To analyze and process signals, we must first understand their basic properties: what types of signals exist, how we can measure their size, and how they respond to common operations like shifting and scaling. These properties provide the foundation for almost every concept in signal processing. Before we explore systems and transforms, we begin by precisely defining and classifying signals themselves.
A continuous-time signal $x(t)$ is represented by an
A discrete-time signal $x[n]$ is represented by an
Notation warning: Some texts use the same bracket notation for both continuous-time and discrete-time signals (i.e., $x(t)$ is continuous-time and $x(n)$ is discrete-time). Furthermore, the book denotes discrete-time using $m$ (common in controls literature) rather than $n$ (common in signal processing literature).
In this course, we will denote continuous-time signal with round brackets $( \cdot)$ and discrete-time signals by square brackets $[ \cdot ]$. Further, we will typically use $t$ to denote continuous-time signals and $n$ to denote discrete-time signals. Note though that the dependent variable does
The continuous-time signals $x(t)$ is even when $$x(-t) = x(t)$$ The continuous-time signals $x(t)$ is odd when $$x(-t) = -x(t)$$ The discrete-time signals $x[n]$ is even when $$x[-n] = x[n]$$ The discrete-time signals $x[n]$ is odd when $$x[-n] = -x[n]$$
A continuous-time signal x(t) is said to be causal when $$x(t) = 0 \quad \textrm{for} \quad t < 0$$ A continuous-time signal x(t) is said to be anticausal when $$x(t) = 0 \quad \textrm{for} \quad t \geq 0$$ A discrete-time signal x[n] is said to be causal when $$x[n] = 0 \quad \textrm{for} \quad n < 0$$ A discrete-time signal x[n] is said to be anticausal when $$x[n] = 0 \quad \textrm{for} \quad n \geq 0$$ Signals that are not causal are also called acausal. Anticausal signals are a type acausal signals.
A continuous-time signal $x(t)$ or discrete-time signal $x[n]$ is said to be periodic if some $T_0$ or $N_0$ exists such that $$x(t) = x(t+m T_0)$$ $$x[n] = x[n+m N_0]$$ where $T_0$ or $N_0$ is finite and $m$ can equal any integer value. The smallest $T_0$ or $N_0$ that satisfies the above conditions is the fundamental period.
The fundamental cyclic frequency is the reciprocal of the fundamental period, such that $$ f_0 = 1/T_0 \quad \textrm{or} \quad f_0 = 1/N_0 \; .$$ All other frequencies in the signal are defined by $$f_{m} = m f_0$$ where $m$ is any integer. These frequencies are known as harmonics.
The fundamental angular frequency is defined by $$ \omega_0 = 2\pi f_0 \; .$$ Angular frequency is notationally convenient since the functions $\cos(\omega_1 t)$ and $\sin(\omega_2 t)$ will have angular frequencies of $\omega_1$ and $\omega_2$, respectively.
Caveat (discrete-time): Some signals that are periodic in continuous time are not periodic in discrete time. For example, $x[n] = \cos(n)$ is not periodic because $\cos(n)$ does not repeat with a rational multiple of $2\pi$.
The energy of a continuous-time signal x(t) $$E_x=\int_{-\infty}^{\infty} |x(t)|^2 dt $$ The energy of a discrete-time signal x[n] is $$E_x=\sum_{n=-\infty}^{\infty} |x[n]|^2 $$
An energy signal is a signal with finite energy (i.e., $0 < E_x < \infty$).
Physical Interpretation: Energy, in this context, does
The power of a continuous-time signal x(t) $$P_x=\lim_{T \rightarrow \infty} \frac{1}{T} \int_{-T/2}^{T/2} |x(t)|^2 dt \; .$$ When the signal is periodic, the power simplifies to $$P_x=\frac{1}{T} \int_{-T/2}^{T/2} |x(t)|^2 dt \; ,$$ where $T$ is a period of the periodic signal. This is equivalent to saying that the power of a periodic signal is equal to the average energy in one period in the signal.
