OE-8.1 Discrete Time Signals & Systems

What this is and why it exists

The sampling theorem is the bridge between the analog world and everything else in this course. Cross it carelessly and you get aliasing, which is not noise and cannot be removed afterwards.

That is the first thing to understand properly. Aliasing is information from above the limit arriving disguised as information below it, and once disguised there is no way to tell it apart.

The other two ideas here are properties rather than operations. Linearity and time invariance are what let convolution and transforms work at all, so checking for them is not box-ticking.

The vocabulary

• Sampling — recording a signal's value at regular instants.
• Sampling rate — how many samples per second.
• Sampling theorem — the rate must exceed twice the highest frequency present.
• Aliasing — high frequencies appearing as low ones after sampling.
• Anti-alias filter — an analog filter removing frequencies before sampling.
• Quantisation — rounding each sample to a representable value.
• Linear system — scaling and adding inputs scales and adds outputs.
• Time invariant — delaying the input only delays the output.
• Impulse response — the output when the input is a single unit impulse.
• Causal — the output never depends on future input.
• Stable — a bounded input always produces a bounded output.

The mental model

The sampling theorem says the rate must be more than twice the highest frequency present in the signal. Note the wording: present, not wanted. A frequency you did not want, above the limit, still folds down and lands somewhere in your band, indistinguishable from a real signal.

That is why the anti-alias filter is analog and sits before the converter. It has to remove those frequencies while they are still separable. Nothing done in software afterwards can undo the folding, because after sampling the alias and the genuine signal are the same numbers.

Digital processing then buys real advantages. Exact repeatability, and no drift with temperature or age. Filters impossible to build from components, and behaviour you can change by changing code. It costs conversion at both ends, finite word length, and processing delay.

Quantisation is the second approximation. Each sample is rounded to a representable value, and that rounding is an error you carry through everything else. More bits, smaller error.

Then the two properties. Linear means scaling and adding inputs scales and adds outputs the same way. Time invariant means the system behaves the same today as yesterday — delay the input and you only delay the output. When both hold, the system is completely described by its impulse response, and its output is the convolution of input with that response. That is an enormous simplification, and it is the reason these two properties are checked first rather than assumed.

Causality says the output cannot depend on future input, which any real-time system must satisfy. Stability says a bounded input gives a bounded output. Both are worth testing rather than hoping for.

What you should now be able to explain or do

State the sampling theorem, with attention to "present" rather than "wanted". Explain aliasing as folding, and say why it cannot be fixed after sampling. Say why the anti-alias filter must be analog and come first. List what digital processing buys and what it costs. Test a system for linearity and time invariance, and say what those two together give you. Test for causality and stability.

Check yourself