PPKEE, Pusat Pengajian Kejuruteraan Elektrik & Elektronik

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-1- [EEE 443]…2/-UNIVERSITI SAINS MALAYSIAFirst Semester Examination2016/2017 Academic SessionDecember 2016/January 2017EEE 443/3 – DIGITAL SIGNAL PROCESSING[PEMPROSESAN ISYARAT DIGIT]Duration : 3 hours[Masa : 3 jam]Please check that this examination paper consists of THIRDTEEN (13) pages of printedmaterial and FIVE (5) pages of Appendix before you begin the examination. English versionfrom page TWO (2) to page SEVEN (7) and Malay version from page EIGHT (8) to pageTHIRDTEEN (13).Sila pastikan bahawa kertas peperiksaan ini mengandungi TIGA BELAS (13) muka suratbercetak beserta Lampiran LIMA (5) mukasurat bercetak sebelum anda memulakanpeperiksaan ini. Versi Bahasa Inggeris daripada muka surat DUA (2) sehingga muka suratTUJUH (7) dan versi Bahasa Melayu daripada muka surat LAPAN (8) sehingga muka suratTIGA BELAS (13).Instructions: This question paper consists of SIX (6) questions. Answer FIVE questions.All questions carry the same marks.[Arahan: Kertas soalan ini mengandungi ENAM (6) soalan. Jawab LIMA soalan. Semuasoalan membawa jumlah markah yang sama]Answer to any question must start on a new page[Mulakan jawapan anda untuk setiap soalan pada muka surat yang baharu].“In the event of any discrepancies, the English version shall be used”.[Sekiranya terdapat sebarang percanggahan pada soalan peperiksaan, versi BahasaInggeris hendaklah diguna pakai].-2- [EEE 443]…3/-ENGLISH VERSION1. (a) An electroencephalogram (EEG) signal is useful in monitoring brain activities.EEG signal contains useful frequencies up to 100 Hz. If we want to process EEGsignal using digital signal processor, analog-to-digital (A/D) conversion needs tobe done.(i) Draw the basic block diagram of an A/D and explain the function of eachblock.(20 marks)(ii) What is the Nyquist rate for this EEG signal?(5 marks)(iii) If we use a sampling rate of 400 samples/s, what is the highest EEGfrequency that can be represented uniquely at this sampling rate?(10 marks)(b) A linear time-invariant (LTI) discrete-time system is built from two blocks. Thesetwo blocks are connected in parallel. The first block has an impulse responseh1(n) while the second block has an impulse response h2(n). The input for thissystem is x(n), whereas the output of this system is y(n).(i) Draw the block diagram for this system.(10 marks)(ii) Determine the output of the system, if:}6,3,1,2,5,4{)(}2,2,2,1{)(}1,2,2,1{)(21nxnhnh(25 marks)(c) Give one of the usages of the autocorrelation. Then, calculate theautocorrelation of signal x(n) and y(n), rxy(l), given:}25,6,16,8,1,1{)(}30,10,20,10{)(nynx(30 marks)-3- [EEE 443]…4/-2. Question 2 is based on Figure 2. This figure shows the ROC for signal x(n), togetherwith its pole-zero plot. In this plot, there are two zeros (i.e., z1 and z2), and two poles(i.e., p1 and p2). The ROC is given as .Figure 2(a) Determine all zeros and poles, in terms of .(10 marks)(b) Give one possible expression for X(z).(15 marks)(c) Give one possible expression for x(n).(15 marks)(d) Analyze the signal:(i) Is the signal stable? Give your reason for your answer.(10 marks)(ii) Is the signal causal? Give your reason for your answer.(10 marks)(iii) If is equal to 0.25, is the signal periodic? If it is periodic, determine itsperiod.(10 marks)(iv) If is equal to 0.25 , is the signal periodic? If it is periodic, determine itsperiod.(10 marks)Re(z)Im(z)ooxxz1 z2p1p200-4- [EEE 443]…5/-(v) Can x(n) be transformed into frequency domain by using Fourier transform?Give the reason for your answer. If x(n) has a Fourier transform pair,determine X().(20 marks)3. (a) Find y(n)=x1(n) 4 x2(n), if x1(n)={1,-1,2,1} and x2(n)={3,4}.(20 marks)(b) Direct implementation of discrete Fourier transform (DFT) requires a lot ofcalculations. To reduce the computational time, the fast Fourier transform (FFT)has been introduced.(i) Draw eight-point FFT.(20 marks)(ii) Draw eight-point inverse FFT (IFFT).