Common Discrete Signals

The Alphabet of Discrete-Time Signals

The previous module established how a continuous signal becomes a sequence of numbers through sampling and quantization. Module 3 begins with a question: what kinds of discrete-time signals actually exist? Just as the Latin alphabet provides a finite set of letters from which all words can be built, a small set of canonical discrete-time signals serves as the vocabulary for everything in DSP.

These fundamental sequences — the unit impulse, the unit step, the exponential, and the sinusoid — are not merely textbook abstractions. They appear constantly in practice: exponentials describe charging and discharging behavior, sinusoids represent every periodic signal, and the unit impulse is the cornerstone of system analysis. Understanding them thoroughly is prerequisite to everything that follows: convolution, the Z-transform, filters, and the Fourier transform all reduce ultimately to how systems respond to these basic signals.

The Unit Impulse: δ[n]

The unit impulse (also called the Kronecker delta or discrete impulse) is the simplest non-trivial discrete-time signal: it equals one at sample index zero and zero at every other index. Despite its simplicity — or perhaps because of it — the unit impulse is the single most important signal in discrete-time analysis.

The impulse can be shifted to any index n₀ by writing δ[n − n₀]: a spike at n = n₀. This shifted impulse selects the value of any sequence at that index: x[n] ⋅ δ[n − n₀] = x[n₀] ⋅ δ[n − n₀]. That sifting property is the reason the impulse is so useful — any sequence can be written as a weighted sum of shifted impulses:

The Unit Step: u[n]

The unit step function is zero for all negative time indices and one for all non-negative indices. It models the turn-on behavior of a system — the signal that “switches on” at n = 0 and stays on. The unit step is deeply connected to the impulse: the impulse is the first difference of the unit step, and the unit step is the running sum (discrete integral) of the impulse.

The unit step is used extensively to describe causal signals — signals that start at some finite time and persist afterward. Multiplying any sequence by u[n − n₀] zeroes out everything before index n₀, effectively “gating” the signal to begin at n₀. This notation appears throughout filter theory and system analysis to indicate that an exponential or sinusoidal sequence is switched on at time zero.

Exponential Sequences

An exponential sequence takes the form x[n] = Aαⁿ, where A is the initial amplitude and α is the base. The behavior depends entirely on the magnitude of α:

|α| < 1: The sequence decays toward zero as n increases. This is the stable case — the energy in the signal diminishes geometrically. A decaying exponential models the impulse response of a stable first-order recursive filter.

|α| = 1: The sequence neither grows nor decays. If α = 1 the signal is a constant (the unit step); if α = −1 it alternates ±A every sample; if α = e^jω it traces a unit circle in the complex plane (the complex sinusoid, discussed below).

|α| > 1: The sequence grows without bound — the unstable case. Such signals arise from unstable systems or diverging iterative processes.

Sinusoidal Sequences

A discrete-time sinusoid is written x[n] = A cos(ωn + φ), where A is amplitude, ω is the discrete-time frequency in radians per sample, and φ is the phase in radians. This is the direct discrete counterpart of the continuous-time sinusoid x(t) = A cos(2πft + φ), with the substitution t = nT_s giving ω = 2πf/f_s = 2πfT_s.

Two properties of discrete-time sinusoids are surprising to engineers accustomed to continuous-time signals. First, a discrete sinusoid at frequency ω is identical to one at ω + 2πk for any integer k — adding 2π to the frequency simply reproduces the same sequence. Second, ω = π is the highest representable frequency in a sampled system, corresponding to the Nyquist frequency f_s/2. At this frequency the sinusoid is +1, −1, +1, −1, … — alternating every sample.

Discrete-Time Frequency vs. Continuous-Time Frequency

Continuous-time frequency f is measured in hertz: it counts full oscillation cycles per second. Discrete-time frequency ω is dimensionless, measured in radians per sample. The two are related through the sampling period T_s = 1/f_s:

This normalization has an important implication: the same discrete sequence x[n] = cos(πn/4) represents a different physical frequency depending on the sampling rate. Sampled at 8 kHz it represents a 1 kHz tone; sampled at 44.1 kHz the same sequence represents a 5.5 kHz tone. The discrete sequence itself carries no intrinsic physical frequency — only the combination of sequence and sampling rate has physical meaning.

Periodic vs. Aperiodic Discrete Signals

A discrete-time signal x[n] is periodic with period N if x[n + N] = x[n] for all n, where N is a positive integer. Not every discrete-time sinusoid is periodic — this is another key difference from continuous-time signals, where every sinusoid is periodic.

A discrete sinusoid x[n] = A cos(ωn + φ) is periodic if and only if the ratio 2π/ω is rational — that is, 2π/ω = N/k for positive integers N and k. The fundamental period is then the smallest such N.

For example: ω = π/4 gives 2π/ω = 8, so the period is 8 samples — perfectly periodic. But ω = 1 radian/sample gives 2π/1 = 2π, which is irrational, so the sinusoid cos(n) is aperiodic: it never exactly repeats. This subtlety matters in the DFT, where we implicitly assume periodicity.