This title slide, named "Signals: Examples & Math," introduces examples and the mathematical representation of signals. It serves as a title slide focused on illustrating signals through practical examples and their mathematical formulations.

Examples and Mathematical Representation of Signals

Source: Section 1.1.1: Signals and Systems

This section header slide introduces "Continuous-Time Signals" (section 1.1.1). It defines them as x(t), where t belongs to the real numbers ℝ and is defined for all real time.

Examples and Mathematical Representation of Signals

1.1.1

Continuous-Time Signals

Definition: x(t) where t ∈ ℝ, defined for all real time

Source: Signals and Systems, Section 1.1.1

Continuous-time signals are represented by x(t), where t is a continuous real-valued variable, and are defined for all real values of time in the form x(t) = f(t), t ∈ ℝ. Examples include speech waveforms and temperature variations, analogous to real-world physical phenomena.

Characteristics of Continuous-Time Signals

• Represented by x(t), where t is continuous
• Examples: speech waveforms, temperature variations
• Mathematical form: x(t) = f(t), t ∈ ℝ
• Defined for all real values of time
• Analogous to real-world physical phenomena

Source: Section 1.1.1: Examples and Mathematical Representation of Signals

The slide presents a diagram of a continuous-time signal x(t) = sin(2πft), defined for all real t values as a smooth, unbroken waveform. It exemplifies signals like audio speech.

Diagram: Continuous-Time Signal Example (Fig 1.1-like)

!Image

• Continuous-time signal x(t) = sin(2πft)
• Defined for all real t values
• Smooth, unbroken waveform
• Example: audio speech signals

Source: Wikipedia search: 'sine wave'

This section header introduces Discrete-Time Signals, specifically section 1.1.1. It describes sequences defined at integer times, denoted as x[n] where n is an integer in ℤ.

Discrete-Time Signals

1.1.1

Discrete-Time Signals

Sequences Defined at Integer Times, x[n] for n ∈ ℤ

Source: Signals and Systems, Section 1.1.1

Discrete-time signals are represented mathematically as x[n], consisting of samples from a continuous-time signal. Real-world examples include digital images (pixels) and stock market data.

Characteristics and Real-World Examples

• Represented mathematically as x[n]
• Examples: digital images (pixels), stock market data
• x[n] = samples of continuous-time signal

Source: Signals and Systems, Section 1.1.1

The slide contrasts continuous-time signals, defined as x(t) = A cos(ωt + φ) for all real t with parameters amplitude A, angular frequency ω, and phase φ, representing smooth waveforms like audio. Discrete-time signals are shown as x[n] = A cos(ωnT + φ) for integer n, sampled from continuous signals at period T for digital processing.

Mathematical Formulations

| x(t) = A cos(ωt + φ)

Defined for all real t ∈ ℝ.

A: amplitude, ω: angular frequency (rad/s), φ: phase shift.

Represents smooth, time-varying signals like audio waveforms. | x[n] = x(nT) = A cos(ωnT + φ)

Defined for integer n ∈ ℤ.

T: sampling period.

Obtained by sampling continuous signal at regular intervals, used in digital processing. |

Continuous-time signals are denoted as x(t) (analog, e.g., speech), while discrete-time signals are x[n] (digital, e.g., stocks), with key distinctions bridging theory to practice. The slide closes by stating signals are decoded and previews exploring signal properties and transformations next.

Summary

<h2>Summary</h2><ul><li>Continuous-time: <b>x(t)</b> (analog signals, e.g., speech)</li><li>Discrete-time: <b>x[n]</b> (digital signals, e.g., stocks)</li></ul><blockquote><i>Key distinctions bridge theory to practice.</i></blockquote><p><b>Closing:</b> Signals decoded!<br><b>Next:</b> Explore signal properties and transformations.</p>