(20 marks)(c) You want to develop a speech recognition system. The discrete-time input toyour system is an aperiodic digital voice signal x(n) with the sample length Lequal to 1800. You found out that x(n) suffers from additive noise. To removethis noise, you plan to do noise filtering process in frequency domain. The noisereduction filter that you want to use has a finite impulse response, with length Mequal to 7. The impulse response of this filter in frequency domain is H(k). Tosave computational time, you are provided with a programming library withfunctions of FFT for 4-point, 16-point, 32-point, 64-point, and 128-points. Byusing appropriate figure, explain how you will process this voice signal.(40 marks)-5- [EEE 443]…6/-4. (a) Assuming that you must design a digital filter for an application in which thephase-distortion is not tolerable, which filter type amongst FIR or IIR will youselect for that application? Explain your answer?(20 marks)(b) Design a bandstop filter that satisfies the following specifications:(i) Estimate the order of the equiripple filter required to meet thesespecifications.(20 marks)(ii) What weighting function, , should be used to design this filter?(10 marks)(iii) What is the minimum number of extremal frequencies that the optimal filtermust have?(20 marks)(c) Design a linear phase digital FIR low-pass filter with the following specifications;using windows design method.(30 marks)5. (a) Determine the coefficients of an IIR filter which are obtained using bilineartransformation from a second-order Butterworth analog prototype filter with a 3-dB cut-off frequency of 3 kHz. The sampling rate for the digital filter is 30,000samples per second.(50 marks)-6- [EEE 443]…7/-(b) A second-order lowpass IIR digital filter with a 3-dB cutoff frequency at hasfollowing transfer function. Design a second-order lowpass filter with a 3-dB cutofffrequency by transforming the above lowpass filter function using a lowpass-to-lowpass spectral transformation.(20 marks)(c) Design a digital low-pass filter using Butterworth and Chebyshev filters that has apassband cutoff frequency with and a stopband cutoff frequency with . The filteris to be designed using the bilinear transformation.(30 marks)6. (a) Given a system with difference equation given below, find a transposed directform II for this system.(10 marks)(b) The unit sample response of an FIR filter is(i) Draw the direct form implementation of this system.(20 marks)(ii) Show that the corresponding system function isand use this to draw a flowgraph that is a cascade of an FIR system with anIIR system.(30 marks)(c) Given a network shown in Figure 6, find the system function and the unit sampleresponse. Then, draw an equivalent direct form II structure.(40 marks)-7- [EEE 443]…8/-Figure 6-oooOOooo-VERSI BAHASA MELAYU1. (a) Isyarat elektroencephalogram (EEG) berguna dalam memantau aktiviti otak.Isyarat EEG mengandungi frekuensi berguna sehingga 100 Hz. Jika kita mahu-8- [EEE 443]…9/-memproses isyarat EEG menggunakan pemproses isyarat digital, penukarananalog-ke-digital (A/D) perlu dilakukan.(i) Lukiskan rajah blok asas A/D dan terangkan fungsi setiap blok tersebut.(20 markah)(ii) Apakah kadar Nyquist bagi signal EEG ini?(5 markah)(iii) Jika kita menggunakan kadar pensampelan 400 sampel/s, apakahfrekuensi tertinggi EEG yang boleh diwakili secara unik pada kadarpensampelan ini?(10 markah)(b) Satu sistem diskret linear masa tak berubah (LTI) dibina daripada dua blok.Kedua-dua blok disambungkan secara selari. Blok pertama mempunyaisambutan dedenyut h1(n) manakala blok kedua mempunyai sambutan dedenyuth2(n). Input bagi sistem ini adalah x(n), manakala output sistem ini ialah y(n).(i) Lukiskan rajah blok bagi system ini.