Signals and Systems

1. Signals Carry Information: The Foundation of Systems

Anything that carries information can be called a signal.

Signals are fundamental. Signals are the lifeblood of any system, carrying the information that drives its behavior. They are single-valued functions of one or more independent variables, such as time, temperature, or position. Understanding signals is crucial for analyzing and designing systems that process and respond to information.

Continuous vs. Discrete. Signals can be broadly classified into continuous-time signals, defined for all instants of time, and discrete-time signals, defined only at discrete instants. Continuous-time signals are often referred to as analog signals, while discrete-time signals are the foundation of digital systems. Human speech, electrical current, and voltage are examples of signals.

Signal Dimensions. Signals can be one-dimensional, depending on a single independent variable, or multi-dimensional, depending on multiple independent variables. This book primarily focuses on one-dimensional signals, providing a foundation for understanding more complex systems.

2. Discrete-Time Signals: Representation and Manipulation

There are four ways of representing discrete-time signals: Graphical representation, Functional representation, Tabular representation, Sequence representation.

Representing discrete data. Discrete-time signals, unlike their continuous counterparts, are defined only at specific points in time. This requires specialized methods for representation, including graphical, functional, tabular, and sequence-based approaches. Each method offers a unique way to visualize and analyze the signal's behavior.

Graphical representation provides a visual depiction of the signal's amplitude at each discrete time point. Functional representation expresses the signal's amplitude as a mathematical function of the discrete time variable 'n'. Tabular representation organizes the signal's values in a table, pairing each time point with its corresponding amplitude. Sequence representation lists the signal's amplitudes in a specific order, often with an arrow indicating the n=0 term.

Operations on discrete signals. Discrete-time signals can be manipulated through operations like summation, multiplication, and scaling. These operations are performed element-wise, combining or modifying the signal's amplitudes at each time point. Understanding these operations is essential for processing and transforming discrete-time signals in various applications.

3. Elementary Signals: Building Blocks of Complex Systems

There are several elementary signals which play vital role in the study of signals and systems.

Standard signals. Elementary signals serve as the fundamental building blocks for constructing more complex signals. These standard signals, such as the unit step, ramp, impulse, and sinusoidal functions, can be combined and manipulated to model a wide range of physical phenomena.

Key elementary signals:

• Unit step function: Represents a sudden change in amplitude
• Unit ramp function: Increases linearly with time
• Unit parabolic function: Increases quadratically with time
• Unit impulse function: Represents an instantaneous burst of energy
• Sinusoidal function: Represents periodic oscillations
• Real exponential function: Represents exponential growth or decay
• Complex exponential function: Represents oscillations with changing amplitude

Modeling physical signals. By understanding the properties and characteristics of these elementary signals, engineers can effectively model and analyze complex systems. These signals are also used as test signals to understand how systems respond to different types of inputs.

4. Systems Defined: Transforming Inputs into Outputs

A system is defined as an entity that acts on an input signal and transforms it into an output signal.

Systems process signals. A system takes an input signal, processes it according to its internal characteristics, and produces an output signal. This transformation can involve amplification, filtering, modulation, or any other operation that modifies the signal's information content.

Input-output relationship. The relationship between the input x(t) and the output y(t) of a system is defined by the system's transfer function. This function describes how the system modifies the frequency components of the input signal to produce the output signal.

System examples. Physical devices like amplifiers, filters, motors, and turbines are all examples of systems. Each system has a unique transfer function that determines its specific behavior and how it processes input signals.

5. Classifying Systems: Understanding System Behavior

Classification of the systems is covered in Chapter 2 along with the numerical examples on determination of the type of a given system.

Categorizing systems. Systems can be classified based on their properties, such as linearity, time invariance, causality, and stability. These classifications help engineers understand and predict how a system will behave under different conditions.

Key system classifications:

• Continuous-time vs. Discrete-time: Based on the nature of the input and output signals
• Linear vs. Non-linear: Based on whether the system obeys the superposition principle
• Time-invariant vs. Time-varying: Based on whether the system's characteristics change over time
• Causal vs. Non-causal: Based on whether the system's output depends on future inputs
• Stable vs. Unstable: Based on whether the system produces a bounded output for every bounded input

Understanding system types. By classifying systems according to these properties, engineers can select the appropriate analysis techniques and design systems that meet specific performance requirements. For example, a stable, linear time-invariant (LTI) system is often preferred for signal processing applications due to its predictable and well-understood behavior.

6. Linear Time-Invariant (LTI) Systems: Properties and Analysis

The filter characteristics of linear time invariant systems, distortionless transmission through linear time invariant systems, signal bandwidth and system bandwidth, various types of filters and time domain and frequency domain criterion for physical realizability are described in Chapter 6.

