Die Druckversion wird nicht mehr unterstützt und kann Darstellungsfehler aufweisen. Bitte aktualisiere deine Browser-Lesezeichen und verwende stattdessen die Standard-Druckfunktion des Browsers.

Signals and Systems

Introduction

Signals and Systems is a foundational subject in electrical engineering, computer engineering, control, communications, acoustics, biomedical engineering, and applied mathematics. It gives you a precise language for describing how information-bearing quantities change and how physical or computational systems transform them. A voltage measured by an oscilloscope, a sequence of audio samples, an image row, an electrocardiogram, a vibration record, and a communication waveform can all be modeled as signals. A filter, communication channel, sensor, controller, or algorithm can be modeled as a system that maps an input signal to an output signal.

At university level, the central goal is not merely to learn formulas. You should learn to move confidently between time-domain, frequency-domain, and transform-domain descriptions; identify properties such as linearity, time invariance, causality, and stability; derive system responses; and decide which representation makes a particular problem easier.

The sine wave is especially important because complex exponentials and sinusoids are eigenfunctions of linear time-invariant systems. Their shape is preserved by an LTI system in steady-state frequency-response analysis, while amplitude and phase may change.

This aiMOOC assumes familiarity with calculus, complex numbers, basic differential equations, and elementary linear algebra. By the end, you should be able to connect mathematical models to engineering interpretations and to analyze both continuous-time and discrete-time systems.

The MIT OpenCourseWare lecture above introduces Signals and Systems through examples from several engineering fields and emphasizes analysis and design through signal-transform properties.

Learning Objectives

After working through this course, you should be able to:

Signal classification: Distinguish continuous-time and discrete-time signals and classify signals by periodicity, energy or power, symmetry, and support.
System properties: Test systems for linearity, time invariance, causality, memory, invertibility, and bounded-input bounded-output stability.
Impulse response: Explain why an LTI system is completely characterized by its impulse response under standard assumptions.
Convolution: Compute and interpret continuous-time convolution integrals and discrete-time convolution sums.
Fourier analysis: Use Fourier series and Fourier transforms to represent signals by frequency content.
Frequency response: Relate sinusoidal inputs, transfer functions, magnitude response, phase response, and Bode plots.
Laplace transform: Use poles, zeros, and regions of convergence to analyze continuous-time systems.
Z-transform: Use the z-transform to analyze discrete-time sequences and LTI systems.
Sampling theorem: Explain sampling, reconstruction, aliasing, anti-alias filtering, and quantization.
Engineering transfer: Select appropriate signal-and-system tools for communications, control, audio, sensing, and data-acquisition problems.

Signals: Mathematical Models of Information

Continuous-Time and Discrete-Time Signals

A continuous-time signal is written as $x (t)$ , where the independent variable can take values on a continuum. A discrete-time signal is written as $x [n]$ , where the independent variable is an integer index. Discrete-time does not automatically mean digital: a discrete-time signal can still have continuous-valued amplitudes. A digital signal usually has both discrete time and quantized amplitude.

Examples of continuous-time signals include microphone voltage, temperature as a function of time, and a displacement measured by a vibration sensor. Examples of discrete-time signals include daily closing prices, sampled audio, and pixel values along a scan line.

A sinusoid may be written as

$x (t) = A \cos (ω_{0} t + ϕ)$

where $A$ is amplitude, $ω_{0}$ is angular frequency in radians per second, and $ϕ$ is phase. For a discrete-time sinusoid,

$x [n] = A \cos (Ω_{0} n + ϕ)$ .

Discrete-time frequency has a special periodicity: frequencies that differ by integer multiples of $2 π$ produce the same sequence.

Elementary Signals and Signal Operations

Important building blocks include the unit impulse, unit step, exponential, sinusoid, and rectangular pulse. The continuous-time unit impulse $δ (t)$ is best understood through its sifting property, while the discrete-time impulse $δ [n]$ is an ordinary sequence equal to one at $n = 0$ and zero elsewhere.

Time shifting replaces $x (t)$ by $x (t - t_{0})$ or $x [n - n_{0}]$ . Time reversal replaces the independent variable by its negative. Time scaling, such as $x (a t)$ , compresses or expands a continuous-time waveform. Amplitude scaling multiplies the signal by a constant.

