T-6.40 P (Problems&olutios, autum 003) Page / 9 T-6.40 P (Problems&olutios, autum 003) Page / 9 T-6.40 igal Processig ystems Exercise material for autum 003 - olutios start from Page 6.. Basics of complex umbers (for example p. 7 / Oppeheim). Euler s formula e jω = cos(ω) + j si(ω) Express the followig complex umbers i Cartesia form (x + jy): a) e jπ b) e j5π/ Express the followig complex umbers i polar coordiates (re jθ, with π < θ π): c) d) + j Usig complex cojugates z = x + jy = r e jθ, z = x jy = r e jθ ad module z = r = (x + y ) /, show that e) zz = r f) (z + z ) = z + z. Eve ad odd fuctios. What is a eve fuctio (Eve), odd (Odd)? ketch a example. Att: Evex(t)} = /[x(t) + x( t)] ja Oddx(t)} = /[x(t) x( t)]. Calculate: a) H(ω) = Evee jω } = /[H(ω) + H( ω)] b) y(t) = Oddsi(4πt)u(t)} 3. ketch the followig sigals ad sequeces aroud origo (t = 0 or = 0). a) x (t) = cos(t π/) b) x [] = si(0.π) c) x 3 [] = si(π) d) x 4 [] = δ[ ] + δ[] + δ[ + ] e) x 5 [] = δ[ ] + δ[0] + δ[] f) x 6 [] = u[] u[ 4] b) x[] = cos( 8 π) c) x[] = cos( π 4 ) + si(π 8 ) cos(π + π 6 ) 6. Cosider two systems ad,, whose iput-outputrelatios are: : y[] = x[] + x[ ] : y[] = x[] 3x[ ] x[ ] a) Express the output for the cascade ja b) Express the output for the parallel ja 7. I Figure there is a discrete-time system, whose output is Is the system y[] = Odx[ + ]} = (x[ + ] x[ ]) x[] y[] Figure : Problem 7: ystem. a) memoryless? b) liear? c) time-ivariat? d) stable? e) causal? 8. The output of a liear time-ivariat system to a iput x (t) is y (t) (see Figure ). Calculate the output of the system with iput x (t). x (t) y (t) 4. Which of the followig cotiuous-time sigals are periodic? Derive the basic period of periodic sigals. 0 t - 0 3 t a) x(t) = 3 cos( 8π 3 t) b) x(t) = e j(πt ) c) x(t) = cos( π 8 t ) 5. Which of the followig discrete-time sequeces are periodic? Derive the basic period of periodic sequeces. a) x[] = 3 cos( 8π 3 ) x (t) 0 3 t Figure : Problem 8: The iput ad output of a liear time-ivariat system.
T-6.40 P (Problems&olutios, autum 003) Page 3 / 9 T-6.40 P (Problems&olutios, autum 003) Page 4 / 9 9. Calculate the covolutio h[] x[] for - 0 3 4 5 a) x[] ja h[] are depicted i Figure 3. (LTI) b) x[] = α u[] h[] = β u[] c) x[] = ( ) u[ 4] h[] = 4 ( ) x[] 0 3 4 5 6 7 3 4 5 6 Figure 3: Problem 9(a): The iput ad impulse respose of the system. 0. ystem properties. Examie, if the system below is a) liear ad/or causal: y[] = a x[] + b x[ ] + abx[ ], where a ad b are real coefficiets b) stabiili ad/or causal: y[] = x[ + ] + 0.5 x[ + ] c) liear ad/or time-ivariat: y[] = x [] = (x[]) d) time-ivariat ad/or stabiili: y[] = x[ ] e) memoryless ad/or ivertible: y[] = x[ ] f) liear ad/or ivertible: y[] = x[] + a, where a is a real coefficiet x x y y a b y * 3 x x a b x 3 Figure 4: howig the liearity. x x D k x = x [-k] y D k y y * = y [-k] Figure 5: howig the time-ivariace. y 3 h[] a) ketch the block diagram of the system. b) Determie the impulse respose h[] of the system. c) What is the respose of the system to iput sequece x[] = ( 3) u[].. Cosider a system defied by the differece equatio y[] y[ ] = x[]. a) ketch the block diagram of the system. b) Determie the impulse respose h[] of the system whe 0 4. What is the impulse respose like with larger values of? c) olve the differece equatio with the iput x[] = ( 3) u[]. 3. uppose we have a cascade coectio of three liear ad time-ivariat (LTI) systems (Figure 6). It is kow that the impulse respose h [] equals h [] = u[] u[ ] ad that the impulse respose of the whole coectio equals the oe show i Figure 7. a) What is the legth of the o-zero portio of the impulse respose h []? Determie the impulse respose h []. b) What is the respose of the system to iput sequece x[] = δ[] δ[ ]? x[] h [] h [] h [] Figure 6: Problem 3: A cascade of three LTI systems. 5 0-0 3 4 5 6 7 8 4 h[] y[] Figure 7: Problem 3: The impulse respose of the cascade system.. Cosider a system defied by the differece equatio y[] = x[] x[ ]. 4. Calculate the covolutio y[] = x[] h[]
