Audio Signal Processing and Coding

PROBLEMS

2.1. Determine the continuous Fourier transform (CFT) of a pulse described by

where u(t) is the unit step function.

2.2. State and derive the CFT properties of duality, time shift, modulation, and convolution.

2.3. For the circuit shown in Figure 2.34(a) and for RC = 1,

1. Write the input-output differential equation.
2. Determine the impulse response in closed-form by solving the differential equation.
3. Write the frequency response function.
4. Determine the steady state response, y(t), for x(t) = sin(10t).

   

   Figure 2.34. (a) A simple RC circuit; (b) input signal for problem 2.3(e), x(t); and (c) input signal for problem 2.3(f).

5. Given x(t) as shown in Figure 2.34(b), find the circuit output, y(t), using convolution.
6. Determine the Fourier series of the output, y(t), of the RC circuit for the input shown in Figure 2.34(c).

2.4. Determine the CFT of . Given, x_s(t) = x(t)p(t), derive the following,

where X(ω) and X_s(ω) are the spectra of ideally bandlimited and uniformly sampled signals, respectively, and ω_s = 2π/T_s. (Refer to Figure 2.8 for variable definitions.)

2.5. Determine the z