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, ...