Introduction to Digital Signal Processing

Transcription

1 Chaptr Introduction to. Introduction.. Signal and Signal Procssing A signal is dfind as any physical quantity which varis with on or mor indpndnt variabls lik tim, spac. Mathmatically it can b rprsntd as a function of on or mor indpndnt variabls. For xampl, th function s(t) = t.(..) dscribs a signal, which varis linarly with th indpndnt variabl t (tim). Spch signal is an xampl, which varis with singl indpndnt variabl. Considr th function s(x, y) = x + 3y + 5xy..(..) This function dscribs a signal, which varis with two indpndnt variabls x and y. Imag is a signal which varis with two indpndnt variabls. Most of th signals ncountrd ar analog in natur i.., thy vary with continuous variabl, such as tim or spac. Procssing of ths signals by analog systms such as a filtrs or frquncy analyzrs or frquncy multiplirs for th purpos of changing thir charactristics or xtracting som dsird information is calld Signal Procssing. In this cas both th input signal and th output signal ar in analog form, which is shown in Fig... Fig.. Analog signal procssing.

2 .. Basic Elmnts of Systms is an altrnativ mthod to procss an analog signal. It rquirs an intrfac btwn an analog input and digital signal procssor, calld an analog-todigital (A/D) convrtr. W should provid anothr intrfac btwn digital signal procssor and an analog output signal calld a digital-to-analog (D/A) convrtr as shown in th Fig... Fig.. Block diagram of basic DSP systm. Th ADC or analog-to-digital (A/D) convrtr contains a samplr, quantizr and an ncodr. Samplr taks an analog signal, sampls it with a prdfind sampling priod and givs out discrt-tim signal, which is discrt in tim domain and continuous (varying) in amplitud. This signal contains diffrnt numbr of amplitud lvls. Quantizr approximats ths diffrnt lvls with fixd numbr of lvls by rounding or truncating th valus. For xampl, if th allowabl signal valus in th digital signal ar intgrs, say to 7, th continuous-amplitud signal will b quantizd into ths intgr valus. Thus th signal valu 5.63 will b approximatd by th valu 6 if th quantization procss is prformd by rounding to th narst intgr or by 5 if th quantization procss is prformd by truncation. Th ncodr convrts ths st of intgr into digital form (i.., binary form). Th digital signal procssor may b a larg programmabl digital computr or a small microprocssor or a hardwird digital procssor to prform th opration on th digital signal i.., on th output of ncodr. Hardwird digital procssor prforms a spcifid st of oprations on th digital signal i.., rconfiguring is difficult with th hardwird machins, whr as programmabl machins provid flxibility to chang th oprations through a chang in th softwar. Th output of DSP block is digital signal. Digital to analog convrtr convrts digital signal into an analog signal, which may not b rquird on som applications lik xtracting information from th radar signal, such as th position of th air-craft and its spd, may simply b printd on papr.

3 Introduction to 3..3 Advantags of ovr Analog Signal Procssing Thr ar many rasons why digital systms ar prfrrd ovr an analog systm. Som of th advantags ar: (a) Nois Immunity: Digital systms ar mor immun to nois compard to an analog systms. (b) Arbitrarily High Accuracy: Tolrancs in analog circuit componnts mak difficult to control th accuracy of an analog systms, whr as digital systms provid much bttr control of accuracy. (c) High Rliability: Digital systms ar mor rliabl compard to an analog systms. (d) Softwar Manipulation: Digital signal procssing oprations can b changd by changing th program in digital programmabl systm, i.., ths ar flxibl systms. () Intgration of Digital Systms: Digital systms can b cascadd or intgratd asily without any loading problms. (f) Storag of Digital Signals: Digital signals ar asily stord on magntic mdia such as magntic tap without loss of quality of rproduction of signal. (g) Transportabl: As digital signals can b stord on magntic taps ths can b procssd off tim i.., ths ar asily transportd. (h) Digital systms ar indpndnt of tmpratur, aging and othr xtrnal paramtrs. (i) Chapr: Cost of procssing pr signal in DSP is rducd by tim-sharing of givn procssor among a numbr of signals. Disadvantag of digital systms is that thy ar not fastr compard to analog systms.. Discrt Tim Signals and Squncs Discrt-tim signals or squncs, which ar discrt in tim domain and continuous in amplitud, can b obtaind by sampling continuous tim or analog signals as shown in Fig..3.

