1.1: Discrete-Time Signals

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Learning Objectives: Discrete-Time Signals

Distinguish discrete-time (DT) signals from continuous-time (CT) signals, and generate new examples of DT signals.
Recognize DT signals vs sampled CT signals inherently.
Graph simple transformations of a discrete-time signal.
Recognize graphs of complex signals as superpositions of simple signals.
Match analytic expressions for exponential signals to visual representations of the signals.

YouTube Videos

DT Signal Fundamentals

Common DT Signals

Basic Operations on DT Signals

DT Periodic Signals

Even and Odd Symmetry

Summary

Definition: Discrete-Time Signals are signals that describe a wide variety of physical phenomena. The information in a signal contains a pattern of some form. Signals can be mathematically represented by functions of one or more independent variables. Time is an example of the independent variable. In fact, time is the actual independent variable in many cases, included signals (i.e., source voltage and capacitor voltage) in circuits, see Fig. 1.1.1.

Figure 1.1 RC Circuit.png

Figure 1.1.1: Simple RC circuit illustrating a source voltage (Vs) and capacitor voltage (Vc) as example discrete-time signals.

Course notation: discrete-time signals $x[n]$

Discrete-time signals are defined only at specific points in time rather than continuously, so their independent variable takes on discrete values. In other words, the discrete-time signal $x[n]$ is only defined for n = 0, ±1, ±2, ±3, .... As a result, $x[n]$ is a discrete sequence.

Where do DT signals come from?

1. Phenomena where the independent variable is inherently discrete, such as the stock market (Fig. 1.1.2).

Figure 1.6.png

Figure 1.1.2: An example of an inherent discrete-time signal showing the weekly Dow-Jones stock market index from July, 2025, to December, 2025.

2. Sampling of continuous-time signals $x(t)$ where the discrete-time signal $x[n]$ represents successive samples of $x(t)$, see Fig. 1.1.3.

Figure 1.7.png

Figure 1.1.3: Example of how a CT signal is sampled to form a DT sequence.

Definition of Discrete-Time average power

\[P = \lim_{N\rightarrow\infty}\left(\frac{1}{2N+1}\right)\sum_{n=-N}^{N}\left|x[n]\right|^{2}\]

Energy signal: finite total energy ($E < \infty)$ $\Rightarrow$ average power is zero.
Power signal: finite average power ($P > 0)$ $\Rightarrow$ total energy is infinite.

Transformations of the Independent Variable

Time shift: $x[n-n_0]$ where $x[n-n_0]$ has the same shape as $x[n]$, but shifted by $n_0$ samples to the right direction.
Time reversal: $x[-n]$ is obtained by reflecting $x[n]$ about the vertical axis (i.e., $n=0$).
Time scaling (discrete-time): Time scaling is more restrictive in DT than CT. Common forms include compressing/expanding via indexing like $x[kn]$ where $k$ is an integer.

(a)

Figure 1.8.png

(b)

(c)

Figures 1.1.4 Illustration of transformation in a discrete-time signal: (a) time shifting; (b) time reversal (middle); and (c) time scaling.

Periodic Signals:

A periodic DT signal $x[n]$ has a period $N$ such that $x[n] = x[n+N]$, for all $n$. If a signal is periodic, then there is a smallest positive integer $N$ (the fundamental period).

Figure 1.15.png

Figure 1.1.5: Example of a periodic signal with a fundamental period of $N=4$.

Even and Odd Signals

Even and Odd signals are defined based on symmetry under time reversal:

Even: $x[-n] = x[n]$
Odd: $x[-n] = -x[n]$ and $x[0]=0$

Figure 1.1.6: Examples of an Odd (Top) and Even (Bottom) signal.

Any signal can be decomposed into an even and an odd part:

\[x_e[n] = \frac{x[n] + x[-n]}{2}\]

\[x_o[n] = \frac{x[n] - x[-n]}{2}\]

Complex exponential

General complex exponential: $x[n] = C\,a^n$, where $C$ and $a$ can be real or complex.

If $0<a<1$, then $|x[n]|$ decays exponentially as $n$ increases.
If $a>1$, then $|x[n]|$ grows exponentially as $n$ increases.
If $a<0$, the sequence alternates sign.
If $a=1$, then $x[n]=C$ (constant).
If $a=-1$, then $x[n]$ alternates between $C$ and $-C$.

Figure 1.24.png

Figure 1.1.7: Examples of 4 exponential signals: (a) $\alpha > 1$; (b) $0 < \alpha < 1$; (c) $-1 < \alpha < 0$; and (d) $\alpha < -1$.

Sinusoidal signal

Sinusoid:

\[x[n] = A\cos(\omega_0 n + \phi).$$.

Figure 1.25.png

Figures 1.1.8: Examples of sinusoidal signals with differing frequencies.

Euler relationship (complex exponential to sinusoid)

\[e^{j\theta} = \cos(\theta) + j\sin(\theta).\]

Figure 1.26.png

Figure 1.1.9: Stem plot of an exponential signal with growth (a) and decay (b).

Discrete-Time Unit Impulse

\[\delta[n] =
\begin{cases}
1, & n=0 \\
0, & n\neq 0
\end{cases}\]

Discrete-Tine Unit step

\[u[n] =
\begin{cases}
1, & n\ge 0 \\
0, & n<0
\end{cases}\]

Note that $δ[n] = u[n] - u[n-1]$

Figure 1.1.10: Stem plot of the DT unit impulse (top panel) and DT unit step (bottom panel).

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