EC-7.6 Sampling, Aliasing and Reconstruction

Work through these

• Sampling as multiplication by a train of impulses

  Taking values at regular intervals can be modelled as multiplying the signal by a regularly spaced set of impulses. Writing it that way is what lets the frequency-domain consequence be worked out rather than asserted.

  MIT RES.6-007: Signals and Systems · Course

• What sampling does in the frequency domain: copies, spaced by the rate

  Multiplying by an impulse train in time convolves in frequency, so the spectrum of the sampled signal is the original spectrum repeated at multiples of the sampling rate. This one picture carries the rest of the topic.

  NPTEL: Principles of Signals and Systems · Course

• The sampling theorem, and the rate it demands

  The copies stay separate only when the rate is more than twice the highest frequency present, and separate copies are what makes recovery possible. The condition is a statement about the copies, not an arbitrary rule.

  NPTEL: Signals and Systems (IIT Kanpur) · Course

• Aliasing, and why an undersampled signal returns as a different one

  Sample too slowly and the copies overlap, so energy from a high frequency lands where a low frequency belongs and cannot be told apart from it afterwards. No amount of later processing undoes it.

  NPTEL: Signals and Systems (IISER Bhopal) · Course

• The anti-alias filter, and why it has to come before the converter

  Because overlap cannot be undone, the high frequencies have to be removed while the signal is still continuous. That is the reason an analog filter sits in front of every converter in practice.

• Reconstruction: the ideal interpolator, and what a real converter does instead

  Recovery in principle means passing the samples through an ideal low-pass filter, which is the smooth interpolation the theorem promises. Real hardware holds each value for one interval and corrects for the resulting droop.