Standard Sequences

In discrete-time signal processing, every complex signal can be built from a small set of fundamental building blocks called standard sequences. Understanding these sequences is not optional. Every concept in DSP, from system analysis to filter design, rests on knowing how impulse, step, exponential, and sinusoidal sequences behave and interact.

Core Concept Explanation

Unit Impulse Sequence

The unit impulse sequence δ[n] is defined as 1 when n = 0 and 0 for all other values of n. It is the discrete-time equivalent of the Dirac delta function. Its most powerful property is the sifting property: any arbitrary sequence x[n] can be expressed as a weighted sum of shifted impulses, written as x[n] = Σ x[k]·δ[n−k]. This decomposition is the foundation of convolution and LTI system analysis.

Unit Step Sequence

The unit step sequence u[n] equals 1 for all n greater than or equal to zero and equals 0 for n less than zero. It models systems that switch on at n = 0, such as a causal filter receiving input. The step and impulse are related through accumulation and differencing: u[n] = Σ δ[n−k] summed from k = 0 to infinity, and δ[n] = u[n] − u[n−1]. These relationships appear frequently in GATE problems involving system response.

Real Exponential Sequence

The real exponential sequence is defined as x[n] = a^n · u[n], where a is a real constant. When |a| < 1, the sequence decays toward zero and the system is bounded-input bounded-output (BIBO) stable. When |a| > 1, the sequence grows without bound and the system is unstable. When a is negative, the sequence alternates in sign. The z-transform of this sequence is 1/(1 − az⁻¹) for |z| > |a|, a result used extensively in filter analysis.

Sinusoidal and Complex Exponential Sequences

The discrete sinusoidal sequence is given by x[n] = A·cos(ω₀n + φ), where ω₀ is the discrete-time angular frequency in radians per sample. An important distinction from continuous-time signals is that a discrete sinusoid is periodic only if ω₀ / (2π) is a rational number. If this ratio is irrational, the sequence never repeats. This is a commonly tested concept in GATE where students confuse continuous and discrete periodicity conditions.

The complex exponential sequence x[n] = e^(jω₀n) = cos(ω₀n) + j·sin(ω₀n) is used to define the DTFT. The magnitude of the complex exponential is always 1, but its phase rotates uniformly. Every sinusoidal sequence can be expressed as a sum of two complex exponentials using Euler's formula, connecting time-domain representations to frequency-domain analysis.

Mathematical Expression

The four standard sequences and their z-transforms are central to discrete system analysis. The impulse has Z-transform equal to 1. The unit step has Z-transform Z/(Z−1) for |Z| > 1. The exponential a^n·u[n] has Z-transform Z/(Z−a) for |Z| > |a|. For sinusoidal sequences, the z-transform is derived using Euler decomposition and partial fractions. The periodicity condition for a discrete sinusoid with frequency ω₀ requires that 2π/ω₀ be an integer or that ω₀/(2π) = p/q where p and q are integers.

Practical Understanding

In practice, the impulse sequence is used to characterize any LTI system through its impulse response h[n]. Once h[n] is known, the output to any input can be computed by convolution. The step sequence models switch-on events and is used to find step response. The exponential sequence models natural system responses such as charging a capacitor or a first-order IIR filter output. Sinusoidal sequences model steady-state responses of filters and are the basis of frequency analysis using the DTFT and DFT.

Given:
ω₀ = π/4 radians/sample, check periodicity. Also find period of x[n] = cos(πn/3).

Why this formula applies:
A discrete sinusoid cos(ω₀n) is periodic with period N if ω₀·N = 2πk for some integer k.
So N = 2πk/ω₀ must be a positive integer.

Formula:
N = 2πk / ω₀  (choose smallest integer k that makes N a positive integer)

Substitution for ω₀ = π/4:
N = 2πk / (π/4) = 8k
For k = 1, N = 8

Substitution for x[n] = cos(πn/3), ω₀ = π/3:
N = 2πk / (π/3) = 6k
For k = 1, N = 6

Calculation:
Both ratios ω₀/2π = 1/8 and 1/6 are rational numbers.

Final Answer:
cos(πn/4) has period N = 8 samples.
cos(πn/3) has period N = 6 samples.

Exam Tip: For GATE, always check if ω₀/(2π) is rational to confirm periodicity. A discrete cosine with irrational frequency like cos(√2·n) is NOT periodic. Also remember u[n] − u[n−1] = δ[n], used in many sequence decomposition problems.

Sequence Properties Summary

• Sifting property: x[n] = Σ x[k]·δ[n−k] allows any sequence to be decomposed into impulses.
• Unit step accumulates impulses; the impulse is the first difference of the step.
• Real exponential a^n·u[n] is stable only when |a| < 1.
• Discrete sinusoid is periodic if and only if ω₀/(2π) is a rational number.
• Complex exponential e^(jω₀n) always has magnitude 1, making it energy-bounded.

Quick Revision

• δ[n] = 1 at n=0, zero elsewhere. Sifting: x[n] = Σ x[k]·δ[n−k].
• u[n] = 1 for n ≥ 0. Relation: δ[n] = u[n] − u[n−1].
• Exponential a^n·u[n]: stable if |a| < 1, unstable if |a| > 1, alternating if a < 0.
• Discrete sinusoid periodic only if ω₀/(2π) is rational. Period N = 2πk/ω₀.
• Z-transform of a^n·u[n] = Z/(Z−a), ROC: |Z| > |a|.
• Common trap: confusing continuous periodicity (always) with discrete periodicity (conditional).
• Complex exponential |e^(jω₀n)| = 1 always; phase rotates at ω₀ radians per sample.