Figure P-1. depict s a system that is designed to detect signals of two different frequenc...

Figure P-1. depict s a system that is designed to detect signals of two different frequencies as in a frequency-shift keying (FSK) modem.

(a) Suppose that the input signal to the system in Fig. P-1 is x(t) = x1(t)= cos(ω1t)where ω1= ωc – ω0with ω0> O. Using the frequency-shift property of Fourier transforms and the Fourier transform of the cosine signal, determine and plot the Fourier transform of the signal w(t).

(b) Now suppose that the lowpass filter (LPF) in Fig. P-1 has frequency response as depicted in Fig. P-2. Determine the smallest value for ωp and the largest value for ωs, such that the output of the filter is

This will give the largest transition region between passband and stopband and therefore the simplest filter to implement.

(c) Show that for the passband and stopband frequencies found in part (b), the overall output is a constant; i.e., .

(d) Now suppose that x(t)= x2(t)= cos(ω2t)where ω2= ωc + ω0and the cutoff frequencies of the filter are the same as found in part (b). Show that the overall output is again a constant but in this case, .

(e) Assume that the input signal can be either x1(t)or x2(t). Give a simple algorithm for determining which signal was used.

(f) Explain how you would implement the system of Fig. P-1 using digital signal processing. Draw a block diagram showing the signal x(t)sampled with an ideal C-to-D converter followed by a discrete-time version of the system in Fig. P-1. For the input signals used in parts (a)–(e), what is the minimum sampling frequency fsamp = 1/Ts, that can be used? For this sampling rate, determine the frequency of the complex-exponential signal  and the normalized cutoff frequencies of the discrete-time lowpass filter that will be needed.