You should become comfortable decomposing complicated signals into shifted and scaled elementary signals. This is particularly useful for piecewise-defined signals and convolution.

Energy, Power, Periodicity, and Symmetry

For a continuous-time signal, energy is

$E = \int_{- \infty}^{\infty} | x (t) |^{2} d t$ .

Average power is

$P = lim_{T \to \infty} \frac{1}{2 T} \int_{- T}^{T} | x (t) |^{2} d t$ ,

when the limit exists. Discrete-time definitions replace the integral by a sum. Many finite-duration signals are energy signals, while nonzero periodic signals typically have finite average power but infinite energy.

A signal is even if $x (- t) = x (t)$ and odd if $x (- t) = - x (t)$ . Any suitable signal can be decomposed into even and odd parts:

$x_{e} (t) = \frac{x (t) + x (- t)}{2}$ , $x_{o} (t) = \frac{x (t) - x (- t)}{2}$ .

These classifications are useful because they can simplify transform calculations and reveal physical structure.

Systems and Their Properties

A system maps an input signal to an output signal. Symbolically, write $y = T {x}$ . The mathematical form might be an algebraic operation, differential equation, difference equation, convolution, state-space model, or algorithm.

Linearity and Time Invariance

A system is linear if it satisfies superposition:

$T {a x_{1} + b x_{2}} = a T {x_{1}} + b T {x_{2}}$ .

It is time invariant if delaying the input by a certain amount causes the output to be delayed by the same amount, without otherwise changing the input-output relationship.

A system that is both linear and time invariant is an LTI system. LTI models are powerful because they combine simple mathematical structure with wide engineering relevance. Many physical systems are approximately LTI over a useful operating range.

Causality, Memory, Invertibility, and Stability

A causal system cannot depend on future input values. For an LTI system, causality requires the impulse response to be zero before time zero in continuous time, or for negative indices in discrete time.

A memoryless system produces its output at a given time using only the input at that same time. A system is invertible if distinct admissible inputs can be recovered from outputs through an inverse system.

The most common stability definition in introductory signals and systems is bounded-input bounded-output stability. A system is BIBO stable if every bounded input produces a bounded output. For an LTI system, a standard necessary-and-sufficient condition is absolute integrability of the continuous-time impulse response or absolute summability of the discrete-time impulse response.

LTI Systems and Convolution

Impulse Response

The impulse response $h (t)$ or $h [n]$ is the output of an LTI system when the input is a unit impulse. Because a general signal can be represented as a weighted superposition of shifted impulses, linearity and time invariance imply that the corresponding output is a weighted superposition of shifted impulse responses.

This idea leads directly to convolution.

Continuous-Time Convolution

For a continuous-time LTI system,

$y (t) = (x * h) (t) = \int_{- \infty}^{\infty} x (τ) h (t - τ) d τ$ .

A graphical convolution method often follows four conceptual operations: keep one signal fixed, reverse the other, shift it by the output time, multiply the overlapping signals, and integrate over the dummy variable. The output changes as the overlap changes.

The MIT OpenCourseWare convolution lecture emphasizes the representation of an input as shifted impulses and the superposition of shifted impulse responses.

Discrete-Time Convolution

For a discrete-time LTI system,

$y [n] = (x * h) [n] = \sum_{k = - \infty}^{\infty} x [k] h [n - k]$ .

If both sequences have finite duration, the sum is finite for every output index. Convolution is commutative, associative, and distributive. These properties let you rearrange cascades and parallel combinations of LTI systems.

A crucial engineering interpretation is that $h$ tells you how the system spreads or shapes each elementary input contribution. A short impulse response produces local dependence in time, while a long impulse response can create long memory.

Differential Equations, Difference Equations, and System Functions

Many continuous-time physical systems are modeled by linear constant-coefficient differential equations, while discrete-time filters and sampled models are often described by linear constant-coefficient difference equations.

For example,

$a_{1} \frac{d y (t)}{d t} + a_{0} y (t) = b_{0} x (t)$

is a first-order continuous-time model. A discrete-time counterpart might be

$y [n] - a y [n - 1] = b x [n]$ .

