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

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 $f(-t) = f(t)$ (cosine function)
  • The Fourier series of an even signal contains only cosine terms
• Odd signals exhibit symmetry about the origin, satisfying $f(-t) = -f(t)$ (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 $\delta(t)$ is a fundamental signal used in time-domain analysis
  • It represents an infinitely short pulse with unit area centered at $t=0$
  • Mathematically, it is defined as $\int_{-\infty}^{\infty} \delta(t) dt = 1$
• The unit step function $u(t)$ represents a signal that switches from 0 to 1 at $t=0$
  • It is related to the unit impulse by $u(t) = \int_{-\infty}^{t} \delta(\tau) d\tau$
• Signal shifting involves delaying or advancing a signal in time
  • A right-shifted signal is represented as $f(t-t_0)$, where $t_0$ is the delay
  • A left-shifted signal is represented as $f(t+t_0)$, where $t_0$ 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 $X(j\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt$
  • 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 $2\pi$
• 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 $y_1(t)$ is the response to $x_1(t)$ and $y_2(t)$ is the response to $x_2(t)$, then the response to $a_1x_1(t) + a_2x_2(t)$ is $a_1y_1(t) + a_2y_2(t)$
• 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 $y(t)$ is the response to $x(t)$, then the response to $x(t-t_0)$ is $y(t-t_0)$
• 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 $x_1(t) = x_2(t)$ for all $t \leq t_0$, then $y_1(t) = y_2(t)$ for all $t \leq t_0$
• 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 $|x(t)| \leq M_x$ for all $t$, then there exists a constant $M_y$ such that $|y(t)| \leq M_y$ for all $t$

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 $y(t) = (x * h)(t) = \int_{-\infty}^{\infty} x(\tau)h(t-\tau) d\tau$
  • The output $y(t)$ is obtained by convolving the input $x(t)$ with the system's impulse response $h(t)$
• The impulse response characterizes the system's behavior and is the output when the input is a unit impulse $\delta(t)$
  • 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