4 4 Fig..3 Obtaining discrt-tim signal from analog signal... Rprsntation of Discrt-Tim Signals Discrt-tim signals can b rprsntd as follows by using th four mthods (i) Graphical Rprsntation (ii) Functional Rprsntation (iii) Tabular Rprsntation (iv) Squnc Rprsntation (i) Graphical Rprsntation: Discrt-tim signals can b rprsntd by a graph whn th signal is dfind for vry intgr valu of n for < n <. This is shown in Fig..4. Fig..4 Graphical rprsntation of a discrt-tim signal. (ii) Functional Rprsntation: Discrt-tim signals can b rprsntd functionally as givn blow, for n =,, for n = xn 3, for n =, lswhr

5 Introduction to 5 (iii) Tabular Rprsntation: Discrt-tim signals can b rprsntd by a tabl as n... x(n) 3 (iv) Squnc Rprsntation: An infinit-duration n signal with th tim as origin (n = ) and indicatd by th symbol, if symbol is not shown in rprsntation, origin is at th bginning of th squnc. x(n)...,, 3, 4,,... x(n) = {, 3,, 4} Hr origin is th first position i.., x() =... Elmntary Discrt-Tim Signals Thr ar som basic signals which play an important rol in th study of discrt-tim signals and systms. Ths ar: (i) Unit-sampl (Cronkar) Squnc, (n) (or) Impuls squnc (ii) Unit-stp Squnc, u(n) (iii) Unit-ramp Squnc, r(n) (iv) Exponntial Squnc (i) Unit-sampl Squnc: This is illustratd in Fig..5, it is dnotd with (n) and is dfind as, n n, n Fig..5 Graphical rprsntation of (n). (ii) Unit-stp Squnc: It is dnotd by u(n) and is dfind as u n This is shown in Fig..6., n, n

6 6 Fig..6 Graphical rprsntation of u(n). (iii) Unit-ramp Squnc: It is dnotd by r(n) and is dfind as r n This is shown in Fig..7. n, for n, for n Fig..7 Graphical rprsntation of r(n). (iv) Exponntial squnc: It is dfind as x(n) = a n for all valus of n. If th paramtr a is ral, thn x(n) is a ral squnc. Fig..8 illustrats this squnc. Fig..8 Graphical rprsntation of xponntial squnc. Exampl.: Lt (n) b an xponntial squnc and lt x(n) and y(n) dnot two arbitrary squncs. Show that n. x n * n. y n n. x n * y n Solution: Givn (n) is an xponntial squnc (n) = a n

7 Introduction to 7 W know that linar convolution of x (n) and x (n) as (* is symbol for linar convolution) * k x n x n x k.x n k In our problm x (n) = (n). x(n) = a n x(n) and x (n) = (n). y(n) = a n y(n) n n xn* xn a xn* a yn Hnc provd k k n k n n n k nk a x k.a y n k k k n x k.a.a.a y n k a x k y n k a x n * a y n a. x n * y n..3 Manipulation of Discrt-Tim Signals Hr w study som simpl modifications on indpndnt variabl (tim) and dpndnt variabl (amplitud of signal). Such modifications ar rquird in DSP tchniqus. Modification of th Indpndnt variabl (tim): This can b don in thr ways. (i) Tim shifting (ii) Folding (iii) Tim scaling (i) Tim Shifting: A signal can b shiftd right sid or dlayd by rplacing n by n k. and is shiftd lft sid or advancd by rplacing n by n + k, whr k is intgr and n is a discrt-tim indx. This is shown in Fig..9.

