Question

A signal x(t) with Fourier transform X(jω) undergoes impulse-train sampling to generate xₚ(t)=

[infinity]
∑ x(nT)δ(t−nT)
n=−[infinity]

​
where T=10⁻⁴ . For each of the following sets of constraints on x(t) and/or X(jω), does the sampling theorem (see Section 7.1.1) guarantee that x(t) can be recovered exactly from xₚ(t)?
X(jω)=0 for ∣ω∣>5000π

Answer 1

The sampling theorem guarantees that x(t) can be recovered exactly from the impulse-train samples because the sampling frequency is greater than the signal's bandwidth.

Step-by-step explanation:

The sampling theorem states that a continuous-time signal can be perfectly reconstructed from its samples if the signal's bandwidth is limited to less than half the sampling frequency. In this case, the signal x(t) has a Fourier transform X(jω) that is zero for ∣ω∣>5000π, which means its bandwidth is limited to 10000π. Since the sampling frequency is 1/T=10^4, which is greater than the signal's bandwidth, we can conclude that x(t) can be recovered exactly from the impulse-train samples.