Filter A is a discrete-time LTI system with its frequency response magnitude and

Question

Filter A is a discrete-time LTI system with its frequency response magnitude and group delay functions shown in Figure 1. A signal x[n], also shown in Figure1, is the sum of three narrowband pulses. In particular, Figure 1 contains the following plots r[n] X(e10)1, the Fourier transform magnitude of x[n] Group delay plot for the system. Frequency response magnitude plot for filter A. . xln In Figures 2 and 3 you are given 8 possible output plots yln]i1,2,..,8. Determine which one of the possible output signals is the output of filter A when the input is x[n]. Provide a justification for your choice.

Explanation / Answer

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First we will study the given plots and derive usefule information:

(a). x(n) consists of three feqeuency pulse. The first one is high frequency, the second one is low freqeuncy and the third pulse is a medium frequency pulse. Note that the output of any LTI system will reproduce these pulses one after the other i.e. their order will not be changed. Only their magnitudes and delays will be changed.

(b). Multiplying the Fourier Transform of the input with the frequency response magntiude of the filter, we note that all frequencies in the input x(n) is reproduced. There are no frequencies in x(n) which falls in the frequency range where frequency response magntiude of the filter = 0. The only thing is that the low frequencies are multiplied by smaller gain. Note the smaller gains from 0.8 till 1.2 on the frequency axis of the filter response.

(c) Note from the group delay plot that the delay for the small frequencies are large and that for the higher frequencies are large.

From the above we can conclude that the output will consist of the three pulses in series, with the biggest delay for the low frequency pulse. Thus the correct output will be y7(n)

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