Under zero initial conditions, transform methods convert differentiation or time shifts into algebraic factors. This makes it possible to define a system function such as $H (s) = Y (s) / X (s)$ or $H (z) = Y (z) / X (z)$ .

Fourier Series and Fourier Transform

Why Frequency-Domain Representation Matters

A time-domain waveform shows when events occur. A frequency-domain representation shows how strongly different sinusoidal or complex-exponential components contribute. Neither view is universally better; the useful view depends on the question.

For periodic continuous-time signals, the Fourier series represents the signal as a sum of harmonically related complex exponentials or sinusoids. For aperiodic signals, the Fourier transform produces a continuous frequency representation.

The square-wave illustration shows how adding odd harmonics improves a Fourier-series approximation. Near jump discontinuities, the overshoot associated with the Gibbs phenomenon does not disappear in height as more terms are added, although it becomes localized.

Fourier Transform

One common continuous-time Fourier-transform convention is

$X (j ω) = \int_{- \infty}^{\infty} x (t) e^{- j ω t} d t$

with inverse

$x (t) = \frac{1}{2 π} \int_{- \infty}^{\infty} X (j ω) e^{j ω t} d ω$ .

The Fourier transform has important properties including linearity, time shifting, frequency shifting, time scaling, differentiation, duality under appropriate conventions, and Parseval-type energy relations.

Most importantly for LTI systems, convolution in time corresponds to multiplication in frequency:

$Y (j ω) = X (j ω) H (j ω)$ .

This is one reason filtering is often easier to understand in the frequency domain.

The diagram connects major Fourier representations for continuous and discrete time. You should notice that sampling or discreteness in one domain is closely related to periodic repetition in the other.

This MIT OpenCourseWare lecture develops the Fourier transform within a full Signals and Systems course.

Frequency Response and Filtering

If an LTI system is driven by a complex exponential $e^{j ω t}$ , the output is the same exponential multiplied by $H (j ω)$ . The magnitude $| H (j ω) |$ describes gain as a function of frequency, while the phase $∠ H (j ω)$ describes phase shift.

Low-pass filters pass low frequencies more strongly than high frequencies; high-pass filters do the opposite. Band-pass and band-stop filters select or reject a frequency range.

The RC low-pass diagram combines a circuit, waveforms, phasors, and a Bode-style view. It illustrates how a physical network corresponds to a system model and a frequency response.

Bode plots use logarithmic frequency and usually express magnitude in decibels. For rational transfer functions, poles and zeros help determine asymptotic slopes, break frequencies, resonances, and phase transitions.

Frequency response gives a compact way to predict how each sinusoidal component of a signal will be modified by an LTI system.

Laplace Transform and Continuous-Time System Analysis

The bilateral Laplace transform is

$X (s) = \int_{- \infty}^{\infty} x (t) e^{- s t} d t$ ,

where $s = σ + j ω$ . Unlike the Fourier transform, the Laplace transform includes a region of convergence that is essential to the complete transform description.

For rational system functions, poles are values of $s$ where the denominator vanishes and zeros are values where the numerator vanishes. Pole locations strongly influence natural modes, transient behavior, resonance, and stability.

For causal rational continuous-time LTI systems, BIBO stability requires the region of convergence to include the imaginary axis; equivalently, under the usual causal rational assumptions, all poles must lie strictly in the left half-plane.

The MIT OpenCourseWare Laplace-transform lecture shows how transform methods connect system equations, modes, and system functions.

Z-Transform and Discrete-Time System Analysis

The bilateral z-transform of a sequence is

$X (z) = \sum_{n = - \infty}^{\infty} x [n] z^{- n}$ .

As with the Laplace transform, the region of convergence matters. The discrete-time Fourier transform can be obtained by evaluating the z-transform on the unit circle when that circle lies within the region of convergence.

For a rational causal discrete-time LTI system, BIBO stability requires all poles to lie strictly inside the unit circle. Pole-zero geometry can therefore give a rapid qualitative understanding of discrete-time behavior.

This MIT OpenCourseWare lecture connects the unit-sample response and the discrete-time system function through the z-transform.

Sampling, Aliasing, and Quantization

Sampling Theorem

Sampling converts a continuous-time signal into a discrete-time sequence by measuring it at separated instants. Ideal uniform sampling can be represented as multiplication by an impulse train.

