Reference no: EM132304527

The Zonama cho - e^Πi is a smart device' that needs to receive voice signals in small room with an echo (i.e., reverberation). The goal is to see if a digital filter can provide needed clarity.

Given that receiving signals in the presence of an echo is a commonplace and practical task, this question looks at a means of understanding and potentially overcoming the echo with a series of filters.

Consider a microphone (receiver) that records a signal m(t) based on the original spoken sound s(t) and its echo, a.s(t - T₀), an attenuated (a < 1 ) and delayed copy of s(t). Thus, m(t) = s(t) + a.s(t - T₀).

The general idea is to sample the received signal m(t) to a discrete sample train m[n] to then filter this via a digital filter (difference equation), h[n] to give an output o[n]. This output is then filtered via an ideal low pass filter of gain K before being reconstructed as an analog output y(t).

Assume that the source sound, s(t), is band limited (|S(jω) = 0 for |ω| > ω* )

(a) Please draw up a system diagram for the process described above. Succinctly explain both (1) why there is an ideal low pass filter at the reconstruction end?, and

(2) Is an ideal low pass filter needed before sampling the signal m(t)?

(b) If T₀ < 7Π/ω* and the system is sampled at a fixed sampling period ( Ts ) that is no more than To (i.e., αT_s = T₀ , α > 1), then determine a difference equation for the filter, h[n] , such that the fmal output y(t) is proportional to s(t).

[Hint: start with a =1, then consider the case for a >> 1 ]

(c) Specify the gain K of the low-pass filter such that the final output y(t) = s(t) [Hint: start with a =1, then consider the case for an arbitrary a ]

(d) Oka do cho - e^Πi, is there aliasing? Suppose the echo gets worse and the delay gets larger. That is, (for this part (d)) assume Π/ω* < T0 < 2Π/ω*. Are we out of luck?

Is there a sampling period T_s, filter gain K, and filter h[n] such that y(t) = s(t)?