8 8 Fig..9 (a) Original squnc (b) Dlayd by on unit vrsion of original squnc (c) Advancd squnc by on unit vrsion of original squnc. (ii) Folding: If indpndnt variabl (tim) n is rplacd by n, thn signal folding (mirror imag) about th tim origin (n = ) taks plac. This is shown in Fig... Fig.. (a) Original squnc (b) Foldd vrsion of original squnc. (iii) Tim Scaling: Tim scaling is prformd by rplacing indpndnt variabl n by kn, whr k is an intgr. This is shown in Fig... Fig.. (a) Original squnc (b) Scald vrsion of original squnc. Modification of th Dpndnt Variabl (Signal Amplitud): Signal amplitud can b modifid in thr ways. (i) Addition of squncs (ii) Multiplication of squncs (iii) Amplitud scaling of squnc

9 Introduction to 9 (i) Addition of Squncs: Th sum of two discrt tim squncs is givn by y(n) = x (n) + x (n), This is shown in Fig.. (a) < n < (ii) Multiplication of Squncs: Th product of two discrt tim squncs is givn by y(n) = x (n). x (n), < n < This is shown in Fig.. (b) (iii) Amplitud Scaling of Squncs: Amplitud scaling of a signal by a constant A is accomplishd by multiplying th valu of vry signal sampl by A. y(n) = A x(n), whr A is ral constant quantity < n < Fig.. (a) Addition of squncs (b) Multiplication of squncs...4 Classification of Discrt-Tim Signals Discrt-tim signals ar classifid basd on numbr of diffrnt charactristics as follows: (i) Enrgy signals and powr signals (ii) Priodic signals and Apriodic signals (iii) Symmtric (Evn) and Antisymmtric (odd) signals.

10 (i) (ii) (iii) Enrgy Signals and Powr Signals Th nrgy E of a signal x(n) is dfind as E xn..(..) n Hr x(n) may b ithr complx or ral valud signal; E may b finit or infinit. If E is finit, thn x(n) is calld nrgy signal. Many signals that posss infinit nrgy hav a finit avrag powr. Avrag powr of a signal x(n) is dfind as N Pav lim xn..(..) N N nn If E is finit, P av = If E is infinit, P av may b ithr finit or infinit if P av is finit, thn x(n) is calld a powr signal. Priodic Signals and Apriodic Signals If a signal x(n) satisfis th condition x(n) = x(n + N), whr N is priod thn th signals is priodic signal, othrwis it is nonpriodic or apriodic signal. Considr a sinusoidal signal cos(ω n + ), it will b priodic only if is an intgral numbr. If is a rational, thn th function will hav a priod longr than. If is not a rational numbr, it will not b priodic at all. Th nrgy of a priodic signal ovr a singl priod, say < n < N, is finit, but nrgy of priodic signal for < n < is infinit. On th othr hand, th avrag powr of th priodic signal is finit. Priodic signals ar powr signals. Evn signals and Odd signals A ral valud signal x(n) is calld symmtric (vn) If x(n) = x( n)..(..3) On th othr hand, a signal x(n) is calld anti symmtric (odd), if x( n) = x(n)..(..4)

11 Introduction to Any ral squnc can b writtn as x(n) x o(n) x (n) whr x o (n) is odd part of x(n) and x (n) is vn part of x(n) x(n) can b writtn as x (n) x(n) x( n)..(..5) and x o (n) x(n) x( n) Similarly for complx squnc Condition for symmtry is x(n) = x*( n) whr * dnots conjugation And for Anti-symmtry is x(n) x *( n) Any complx squnc can b writtn as x(n) x o(n) x (n) whr x (n) x(n) x*( n) (..6) and x o (n) x(n) x*( n) Exampl.: Show that th vn and odd parts of a ral squnc ar, rspctivly, vn and odd squncs. [JNTU ] Solution: Lt x(n) b ral squnc, which can b writtn as x(n) x (n) x (n) o whr x o (n) is odd part of x(n) and x (n) is vn part of x(n) W know that x (n) x(n) x( n) and x o (n) x(n) x( n) W hav to show that vn part (x (n)) of x(n) is a vn squnc i.., it should satisfy x(n) x( n) Considr x (n) x(n) x( n)

12 x(n) x( n) x( n) x(n) tak x ( n) x( n) x(n) Hnc provd. Similarly w hav to show that odd part (x o (n)) of x(n) is a odd squnc i.., it should satisfy x(n) o x( n) o Considr x o (n) x(n) x( n) tak x o ( n) x( n) x(n) x(n) x( n) x( n) x(n) Hnc provd. o o.3 Linar Shift Invariant Systm, Stability and Causality Bfor going to discuss linar shift invariant systms, stability and causality, lt us dfin discrt tim systm. A discrt tim systm is a dvic or an algorithm that oprats on a discrt tim signal, calld th input or xcitation, according to som wll-dfind rul, to produc anothr discrt tim signal calld th output or rspons of th systm. W say that th input signal x(n) is transformd by th systm into a signal y(n). Ths two can b rlatd as y(n) H[x(n)]..(.3.) This is shown graphically in Fig..3. Fig..3 Discrt tim systm.

