Signal Classification

Signals can be of finite or infinite duration. Finite duration signals are called time-limited. Signals of semi-infinite extent may be right-sided if they are zero for \(t < \alpha\) (where \(\alpha\) is finite) or left-sided if they are zero for \(t > \alpha\). Signals that are zero for \(t < 0\) are often called causal.

Piecewise continuous signals possess different expressions over different intervals. Continuous signals, such as \(x(t) = \sin(t)\), are defined by a single expression for all time.

Periodic signals are infinite-duration signals that repeat the same pattern endlessly. The smallest repetition interval is called the period \(T\).

\(\boxed{x_{p}(t) = x_{p}(t \pm nT)}\)

where \(x_{p}(t)\) is the periodic signal and \(n\) is an integer.

One-sided or time-limited signals can never be periodic.

Links to this page
  • Sinusoids

    A periodic function is one that satisfies \(f\left(t\right) = f\left(t + nT\right)\), for all \(t\) and for all integers \(n\).

  • Signal Measures

    All time-limited functions of finite amplitude have finite absolute area. The criterion of absolute integrability is often used to check for system stability or justify the existence of certain transforms.

    Signals with finite energy \((0 < E < \infty)\) are called an energy signal or square integrable. Energy signals have zero signal power \((P = 0)\) since finite energy is averaged over all (infinite) time. All time-limited signals of finite amplitude are energy signals.

    Periodic signals are characterized by several measures:

  • Parseval’s Theorem

    Parseval’s theorem shows that the energy associated with a non-periodic signal is spread over the entire frequency spectrum, whereas the energy of a periodic signal is concentrated at the frequencies of its harmonic components.

  • Fourier Transform

    Fourier series enable us to represent a periodic function as sum of sinusoids and to obtain the frequency spectrum from the series. The Fourier transform allows us to extend the concept of a frequency spectrum to non-periodic functions.

  • Fourier Series

    Since the Fourier series describes the periodic signal \(f(t)\) as a sum of sinusoids at different frequencies, its signal power also equals the sum of the power in each sinusoid.

    The (trigonometric) Fourier series of a periodic function \(f(t)\) is a representation that resolves \(f(t)\) into a DC component and an AC comprising an infinite series of harmonic sinusoids.

  • Discrete Signals

    The period \(N\) is the smallest number of samples that repeats. Unlike its analog counterpart, the period \(N\) of discrete signals is always an integer.

  • Continuous Convolution

    For two causal signals \(x(t)\,u(t)\) and \(h(t)\,u(t)\), the product \(x(\lambda)\,u(\lambda) \cdot h(t - \lambda)\,u(t - \lambda)\) is non-zero only over the range \(0 \leq \lambda \leq t\). Since both \(u(\lambda)\) and \(u(t - \lambda)\) are unity in this range, the convolution integral simplifies to

#signal