T-6.40 P (Problems&olutios, autum 003) Page 5 / 9 T-6.40 P (Problems&olutios, autum 003) Page 6 / 9 where ( ) x[] = 3 u[ ] + u[] 3 ( ) h[] = u[ + 3] 4 b) x [] = si(π/) + cos(π/4). Periodic sigal x(t), whose fudametal period is is defied: t, 0 t x(t) = t, < t a) directly by usig the defiitio of covolutio, b) by usig the distributive property of covolutio ((x +x ) h = (x h)+(x h)). 5. uppose we have a LTI system whose output y[] ad iput x[] ca be characterized by the differece equatio y[] y[ ] + y[ 3] = x[] 5x[ 4]. a) Verify, whether ca be represeted as a cascade of two causal LTI systems ja defied as : y [] = x [] 5x [ 4] a) Fid Fourier coefficiet a 0. What does it represet? b) What is the Fourier series of the derivate dx(t) dt c) Use the result from b) ad differetiatio property of Fourier series (from the table: dx(t) dt... jkω 0 a k ) ad fid the Fourier coefficiets of x(t).. Fid Fourier coefficiets for the followig discrete-time sigals. a) x[] is like i Figure 8 x[] : y [] = y [ ] y [ 3] + x [] b) ketch the block diagram of. c) ketch the block diagram of. d) ketch the block diagram of a cascade cosistig of the systems ad (i that order). e) ketch the block diagram of a cascade cosistig of the systems ad (i that order). 6. Examie systems ad with complex iput e jπ/6. Does the iformatio prove that the system or is ot a LTI (ectio 3.)? : e jπ/6 e j3π/7 : e jπ/6 0.3 e jπ/6 7. Cosider Fourier coefficiets a 0 = 0, a = a =, a = a = 0, a 3 = a 3 =, a k = 0 for other k. Form x(t) with a sythesis equatio, whose legth of basic period is 4. (3.3) 8. Fid Fourier coefficiets a) x (t) = e jω0t, (complex sigal) b) x (t) = cos(πt) + cos(3πt), (real sigal) 9. Cosider Fourier coefficiets a 0 =, a = a =, a = a = with basic period N = 5. Form x[]. (3.6) 0. Fid fudametal agular frequecies ad Fourier coefficiets -6 0 6 - Figure 8: Problem : discrete-time sigal. b) x[] = si(π/3) cos(π/) 3. uppose we kow the followig iformatio about a sigal x(t): i) x(t) is real. ii) x(t) is periodic with period T = 6. iii) a k = 0, for k = 0 ad k >. iv) x(t) = x(t 3). 3 3 x(t) dt =. v) 6 vi) a is real ad positive. how that x(t) = A cos(bt + C) ad determie the costats A, B, ad C. 4. ketch the amplitude resposes of the followig filter types: a) lowpass filter c) badstop filter b) highpass filter d) badpass filter a) x [] = cos(π/3)
T-6.40 P (Problems&olutios, autum 003) Page 7 / 9 T-6.40 P (Problems&olutios, autum 003) Page 8 / 9 5. Cosider a cotious periodic sigal x(t), defied as: a) ketch the sigal x(t) i time-domai. x(t) = cos(πt) + 0.3 cos(0πt). b) Determie the fudametal agular frequecy ad Fourier coefficiets a k of the sigal x(t). c) The sigal x(t) is filtered with a ideal lowpass filter havig the cut-off frequecy ω c = 0π. ketch the filtered sigal. d) The sigal x(t) is filtered with a ideal highpass filter havig the cut-off frequecy ω c = 0π. ketch the filtered sigal. 6. Cosider a mechaical system displayed i Figure 9. The differetial equatio relatig velocity v(t) ad the iput force f(t) is give by Bv(t) + K v(t)dt = f(t). a) Assumig that the output is f s (t), the compressive force actig o the sprig, write the differetial equatio relatig f s (t) ad f(t). Obtai the frequecy respose of the system ad argue that it approximates that of a lowpass filter. b) Assumig that the output is f d (t), the compressive force actig o the dashpot, write the differetial equatio relatig f d (t) ad f(t). Obtai the frequecy respose of the system ad