13 Introduction to 3.3. Basic Building Blocks of a Discrt Tim Systm Thr ar thr basic building blocks of a discrt tim systm. (i) Addrs or summing lmnt (ii) Multiplirs (iii) Dlay Elmnts (i) Addrs: It prforms addition of two or mor discrt-tim signals as shown in Fig..4(a). (ii) Multiplirs: Thr ar two typs of multiplirs (a) constant multiplir (b) signal multiplir. A signal multiplir prforms multiplication of two or mor discrt-tim signals as shown in Fig..4(b). A constant multiplir prforms multiplication of a discrt-tim signal with a scalar quantity as shown in Fig.4(c). (iii) Dlay Elmnts: Thr ar two typs of dlay lmnts (a) positiv dlay lmnt, which is indicatd by Z (b) ngativ dlay lmnts, which is indicatd by Z (or) advanc lmnt. Positiv and ngativ dlay lmnts provid dlay as shown in Fig..4(d) and () rspctivly. (a) (b) (c) (d) () Fig..4 (a) Addr (b) A signal multiplir (c) A constant multiplir (d) A unit dlay lmnt () A unit advanc lmnt..3. Classification of Discrt-Tim Systms Discrt-tim systms can b classifid into (i) Static (mmory lss) systms and Dynamic (systms with mmory )systms.

14 4 (ii) Tim-invariant and Tim-varying systms (iii) Linar systms and Non-linar systms (iv) Causal systms and Non-causal systms (v) Stabl systms and Unstabl systms. (i) Static and Dynamic Systms: Static systms ar also calld mmory lss systms. A discrt-tim systm is calld mmory lss systm if its output at any instant n dpnds at most on th input at th sam instant, but not on past or futur valus of input sampls. Exampl: y(n) A.x(n) y(n) n x(n) Ax (n) On th othr hand, output of a systm which dpnds on past or futur sampls of th input signal is calld dynamic systm. It is also calld a systm with mmory. Ths systms rquir mmory for storag for futur and past sampls of input signal. Exampl: y(n) x(n) x(n ) x(n ) Both systms ar static systms is a dynamic systm. (ii) Tim-invariant and Tim-varying systms: A systm is calld tim-invariant if its input-output charactristics do not chang with tim. If th rspons to a dlayd input, and th dlayd rspons ar qual thn th systm is calld tim-invariant systm (or) shift invariant systm. Th rspons to a dlayd input is dnotd by y(n, k) and th dlayd rspons is dnotd by y(n k). If both rsponss y(n,k) and y(n k) ar qual thn th systm is calld tim-invariant systm. If both rsponss ar not qual thn th systm is calld tim-varying systm. Exampl.3: Chck th following systm for Tim-invarianc y(n) x(n) n x(n ) Solution: Th rspons to a dlayd input is y(n, k) x(n k) n x(n k ) Th dlayd rspons is y(n k) x(n k) (n k)x(n k ) both rsponss ar not qual i.., y(n,k) y(n k) Thrfor th givn systm is not a Tim-invariant systm. It is a Tim-varying systm.

15 Introduction to 5 Exampl.4: Chck th following systm for Tim-invarianc y(n) x(n) x(n ) Solution: Th rspons to a dlayd input is y(n, k) x(n k) x(n k) Th dlayd rspons is y(n k) x(n k) x(n k ) both rsponss ar qual. Hnc givn systm is a Tim-invariant systm. (iii) Linar and Non-linar Systms: A systm which satisfis suprposition principl is calld a linar systm. A systm which dos not satisfy suprposition principl is trmd as a non-linar systm. Suprposition principl is statd as Rspons of th systm to a wightd sum of input signals b qual to th corrsponding wightd sum of rsponss of th systm to ach of th individual input signals. A systm is linar if and only if H[ax (n) bx (n)] ah[x (n)] b H[x (n)]..(.3.) whr x (n) and x (n) ar arbitrary input signals and a and b ar arbitrary constants. Exampl.5: Chck th following systms for Linarity. (i) y(n) x(n ) (ii) yn Solution: (i) Th corrsponding outputs for two discrt-tim squncs x(n) and x(n) ar y(n) x(n ) y(n) x(n) A linar combination of two input squncs rsults in th output y 3(n) H[x 3(n)] H[a x (n) b x (n)] x(n) 3 x(n) a x(n) b x(n)..(i) A linar combination of th two outputs rsults in th output a y (n)+b y (n) a x (n ) b x (n )..(ii) Sinc both outputs ar qual, th systm is linar. (ii) Th corrsponding outputs for two discrt-tim squncs x (n) and x (n) ar