The power of a discrete-time signal x[n] is $$P_x=\lim_{N \rightarrow \infty} \frac{1}{2N + 1} \sum_{n=-N}^{N} |x[n]|^2 \; .$$ When the signal is periodic, the power simplifies to $$P_x=\frac{1}{N} \sum_{n=0}^{N-1} |x[n]|^2 \; ,$$ where $N$ is a period of the periodic signal. This is equivalent to saying that the power of a periodic signal is equal to the average energy in one period in the signal.
A power signal is a signal with finite power (i.e., $0 < P_x < \infty$).
Physical Interpretation: Power, in this context, does
The continuous-time signal $y(t) = x(t-T)$ is the signal $x(t)$ shifted to the
The continuous-time signal $y(t) = x(t+T)$ is the signal $x(t)$ shifted to the
The discrete-time signal $y[n] = x[n-N]$ is the signal $x[n]$ shifted to the
The discrete-time signal $y[n] = x[n+N]$ is the signal $x[n]$ shifted to the
The continuous-time signal $y(t) = x(a t)$ is the signal x(t)
The continuous-time signal $y(t) = x(t / a)$ is the signal x(t)
Scaling only works when $a$ is an integer.
The discrete-time signal $y[n] = x[a n]$ is the signal x[n]
The discrete-time signal $y[n] = x[n / a]$ is the signal x[n]
The continuous-time signal $y(t) = x(-t)$ is the time-reversed signal of $x(t)$.
The discrete-time signal $y[n] = x[-n]$ is the time-reversed replica of signal $x[n]$.
The Dirac $\delta$ signal (continuous-time impulse signal) is defined by $$ \delta(t) = \left\{ \begin{eqnarray} \infty &\quad& \textrm{for} \quad t = 0 \\ 0 &\quad& \textrm{for} \quad t \neq 0 \end{eqnarray} \right. $$ where $$\int_{-\infty}^{\infty} \delta(t) dt = 1 $$
The Kronecker $\delta$ signal (discrete-time impulse signal) is defined by $$ \delta[n] = \left\{ \begin{eqnarray} 1 &\quad& \textrm{for} \quad n = 0 \\ 0 &\quad& \textrm{for} \quad n \neq 0 \end{eqnarray} \right. $$
The properties of impulse signals:
The continuous-time step function $u(t)$ is defined by $$ u(t) = \int_{-\infty}^{t} \delta(\tau) d\tau = \left\{ \begin{eqnarray} 1 &\quad& \textrm{for} \quad t \geq 0 \\ 0 &\quad& \textrm{for} \quad t < 0 \end{eqnarray} \right. $$
The discrete-time step function $u[n]$ is defined by $$ u[n] = \sum_{k = -\infty}^{n} \delta[k] = \left\{ \begin{eqnarray} 1 &\quad& \textrm{for} \quad n \geq 0 \\ 0 &\quad& \textrm{for} \quad n < 0 \end{eqnarray} \right. $$
The properties of the Heaviside step functions:
The continuous-time cosine and sine signals are defined by $$ x_1(t) = \cos(\omega_0 t) \quad , \quad x_2(t) = \sin(\omega_0 t) $$
The properties of the continuous-time cosines or sines:
A continuous-time exponential signal is defined by $$ x(t) = e^{a_0 t} u(t) $$
The properties of the exponential signals are:
The continuous-time complex exponential signal is defined by $$ \begin{eqnarray} x(t) &=& e^{j \omega_0 t} \\ &=& \cos(\omega_0 t) + j \sin(\omega_0 t) \end{eqnarray}$$
The properties of the continuous-time complex exponential:
The continuous-time general exponential signal is defined by $$ \begin{eqnarray} x(t) &=& e^{(j \omega_0 + a_0)t} u(t) \\ &=& \left[ e^{a_0 t} \cos(\omega_0 t) + j e^{a_0 t} \sin(\omega_0 t) \right] u(t) \end{eqnarray}$$
The properties of the general exponential signals are: