Intro to Electrical Engineering

🔌Intro to Electrical Engineering Unit 18 – Continuous-Time Signals & Systems

Continuous-time signals and systems form the foundation of electrical engineering. This unit explores how signals represent physical quantities over time and how systems process these signals. We'll dive into signal types, time and frequency domain analysis, and key system properties. Understanding these concepts is crucial for analyzing and designing electrical systems. We'll cover Fourier analysis, convolution, and applications in signal processing, communications, and control systems. These tools enable engineers to manipulate and interpret signals in various real-world scenarios.

Key Concepts and Definitions

  • Signals represent physical quantities that vary over time (voltage, current, temperature)
  • Systems process input signals to produce output signals based on their characteristics
    • Linear systems satisfy the properties of superposition and homogeneity
    • Time-invariant systems produce the same output for a given input regardless of the time shift
  • Continuous-time signals are defined for all real values of time and are represented by functions of a continuous variable
  • Discrete-time signals are defined only at discrete time instants and are represented by sequences of numbers
  • Periodic signals repeat their values at regular intervals (sinusoidal waves)
  • Aperiodic signals do not exhibit repeating patterns (exponential decay)
  • Energy signals have finite energy and are square-integrable (transient signals)
  • Power signals have finite average power over an infinite time interval (periodic signals)

Signal Types and Properties

  • Deterministic signals can be described by mathematical functions and have no uncertainty in their values (sinusoids, exponentials)
  • Random signals exhibit unpredictable behavior and are characterized by probability distributions (noise)
  • Even signals exhibit symmetry about the vertical axis, satisfying f(t)=f(t)f(-t) = f(t) (cosine function)
    • The Fourier series of an even signal contains only cosine terms
  • Odd signals exhibit symmetry about the origin, satisfying f(t)=f(t)f(-t) = -f(t) (sine function)
    • The Fourier series of an odd signal contains only sine terms
  • Causal signals are zero for negative time instants and have a starting point (unit step function)
  • Non-causal signals have non-zero values for negative time instants (sinc function)
  • Stable signals remain bounded for all time instants when the input is bounded (decaying exponential)
  • Unstable signals grow without bound for a bounded input (growing exponential)

Time-Domain Analysis

  • Time-domain analysis involves examining signals as functions of time
  • The unit impulse function δ(t)\delta(t) is a fundamental signal used in time-domain analysis
    • It represents an infinitely short pulse with unit area centered at t=0t=0
    • Mathematically, it is defined as δ(t)dt=1\int_{-\infty}^{\infty} \delta(t) dt = 1
  • The unit step function u(t)u(t) represents a signal that switches from 0 to 1 at t=0t=0
    • It is related to the unit impulse by u(t)=tδ(τ)dτu(t) = \int_{-\infty}^{t} \delta(\tau) d\tau
  • Signal shifting involves delaying or advancing a signal in time
    • A right-shifted signal is represented as f(tt0)f(t-t_0), where t0t_0 is the delay
    • A left-shifted signal is represented as f(t+t0)f(t+t_0), where t0t_0 is the advance
  • Signal scaling involves multiplying a signal by a constant factor
    • Amplitude scaling changes the signal's magnitude without affecting its shape
    • Time scaling compresses or expands the signal along the time axis

Frequency-Domain Analysis

  • Frequency-domain analysis involves examining signals as functions of frequency
  • The Fourier transform decomposes a signal into its frequency components
    • It maps a time-domain signal to its frequency-domain representation
    • The forward Fourier transform is given by X(jω)=x(t)ejωtdtX(j\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt
    • The inverse Fourier transform recovers the time-domain signal from its frequency-domain representation
  • The spectrum of a signal represents its frequency content
    • The magnitude spectrum shows the amplitude of each frequency component
    • The phase spectrum shows the phase shift of each frequency component
  • Bandwidth refers to the range of frequencies present in a signal
    • Narrowband signals have a small range of frequencies (sinusoidal signal)
    • Wideband signals have a large range of frequencies (square wave)
  • Filtering involves selectively attenuating or amplifying specific frequency components of a signal
    • Low-pass filters allow low frequencies to pass while attenuating high frequencies
    • High-pass filters allow high frequencies to pass while attenuating low frequencies
    • Band-pass filters allow a specific range of frequencies to pass while attenuating others

Fourier Series and Transforms

  • Fourier series represent periodic signals as a sum of sinusoidal components
    • The Fourier series coefficients determine the amplitude and phase of each component
    • The fundamental frequency is the lowest frequency in the series and is related to the signal's period
  • The Fourier transform extends the concept of Fourier series to aperiodic signals
    • It decomposes a signal into a continuous spectrum of frequencies
    • The forward Fourier transform maps a time-domain signal to its frequency-domain representation
    • The inverse Fourier transform recovers the time-domain signal from its frequency-domain representation
  • The Discrete-Time Fourier Transform (DTFT) is used for discrete-time signals
    • It represents a discrete-time signal as a continuous function of frequency
    • The DTFT is periodic with a period of 2π2\pi
  • The Discrete Fourier Transform (DFT) is a sampled version of the DTFT
    • It represents a finite-length discrete-time signal as a finite sequence of frequency components
    • The Fast Fourier Transform (FFT) is an efficient algorithm for computing the DFT

System Characteristics and Properties

  • Linearity is a property of systems where the output is proportional to the input
    • For a linear system, the response to a sum of inputs is equal to the sum of the responses to each input individually
    • Mathematically, if y1(t)y_1(t) is the response to x1(t)x_1(t) and y2(t)y_2(t) is the response to x2(t)x_2(t), then the response to a1x1(t)+a2x2(t)a_1x_1(t) + a_2x_2(t) is a1y1(t)+a2y2(t)a_1y_1(t) + a_2y_2(t)
  • Time-invariance means that the system's response does not depend on the absolute time
    • Shifting the input in time results in an equivalent shift in the output
    • Mathematically, if y(t)y(t) is the response to x(t)x(t), then the response to x(tt0)x(t-t_0) is y(tt0)y(t-t_0)
  • Causality implies that the system's output depends only on the current and past inputs
    • A causal system cannot respond to future inputs
    • Mathematically, if x1(t)=x2(t)x_1(t) = x_2(t) for all tt0t \leq t_0, then y1(t)=y2(t)y_1(t) = y_2(t) for all tt0t \leq t_0
  • Stability ensures that the system's output remains bounded for bounded inputs
    • A stable system's response to a bounded input is also bounded
    • Mathematically, if x(t)Mx|x(t)| \leq M_x for all tt, then there exists a constant MyM_y such that y(t)My|y(t)| \leq M_y for all tt

Convolution and System Response

  • Convolution is a mathematical operation that describes the relationship between the input and output of a linear time-invariant (LTI) system
    • It is denoted by the symbol * and is defined as y(t)=(xh)(t)=x(τ)h(tτ)dτy(t) = (x * h)(t) = \int_{-\infty}^{\infty} x(\tau)h(t-\tau) d\tau
    • The output y(t)y(t) is obtained by convolving the input x(t)x(t) with the system's impulse response h(t)h(t)
  • The impulse response characterizes the system's behavior and is the output when the input is a unit impulse δ(t)\delta(t)
    • It represents the system's response to an infinitesimally short input
    • The impulse response fully describes an LTI system
  • The convolution integral can be interpreted as a weighted sum of scaled and shifted impulse responses
    • Each input value is multiplied by the impulse response shifted by the corresponding time instant
    • The output is the sum of these scaled and shifted impulse responses
  • Convolution in the time domain corresponds to multiplication in the frequency domain
    • The Fourier transform of the convolution of two signals is equal to the product of their Fourier transforms
    • This property simplifies the analysis of LTI systems in the frequency domain

Applications in Electrical Engineering

  • Signal processing techniques are used to analyze, modify, and extract information from signals
    • Filtering removes unwanted frequency components (noise reduction, signal enhancement)
    • Modulation encodes information onto a carrier signal for transmission (AM, FM, digital modulation)
    • Demodulation recovers the original information from the modulated signal
  • Communication systems rely on signal processing to transmit information efficiently and reliably
    • Analog communication systems use continuous-time signals (radio, television)
    • Digital communication systems use discrete-time signals (mobile phones, internet)
    • Modulation and demodulation techniques are employed to convert between baseband and passband signals
  • Control systems use feedback to regulate and stabilize the behavior of dynamic systems
    • The system's output is compared with a desired reference, and the error signal is used to adjust the input
    • Transfer functions describe the input-output relationship of control systems in the frequency domain
    • Stability analysis ensures that the closed-loop system remains stable and responsive
  • Audio and speech processing involve the analysis, synthesis, and manipulation of acoustic signals
    • Fourier analysis is used to identify the frequency components of audio signals
    • Filtering techniques are employed to remove noise, enhance specific frequencies, or create audio effects
    • Speech recognition systems use signal processing to extract features and classify speech patterns


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© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.