Consider a continuous-time LTI system with input a (t) and output y(t), as shown

Question

Consider a continuous-time LTI system with input a (t) and output y(t), as shown below: Unknown LTI System y(t) signal 1 (t) shown in Figure 6.1, the output is y1(t) shown in the same figure. Note that ac1(t) and yl (t) are periodic (they repeat outside the interval shown). x1 (t) 0.5 0.5 0.6 0.7 0.8 0.5 0.9 0.1 0.2 0.3 0.4 Time (seconds) Figure 6.1: Input/output pair for the system. Figure 6.2 shows the frequency responses of six systems, H1 w) through H6(jw). These frequency responses are purely real (a) Which of the frequency responses shown in Figure 6.2 could be the frequency response of the unknown LTI system? Hint: There are at least two possibilities.

Explanation / Answer

(a) The given signal has a frequency of 4pi. The magnitude of the input signal is reduced by 1/2 when the input signal passes through the system H(jw). Further, it is given that the frequency response of the system is purely real, Hence, although theoratically, negative frequencies may be present in graphical representation of the frequency respose, only the positive frequencies are of practcal importance.

Hence, the options H1, H2, H3, and H4 allow the frequency of 4pi to pass through then and with a gain factor of 1/2. Although the system H3 shows only the positive frequency response, its okay because we are given a real system and and real signals.

Options H5 and H6 offer a gain factor of 2 which is incorrect for the given input and output signals as we can clearly see from the graph that the magnitude of input x is reduced by 1/2 in the output y.

(b) Yes. The four possible systems will give different responses to different frequencies. Hence, the input signal should include at least one such frequency to which the systems H1 to H4 react differently.

In order to differetiate between H1 and H2, we need to include a frequency between 0 to 2pi. In order to differentiate between H1 and H3, we need to include a frequency between 2pi to 3pi.

Hence, the input signal x2 = 1 + cos(2pi*t), i.e., a sum of frequencies 0 and 2pi.

If the system is H1, the output signal y2 = 0.5*cos(2pi*t)

If the system is H2, the output signal y2 = 0.5*(1+cos(2pi*t))

If the system is H3, the output signal y2 = 0

If the system is H4, the output signal y2 = 1+(0.25*cos(2pi*t))

Thus we can identify the system as aone out of H1, H2, H3, and H4.

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