16 6 y(n) = y(n) = x(n) x (n) A linar combination of x (n) and x (n) rsults in th output. 3 3 ax (n)+bx (n) y (n) = H [x (n)] H[a x (n) b x (n)]..(iii) Linar combination of th two outputs rsults in th output x (n) x (n) a y (n) b y (n) a b..(iv) Hr both outputs i.., quations (iii) and (iv) ar not qual, hnc th systm is non-linar. (iv) Causal and Non-causal Systms: A systm whos prsnt output dpnds only on prsnt and past inputs, but not on futur inputs is calld a causal systm. If a systm rspons dpnds on futur valus, thn it is a non-causal systm. Exampl: (i) y(n) x(n) x(n ) Sinc rspons dpnds upon a futur valu (x(n )), it is a non-causal systm. (ii) y(n) x(n) x(n ) Sinc rspons dos not dpnd upon futur valus, it is a causal systm. (iii) y(n) x( n) Tak n =, thn y( ) = x(), which is a futur valu i.., it dpnds on futur valus. Hnc th systm is a non causal systm. (iv) y(n) x(n ) and y(n) x(n) Tak n =, thn y() = x(4) in both th systms which is a futur valu. Hnc both systms ar non causal systms. (v) Stabl and Unstabl Systms: A systm which producs boundd (finit) output for a boundd (finit) input is calld as stabl systm, othrwis it is calld as an unstabl systm. Exampls will b discussd in th sction Rprsntation of Discrt Tim Signal as Summation of Impulss Graphical rprsntations of impuls squncs and its shiftd vrsions ar shown in Fig..5 (a) (b) and (c). Lt us considr a squnc

17 Introduction to 7 x(n),,,3 which is shown graphically in fig.5 (d) (a) (b) (c) (d) Fig..5 (a) Impuls squnc (b) Shiftd towards right (c) Shiftd towards lft (d) A gnral squnc. product of x(n) and (n) givs x() i.., x(n) (n) = x() x() (n) = x() similarly x(n) (n ) = x() x() (n ) = x() x(n) (n ) = x() x() (n ) = x() x(n) (n + ) = x( ) x( ) (n + ) = x( ) This can b writtn as xn xk nk..(.3.3) k Exampl: Rprsnt th squnc x(n),3,5 Solution is x(n) = (n + ) + 3 (n) + 5 (n ) as sum of impuls squncs.3.4 Rspons of Linar Tim Invariant (LTI) Systm Fig..6 shows an LTI systm with an xcitation x(n) and rspons y(n). Fig..6 A discrt tim LTI systm.

18 8 Unit sampl rspons (or) Impuls rspons: It is dfind as rspons of a systm whn input signal is impuls squnc, impuls rspons is dnotd with h (n) i.., whn x(n) = (n) y(n) = h(n) Sinc it is a tim invariant systm, whn impuls squnc is dlayd by k, rspons i.., h (n) should b dlayd by k. (n k) h(n k) This also can b writtn as h(n k) = H [(n k)] Rspons of an LTI (Linar Tim Invariant) systm is y(n) = H [x(n)]..(.3.4) w known that x(n) can b rprsntd as sum of impuls squnc as in qn. (.3.3). i.., xn xknk k Substitut this in th quation () y n H x k nk k k x k H n k yn xkhnk..(.3.5) k which is an quation for convolution of x (n) and h (n). Hnc th rspons of an LTI systm is convolution of input squnc and impuls rspons. Exampl.6: Considr a discrt linar shift invariant systm with unit sampl rspons h(n). If th input x(n) is a priodic squnc with priod N i.., x(n) = x(n + N), show that th output y(n) is also a priodic squnc with priod N. Solution: Givn x(n) = x(n + N) W known th rspons of an LTI (or) Linr shift invariant systm as y n x k h n k k also yn hxxnk k considr ynn hkxnnk k

