Signals, Systems and Inference

Prentice Hall
Alan V. Oppenheim / George C. Verghese  
Total pages
April 2015
Related Titles


For upper-level undergraduate courses in deterministic and stochastic signals and system engineering

An Integrative Approach to Signals, Systems and Inference

Signals, Systems and Inference is a comprehensive text that builds on introductory courses in time- and frequency-domain analysis of signals and systems, and in probability. Directed primarily to upper-level undergraduates and beginning graduate students in engineering and applied science branches, this new textbook pioneers a novel course of study. Instead of the usual leap from broad introductory subjects to highly specialized advanced subjects, this engaging and inclusive text creates a study track for a transitional course.

Properties and representations of deterministic signals and systems are reviewed and elaborated on, including group delay and the structure and behavior of state-space models. The text also introduces and interprets correlation functions and power spectral densities for describing and processing random signals. Application contexts include pulse amplitude modulation, observer-based feedback control, optimum linear filters for minimum mean-square-error estimation, and matched filtering for signal detection. Model-based approaches to inference are emphasized, in particular for state estimation, signal estimation, and signal detection.

The text explores ideas, methods and tools common to numerous fields involving signals, systems and inference: signal processing, control, communication, time-series analysis, financial engineering, biomedicine, and many others. Signals, Systems and Inference is a long-awaited and flexible text that can be used for a rigorous course in a broad range of engineering and applied science curricula.



Signals, Systems and Inference facilitates learning with the following features.


A text structure that is highly organized and easy to navigate

  • The text is divided into four major parts:
    • Chapters 1-3 present a review of the assumed prerequisite notions in signals and systems, and apply these to digital communication by pulse amplitude modulation.
    • Chapters 4-6 treat space-state models,
      • concentrating on the single-input single-output LTI case;
      • introducing the idea of model-based inference;
      • examining associated feedback control strategies.
    • Chapters 7-9 provide a review of assumed prerequisites in probability, including estimation and hypothesis testing for static random variables.
    • Chapters 10-13 explore wide-sense stationary random signals and their processing by LTI systems for various applications.
      • The properties and interpretations of correlation functions and power spectral densities are developed in Chapters 10-11,and used in the remaining chapters to study canonical inference problems in signal estimation and signal detection.
      • Chapter 12 focuses on Wiener filtering, or linear minimum mean square error signal estimation.
      • Chapter 13 emphasizes signal detection problems for which the optimum solutions involve matched filtering.


Thorough and interesting chapters full of information

  • An exploration of fundamental material in an interesting and engaging manner.
  • Further Reading sections at the end of each chapter help students gain further knowledge of the subject matter.
  • Basic, Advanced, and Extension problems that review chapter material and ask the students to test and apply their knowledge of the subject.


A flexible approach to a broad course of study

  • Since there is more material in this text than can comfortably be taught in a one-semester course, the text allows for different routes of instruction that emphasize various paths of study.
    • Chapters 4-6 can be omitted or only briefly addressed in courses oriented towards communication and signal processing.
    • Chapters 3, 9 and 13 can be considered optional for courses with more of a control orientation.

A course that includes core material from every chapter can be taught with two weekly lectures and associated small group discussions over an approximately 13-week semester.

