Flat top vs natural sampling; Nyquist rate of sin(800πt)/πt + 10cos2000t

Flat-top sampling

In flat-top sampling, each sample value $g(nT_s)$ is held constant for a pulse width $\tau$ (sample-and-hold). The PAM wave is

$$s(t) = \sum_n g(nT_s)\,h(t - nT_s), \qquad S(f) = f_s H(f)\sum_n G(f - nf_s)$$

where $h(t)$ is a rectangular pulse of width $\tau$ and $H(f) = \tau\,\text{sinc}(f\tau)\,e^{-j\pi f\tau}$. The spectrum copies are shaped by $H(f)$, so high message frequencies are slightly attenuated (aperture effect), corrected by an equalizer $1/H(f)$ at the receiver.

Advantages of flat-top over natural sampling

So flat-top sampling is preferred in practical PCM/PAM systems because the held value can be digitized and transmitted easily.

Nyquist rate and interval

$$x(t) = \frac{\sin(800\pi t)}{\pi t} + 10\cos(2000t)$$

• First term: $\sin(2\pi Wt)/(\pi t)$ is an ideal low-pass (sinc) signal with flat spectrum up to $W$. Here $2\pi W = 800\pi$, so $W_1 = 400$ Hz.
• Second term: $\omega = 2000$ rad/s, so $f_2 = 2000/(2\pi) = 318.31$ Hz (the argument has no $\pi$).
• Highest frequency: $f_{max} = 400$ Hz.

$$\begin{aligned} f_{Nyquist} &= 2f_{max} = 2 \times 400 = 800\ \text{Hz} \\ T_{Nyquist} &= \frac{1}{800} = 1.25\ \text{ms} \end{aligned}$$

Answer: Nyquist rate = 800 Hz (samples/s); Nyquist interval = 1.25 ms.