19 Introduction to 9 dlay it by k givn xnxn N thn xnk xnn k ynn hxxnk k yn Hnc th output is also priodic whn input is priodic.3.5 Stability of an LTI Systm Lt x(n) is input squnc, assum that it is finit with a valu M x. Rspons of an LTI systm is y n x k h n k k k Tak absolut on both sids k h k x n y n h k x n k k x k M h k..(.3.6) According to dfinition for stability, for a finit input squnc, systm should produc finit output. From qn (.3.6), to gt finit output, hk k hk must b finit k Hnc an LTI systm is stabl if its impuls rspons (or) unit sampl rspons is absolutly summabl. Exampl.7: Tst th stability of th following systms (i) y(n) = x( n ) (ii) y(n) = n x(n) Solution: W know that whn x(n) = (n), th output y(n) = h(n)

20 (i) h(n) = ( n ) n h ; n = h n 3 h ; n = h n h ; n = h (ii) n n h n h n Systm is stabl h(n) = n (n) n h; n = h n h ; n = h n h; n = h n h n Systm is stabl Exampl.8: Dtrmin th rang of valus of th paramtr a for which th LTI systm with impuls rspons h(n) = a n u(n) is stabl. Solution: Condition for a systm to b stabl is n n h n n a a a This infinit sris convrgs to if a a

21 Introduction to Rang of valus of paramtr a is a Exampl.9: Dtrmin th rang of valus of a and b for which LTI systm with n a, n impuls rspons hn is stabl n b, n Solution: Put n = in first sris n h n b a n n n a n nb n n n a a b b a a b b b and a < if b b a b and a < b Rang of valus of a and b ar b and a < Exampl.: A unit sampl rspons of a linar systm is givn by hnnba n, n, n For what valus of a and b th systm will b stabl? nb a n a b a Solution: n n n finit n n n 3 a a 3a b a a a a 3a b a b a if a < and b must finit to bcom sris a a

22 Valus of a and b ar a < and b <.4 Linar-Constant Cofficint Diffrnc Equations W know that continuous tim systms ar dscribd by diffrntial quations. But discrt-tim systms ar dscribd by diffrnc quations. Input-output rlation of N th ordr discrt-tim systm can b writtn as N M a k ynk b x xnk..(.4.) k k whr y(n) is output x(n) is input and a k and b k ar constant cofficints. Ordr of th systm is dtrmind by L.H.S summation sinc input-output rlation is linar with constant cofficints, this quation is calld Linar-constant cofficint diffrnc quation for N th ordr. Thr ar two mthods by which diffrnc quations can b solvd. Dirct Mthod: This mthod is dirctly applicabl in th tim domain. W ar not discussing this mthod. Indirct Mthod: It is also calld z-transform mthod. This mthod will b discussd in th chaptr 3..5 Frquncy Domain Rprsntation of Discrt-Tim Systms and Signals.5. Frquncy Domain Rprsntation of Discrt-Tim Systm Systm function (or) transfr function of a systm can b obtaind by taking Z-transform (for Z-transform rfr chaptr 3) of impuls rspons h(n) i.., systm function = H(z) = n h(n)z.(.5.) n systm function can also b dfind as ratio of z-transform of rspons to z-transform of input with zro initial conditions. i.., H(z) = Y(z).(.5.) X(z) Frquncy rspons of a systm can b obtaind just by putting z = j in quation (.5.).