Table of Contents


The Cover




1. Signals and Systems

1.1 Signals, Systems, Models, and Properties

            1.1.1 System Properties

1.2 Linear, Time-Invariant Systems

            1.2.1 Impulse-Response Representation of LTI Systems

            1.2.2 Eigenfunction and Transform Representation of LTI Systems

            1.2.3 Fourier Transforms

1.3 Deterministic Signals and Their Fourier Transforms

            1.3.1 Signal Classes and Their Fourier Transforms

            1.3.2 Parseval’s Identity, Energy Spectral Density, and Deterministic Autocorrelation

1.4 Bilateral Laplace and Z-Transforms

            1.4.1 The Bilateral z-Transform

            1.4.2 The Bilateral Laplace Transform

1.5 Discrete-Time Processing of Continuous-Time Signals

            1.5.1 Basic Structure for DT Processing of CT Signals

            1.5.2 DT Filtering and Overall CT Response

            1.5.3 Nonideal D/C Converters

1.6 Further Reading


            Basic Problems

            Advanced Problems

            Extension Problems


2. Amplitude, Phase, and Group Delay

2.1 Fourier Transform Magnitude and Phase

2.2 Group Delay and the Effect of Nonlinear Phase

            2.2.1 Narrowband Input Signals

            2.2.2 Broadband Input Signals

2.3 All-Pass and Minimum-Phase Systems

            2.3.1 All-Pass Systems

            2.3.2 Minimum-Phase Systems

            2.3.3 The Group Delay of Minimum-Phase Systems

2.4 Spectral Factorization

2.5 Further Reading


            Basic Problems

            Advanced Problems

            Extension Problems


3. Pulse-Amplitude Modulation

3.1 Baseband Pulse-Amplitude Modulation

            3.1.1 The Transmitted Signal

            3.1.2 The Received Signal

            3.1.3 Frequency-Domain Characterizations

            3.1.4 Intersymbol Interference at the Receiver

3.2 Nyquist Pulses

3.3 Passband Pulse-Amplitude Modulation

            3.3.1 Frequency-Shift Keying (FSK)

            3.3.2 Phase-Shift Keying (PSK)

            3.3.3 Quadrature-Amplitude Modulation (QAM)

3.4 Further Reading


            Basic Problems

            Advanced Problems

            Extension Problems


4. State-Space Models

4.1 System Memory

4.2 Illustrative Examples

4.3 State-Space Models

            4.3.1 DT State-Space Models

            4.3.2 CT State-Space Models

            4.3.3 Defining Properties of State-Space Models

4.4 State-Space Models from LTI Input-Output Models

4.5 Equilibria and Linearization of Nonlinear State-Space Models

            4.5.1 Equilibrium

            4.5.2 Linearization

4.6 Further Reading


            Basic Problems

            Advanced Problems

            Extension Problems


5. LTI State-Space Models

5.1 Continuous-Time and Discrete-Time LTI Models

5.2 Zero-Input Response and Modal Representation

            5.2.1 Undriven CT Systems

            5.2.2 Undriven DT Systems

            5.2.3 Asymptotic Stability of LTI Systems

5.3 General Response in Modal Coordinates

            5.3.1 Driven CT Systems

            5.3.2 Driven DT Systems

            5.3.3 Similarity Transformations and Diagonalization

5.4 Transfer Functions, Hidden Modes, Reachability, and Observability

            5.4.1 Input-State-Output Structure of CT Systems

            5.4.2 Input-State-Output Structure of DT Systems

5.5 Further Reading


            Basic Problems

            Advanced Problems

            Extension Problems


6. State Observers and State Feedback

6.1 Plant and Model

6.2 State Estimation and Observers

            6.2.1 Real-Time Simulation

            6.2.2 The State Observer

            6.2.3 Observer Design

6.3 State Feedback Control

6.3.1 Open-Loop Control

            6.3.2 Closed-Loop Control via LTI State Feedback

            6.3.3 LTI State Feedback Design

6.4 Observer-Based Feedback Control

6.5 Further Reading


            Basic Problems

            Advanced Problems

            Extension Problems


7. Probabilistic Models

7.1 The Basic Probability Model

7.2 Conditional Probability, Bayes’ Rule, and Independence

7.3 Random Variables

