Question

PROBLEM 7.3*: The diagram in Fig. 2 depicts a cascade connection of two linear time-invariant (LTI) systems; i.e., the output of the first system is the input to the second system, and the overall output is the output of the second system. [n] yi[n] y[n] LTI System #1 hin] LTI System #2 h2[1] Figure 2: Cascade connection of two LTI systems. (a) Suppose that System #1 is a "blurring" filter described by the following equation y1 [n] =arn-k] k=0 and System #2 is described by the impulse response hu[n] = 8[n] - a[n - 1] where a 70 is a real number. Determine the impulse response sequence, hin, of the first system. (b) Obtain a single equation as shown below that relates the output y[n] to input on in Fig. 2. Determine the filter order M and express the nonzero coefficients {bk} as a function of a and M y[n] = bxr[n- k] k=0 (c) Suppose that the input rn is non-zero only for 9<n<23. Where will the output start, i.e., what is the index of the first nonzero y In? Where does yn end, i.e., determine the index of the last non-zero value in the output sequence y[1]? (d) Determine the total length of the output signal y[n] (in samples) over its nonzero region. Explain by relating the nonzero output length to the nonzero input length and the length of the nonzero part of hn. (e) Give numerical values of the impulse response of the overall system h[k] which is the same as the filter coefficients bk in part (b) for the specific case where a = 0.6.

Answer 1

Sola) Viim ak man-k'). (n) = n(njt a n(n1)+ at nln-1} at qe n. 37 taking a truns for both side Y (2) = X(2) { 14 921 4622 4923).-0 H, (2)= 11 (2) = 14921 4 922 2493 23. XC2) taking inverno z transform both side. hi [n] = sinjt as [n-1) + a2 s (n.23+ a3 8(n-3) b) Given he(nga sing-a (n-1).. both system connected in cascade so heq [no= h, (n] *h2 (n] Hea (2) = H, (2) M₂ (2) CD. In can 11 H₂ (2) = (1-a zl) Hea (Z) =114 azit a 2-2 Q3 2-3) (I- a2-) - a 4-1 (- 942-4) 19 = Y (2)= Y (2) = Hc2J X (2) (1-a9 2 ) X (2) hang = (sen) - a* sena))) = 1) T CS Scanned with CamScanner

c) 4 so nen] is non-aro g sn 223 8 = 4 so output will start output will and from 13 cut 27 and (n) = hing * nin). total length of eg g from ot9 = 9 till 4+27= 31 length = 22 total Heq (2) 2 hea (no = - at 2-4 dini- a2d (no). ICS Scanned with CamScanner