23 Introduction to 3 i.., H( j ) = H() = jn h(n).(.5.3) n Magnitud spctrum of a systm is obtaind by taking modulus of H( j ) i.., H( j ) Phas spctrum of a systm is obtaind by H( i ) tan.(.5.4) H( r ) whr H i () = Imaginary part of H() H r () = Ral part of H().5. Frquncy Domain Rprsntation of Discrt-Tim Signals Lt us considr any discrt-tim squnc say x(n) Frquncy domain rprsntation of this squnc can b obtaind by taking z-transform of x(n) and putting z = j i.., X(z) = n x(n) z whr x(z) is z-transform of x(n) put z = j X j X( n n x(n) jn which is frquncy domain rprsntation of x(n). Exampl.: An LTI systm has unit sampl rspons h(n) = u(n) u(n N). Find th amplitud and phas spctra. Solution: h(n) =, n = to N from th figurs shown H(z) = N n. z n

24 4 N n z z z N n a a n a Frquncy rspons can Finit b obtaind Gomtric by Sris putting z = j jn j jn jn jn j j j N j sin (N ) sin Magnitud spctra is N sin H( ) sin Phas spctra is H( ) (N ) n N N Rviw Qustions. Pick th signal which varis with singl indpndnt variabl. (a) Spch (b) Imag (c) S = x + 6xy (d) S = 3x y Ans: [b]. Pick th signal which varis with two indpndnt variabls. (a) Imag (b) S = t (c) Spch (d) S = t + 3t 3 Ans: [c]

25 Introduction to 5 3. Which of th following is not an analog systm? (a) Frquncy analyzrs (b) Analog filtrs (c) Frquncy multiplirs (d) Programmabl machins Ans: [d] 4. Which of th following is not a block in basic DSP systm? (a) ADC (b) Digital signal procssor (c) Analog signal procssor (d) DAC Ans: [c] 5. Which of th following is not a part of an analog-to-digital convrtr? (a) Samplr (b) Dcodr (c) Encodr (d) Quantizr Ans: [b] 6. Th following is th disadvantag of digital systms (a) Cost (b) Spd (c) Transportability (d) Nois immunity Ans: [b] 7. Discrt-tim signal is (a) Discrt both in tim and amplitud (b) Discrt in tim and continuous in amplitud (c) Continuous in tim and discrt in amplitud (d) All th abov Ans: [b] 8. Digital signal is (a) Discrt both in tim and amplitud (b) Continuous in tim and discrt in amplitud (c) Discrt in tim and continuous in amplitud (d) All th abov Ans: [a] 9. Discrt-tim signal can b rprsntd by (a) Graphical mthod (b) Functional mthod (c) Squnc mthod (d) All th abov Ans: [d]. Th othr nam of unit impuls squnc (a) Unit-sampl squnc (b) Unit-stp squnc (c) Unit ramp squnc (d) All th abov Ans: [a]. Unit stp squnc is dfind as (a) x(n) = (c) x(n) = n n n n n (b) x(n) = (d) x(n) = n n n n Ans: [b]

26 6. Unit impuls is dfind as (a) x(n) = n n n (c) x(n) = n 3. Unit ramp is dfind as (a) x(n) = n n (b) x(n) = (d) x(n) = (b) x(n) = n n n n n n Ans: [a] (c) n n x(n) = (d) x(n) = n n Ans: [c] 4. Exponntial squnc a n is dcaying whn (a) < a < (b) a > (c) a < (d) < a < Ans: [a] 5. Discrt-tim signal can b modifid by modifying indpndnt variabl using th following mthods. (a) Tim shifting (b) Folding (c) Tim-scaling (d) All th abov Ans: [d] 6. A signal x(n) can b shiftd right sid by rplacing n with (a) n + k (b) n k (c) n k (d) nk Ans: [b] 7. A signal x(n) can b shiftd lft sid by rplacing n with (a) n + k (b) n k (c) n k (d) nk Ans: [a] 8. A signal x(n) will b foldd if n is rplacd by (a) n (b) n (c) n + k (d) n k Ans: [a] 9. Discrt-tim signal can b modifid by modifying dpndant variabl using th following mthods (a) Addition of squncs (b) Multiplication of squncs (c) Amplitud scaling of squnc (d) All th abov Ans: [d]. Enrgy of a signal is (a) E = x(n) (b) E = x(n) N n n N n N (c) E = Lim N x(n) (d) E = x(n) Ans: [a] N N N n N

