Basic Continuous-Time and Discrete-Time Signals

Course Overview and Institutional Context

• Institutional Information:

  • University: Malawi University of Science and Technology (MUST)
  • Department: Department of Engineering
  • Course Code: BME-SCSY-3100 / SISY-3100 MIT - 2026

• Core Review Topics:

  • Signals and Systems fundamentals
  • Continuous-Time (CT) versus Discrete-Time (DT) signals
  • Essential signal properties and mathematical classifications

Continuous-Time Unit Step Function

• Definition of the Unit Step Function:
  • The unit step function $u(t)$, also referred to as the Heaviside unit step function, is mathematically defined as:

$u(t) = \begin{cases} 1 & t > 0 \\ 0 & t < 0 \end{cases}$

• Discontinuity property: The function is discontinuous at $t = 0$, and its value at $t = 0$ is explicitly undefined.

• Definition of the Shifted Unit Step Function:
  • The shifted (time-delayed) unit step function $u(t - t_0)$ is defined as:

$u(t - t_0) = \begin{cases} 1 & t > t_0 \\ 0 & t < t_0 \end{cases}$

Continuous-Time Unit Impulse Function

• Definition of the Dirac Delta Function:
  • The unit impulse function $\delta(t)$, also called the Dirac delta function, plays a central role in system analysis and mathematical signal processing.
  • Mathematical definition:

$\delta(t) = \begin{cases} 0 & t \neq 0 \\ \infty & t = 0 \end{cases}$

• Area constraint condition:

$\int_{-\varepsilon}^{\varepsilon} \delta(t)\,dt = 1$

• Limiting Approximation:
  • The Dirac delta function $\delta(t)$ can be conceptualized as the limiting form of a rectangular pulse centered around $t = 0$ with width $\varepsilon$ and height $\frac{1}{\varepsilon}$, as $\varepsilon \rightarrow 0$. This construction guarantees unity area under the curve regardless of how small $\varepsilon$ becomes.

• Shifted (Delayed) Impulse Function:
  • The delayed unit impulse function $\delta(t - t_0)$ satisfies the sampling property for any regular test function $\phi(t)$ continuous at $t = t_0$:

$\int_{-\infty}^{\infty} \phi(t)\delta(t - t_0)\,dt = \phi(t_0)$

• Graphical Representation of CT Impulses:

Continuous-Time Complex and Real Exponential Signals

• Complex Exponential Signal:
  • Formulated as:

$x(t) = e^{j\omega_0 t}$

• Applying Euler's formula expands the complex exponential signal into its trigonometric components:

$e^{j\omega_0 t} = \cos(\omega_0 t) + j\sin(\omega_0 t)$

• The real part is $\text{Re}\{x(t)\} = \cos(\omega_0 t)$.
• The imaginary part is $\text{Im}\{x(t)\} = \sin(\omega_0 t)$.
• Periodicity Property: The continuous-time complex exponential $x(t) = e^{j\omega_0 t}$ is periodic for any value of $\omega_0$. Its fundamental period $T_0$ is given by:

$T_0 = \frac{2\pi}{\omega_0}$

• General Complex Exponential Signals:
  • Let $s = \sigma + j\omega$ be a complex frequency variable. The general complex exponential signal is defined as:

$x(t) = e^{st} = e^{(\sigma + j\omega)t} = e^{\sigma t}e^{j\omega t} = e^{\sigma t}\cos(\omega t) + j e^{\sigma t}\sin(\omega t)$

• Real component: $e^{\sigma t}\cos(\omega t)$
• Imaginary component: $e^{\sigma t}\sin(\omega t)$
• Behavior determined by $\sigma$:
  • If $\sigma > 0$: The signal amplitude grows exponentially, forming an exponentially increasing sinusoidal signal.
  • If $\sigma < 0$: The signal amplitude decays exponentially, forming an exponentially decreasing sinusoidal signal.

• Real Exponential Signals:
  • When $s = \sigma$ (a purely real number), the general complex exponential simplifies to a real exponential signal:

$x(t) = e^{\sigma t}$

• Growth and decay regimes:
  • $\sigma > 0$: Growing real exponential signal.
  • $\sigma < 0$: Decaying real exponential signal.

Continuous-Time Sinusoidal Signals

• General Expression:
  • A continuous-time sinusoidal signal is expressed as:

$x(t) = A\cos(\omega_0 t + \theta)$

• Parameters:
  • $A$: Amplitude (real constant)
  • $\omega_0$: Radian frequency in radians per second ($\text{rad/s}$)
  • $\theta$: Phase angle in radians ($\text{rad}$)

• Period and Frequency Relationships:
  • Fundamental Period $T_0$:

$T_0 = \frac{2\pi}{\omega_0}$

• Fundamental Frequency $f_0$ (in hertz, $\text{Hz}$):

$f_0 = \frac{1}{T_0} = \frac{\omega_0}{2\pi}\quad\text{Hz}$

• Fundamental Angular Frequency $\omega_0$:

$\omega_0 = 2\pi f_0 = \frac{2\pi}{T_0}$

• Euler's Representation of Sinusoids:
  • Sinusoidal signals can be expressed using complex exponentials via Euler's formula:

$x(t) = \text{Re}\{A e^{j(\omega_0 t + \theta)}\}$

• Using the imaginary part notation $\text{Im}\{\cdot\}$:

$\text{Im}\{A e^{j(\omega_0 t + \theta)}\} = A\sin(\omega_0 t + \theta)$

Discrete-Time Unit Step and Unit Impulse Sequences

• The Discrete Unit Step Sequence:
  • Defined as:

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

• Critical Difference from CT Step Function: The value of $u[n]$ at $n = 0$ is explicitly defined and equal to $1$ (unity), unlike the continuous-time unit step function $u(t)$ which is undefined at $t = 0$.