argue that it approximates that of a highpass filter. b) Express the F-trasform of a) with sic-fuctio, sic(θ) = 8. Calculate Fourier-trasforms for the followig sigals ad impulse resposes. Use the result from ) ad tables 4. ad 4.. (time shiftig, liearity, differetiatio i time)., 0 < t < a) x(t) = 0, elsewhere, 0 < t < 3, < t < b) x(t) =, < t < 4 0, elsewhere c) h(t) = e (t ) u(t ) t, < t < d) x(t) = 0, elsewhere 9. Covolutio property (p. 34) y(t) = h(t) x(t) Y (jω) = H(jω)X(jω) Calculate the covolutio h (t) h (t) of impulse resposes h (t) = e 0.5t u(t) ad h (t) = e t u(t) usig the covolutio property of F-trasform (multiplicatio of trasforms, iverse trasform back to time domai). 30. Multiplicatio property (p. 3) si πθ πθ f(t) v(t) K r(t) = s(t)p(t) R(jω) = [(jω) P(jω)] π a) Let X(jω) i Figure 0 be the spectrum of x(t). X(jω) B - ω Figure 9: Problem 6: The mechaical system. 7. Fourier-trasform, calculatig itegrals, ad sic-fuctio. Examples 4.4 ad 4.5 i the book. a) Calculate Fourier-trasform for a sigal, t < T x(t) = 0, t > T Figure 0: Problem 30: pectrum, Fourier-trasform of the sigal x(t). Draw the spectrum Y (jω) of y(t) = x(t)p(t), whe i) p(t) = cos(t/), ω = 0.5, T = 4π ii) p(t) = cos(t), ω =, T = π iii) p(t) = cos(t), ω =, T = π iv) impulse trai p(t) = + = δ(t π), T = π, ω =, first fid the F-coefficiets of p(t) (p. 99, example 4.8)
T-6.40 P (Problems&olutios, autum 003) Page 9 / 9 b) Calculate the F-trasform for the sigal x(t) = ( si(πt) πt )( ) si(π(t )) π(t ) 3. Represetig sigals with the discrete-time Fourier trasform. Calculate the Fourier trasforms of the followig sigals: a) x[] = ( ) u[ ] b) x[] = δ[ ] + δ[ + ] 3. Properties of the discrete-time Fourier trasform (see Table 5. p. 39 i the course book). Give that X(e jω ) is the Fourier trasform of sigal x[], express the Fourier trasforms of the followig sigals i terms of X(e jω ). a) x [] = x[ ] + x[ ] b) x [] = (x [ ] + x[]) c) x 3 [] = ( ) x[] 33. uppose X(e jω ) is the Fourier trasform of sigal x[] show i Figure. T-6.40 P (Problems&olutios, autum 003) Page 0 / 9 4) +π π X(ejω )dω = 0. 5) X(e jω ) is periodic. 6) X(e j0 ) = 0. ( ) ( ) a) x[] = u[]. b) x[] =. ( c) x[] = δ[ ] + δ[ + ]. d) x[] = si π ). e) x[] = δ[ ] + δ[ + 3]. f) x[] = δ[ ] δ[ + ]. a) b) c) x[] d) e) f) -3 7 - - 3 4 5 6 - Figure : Problem 33: A discrete sigal x[]. a) Calculate X(e j0 ). b) Fid X(e jω ). c) Calculate +π π X(e jω )dω. d) Fid X(e jπ ). e) Determie ad sketch the sigal, whose F-trasfrom is ReX(e jω )}. +π +π f) Determie X(e jω ) dω ja dx(e jω ) dω dω. π π Note! You do t have to determie the Fourier trasform X(e jω ) itself i ay of the problems above. 34. Determie which of the sigals (a) - (f) i Figure satisfy the followig coditios ) ReX(e jω )} = 0. ) ImX(e jω )} = 0. 3) There exists a real-valued α so that e jαω X(e jω ) is real. Figure : Problem 34: (a) - (f). 35. uppose X(e jω ) ad G(e jω ) are the Fourier trasforms of sigals x[] ad g[], respectively. Additioally, we have the followig equatio is valid: π π π X(e jθ )G(e j(ω θ) )dθ = + e jω a) Give that x[] = ( ), determie a discrete sigal g[] whose Fourier trasform G(e jω ) satisfies the above equatio. Does other solutios for g[] exist? b) Cosider the same problem for x[] = ( ) u[]. 36. Let there be a system with frequecy respose H(e jω ) = + e jω. Frequecy respose ca be decomposed to amplitude ad phase resposes H(e jω ) = H(e jω ) e j argh(ejω )}. a) ketch H(e jω ) i complex plae, whe ω gets values of 0..π. b) Calculate the amplitude respose H(e jω ) (absolute value of a complex umber) ad sketch it i rage 0..π. (Frequecy i x-axis. I case of filters, their maximum value is scaled to uity.) c) Calculate phase respose argh(e jω )} (agle of a complex umber) ad sketch it i rage 0..π.