27 Introduction to 7. Avrag powr of a signal is (a) Pav Lim x(n) N N n N N (b) P av N n N x(n) (c) Pav x(n) (d) Pav x(n) n N n N Ans: [a]. Enrgy singal s avrag powr is (a) Infinit (b) Zro (c) Cannot b dtrmind (d) Non Ans: [b] 3. Powr signal s is nrgy is (a) Infinit (b) Zro (c) Cannot b dtrmind (d) Non Ans: [a] 4. A priodic signal satisfis th condition (a) x(n) x(n N) (b) x(n) x(n N) (c) x(n) x( n) (d) xn x n Ans: [a] 5. A sinusoidal signal cos n will b priodic only if is an/a (a) Intgr (b) Irrational (c) Infinit (d) Zro Ans: [a] 6. A ral signal x(n) is calld symmtric if (a) x(n) x(n N) (b) x(n) x( n) (c) x(n) x( n) (d) x(n) x(n N) Ans: [b] 7. A ral signal x(n) is calld anti symmtric if (a) x(n) x(n N) (b) x(n) x( n) (c) x(n) x( n) (d) x(n) x(n N) Ans: [c] 8. A complx signal x(n) is calld symmtric if (a) x(n) x(n N) (b) x(n) x *( n) (c) x(n) x( n) (d) x(n) x( n) Ans: [b] 9. A complx signal x(n) is calld odd signal if (a) xnx n (b) xn x n (c) xnx n (d) xn x n Ans: [a]

28 8 3. Th othr nam of LTI systm is (a) LTV (b) LSI (c) TV (d) Non Ans: [b] 3. Positiv dlay lmnt is (a) Z (b) Z + (c) Z (d) Non Ans: [a] 3. Ngativ dlay lmnt is (a) Z + (b) Z (c) Z (d) Non Ans: [a] 33. Th othr nam of static systm (a) Causal (b) Mmory lss (c) Tim variant (d) Stabl Ans: [b] 34. Th othr nam of Dynamic systm (a) Causal (b) Mmory lss (c) Tim variant (d) Systm with mmory Ans: [d] 35. Linar systm should satisfy (a) Stability condition (b) yn k xn k (c) Suprposition principl (d) Non Ans: [c] 36. Tim invariant systm should satisfy (a) Stability condition (b) yn k xn k (c) Suprposition principl (d) Non Ans: [b] 37. Causal systm rspons dpnds upon (a) Futur input (b) Prsnt input, past input, futur input (c) Past input, futur input (d) Prsnt input, past input Ans: [d] 38. Pick a causal systm (a) yn x n (b) yn xn (c) yn xn y n x n (d) Ans: [c]

29 Introduction to Pick non causal systm (a) yn x n (b) yn = x n + xn (c) yn xn (d) yn xn 4. Pick non causal systm y n x n Ans: [a] (a) (b) yn = xn + xn (c) ynxnxn (d) yn xn Ans: [b] 4. A systm is said to b unstabl if it givs.. output, for a finit input (a) Finit (b) Zro (c) Infinit (d) On Ans: [c] 4. Condition for a systm to b stabl is (a) Impuls rspons should b absolutly summabl (b) (c) (d) n N nn N h n h n hn Ans: [a] nn 43. Rprsnt th squnc xn,, 3, 4 as sum of impulss (a) nn3 n 4n 3 (b) nn3 n 4n (c) nn3 n 4n (d) nnn n Ans: [b] 44. Rspons of an LTI systms is (a) Multiplication of input and impuls rspons (b) Subtraction of input and impuls rspons (c) Addition of input and impuls rspons (d) Convolution of input and impuls rspons Ans: [d] 45. Discrt-tim systms ar dscribd by (a) Diffrntial quations (b) Diffrnc quations (c) Linar quations with variabl cofficints (d) Non Ans: [b]

30 3 46. Continuous-tim systms ar dscribd by (a) Diffrntial quations (b) Diffrnc quations (c) Linar quations with variabl cofficint (d) Non Ans: [a] 47. Solution of diffrnc quations can b obtaind by (a) Laplac transform (b) Fourir transform (c) Z-transform (d) Non Ans: [c] 48. Solution of diffrntial quations can b obtaind by (a) Laplac transform (b) Fourir transform (c) Z-transform (d) Non Ans: [a] 49. Rlation btwn systm function and impuls rspons is H z h n (a) n (b) Hz hnz n (c) H (z) = Laplac Transform of h (n) (d) Non Ans: [b] N 5. Finit Gomtric sris a n is (a) (c) N a a N a a n (b) (d) N a a a a N Ans: [b]