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 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 some $m$ 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 for discrete-time signals: Note that determining periodicity for discrete-time signals is more complicated than it may initially seem. Some signals that are periodic in continuous-time do not satisfy periodicity in discrete-time (for example, $x[n] = \cos(n)$ does
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)
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: