) 3
21 35 8 Y 3 X Y 3 X ( z) 8 0 Y3 X ( ) 4 Y 35 H 3 X 8 35 Y Y 8 3 Let Y 8 H 3 X 8 Y Y X 3 Y3 0 H3 X 4 Y3 Y3 0X 4 5. Obtain cascade form realization for the given transfer function H( ) 4 H( ) x z z 3 z
22 H ( z) H ( z) H ( z) H ( z) H ( z) 4 Y ( z) H( z) X( z) Y( z) Y( z) X ( z) Y z X z z ( ) ( ) ( ) H Y Y Y W Y. Y Y W W Y 4 Y and W 4
23 Y W W W 4 W Y W W 4 Y W W W 6. Obtain the parallel form realization for the system described by the difference equation. 3 3 y n y n y n y n H xn xn +x n A B C H( ) 8 4 A B C / / / H( ) By practical fraction expansion are can write
24 A 3 C A B C A A B B C C A+B= 7 /4 7 B A 4 A B C A C 9 / 8 4 C 9 / 8 A 4 C 9 A A A A A A A B A C A 8 4
25 5 A B C H Y 4 8 H( ) X 4 Y ( z) X ( z) Y ( z) Y ( z) 5 Y ( z) 4 X 4 8 Y ( z) X Y Y H and H X X 5 Y Y X Y X Y 4 4 Y Y Y X X Y X 8 X Y Y - +8
26 The parallel structure of H() is obtained by connecting the direct form structure of H () and H (z) 7. For the analog transfer function H() s impulse invariance method Assume T= sec. S S Determine H() using Given H() s H() s S S S S A s S Ss S using partial fraction B S S S ie H( S) s s s S ( ) Using impulse invariance technique we have N N AK AK H ( S) then H ( S) S P S P T K K K K p S-P e Tz - K For T= H()= - -e -e z H( z) Using impulse invariance with T= sec determine H() if H(S) = s s
27 L H ( S) L S S L L S S a S L - L - B e S a B s a s a b e at e at at cosbt sin bt L L 4 s+ s+ t sin ze t Let t = nt nt nt h( nt ) e sin if T= nt nt h( nt ) e sin H ( z) z h( n) e Sin e cos e 9.What is a recursive and non-recursive system? A system whose output y(n) depends on any number of past output values such
28 as y(n-), y(n-), y(n-3),.. is called a recursive system. It can be generally expressed as y( n) F y( n ), y( n ),..., y( n N), x( n), x( n ), x( n ),...., x( n M ) Whereas, if the output of a system, y(n), depends only on present and past inputs, i.e. not depending upon past outputs, is called non-recursive system. It can be generally expressed as y( n) F x( n), x( n ), x( n ), x( n 3),...., x( n M ) For example IIR is a recursive system, whereas FIR is a non-recursive system. Block diagrammatically, the recursive system can be shown as below. x(n) F[y(n-),y(n-),.....,y(n-N),x(n), x(n-),..,x(n-m) y(n) z In similar manner, block diagrammatic view of non-recursive system can be shown as below. x (n) F[x(n), x(n-), y(n)..,x(n-m) 0.What are the basic building blocks for block diagram representation of discrete-time systems? The basic building block for block diagram representation of discrete-time system is as follows. a. Adder. It performs the addition of two sequences. x ( n) y( n) x ( n) x ( n) x ( n)
29 b. Multiplier. This multiply two sequences. x ( n) y( n) x ( n). x ( n) x ( n) c. Delay. This simply delay the signal by one sample. x(n) z y ( n) x( n ).Compare FIR and IIR filters. FIR. The FIR filter is defined as H ( z) N n0 h( n) z n Therefore, the Finite Impulse Response filter is the functions of present input and finite numbers of past input samples.. It is a non-recursive filter, because this does not depend upon past output sample. 3. It is a linear phase filter. 4. Stable. 5. Less noise. 6. Open loop system. IIR. The IIR is defined as Y ( z) H ( z) X ( z) N b k k 0 M k Therefore, the Infinite Impulse Response filter is the functions of present input and past input samples and past output signal samples.. It is a recursive filter. a z k k z k 3. It is not a linear phase filter. 4. Not stable. 5. More noise. 6. Close loop system.
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