Assuming that a continuous-time $x(t)$ having its frequency content in the frequency band $1612\leq|F|\leq2015(Hz)$ is sampled with the sampling rate $Fs=806$ samples per second. Which of the following sampling methods can be used to sample $x(t)$ such that this signal can be uniquely recovered from its samples?

A. Uniform first-order sampling

B. Uniform second-order sampling

C. None of the above

I have searched a lot for the concept of uniform $n^{th}$-order sampling here but haven't found anything useful or related, tbh, Idk the concepts of those sampling techniques.

Can someone provide me full explanation for this mcq, please!


1 Answer 1


B): $f_s \geq 2B = 2\cdot(2015-1612) = 2\cdot 403 = 806$.

"Second-order" bandpass sampling is described in this paper (or older, here). It's sampling $x(t)$ at a lower sampling rate, $M$ times, each with a different offset - then keeping only (carefully selected) 2 of the $M$ sequences:

$$ \begin{align} x_A(n) &= x(n / f_0) \\ x_B(n) &= x(n / f_0 + k) \\ x(n) &= x_A(n) + x_B(n) \end{align} $$

The trick is in appropriately choosing $f_0$ and $k$. Graphically, $x(t)$'s spectrum is

enter image description here

and it sampled at $f_0 = 2B$ is (showing one half of one of the sequences)

enter image description here

(Sampling in time <=> Periodizing in frequency). From above it's clear that an appropriate combination of samplings of $x(t)$ at a rate $f_s \geq 2B$ will uniquely represent $x(t)$. This theoretical minimum average sampling rate, $2B$, is called the Nyquist-Landau rate.

The motivation is described in paper introduction:

Uniform sampling of $x(t)$ at the Nyquist rate $2(f_L + B)$ will be impractical when the frequency is high because this will increase the power consumption of the ADC applied in the sampling operation, resulting in reduced overall system efficiency.

But don't be mislead; the Nyquist frequency imposes a fundamental limit on representing variations with finite number of samples. A consequence is, to recover $x(t)$, we require specifically designed interpolation functions that use the knowledge of $f_L$ and $B$. The dependence on $B$ follows directly from inability to represent a greater range of variations with fewer samples.

  • $\begingroup$ thank you a lot for clearing it up for me this concept. I think I have understood the question now!:) $\endgroup$ Jul 31, 2021 at 16:11
  • 1
    $\begingroup$ @ViệtNguyễn Glad it helped. A nice resource on sampling and SP fundamentals, here. $\endgroup$ Jul 31, 2021 at 16:30
  • $\begingroup$ nice,I will look into it, thank you! $\endgroup$ Aug 1, 2021 at 15:58

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