Sr WoVA CEU OF cnaNcERNG 8 TECHNOLOGY, RUOHUCAR course MATERA (QUESTON int wn Part \~2 Marks 1, Define DFT and IDF (or) what are the analysis and synthesis equations of DFT DET (Analysis Equation) Ktbe B eeme feo DFT (Synthesis Eaton) — np efy p PEON Ze NL No ane: xeme SF xtke IN Keo,s)2,-+ > Nol ko ate the properties DFT Perey 3 Linear ad symety Matpitation of BETS Citar cnvohion ‘Time reversal (Circular tine shift and requeney shit Complex conjugate Circle conelation 3. Define cirelar convolution Le xi(a) and x0) ate faite duration sequenees both of length N with DFTS Xi (K) and X2 (. IEX(k) = Xi) Xk) tha the sequence x(a) can be abiined by circular convolution defined as 4. How to obtain the output sequence of lucae eonvolui Consider wo finite duration sequences x(n) and (0) of duration L samples and M samples ‘The linear convolution of these two Sequences produces an output sequente of duration L+M-1 samples. Whereas, the circular convolution of x(n) and a) give N samples where Nemax(L.M).ia ‘order fo obit the number of samples in cicular convolution equal to L*M-1, bath x(a) and h(n) ‘must be appended wilh appropriate number of zro valued samples. In other words by increasing the length of the sequences x (a) and ha) to L+N-1 poins and then circularly convolving the resulting Sequences we tain te same resulta hat of line convolton, S What is zero paddiag? What are its uses? Let the sequonce x (a) has a length L. If we want to find the N-point DET (NL) of the sequence x(n), we have to all (N-L) zor othe sequence x(n), This is known a zero pang “The wes of zero pading are ‘© Wecan gt bor dispiy ofthe Frequency spectrum, ‘© With zero padding the DFT can be used ia liner fiesing 6. Define sectional convolution. Ifthe daa sequence X (0) is of log duration itis very dificult o obtain the output sequence (o) doe to limited memory ofa digital computer. Therefore, the data sequence ts divided up into Smaller sections. These sections ae provessed separately one st tine and controlled le 0 et the output Tearoom Sova riocaan one ee eeeme YONA COUEGEOF ENGNESRNG KTCINOLEGY, UDA cous ATEN (CEST A 11. What are the wo mies used forthe sectional eowvotution? "The two methods usd forthe sectional convolution are 1) The overlapadd method and 2) overap-save method. 8. Whatis overiap-add method? Tn his mcd the sizeof the input dats block x(n) sf. To each data block we-append M-1 zeros and perform N point eculr convolution of x (0) ad ha). Sie each data block i terminated ‘ih Mc sors the lst M1 pots fom each output block mst be overlapped and add to fist M- 1 points ofthe suceeding blacks This method is ealled overlap-add method. 9, What is overtap-save method? Tn this ncthod, the data Sequenee i divided into N point sesons x(n). Each section contains the last M1 dats pints of the previows scton, followed by L sew data points to form a data ‘uence of enath N-L#NEI fn rear convolution of x(n) wih. (0) the Fist Met points will ot see lin the Tinar convolution of (a) and Mp) because of alising, the remaining points will bre with Linear convolution, Hence we diva the ist (M1) pots of filtered setion (0) Nh {This process is repeated forall sections and he filtered section» ar butted together. 10, Why FFT is weeded? "fn dive! evaluation DET requires Nesomplex muliplicatis and N2—N complex aditons. “Ths for large values of N diet evakstion of the DFT is dificult, By using FFT algoritim, the umber of complex compatations can be reduce. Therefore, we ue FFT. 11, What is FFT? "The Fast Fourier Tiansform isan algorithm used to compuic the DFT. It makes use ofthe symmetry aid perilicty properties of twiddle factor to effectively reduce the DFT computation ‘Bae Ir 2 based on the fundamental principle of decomposing the emputation of DFT of a sequence fof ength N into sucessively smaller DFTs 12, How many multiplications and addiions ave required to cou pute N point DFT using radix~ 2 FFT? The number of multiplications and addons required to coupateN point DFT using mdix-2 FFT areN loge and N2 lop2N respestively 13, What is meat by vadie2 FFT “The FET algo is mest eficiot in calculating N pois DFT. Ifthe number of ouput points N ean be expressed asa power oF thats N= 2, where Ms an Steger, them this algoitim {sknown as radi-2 algorithm, ‘Decimation nTime algorithm i used to calelate the DFT of an N point sequence. The idea isto break the N point sexuenee ino Ovo sequences, the DFTS of wich ean be combined to give the DET ofthe origtal N poi sequence, This algoridin is called DIT because the sequence x (2) is often splited into smaller subsequences. 15, What is DIF algorihn? it isa popular fxm ofthe FFT algo, In this the ouip.t ssquence X (k i divided into smaller and sale subsequences thai why the ame Decinati. - In- Frequeney 16, What are the applications of FFT al “The applications of FFT algorit i eeSI WOYACDLEGE OF eNGeERNG A TECMOLOGY, RUA NCAR ‘course MATIN UESTON aA) Linear fering Corson Spectra analysis 17, Why the computations ia FFT algorithus ssa to be in place? Once the utterly operation fs perfored on pair of complex numbers (a,b) to produce (A, ), there is no nee to save the input pas. We ean stor the result (A,B) inthe same location a a, Dy. Since the same storage locations are used tvoughout the computation, we sey that te ‘computations are done in place 18 Distinguish between Haeur convolution an circular cowvolution of two sequences. ar convotution Circular convolution TENG) i Sequense of Lumber of samples | ITXC) a sequence of L number OF sampler and hia) with M number of samples, afr | and h(a) will M samples, after convolution convolatc (a) will have N=L¥M-I samples. | y(a) wll ave Nemax(LeM) samples, an be wed f find the response OF Tne | Weannot be used To Find the eoponse ofa REY Site Zero padding & wot nessssary To Tind the | Zero pada necessary tad the SPORE response of ica fk, 419. What are ie differences und similarities betwsen DIF and DIT algorithms? Differences 1) The input is bit reversed while the output sin nat ‘output is bit reversed while the input isin natural order 2) The DIF butcely is silly diferent fom the DIT buttery, the difference being thatthe complex muliplcaion kes ploe afer the add-subiract operation in DIF. ‘Sinitariien Both algorithms require some number of opcrations to compute the DET. Both algoeithns can be ‘done in place and both need to perform bit reversal at some place during the computation al order for DIT, whereas for DIF the 20. What are differences between overlap-se and overlap-add methods. ‘Overiap-aave method —_—Overarad method i method the sizeof te input date Tihs method the ste of the input ata block is LMI block i ach data block consists ofthe last NI ach data Dock iL. ponts and we append data point ofthe previous data block MeL zeros to compute N point DFT followed by L nev data points Th each output book N-I points ae Tn is no compton doe w aiasingas somrupted due aliasing ts cislar linear convolution is performed using somolaton Is pled convolution To form the opt sequsive the Fat To foum te oupar sequence te ast Mal datapoints are discarded in each [Ml points fom each ouput block is added ‘output block and th remaining daa are othe first MI points of the succeeding {ited together block. 21. Distinguish between fourier series aud Fourier Transform Fourie Series ae “Fourier transform Gives the Raronic eowtent of a perioie tine | Gives The Trequeney “infomation ora a ——smvovACOUEGEOFGIEERNG A TECINOLOY. RUDIGER unsere avesnoN eae) aneion aperiodi Sanat Diserete Frequeney Spaccam Conlin Fregu- ay Spectrum PART 8-16 Macks 1: (@) Determine the output response y(n) i444) = {1b 3a 1) by using () Linear convolotion Circular coavoltion ani ii Cicular convolution ws. 2er0 passing, (12) {@) Explain any two properties of Disrete Fourier Transfoon. @ 2. Using Hinear convolution find y (a) = x (a) (a) forthe sequences x (a) = (12123-2-3e 11,121) and ho) ~ (1.2) Compare the result by solving the problem using ‘i Overlap save method and i) Overlap ~ add eto. To) 5. Desert the desination in tine (DIT] eadis-2 PFT alortinfo termine N-point Drr. 09) 4. An Spoint discrete time sequence i gven by x0) sila}. Compute the &-point DFT (of (a) using fadiv2 FET algo, (6) 5. (@ Compute the 4-point DET and FET-DIT for the sequence x0) = {Tay 3} and What are the ise steps for & point FET-DIT algo con ptstion? (12) (@) What isthe mivataae ofradis-2 FFT algoriia in cmpariso, wi the clasical DFT method? ® 6. (4) Perform circular cowvoluton ofthe two soquences graphically x1(a) = (21.2.1) and x2(a)> 11234) 6) (Find the DFT ofa sequence by tu) 1.23442.21) using DIY algorithm. a9) 17. (a) Explain the decimation in feguensy ris-2 FFT alzorithm «evaluating N-point DET ofthe ven sequence, Draw the signal flow prop for N=. (2) (6) Find the IDET of y (0) = (1,041.0) ® £8. (@) Find the circular convolution ofthe sequences x1(a)> {1.23} nd 2(a)= 43.61} 8) (6) Write the properties of DFT and explain. ® 9 (a) Draw the 8-point Mow diagram of rac? DIF-FFT algorth, ® (@) Find the DFT ofthe sequence x(0)~ (2.3.4.5) using the shoves, gorithm. © 10.(a) What are te diferences and similares between DIT and LiF FET algorithms? (6) () Compute the Spoint IDET of the ssqyence x(k) = (7, - 70790.207, 3 0.70730-707, 1, 10.2074j0:707j-0.707490.707} using DIY algorithm, 10) 1, @ Compute tne $-point DFT ofthe soyuence x (0)= {0.05.0 505,0.000) using radixe2 DIT algoritin ® (@) Find the IDF of the sequense x (k) = (41-2414 0,1j0414.0.14,414,0,14)2414) using DIF algorithm, ® 12, Compt the 8-point DFT ofthe sequence tn) = 100 57 G,ouenwise by using DIT,DIF slgoiuns 09) Tesroamncw mocesncunieIp Sica el hb hag pal DY wendeL 12.319 armel hemegn V1) VY Ferd cemvotution Ccireutecr) a cenceriic eirele me'thed 7 li Beth Aeqventen must be Of ATE lorgm wa not by 2x0 padeling metned, add 2c to the Aryven cer (7 Graph all the pem& % x1en> bm tro outer Circle to the Countarclockwire dincetion Uv Graph a) the pein’ % matm on eotinnes. 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