7.4 Probability Distributions

7.5 Jointly Distributed Random Variables

7.6 Expectations, Moments, and Variance

7.7 Correlation and Covariance for Bivariate Random Variables

7.8 A Vector-Space Interpretation of Correlation Properties

7.9 Further Reading


            Basic Problems

            Advanced Problems

            Extension Problems


8. Estimation

8.1 Estimation of a Continuous Random Variable

8.2 From Estimates to the Estimator

            8.2.1 Orthogonality

8.3 Linear Minimum Mean Square Error Estimation

            8.3.1 Linear Estimation of One Random Variable from a Single Measurement of Another

            8.3.2 Multiple Measurements

8.4 Further Reading


            Basic Problems

            Advanced Problems

            Extension Problems


9. Hypothesis Testing

9.1 Binary Pulse-Amplitude Modulation in Noise

9.2 Hypothesis Testing with Minimum Error Probability

            9.2.1 Deciding with Minimum Conditional Probability of Error

            9.2.2 MAP Decision Rule for Minimum Overall Probability of Error

            9.2.3 Hypothesis Testing in Coded Digital Communication

9.3 Binary Hypothesis Testing

            9.3.1 False Alarm, Miss, and Detection

            9.3.2 The Likelihood Ratio Test

            9.3.3 Neyman-Pearson Decision Rule and Receiver Operating Characteristic

9.4 Minimum Risk Decisions

9.5 Further Reading


            Basic Problems

            Advanced Problems

            Extension Problems


10. Random Processes

10.1 Definition and Examples of a Random Process

10.2 First- and Second-Moment Characterization of Random Processes

10.3 Stationarity

            10.3.1 Strict-Sense Stationarity

            10.3.2 Wide-Sense Stationarity

            10.3.3 Some Properties of WSS Correlation and Covariance Functions

10.4 Ergodicity

10.5 Linear Estimation of Random Processes

            10.5.1 Linear Prediction

            10.5.2 Linear FIR Filtering

10.6 LTI Filtering of WSS Processes

10.7 Further Reading


            Basic Problems

            Advanced Problems

            Extension Problems


11. Power Spectral Density

11.1 Spectral Distribution of Expected Instantaneous Power

            11.1.1 Power Spectral Density

            11.1.2 Fluctuation Spectral Density

            11.1.3 Cross-Spectral Density

11.2 Expected Time-Averaged Power Spectrum and the Einstein-Wiener-Khinchin Theorem

11.3 Applications

            11.3.1 Revealing Cyclic Components

            11.3.2 Modeling Filters

            11.3.3 Whitening Filters

            11.3.4 Sampling Bandlimited Random Processes

11.4 Further Reading


            Basic Problems

            Advanced Problems

            Extension Problems


12. Signal Estimation

12.1 LMMSE Estimation for Random Variables

12.2 FIR Wiener Filters

12.3 The Unconstrained DT Wiener Filter

12.4 Causal DT Wiener Filtering

12.5 Optimal Observers and Kalman Filtering

            12.5.1 Causal Wiener Filtering of a Signal Corrupted by Additive Noise

            12.5.2 Observer Implementation of the Wiener Filter

            12.5.3 Optimal State Estimates and Kalman Filtering

12.6 Estimation of CT Signals

12.7 Further Reading


            Basic Problems

            Advanced Problems

            Extension Problems


13. Signal Detection

13.1 Hypothesis Testing with Multiple Measurements

13.2 Detecting a Known Signal in I.I.D. Gaussian Noise

            13.2.1 The Optimal Solution

            13.2.2 Characterizing Performance

            13.2.3 Matched Filtering

13.3 Extensions of Matched-Filter Detection

            13.3.1 Infinite-Duration, Finite-Energy Signals

            13.3.2 Maximizing SNR for Signal Detection in White Noise

            13.3.3 Detection in Colored Noise

            13.3.4 Continuous-Time Matched Filters

            13.3.5 Matched Filtering and Nyquist Pulse Design

            13.3.6 Unknown Arrival Time and Pulse Compression

13.4 Signal Discrimination in I.I.D. Gaussian Noise

13.5 Further Reading


            Basic Problems

            Advanced Problems

            Extension Problems