Sampling Theorem

Summary

The lecture introduces sampling and derives the sampling theorem, explaining when a continuous-time signal can be recovered from its samples.

Main Ideas

• Why sampling is needed: Real-world information is carried by continuous-time (analog) signals, while digital systems process discrete-time signals. Sampling converts a continuous-time signal into a discrete-time representation.
• Message signal and band limitation: The signal is represented as (m(t)), with spectrum (M(\Omega)). It is assumed to be band-limited: its spectrum is nonzero only between (-\Omega_m) and (+\Omega_m), where (\Omega_m) is the maximum angular frequency in the signal.

• How sampling is modeled: A sampler is modeled as multiplying (m(t)) by a periodic impulse train: [ c(t)=\sum_{n=-\infty}^{\infty}\delta(t-nT_s), ] where (T_s) is the sampling period. The sampled signal is [ s(t)=m(t)c(t). ] Each impulse in (s(t)) has a weight equal to the value of (m(t)) at that sampling instant.

• Effect of sampling in the frequency domain: Multiplication in time corresponds to convolution in frequency. Sampling therefore produces repeated, shifted copies of the original spectrum (M(\Omega)), spaced by the sampling angular frequency [ \Omega_s=\frac{2\pi}{T_s}. ]

Sampling Theorem and Recovery Condition

The spacing between spectral copies determines whether the original signal can be recovered:

• If (\Omega_s>2\Omega_m): The copies are separated, leaving a gap called the guard band.
• If (\Omega_s=2\Omega_m): Adjacent copies just touch but do not overlap.
• If (\Omega_s<2\Omega_m): The copies overlap. This overlap prevents accurate recovery of the original signal.

The lecture states the sampling theorem as follows: a band-limited signal can be represented by its samples and recovered when [ \boxed{\Omega_s\geq 2\Omega_m}. ]

Why Band Limitation Matters

If a signal is not band-limited, its spectrum extends beyond a finite frequency range. Repeated copies of that spectrum will overlap when sampled, so the original continuous-time signal cannot be accurately recovered by the method described.

Speaker or Source

• Neso Academy narrator/instructor — presents the lecture and explains the sampling process, frequency-domain analysis, and sampling theorem.