Discrete Signals
Before building systems, you need to know the signals. A small set of canonical sequences — impulse, step, exponential, sinusoid — forms the vocabulary of all discrete-time DSP.
One at n = 0, zero everywhere else. Any sequence can be written as a sum of scaled, shifted impulses — this decomposition is the key to LTI system analysis.
Zero before n = 0, one from n = 0 onward. It models a signal that turns on at time zero. The step is the running sum of the impulse; the impulse is the first difference of the step.
Multiply any signal by u[n−n₀] to “gate” it on at index n₀.
The behavior of x[n] = Aαn depends entirely on |α|.
When α = rejω, the exponential Arnejωn is the eigenfunction of every LTI system. Its magnitude r controls stability; its angle ω sets the oscillation frequency.
Taking the real part gives a decaying (or growing) sinusoid.
x[n] = A cos(ωn + φ) — where ω is radians per sample. Adding 2π to ω gives the identical sequence: all distinct frequencies live in [−π, π].
ω = π corresponds to the Nyquist frequency f_s/2 — the fastest possible oscillation in a sampled system.
Not every discrete sinusoid is periodic! A sinusoid at ω is periodic only if 2π/ω is rational. Otherwise it never exactly repeats — a key difference from continuous time.
ω = π/4 → period N = 8. ω = 1 rad/sample → aperiodic (2π is irrational).
- δ[n] = 1 at n = 0 only; any signal = sum of scaled shifted impulses
- u[n] = 1 for n ≥ 0; δ[n] = u[n] − u[n−1]
- Aαn: decays |α| < 1, constant |α| = 1, grows |α| > 1
- Complex exponentials are eigenfunctions of LTI systems
- Discrete frequency ω = 2πf/fs; unique range is [−π, π]
- Discrete sinusoid is periodic only if 2π/ω is rational