Continuous and Discrete Time
Two independent splits, constantly confused: one is about the variable, the other about the value — and all four combinations exist.
Skip to the animationContinuous and discrete describe the independent variable; analogue and quantised describe the value — two independent axes that give four combinations, all of which physically exist, and only one of which deserves the word digital.
The variable
A continuous-time signal x(t) has a value at every instant. A discrete-time signal x[n] is a sequence, defined only at integer n — and n is an index, not a time. The notation carries the distinction: parentheses for continuous, square brackets for discrete.
x[2.5] is not zero — it does not exist. A discrete-time signal is a list of numbers. Those numbers may have come from sampling something, but the sequence carries no memory of what happened between them, and recovering it needs the sampling theorem's extra hypothesis.
The value, which is a separate question
| Analogue value | Quantised value | |
|---|---|---|
| Continuous time | A microphone's output | A digital logic waveform |
| Discrete time | A sample-and-hold output, inside an ADC | A WAV file — the only truly *digital* case |
All four boxes are occupied. The sample-and-hold case is a real physical state, held for microseconds before the quantiser sees it — students often insist it cannot exist.
Why courses split on time, not value
- Quantisation adds an error, modelled as additive noise and made negligible with more bits. A nuisance, not a change of theory.
- Discretisation changes the mathematics: integrals become sums, derivatives become differences, Laplace becomes the Z transform.
That is why a syllabus is labelled "continuous-time" or "discrete-time" and never "analogue" or "digital".
Operations that look the same and are not
A continuous signal shifts by any real t₀. A sequence shifts by an integer number of samples — a half-sample shift requires interpolation, which is an approximation rather than an operation.
Time scaling is worse. x(2t) loses nothing; x[2n] discards every other sample and cannot be undone. Downsampling is destructive in a way that has no continuous-time counterpart.
Why the continuous theory is still taught first
Physical signals are continuous and every processor is discrete, so a sampler sits at every boundary. The continuous results mirror into discrete ones — convolution, Fourier analysis and stability all reappear with sums for integrals — so it is one set of ideas learned twice rather than two sets learned once.
The numbers you will be asked for
- Continuous shift
x(t − t₀)
t₀ may be any real number.
- Discrete shift
x[n − k]
k must be an integer number of samples.
- Discrete exponential
aⁿ
Decays for |a| < 1 — the origin of the unit circle criterion.
- Quantisation levels
L = 2^b
b bits give L levels; SNR improves about 6 dB per bit.
Advantages and disadvantages
Advantages
- Discrete signals are exactly storable and reproducible.
- Discrete processing is programmable rather than built from components.
- The discrete impulse is an ordinary sequence, needing no distribution theory.
- Continuous results carry over with sums replacing integrals.
Disadvantages
- Sampling discards everything between samples unless the band-limit condition holds.
- Fractional shifts and time scaling need interpolation or destroy information.
- Quantisation always adds error, however many bits are used.
- The two axes are constantly conflated, which is the subject's most common confusion.
Watch it work
Check yourself
question 1 / 4
One question at a time. Pick an answer to see why it is right or wrong, then move on — there is no score to keep and nothing is saved.