(10 markah)(ii) Tentukan keluaran sistem, jika:}6,3,1,2,5,4{)(}2,2,2,1{)(}1,2,2,1{)(21nxnhnh(25 markah)(c) Berikan satu daripada kegunaan-kegunaan autokorelasi. Kemudian, kiraautokorelasi signal x(n) dan y(n), rxy(l), diberikan:}25,6,16,8,1,1{)(}30,10,20,10{)(nynx(30 markah)-9- [EEE 443]…10/-2. Soalan 2 adalah berdasarkan Rajah 2. Rajah ini menunjukkan ROC bagi isyarat x(n),bersama dengan plot kutub-sifar. Dalam plot ini, terdapat dua sifar (iaitu z1 dan z2),dan dua kutub (iaitu p1 dan p2). ROC diberikan sebagai .Rajah 2(a) Tentukan kesemua sifar dan kutub, dalam terma 0.(10 markah)(b) Berikan satu ungkapan yang mungkin bagi X(z).(15 markah)(c) Berikan satu ungkapan yang mungkin bagi x(n).(15 markah)(d) Analisa signal tersebut:(i) Adakah signal tersebut stabil? Berikan alasan kepada jawapan anda.(10 markah)(ii) Adakah signal tersebut kausal? Berikan alasan kepada jawapan anda.(10 markah)(iii) Jika  0 bersamaan 0.25, adakah signal tersebut berkala? Jika ianyaberkala, tentukan tempoh kalaan tersebut.(10 markah)(iv) Jika  0 bersamaan 0.25  , adakah signal tersebut berkala? Jika ianyaberkala, tentukan tempoh kalaan tersebut.(10 markah)Re(z)Im(z)ooxxz1 z2p1p200-10- [EEE 443]…11/-(v) Bolehkah x(n) diubah ke domain frekuensi menggunakan jelmaan Fourier?Berikan alasan kepada jawapan anda. Jika x(n) mempunyai pasanganjelmaan Fourier, tentukan X().(20 markah)3. (a) Dapatkan y(n)=x1(n) 4 x2(n), jika x1(n)={1,-1,2,1} dan x2(n)={3,4}.(20 markah)(b) Implementasi terus jelmaan Fourier diskret (DFT) memerlukan banyak pengiraan.Untuk mengurangkan masa pengiraan, jelmaan Fourier pantas (FFT) telahdiperkenalkan.(i) Lukiskan lapan-titik FFT.(20 markah)(ii) Lukiskan lapan-titik FFT songsang (IFFT).(20 markah)(c) Anda mahu membangunkan satu sistem pengecaman pertuturan. Masukanmasa diskret kepada sistem anda adalah isyarat suara digital tak berkala x(n)dengan panjang sampel L bersamaan 1800. Anda mendapati bahawa x(n)mempunyai hingar tambah. Untuk membuang hingar ini, anda merancang untukmelakukan proses penapisan hingar dalam domain frekuensi. Penapispengurangan hingar yang anda mahu gunakan mempunyai sambutan dedenyut,dengan panjang M sama dengan 7. Sambutan dedenyut penapis ini dalamdomain frekuensi ialah H(k). Untuk menjimatkan masa pengiraan, andadisediakan dengan sebuah perpustakaan pengaturcaraan dengan fungsi FFTuntuk 4-titik, 16-titik, 32-titik, 64-titik, dan 128-titik. Dengan menggunakan rajahyang sesuai, terangkan bagaimana anda akan memproses isyarat suara ini.(40 markah)-11- [EEE 443]…12/-4. (a) Dengan mengandaikan bahawa anda perlu mereka bentuk penapis digital untuksebuah aplikasi di mana fasa-penyelewengan tidak boleh dipertimbangkan,apakah jenis penapis di kalangan FIR atau IIR yang anda akan pilih untukaplikasi tersebut? Terangkan jawapan anda?(20 markah)(b) Reka bentuk penapis jalur henti yang memenuhi spesifikasi berikut:(i) Anggarkan tertib penapis sama riak yang diperlukan untuk memenuhispesifikasi ini.(20 markah)(ii) Apakah fungsi pemberat, , yang boleh digunakan untuk mereka bentukpenapis ini?(10 markah)(iii) Apakah bilangan minimum frekuensi extremal yang penapis optimum mestiperolehi?(20 markah)(c) Reka bentuk satu fasa lelurus penapis digital FIR laluan rendah denganspesifikasi berikut:menggunakan kaedah tetingkap.(30 markah)-12- [EEE 443]…13/-5. (a) Tentukan pekali penapis IIR yang diperolehi dengan menggunakan transformasidwilelurus daripada tertib-kedua prototaip penapis analog Butterworth dengan 3-dB frekuensi potong 3 kHz. Kadar pensampelan untuk penapis digital ialah30,000 sampel sesaat. Tentukan pekali penapis IIR ini.(50 markah)(b) Satu tertib-kedua penapis digital IIR laluan rendah dengan frekuensi potong 3-dBpada mempunyai fungsi pindah berikut. Reka bentuk satu tertib-kedua penapislaluan rendah dengan 3-dB frekuensi potong dengan mengtransformasikanfungsi penapis laluan rendah di atas dengan menggunakan transformasispektrum laluan rendah ke laluan rendah.(20 markah)(c) Reka bentuk penapis digital laluan rendah menggunakan penapis Butterworthdan Chebyshev yang mempunyai frekuensi potong jalur lulus dengan danfrekuensi potong jalur henti dengan . Penapis ini mestilah direka bentukmenggunakan transformasi dwilelurus.