For a continuous-time signal bandlimited to a highest frequency $f_{max}$ , perfect reconstruction from ideal uniform samples is possible when the sampling frequency satisfies

$f_{s} > 2 f_{max}$ .

The quantity $2 f_{max}$ is often called the Nyquist rate for that signal. In engineering practice, one normally samples above the theoretical minimum and uses an analog anti-alias filter before the sampler.

The animation illustrates how reconstruction improves as sample density increases relative to signal bandwidth.

The MIT OpenCourseWare sampling lecture connects continuous phenomena, discrete samples, reconstruction, and anti-aliasing.

Aliasing

If the sampling rate is too low for the signal bandwidth, different continuous-time frequencies can generate the same sample sequence. This ambiguity is aliasing. Once aliasing has occurred in the sampled data, ideal digital post-processing cannot generally determine which of the aliased analog frequencies was originally present.

This Neso Academy video gives an additional explanation of the sampling theorem and the relation between sampling rate and signal frequency.

Quantization and Digital Representation

Sampling discretizes the independent variable. Quantization discretizes amplitude. An analog-to-digital converter typically performs both operations and then encodes quantized levels as binary words.

Quantization introduces error because a continuum of possible amplitudes is mapped to a finite set of levels. Increasing the number of bits generally reduces quantization step size for a fixed full-scale range, but real converters also have noise, nonlinearity, clock jitter, bandwidth limits, and other nonidealities.

Connections Across Domains

A major learning goal in Signals and Systems is to see that different tools describe the same system from different viewpoints.

View	Core object	Typical operation	Useful question
Time domain	$x (t)$ or $x [n]$	Convolution	What happens at each time or sample?
Frequency domain	$X (j ω)$ or $X (e^{j Ω})$	Multiplication	Which frequencies are passed, attenuated, or shifted in phase?
Laplace domain	$X (s)$	Algebra with poles and zeros	What are the modes, transients, causality conditions, and stability properties?
Z domain	$X (z)$	Algebra with poles and zeros	How does a discrete-time system behave and when does its frequency response exist?

For an LTI system, these viewpoints reinforce one another. The impulse response determines convolution in time. Its transform is the system function. Evaluating that system function on the appropriate frequency axis or unit circle gives frequency response when the relevant region of convergence includes that contour.

Worked Analysis Patterns

Pattern: Testing Linearity

Given a system $T$ , start with two arbitrary inputs $x_{1}$ and $x_{2}$ and constants $a$ and $b$ . Compare $T {a x_{1} + b x_{2}}$ with $a T {x_{1}} + b T {x_{2}}$ . A single counterexample proves nonlinearity. Do not confuse time invariance with linearity; they are independent properties.

Pattern: Testing Time Invariance

Compute the response to a delayed input and compare it with a delayed version of the original output. If $T {x (t - t_{0})} = y (t - t_{0})$ for all relevant inputs and delays, the system is time invariant. A coefficient such as $t$ multiplying the input often signals explicit time dependence and therefore time variance.

Pattern: Finding Output by Convolution

First identify the support of $x$ and $h$ . Then determine for which shifts the two signals overlap. For piecewise functions, this often produces a piecewise output. Use symmetry and support before integrating; these qualitative observations reduce algebra and provide a check on the result.

Pattern: Using a Transfer Function

For a rational transfer function, factor numerator and denominator, identify poles and zeros, and state the region of convergence when needed. Then connect pole-zero locations to causality and stability. If the frequency response exists, evaluate the system function on the frequency axis or unit circle and interpret magnitude and phase.

Pattern: Checking a Sampling Plan

Determine the highest frequency that must be preserved, not merely the most visually obvious frequency. Include modulation products and noise bandwidth when relevant. Select an anti-alias filter and a sampling rate with practical margin. Then consider quantization resolution, dynamic range, clock quality, and reconstruction requirements.

Applications

Signals and Systems provides a common framework across many engineering fields.

