A DTMF signal with duration T seconds can be modeled as a sum of two cosine signals with frequencies frand fc, multiplied by a rectangle function. For example, the DTMF signal for "1" is g1(t) = rect (t/T) * [cos(2pi*687t)+cos(2pi*1209t)] Questions:
1). What is the expression for the Fourier transform of a DTMF signal?
2). Sketch the amplitude spectrum of a DTMF signal with infinite duration, i.e., with T = infinity.
3). Sketch the amplitude spectrum of a DTMF signal with finite duration. You should make this sketch by hand, using the Fourier transform expression obtained in part 1). What happens to the DTMF spectrum when T is "too small"?
4). What would you estimate as the smallest value of signal duration T so that the spectra of the 12 DTMF signals are distinguishable from each other?