T-6.40 P (Problems&olutios, autum 003) Page / 9 T-6.40 P (Problems&olutios, autum 003) Page / 9 d) Decibels are ofte used. The trasform is 0 log 0 H(e jω ). ketch amplitude respose of b) i decibel-scale. e) Group delay is the egatio of the derivate of phase respose τ(ω) = d dω argh(ejω )}. Calculate τ(ω). 37. There is a frequecy respose H(e jω ) of a discrete-time sequece i rage 0..π i Figure 3. Plotted with Matlab commad freqz. Magitude (db) Phase (degrees) 0 0 40 60 80 00 0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Normalized Agular Frequecy ( π rads/sample) 400 00 0 00 400 0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Normalized Agular Frequecy ( π rads/sample) Figure 3: Problem 37: H(e jω ): amplitude respose ad phase respose a) ketch the amplitude ad phase respose of H(e jω ) i rage π..π. b) Why ca you draw the result? (Hit: H(e jω ) = x[] e jω, examie with ω = π ja ω = 5π.) c) Why does the correspodig ot work for cotiious-time Fourier-trasforms (Bode diagram)? 38. Covolutio i time domai correspods multiplicatio of trasforms i frequecy domai (see Figure 4). x[] X(e jω) h[] H(e jω) y[] Y(e jω) Figure 4: Problem 38: y[] = h[] x[] Y (e jω ) = H(e jω )X(e jω ) Let us kow a LTI system whose impulse fuctio is h[] = ( δ[] + δ[ ] δ[ ]). 3 a) Draw a block diagram of (i time domai), where x[] comes from left ito the system (delay, sum,...) ad the output is y[]. b) Fid H(e jω ) ad sketch amplitude respose H(e jω ). What is the filter like? c) The respose y[] ca be gotte either with covolutio x[] h[] i time domai or with iverse trasform of multiplicatio of their Fourier trasforms F F x[]} F h[]}}. ketch the output y[], whe the system is fed with sequeces (time t = 0) i) costat sequece, amplitude A, A, A, A,...} ii) sequece A, A, A, A, A, A, A, A, A, A, A, A,...} iii) periodic sequece A, A, A, A,...} d) Ca the system be realized? Is it causal ad/or stable? 39. The frequecy respose of a cotiuous time, causal ad stabile LTI-system is H(jω) = jω + jω a) how that H(jω) = A, where A is costat. Calculate A. b) What is the filter like? c) Which of the followig is true for the group delay τ(ω) = d(arg H(jω))/dω: i) τ(ω) = 0, ku ω > 0 ii) τ(ω) > 0, ku ω > 0 iii) τ(ω) < 0, ku ω > 0 40. Cosider a cotiuous liear time ivariat system, with frequecy respose (j arg H(jω)) H(jω) = H(jω) e ad the impulse respose h(t) is real. To iput x(t) = cos(ω 0 t + φ 0 ) the output is y(t) = Ax(t t 0 ), where A is a o-egative real umber ad t 0 is a time delay. a) Calculate A usig H(jω 0 ). b) Calculate t 0 usig the phase arg H(jω 0 ). 4. ketch the phase-magitude represetatio of F-trasform of x[] = cos(0.π)+ cos(0.05π)+ 0. ɛ[], where ɛ[] is gaussia white oise. 4. Liear ad oliear phase. Examie the sequeces x = cos(0.π ) ad x = cos(0.05π ) ad the sum of these x 3 [] = x [] + x []. a) Draw the sequeces x, x ad x 3. b) Let there be a system, whose group delay is costat τ (ω) = 3 ad amplitude. ketch the output for both sequeces. Draw also x 3. c) Let there be a system, whose phase is oliear. Group delay is τ (0.05π) =, τ (0.π) = 5 ad amplitude costat. ketch the output for both sequeces. Draw also x 3. Compare results. 43. Whe desigig highpass or badpass filters, the stadard method is to specify a lowpass filter havig the desired characteristics ad the covert it to a HP or BP filter. With this approach, we ca use the lowpass desig algorithms to desig all filter types. Let us cosider a discrete-time lowpass filter with a impulse respose h lp [] ad a frequecy respose H lp (e jω ). The we modulate the impulse respose with the sequece ( ) so that h hp [] = ( ) h lp []. a) Defie H hp (e jω ) usig H lp (e jω ). how that H hp (e jω ) is the frequecy respose of a highpass filter.