!Discrete unit step sequence u[n]

• The Shifted Unit Step Sequence:
  • Defined as:

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

!Shifted discrete unit step sequence u[n - k]

• The Discrete Unit Impulse (Unit Sample) Sequence:
  • Defined mathematically as:

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

!Discrete unit impulse sequence delta[n]

• The Shifted Unit Impulse Sequence:
  • Defined mathematically as:

$\delta[n - k] = \begin{cases} 1 & n = k \\ 0 & n \neq k \end{cases}$

!Shifted discrete unit impulse sequence delta[n - k]

Discrete-Time Complex Exponential Sequences and Periodicity

• Formulation:
  • A discrete-time complex exponential sequence is given by:

$x[n] = e^{j\Omega_0 n}$

• By Euler's formula:

$e^{j\Omega_0 n} = \cos(\Omega_0 n) + j\sin(\Omega_0 n)$

• Real part: $\text{Re}\{x[n]\} = \cos(\Omega_0 n)$

• Imaginary part: $\text{Im}\{x[n]\} = \sin(\Omega_0 n)$

  • Periodicity Condition for Discrete Complex Exponentials:

• For $e^{j\Omega_0 n}$ to be periodic with integer period $N > 0$, the frequency $e^{j\Omega_0 (n + N)} = e^{j\Omega_0 n} e^{j\Omega_0 N}$ requires $e^{j\Omega_0 N} = 1$. Thus, $\Omega_0 N$ must be an integer multiple of $2\pi$:

$\frac{\Omega_0}{2\pi} = \frac{m}{N} \quad (m = \text{positive integer})$

• Fundamental Rule: The discrete sequence $e^{j\Omega_0 n}$ is not periodic for all arbitrary values of $\Omega_0$. It is periodic if and only if $\frac{\Omega_0}{2\pi}$ is a rational number.

• This behavior directly contrasts with the continuous-time complex exponential $e^{j\omega_0 t}$, which is periodic for any value of $\omega_0$

  • Fundamental Period $N_0$ Computation:

• Assuming $\Omega_0 \neq 0$, and that $m$ and $N$ share no common factors, the fundamental period $N_0$ is given by:

$N_0 = m\left(\frac{2\pi}{\Omega_0}\right)$

• Distinctness of Discrete Frequencies:
  • Continuous-time exponentials $e^{j\omega_0 t}$ are distinct for all distinct values of $\omega_0$.
  • Discrete-time exponentials $e^{j\Omega_0 n}$ separated by integer multiples of $2\pi$ are completely indistinguishable (identical), because:

$e^{j(\Omega_0 + 2\pi m)n} = e^{j\Omega_0 n} e^{j 2\pi m n} = e^{j\Omega_0 n}$

General and Real Discrete-Time Exponential Sequences

• General Discrete Complex Exponential Sequence:
  • Defined as:

$x[n] = C \alpha^n$

• Where $C$ and $\alpha$ are general complex numbers. The standard complex exponential $e^{j\Omega_0 n}$ is a special case where $C = 1$ and $\alpha = e^{j\Omega_0}$.

  • Real Discrete Exponential Sequences:

• If both $C$ and $\alpha$ are real numbers, $x[n]$ is a real exponential sequence. Four distinct qualitative behaviors occur depending on the value of $\alpha$:

1. $\alpha > 1$: Monotonically growing positive sequence.

2. $0 < \alpha < 1$: Monotonically decaying positive sequence.

3. $-1 < \alpha < 0$: Alternating-sign exponentially decaying sequence.

4. $\alpha < -1$: Alternating-sign exponentially growing sequence.

• Boundary Conditions:
  • If $\alpha = 1$: $x[n] = C$ (a constant sequence).
  • If $\alpha = -1$: $x[n]$ alternates strictly between $+C$ and $-C$.

Discrete-Time Sinusoidal Sequences

• General Formula:
  • A discrete-time sinusoidal sequence is expressed as:

$x[n] = A\cos(\Omega_0 n + \theta)$

• Units: Since sample index $n$ is dimensionless, both $\Omega_0$ and $\theta$ carry units of radians.
• Complex representation:

$x[n] = \text{Re}\{A e^{j(\Omega_0 n + \theta)}\}$

• Comparative Examples of Periodicity in Discrete Sinusoids:

  • Case 1: Periodic Sequence $x[n] = \cos\left(\frac{\pi}{6} n\right)$
  • Angular frequency ${\Omega_0 = \frac{\pi}{6}}$.
  • Testing periodicity ratio:

$\frac{\Omega_0}{2\pi} = \frac{\pi/6}{2\pi} = \frac{1}{12}$

- Since $\frac{1}{12}$ is a rational number ($m = 1, N = 12$), the sequence is periodic with a fundamental period of $N_0 = 12$ samples.

!Periodic discrete sinusoidal sequence x[n] = cos(pi n / 6)

• Case 2: Non-Periodic Sequence $x[n] = \cos\left(\frac{n}{2}\right)$
  • Angular frequency ${\Omega_0 = \frac{1}{2}}$.
  • Testing periodicity ratio:

$\frac{\Omega_0}{2\pi} = \frac{1/2}{2\pi} = \frac{1}{4\pi}$

- Because $\frac{1}{4\pi}$ is an irrational number, no integer $N$ exists such that $\cos\left(\frac{n+N}{2}\right) = \cos\left(\frac{n}{2}\right)$ for all $n$. Therefore, the sequence is **non-periodic**.

!Non-periodic discrete sinusoidal sequence x[n] = cos(n / 2)