T-61.140 Signaalinkäsittelyjärjestelmät T-61.140 Signal Processing Systems 2nd MTE Fri, 3.5.2002, 12-15, M Paikalla N=89 (ilmoittautuneita n. 150) Joitakin ratkaisuja / Some solutions: 1. Väitteet / Statements, Oikein, Väärin / True, False 1a) V / F: product of, not sum of 1b) V / F: phase is also important 1c) O / T: (-1)^n <-F-> e^(j pi) --> LP--(-1)^n--> HP, HP--(-1)^n--> LP 1d) V / F: Cont.time FT is aperiodic, Discr.time FT is 2pi-periodic 2. Termit / Terms. B-kohtaan vaaditaan ainakin pari asiaa. 2a) Jako mahdollisista arvoista "ajan" ja amplitudin suhteen. / Division using "granularity", which values are possible in "time" and amplitude. 2b) Voi ajatella esimerkiksi signaalin tallentamisen ja toistamisen tai signaalia käsittelevän järjestelmän kannalta. / Using digital signal and digital systems. SEE below MTE 15.5. 3. LTI-system 3a) of FIR-type, no feedback 3b) H(e^jw) = 0.25 - 0.5 e^-jw + 0.25 e^-j2w 3c) HP 3d) detects changes 4. Sampling 4a) Three peaks at 150, 350 and 450 Hz. 4b) f_s0 = 900 Hz 4c) fs=380 Hz 4c) 150 Hz -> 150 Hz, 350 Hz -> (fs-350) = 30 Hz 450 Hz -> (450-fs) = 70 Hz Spectrum periodic with fs = 380 Hz. Nyqvist frequency fs/2 = 190 Hz, highest frequency to be observed. 5A) y[n] = 3 y[n-1] - 3 y[n-2] + y[n-3] Initial values of delay registers: three adjacent squares (n-1)^2, (n-2)^2, (n-3)^2. You can solve the problem, for example, with a linear equation group. I guess that also second order systems with non zero input will be approved. 2nd MTE & Exam, Wed 15.5.2002, 9-12 C, L Paikalla MTE N = 64 (total 89+64=153), Exam N = 40 1a) alpha == 0 --> system linear (when no constants) & time-invariant 1b) Yes, Sigma |h[n]| < oo, |x[n]| <= B, --> |y[n]| <= 28 B 1c) No, e.g. n=-2, y[-2] = x[2] (future prediction!) 1d) Yes, T= 9pi 1e) No, not any multiple of 9pi/2 in Z 1f) N1 = 12, N2 = 18 --> N=36 2a) deconvolution... 2b) h1[n] = delta[n-1] + delta[n-2] 2c) y[n] = d[n-2] - 4d[n-3] + d[n-4] + 6d[n-5] 2d) h[n] == hat{h}[n] for LTIs 3a) TRUE, see formula... 3b) FALSE, FIR is stable, but it does NOT have any feedback 3c) TRUE, T = 1/f... 3d) FALSE, s[n] = SIGMA h[k], h_1[n] = 0.9^n --> s_1[n] = {1, 1.9, 2.71, ..., -> 10 } h_2[n] = 0.1^n --> s_2[n] = {1, 1.1, 1.11, ..., -> 1.11... } 4a) x(t), t in R x[n] in R, n in Z x[n] in Q, n in Z 4b) Digital signal, Digital system, ... SEE above MTE 3.5. 5a) h[0] = 1, h[1] = -1.8, ... 5b) H(e^jw) = 1/(1 +0.8 e^-jw) - e^-jw(1/(1 + 0.8 e^-jw)) = (1 - e^jw) / (1 + 0.8 e^jw) 5c) HP 5d) H(e^jw) = Y(e^jw)/X(e^jw) = (1 - e^jw) / (1 + 0.8 e^jw) --> Y(e^jw) (1 + 0.8 e^jw) = X(e^jw) (1 - e^jw) --> y[n] + 0.8 y[n-1] = x[n] - x[n-1] 6a) Yes, f0 = 1kHz, f1 = 3f0, f2= 4f0, f3=5f0, f4=14f0 6b) fs=28 kHz 6c) peaks at _2_, 3, 4, and 5 kHz 6d) No influence at important channel 7Ab) LP 7Ac) y[n] = 0.5 x[n] + 0.5 x[n-2] H(e^jw) = 0.5 + 0.5 e^-j2w 7Ad) BS 7B) Types, ideal filters, cut-off frequency, examples, ...