Sampling Theorem

The lecture introduces sampling as the process of converting continuous time signals to discrete time signals, which is essential for digital technologies. All real-life signals are analog (continuous), but digital systems require discrete signals for processing. Sampling allows this conversion and, under certain conditions, enables the recovery of the original continuous signal.

The process begins with a continuous time 'message signal' (Mt) that carries information. This signal is assumed to be 'band-limited,' meaning its Fourier transform (spectrum) is non-zero only within a finite range of frequencies, from -Ωm to +Ωm. Ωm represents the maximum frequency component of the message signal, a crucial parameter for the sampling theorem.

The message signal Mt is fed into a 'sampler,' which acts as a multiplier. It multiplies Mt with a periodic impulse train (Ct). This impulse train consists of unit impulses at intervals of Ts, known as the sampling period. The angular frequency of this impulse train is Ωs = 2π/Ts, referred to as the sampling frequency.

Through convolution, the spectrum of the sampled signal SΩ is shown to be a series of shifted versions of the original message signal's spectrum MΩ. These shifted spectra are centered at integer multiples of the sampling frequency Ωs. Specifically, SΩ is a sum of M(Ω - nΩs) terms, scaled by 1/Ts.

The video illustrates the spectral representation of SΩ. It highlights that if Ωs - Ωm > Ωm, or Ωs > 2Ωm, there is a clear 'guard band' or gap between adjacent spectral repetitions, preventing overlapping. If Ωs = 2Ωm, the spectra touch but do not overlap. In both these scenarios, the original continuous signal can be recovered without distortion.

The critical condition for recovering the continuous signal is provided by the sampling theorem: the sampling frequency (Ωs) must be greater than or equal to twice the maximum frequency component (Ωm) of the message signal (Ωs ≥ 2Ωm). If Ωs < 2Ωm, the shifted spectra overlap, leading to 'aliasing,' which makes accurate signal recovery impossible. This explains why band-limited signals are crucial, as non-band-limited signals would inevitably cause overlapping, preventing recovery.

The multiplication yields a 'sampled signal' (St), which is a series of impulses. The weight or strength of each impulse in St corresponds to the instantaneous value of the original message signal Mt at the sampling instant. The video then derives the Fourier transform (spectrum) of this sampled signal (SΩ) by convolving the Fourier transform of Mt (MΩ) with the Fourier transform of Ct (CΩ).