Communication systems: Modulation, channels, filtering, equalization, and bandwidth are naturally described in frequency and transform domains.
Control systems: Poles, zeros, feedback, stability, and transient response connect system models to physical behavior.
Audio signal processing: Sampling, filtering, convolution, spectra, and reconstruction appear in recording, synthesis, effects, and compression.
Biomedical engineering: ECG, EEG, imaging, and sensor signals require careful filtering, sampling, and interpretation.
Mechanical vibration: Modal frequencies, resonance, damping, and sensor responses can be modeled with LTI concepts.
Computer engineering: Discrete-time systems, difference equations, digital filters, and transforms support real-time embedded processing.
Image processing: Two-dimensional extensions of convolution and Fourier analysis underpin blur, sharpening, denoising, and imaging systems.

Common Misconceptions

Discrete-time is not identical to digital. A sampled sequence may still have continuous-valued amplitudes.

A high sampling rate does not remove aliasing that occurred before sampling. The anti-alias decision belongs before digitization.

A transfer-function formula alone is not always a complete transform description. The region of convergence can distinguish signals or systems with the same algebraic ratio.

Linearity and time invariance are separate tests. A system may have one property without the other.

Convolution is not ordinary pointwise multiplication. In time-domain LTI analysis, convolution combines shifted versions across the entire support.

Stability depends on the system model and its region of convergence. Pole locations must be interpreted together with causality or ROC assumptions.

Interactive Tasks

Quiz: Test Your Knowledge

What property allows scaled input responses to be added for a linear system? (Superposition) (!Periodicity) (!Quantization) (!Aliasing)

Which object completely characterizes an LTI system under standard assumptions? (Impulse response) (!Sampling period) (!Carrier frequency) (!Quantization level)

What operation gives the output of an LTI system from its input and impulse response? (Convolution) (!Differentiation) (!Quantization) (!Interpolation)

What does the magnitude of a frequency response describe? (Frequency dependent gain) (!Time delay only) (!Sample index) (!Input duration)

What phenomenon occurs when distinct continuous frequencies become indistinguishable after undersampling? (Aliasing) (!Linearity) (!Causality) (!Superposition)

For a causal rational continuous time system which pole location is consistent with BIBO stability? (Left half plane) (!Right half plane) (!Imaginary axis) (!Everywhere equally)

For a causal rational discrete time system where must stable poles lie? (Inside unit circle) (!Outside unit circle) (!Only at origin) (!On unit circle)

Which transform is especially associated with discrete time pole zero analysis? (Z transform) (!Cosine rule) (!Gradient theorem) (!Taylor test)

What operation in the frequency domain corresponds to convolution in time? (Multiplication) (!Integration) (!Quantization) (!Sorting)

What should usually precede an analog to digital converter to limit out of band content? (Anti alias filter) (!Power amplifier) (!Integrator only) (!Phase detector)

Memory Game

Impulse response	Output produced by a unit impulse in an LTI system
Linearity	Property defined by additivity and homogeneity
Causality	Dependence only on present and past input values
Convolution	Time domain operation that combines an input with an impulse response
Spectrum	Representation of signal content across frequency
Aliasing	Frequency ambiguity created by insufficient sampling
Quantization	Mapping amplitudes to a finite set of levels

Drag and Drop

Match the correct terms.	Topic
Impulse response	Complete time domain description of an LTI system
Frequency response	Gain and phase as functions of frequency
Laplace transform	Continuous time transform using a complex frequency variable
Z transform	Discrete time transform using a complex variable
Sampling theorem	Condition for ideal reconstruction of a bandlimited signal

...

Crossword Puzzle

Linearity	Which system property is defined by additivity and homogeneity?
Causality	Which property forbids dependence on future input values?
Convolution	Which operation combines an input with an LTI impulse response?
Spectrum	What one word names a representation of content across frequency?
Aliasing	What distortion makes different analog frequencies produce indistinguishable samples?
Stability	Which property requires bounded inputs to produce bounded outputs under the BIBO definition?

LearningApps

Cloze Text

Open-Ended Tasks

Easy

Signal photo essay: Find four real-world devices that measure or process signals, photograph or sketch them, and explain what each input and output represents.
Waveform notebook: Create a one-page illustrated reference showing a sinusoid, unit step, impulse concept, and rectangular pulse with labels for amplitude, time shift, and duration.
System property examples: Invent one simple system for each of linearity, nonlinearity, time invariance, and time variance, then justify your classifications in clear mathematical language.
Engineering interview: Interview a student, technician, engineer, musician, or researcher about one practical use of filtering or sampling and summarize the connection to this course.

