If x_in(t) = sin(2$$\pi $$ × 4000t) + 0.75sin(2$$\pi $$ × 5000t + $$\frac{\pi }{4}$$) is sampled with F_s = 16000 Hz. Calculate X(0) if $$X\left( m \right) = \sum\nolimits_{n = 0}^{N - 1} {x\left( n \right){e^{ - \frac{{j2\pi nm}}{N}}}} .$$      When N = 8, where x(n) = x_in(nt_s)

The Fourier transform of a real valued time signal has

The RMS value of a rectangular wave of period T, having a value of +V for a duration, T₁(< T) and -V for the duration, T - T₁ = T₂ equals

A continuous time function x(t) is periodic with period T. The function is sampled uniformly with a sampling period T_s. In which one of the following cases is the sampled signal periodic?

The phase response of a passband waveform at the receiver is given by
$$\varphi \left( f \right) = - 2\pi \alpha \left( {f - {f_c}} \right) - 2\alpha \beta {f_c},$$
where f_c is the centre frequency, and α and β are positive constants. The actual signal propagation delay from the transmittance to receiver is