What is the Nyquist rate (in Hz) for the signal
𝑥
(
𝑡
)
=
2
sinc
2
(
10
𝑡
)
?

Question

What is the Nyquist rate (in Hz) for the signal 
𝑥
(
𝑡
)
=
2
sinc
2
(
10
𝑡
)
?

BlackTom AI · Accepted Answer

The question asks for the Nyquist rate of the given time-domain signal.

First, identify the time-domain form: x(t) = 2 sinc^2(10 t). Here, sinc is the normalized sinc, typically defined as sinc(u) = sin(pi u) / (pi u).

Under this definition, sinc(10 t) has a spectrum that is bandlimited, with its main spectral content centered at 0 and extending predominantly up to a cutoff proportional to 10 Hz. In many standard treatments, the frequency support of sinc(10 t) is effectively limited to |f| ≤ 10 Hz in terms of where the main energy lies, with the idealized interpretation that the signal does not require frequency components beyond this scale for its essential shape.

When we square the time-domain signal, the spectrum is convolved with itself in the frequency domain. This operation spreads energy, but the overall bandwidth of the resulting signal remains tied to the original scale set by the 10 t argument; the dominant, practically significant bandwidth does not extend beyond about 10 Hz, though there are higher-frequency sidelobes. For the purpose of determining a safe sampling rate (Nyquist rate) that preserves the essential content, we use the principal bandwidth dictated by the sinc(10 t) term, which is 10 Hz.

The Nyquist rate is defined as twice the highest frequency present in the signal. If the effective maximum frequency content is 10 Hz, then the Nyquist rate is 2 × 10 Hz = 20 Hz.

In summary, based on the standard interpretation of the normalized sinc and its scaling, the signal bandwidth is about 10 Hz, so sampling at 20 Hz ensures proper reconstruction without aliasing.

Note: Some treatments emphasize the full, mathematical spectral extent of sinc^2 and would discuss convolution broadening, but for practical Nyquist-rate determination with the given form, 20 Hz is the appropriate result.

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What is the Nyquist rate (in Hz) for the signal 𝑥 ( 𝑡 ) = 2 sinc 2 ( 10 𝑡 ) ?

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