a) Consider two LTI systems with impulse responses h(t) and g(t), respectively, and supp...

a) Consider two LTI systems with impulse responses h(t) and g(t), respectively, and suppose that these systems are inverses of one another. Suppose also that the systems have frequency responses denoted by H(jω) and G(jω), respectively. What is the relationship between H(jω) and G(jω)?

b) Consider the continuous-time LTI system with frequency response

(i) Is it possible to find an input x(t) to this system such that the output is as depicted in Figure P4.50? If so, find x(t). If not, explain why not.

(ii) Is this system invertible? Explain your answer.

(e) Consider an auditorium with an echo problem. As discussed in Problem 2.64, we can model the acoustics of the auditorium as an LTI system with an a pulse response consisting of an impulse train, with the kth impulse in the train corresponding to the kth echo. Suppose that in this particular case the impulse response is

where the factor represents the attenuation of the kth echo. In order to make a high-quality recording from the stage, the effect of the echoes must be removed by performing some processing of the sounds sensed by the recording equipment. In Problem 2.64, we used convolutional technique' to consider one example of the design of such a processor (for a different acoustic model). In the current problem, we will use frequency-domain technique' Specifically, let G(jω) denote the frequency response of the LTI system to be used to process the sensed acoustic signal. Choose G(jω) so that the echoes are completely removed and the resulting signal is a faithful reproduction of the original stage sounds.

(d) Find the differential equation for the inverse of the system with impulse response

(e) Consider the LTI system initially at rest and described by the differential equation

The inverse of this system is also initially at rest and described by a differential equation. Find the differential equation describing the inverse, and find the impulse responses h(t) and g(t) of the original system and its inverse.