Standard

Convolution experiment: Use a spreadsheet, programming language, or graphing tool to convolve two finite discrete sequences and compare the numerical result with a hand calculation.
Audio spectrum study: Record a short noncopyrighted sound, inspect its waveform and spectrum with suitable software, and explain which time-domain and frequency-domain features correspond.
Sampling video: Produce a short explanatory video that demonstrates how a sinusoid can be sampled correctly and incorrectly, including a visual example of aliasing.
Laboratory visit: Visit an electronics, acoustics, robotics, telecommunications, or measurement laboratory and document three examples of signals, systems, sensors, filters, or data converters.

Advanced

Filter design project: Design and simulate an analog or digital low-pass filter for a stated specification, justify your architecture, and compare predicted and simulated magnitude and phase responses.
Pole zero investigation: Choose two rational system functions with contrasting pole locations, analyze causality and stability assumptions, and verify their time-domain behavior computationally.
Sampling chain design: Propose a complete acquisition chain for a physical sensor, including anti-alias filtering, sampling rate, quantization resolution, expected bandwidth, and reconstruction or digital-processing needs.
Research replication: Select an open technical paper or textbook example involving convolution, Fourier analysis, sampling, or system identification, reproduce one result, and explain assumptions, limitations, and discrepancies.

Learning Assessment

Model selection assessment: Given three descriptions of a physical process, decide whether a continuous-time, discrete-time, or hybrid model is most appropriate and defend each choice.
LTI reasoning assessment: Analyze a proposed system for linearity, time invariance, causality, and BIBO stability, showing either a proof or a counterexample for each property.
Convolution transfer assessment: Predict the shape and support of a convolution before calculating it, then compare the qualitative prediction with the exact result.
Frequency domain assessment: Given a signal spectrum and a filter response, determine the output spectrum and explain what changes you expect in the time-domain waveform.
Transform domain assessment: Interpret a rational system function through poles, zeros, and region of convergence, then infer consequences for causality, stability, and natural modes.
Sampling design assessment: Evaluate a proposed sampling and quantization plan for a sensor system, identify failure risks, and recommend technically justified improvements.

Evidence of Learning

Knowledge: You can explain signal classes, system properties, impulse response, convolution, Fourier representations, frequency response, Laplace and z transforms, sampling, aliasing, and quantization using correct technical vocabulary.

Analytical skills: You can test system properties, perform convolution, manipulate transform representations, interpret poles and zeros, and connect time-domain and frequency-domain conclusions.

Practical products: Your evidence may include annotated waveforms, simulations, filter plots, recorded measurements, code or spreadsheets, laboratory notes, explanatory videos, and short technical reports.

Reasoning: You can state assumptions, select an appropriate representation, check units and limiting cases, compare analytic and numerical results, and identify when a mathematical model is only an approximation.

Transfer: You can apply Signals and Systems ideas to unfamiliar contexts such as communications, sensing, control, audio, biomedical data, vibration, image processing, or embedded systems.

OERs on the Topic

A comprehensive open university-level reference is available through MIT OpenCourseWare: MIT 6.003 Signals and Systems. Another open MIT course by Alan V. Oppenheim provides a broad treatment of analog and digital signals and systems: Signals and Systems.

Linked Learning Areas

Signals and Systems

Key Connections for Further Study

Digital signal processing: Extends discrete-time analysis toward practical filter design, spectral estimation, multirate processing, and implementation.
Communication theory: Uses modulation, filtering, sampling, and spectra to represent information transfer through channels.
Control theory: Builds on system functions, poles, zeros, feedback, stability, and transient response.
Probability and random processes: Generalizes deterministic signal concepts to noise, stochastic signals, correlation, and power spectral density.
Linear algebra: Provides the language of vector spaces, eigenfunctions, state-space systems, and numerical signal representations.
Differential equations: Connects physical dynamics to continuous-time system models and natural modes.
Complex analysis: Deepens understanding of contour behavior, poles, zeros, transforms, and regions of convergence.
Data acquisition: Connects sensors, anti-alias filters, converters, sampling clocks, quantization, and digital processing.