T-6.40 P (Problems&olutios, autum 003) Page 3 / 9 T-6.40 P (Problems&olutios, autum 003) Page 4 / 9 b) how that modulatig the impulse respose of a discrete-time HP filter with the sequece ( ) produces a LP filter. 44. The behavior of a liear time-ivariat system is give by dy(t) dt + y(t) = x(t). a) Determie the frequecy respose of the system ad sketch the correspodig Bode plot. b) What is the group delay of the system? c) Calculate the Fourier trasform of the output of the system give the iput x(t) = e t u(t). d) Calculate the output of the system give the iput from c) usig the partial fractio decompositio. 45. uppose a o-ideal cotious-time LP filter whose frequecy respose is H 0 (jω), impulse respose h 0 (t), ad step respose s 0 (t). The cutoff frequecy of the filter is ω 0 = π 0 rad/s ad the rise time of the step respose (the amout of time i which the step respose rises from 0% of its fial value to 90% of it) is τ r = 0 secods. Let us implemet a ew filter with the frequecy respose where a is the scalig factor. H lp (jω) = H 0 (jaω), a) Defie a so that the cutoff frecuecy of H lp (jω) is ω c. b) Preset the impulse respose h lp (t) of the ew system with ω c, ω 0, ad h 0 (t). c) Defie the step respose of the ew system. d) How do the cutoff frequecy ad the rise time of the ew system relate to each other? 46. A causal ad stable LTI system is defied with the followig differece equatio: Determie (a) the frecuecy respose H(e jω ) (b) the impulse respose h[] y[] 6 y[ ] y[ ] = x[] 6 47. Cosider a ideal badpass filter with the frequecy respose, ω c ω 3ω c H(jω) = 0, otherwise (a) Give that h(t) is the impulse respose of the filter, defie a fuctio g(t) so that h(t) = ( si ω ct )g(t) πt (b) How does the impulse respose chage if the cutoff frequecy ω c is icreased? 48. Explai the terms briefly: samplig process, impulse trai, prefilterig, recostructio of sigal, zero order hold, aliasig 49. how that a periodic impulse trai p(t) ca be expressed as a Fourier series p(t) = p(t) = T = k= δ(t T) e j(π/t)kt where Ω T = π/t is samplig agular frequecy. I other words, express p(t) as Fourier series ad fid Fourier-coefficiets for p(t)! 50. amplig ad aliasig a) Fid a value for agular frequecy θ which satisfies si(θ ) 0 0 / What is θ i geeral? b) Cosider a cotiuous time periodic sigal x(t) = si(πf t) + si(πf t) si(πf 3 t), t 0 0, t < 0 where f =00 Hz, f =300 Hz ad f 3 =700 Hz. The sigal is sampled usig frequecy f s, i other words, T = /f s, p(t) = f s k= ej(πfs)kt. Thus, a discrete sigal x[] = x p (t) = x(t) is obtaied. ketch the magitude of the Fourier spectrum of x(), the sampled sigal, whe f s equals to (i) 500 Hz (ii) 800 Hz (iii) 400 Hz. (Hit: a sampled siusoid ca be see as a peak i the Fourier spectrum.) c) With low samplig frequecies high frequecies of the sigal aliased to low frequecies. Whe does this happe? How it is see i the recostructed sigal? 5. igal recostructio from samples a) Draw a arbitrary badlimited X(jω) whose biggest agular frequecy is ω M.
T-6.40 P (Problems&olutios, autum 003) Page 5 / 9 b) ample the sigal with samplig agular frequecy ω s > ω M. Draw the spectrum of sampled X p (jω). c) The sequece ca be filtered. Recostruct the sigal usig a ideal lowpass filter H(jω), whose cut-off frequecy ω c is ω M < ω c < 0.5 ω s. Draw the spectrum of recostructed X r (jω) = X p (jω)h(jω). d) Express the equatio of c) i time domai x r (t) = x p (t) h(t) = k= x p(k)h(t kt) What is the impulse respose h(t) of the ideal lowpass filter H(jω) of c). 5. Why is it useful to sample a cotiuous sigal ad process it digitally? What problems occur whe usig too low samplig frequecy. What about if samplig frequecy is 000 times bigger tha highest frequecy i sigal?