(30 markah)6. (a) Diberi sistem dengan persamaan perbezaan diberikan di bawah, cari satu bentuklangsung II untuk sistem ini.(10 markah)(b) Sambutan unit sampel penapis FIR adalah(i) Lukis bentuk langsung perlaksanaan sistem ini.(20 markah)(ii) Tunjukkan bahawa fungsi sistem bagi sistem yang sama adalah-13- [EEE 443]…14/-dan gunakan fungsi ini untuk melukis gambarajah aliran yang latasistemnya daripada sistem FIR dengan IIR.(30 markah)(c) Diberi rangkaian yang ditunjukkan dalam Rajah 6, cari fungsi sistem dansambutan sampel unit. Kemudian, lukiskan satu struktur bentuk langsung II yangsetara.(40 markah)Rajah 6-oooOOooo-Appendix A/Lampiran A-14- [EEE 443]…15/-Table 1: Summary of analysis and synthesis formulasContinuous-time signal Discrete-time signalsTime-domain Frequency-domain Time-domain Frequency-domainPeriodicsignalsFourierseries dtetxTTc tkFjaTppkp02)(1  ktkFjka ectx 02)(  10/2)(1NnNknjk enxNc  10/2)(NkNknjk ecnxContinuous andperiodicDiscrete and aperiodic Discrete and periodic Discrete and periodicAperiodicsignalFouriertransform   dtetxFX Ftjaa 2)()(  dFeFXtx Ftjaa 2)()( nnjenxX  )()(   2)(21)( deXnx njContinuous andaperiodicContinuous andaperiodicDiscrete and aperiodic Continuous andperiodicDiscrete Fourier Transform (DFT):1,...,2,1,0,)()(10/2  NkenxkXNnNknj Inverse Discrete Fourier Transform (IDFT):1,...,2,1,0,)(1)(10/2  NnekXNnxNkNknj Table 2: Some common z-transform pairs.Signal, x(n) z-Transform, X(z) ROC1 )(n 1 z All2 )(nu111 z1|| z3 )(nuan111 az|||| az 4 )(nunan  2111  azaz |||| az 5 )1(  nuan111 az|||| az 6 )1(  nunan  2111  azaz |||| az 7 )()(cos 0 nun20101cos21cos1zzz 1|| z8 )()(sin 0 nun20101cos21sin zzz 1|| z9 )()cos( 0 nunan 220101cos21cos1zaazaz |||| az 10 )()sin( 0 nunan 220101cos21sin zaazaz |||| az 1/5Table 3: Properties of the z-transform.Property Time domain z-domain ROC-15- [EEE 443]…16/-Notation)()()(21nxnxnx)()()(21zXzXzX ROC: r2 < |z| <r1ROC1ROC2Linearity )()( 21 nbxnax  )()( 21 zbXzaX  At least the intersection of ROC1 andROC2Time-shifting )( knx  )(zXz k That of X(z) except z=0 if k>0 andz=∞ if k<0.Scaling in the z-domain )(nxan )( 1zaX  |a|r2 < |z| <|a|r1Time reversal )( nx  )( 1zX (1/r1)< |z| <(1/r2)Conjugation )(* nx *)(* zX ROCReal part )}(Re{ nx  *)(*)(21 zXzX  Includes ROCImaginary part )}(Im{ nx  *)(*)(21 zXzXj  Includes ROCDifferentiation in the z-domain)(nnxdzzdXz )( r2 < |z| <r1Convolution )()( 21 nxnx  )()( 21 zXzX  At least the intersection of ROC1 andROC2Correlation )()()( 2121 lxlxlr xx  )()()( 12121  zXzXzR xx At least the intersection of ROC ofX1(z) and ROC of X2(z-1)Initial value theorem causal  )( If nx )(lim)0( zXxz Multiplication )()( 21 nxnx dvvvzXvXj C121 )(21   At least, r1l r2l < |z| < r1v r2vParseval’s relationdvvvXvXjnxnxCn1**21*211)(21)()(   2/5Appendix B/Lampiran Bi. Information on FIR filter design using common windows-16- [EEE 443]…17/-Table A.1 Some common windowsTable A.2 The Peak side-lobe amplitude of some common windows and theapproximate transition width and stopband attenuation of an Nth-Order low-pass filter designed using the given windowii. Information on FIR and IIR filter designTable A.3 Equations used to design FIR and IIR filtersUnit sample response foran ideal lowpass filterEquiripple filterDiscrimination FactorSelectivity FactorButterworthwhere,3/5-17- [EEE 443]…18/-The poles:The order:ChebyshevSystem function of a type I Chebyshev filterwhere,The order:EllipticwhereThe order:wherewhereiii. Information on bilinear and frequency transformation4/5-18- [EEE 443]…19/-Table A.4 Information used for bilinear and frequency transformationBilinear Z-transformPre-warped analog frequencyTable A.5 The transformation of an analog low-pass filter with a 3-dB cutoff frequencyto other frequency selective filtersTable A.6 The transformation of a digital low-pass filter with a cutoff frequency toother frequency selective filters5/5

EEE 443/3 DIGITAL SIGNAL PROCESSING [PEMPROSESAN ISYARAT DIGIT] (DECEMBER 2016 / JANUARY 2017)

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