LTI systems are fundamental. Linear Time-Invariant (LTI) systems are a cornerstone of signal processing and system analysis. Their behavior is governed by the principles of linearity and time invariance, making them predictable and easier to analyze.

Key properties of LTI systems:

• Commutativity: The order of input and impulse response doesn't affect the output.
• Distributivity: The response to a sum of inputs is the sum of the responses to each input.
• Associativity: Cascading two LTI systems is equivalent to a single LTI system.
• Stability: Bounded inputs produce bounded outputs.
• Causality: The output depends only on present and past inputs.

Impulse response. The impulse response, h(t), is the system's output when the input is a unit impulse. It completely characterizes the behavior of an LTI system. The output for any arbitrary input can be found by convolving the input with the impulse response.

7. Signal Transmission: Distortion, Bandwidth, and Filters

Transmission of signals through linear systems is very important.

Signal fidelity. When a signal passes through a system, its shape can be altered, leading to distortion. Understanding the conditions for distortionless transmission is crucial for preserving signal integrity.

Key concepts in signal transmission:

• Distortionless transmission: The output is an exact replica of the input, possibly with a change in amplitude and time delay.
• Bandwidth: The range of frequencies that contain most of the signal's energy.
• Filters: Frequency-selective networks that allow certain frequencies to pass while attenuating others.

Ideal filters. Ideal filters have sharp cutoff characteristics, perfectly passing signals within their passband and completely rejecting signals outside it. However, ideal filters are non-causal and physically unrealizable. Practical filters approximate these characteristics with varying degrees of accuracy.

8. Convolution and Correlation: Comparing and Combining Signals

Convolution is a mathematical way of combining two signals to form a third signal.

Convolution combines signals. Convolution is a mathematical operation that combines two signals to produce a third signal, representing the system's output. It is a fundamental tool for analyzing LTI systems and understanding how they modify input signals.

Correlation measures similarity. Correlation, similar to convolution, compares two signals to determine the degree of similarity between them. It is widely used in communication engineering for tasks like signal detection and synchronization.

Types of correlation:

• Cross-correlation: Measures the similarity between two different signals.
• Autocorrelation: Measures the similarity between a signal and its time-delayed version.

Applications of correlation. Correlation techniques are used in radar, sonar, and digital communications to detect and extract signals from noisy environments. Autocorrelation is particularly useful for identifying periodic components in signals.

9. Fourier Series: Analyzing Periodic Signals

Periodic signals can be easily analysed using Fourier series.

Decomposing periodic signals. Fourier series provide a powerful tool for analyzing periodic signals by decomposing them into a sum of harmonically related sinusoids. This representation allows engineers to understand the frequency content of a signal and design systems that interact with it effectively.

Forms of Fourier series:

• Trigonometric form: Expresses the signal as a sum of sines and cosines.
• Cosine form: Expresses the signal as a sum of cosines with phase shifts.
• Exponential form: Expresses the signal as a sum of complex exponentials.

Dirichlet conditions. For a periodic signal to be represented by a Fourier series, it must satisfy certain conditions known as Dirichlet's conditions. These conditions ensure that the signal is well-behaved and that the Fourier series converges to the signal.

10. Fourier Transform: From Time to Frequency Domain

Signal analysis becomes very easy in frequency domain.

Transforming signals. The Fourier transform extends the concept of Fourier series to aperiodic signals, providing a way to analyze their frequency content. It transforms a signal from the time domain to the frequency domain, revealing the amplitudes and phases of its constituent frequencies.

Fourier transform pairs. The Fourier transform and its inverse form a transform pair, allowing signals to be converted back and forth between the time and frequency domains. This duality is essential for analyzing and manipulating signals in both domains.

Properties of Fourier transform:

• Linearity: The transform of a sum of signals is the sum of their transforms.
• Time shifting: Shifting a signal in time introduces a linear phase shift in the frequency domain.
• Time scaling: Compressing or expanding a signal in time affects its frequency spectrum.
• Convolution: Convolution in the time domain corresponds to multiplication in the frequency domain.

11. Laplace Transforms: A Powerful Tool for System Analysis

Laplace transform is a very powerful mathematical technique for analysis of continuous-time systems.

Solving differential equations. Laplace transforms provide a powerful method for solving differential equations that describe LTI systems. By transforming the differential equation into an algebraic equation, the solution process becomes much simpler.

Key concepts in Laplace transforms:

• Region of Convergence (ROC): The range of complex frequencies for which the Laplace transform converges.
• Transfer function: The ratio of the Laplace transform of the output to the Laplace transform of the input.
• Impulse response: The system's output when the input is a unit impulse.

Applications of Laplace transforms. Laplace transforms are widely used in circuit analysis, control systems, and signal processing to analyze system stability, design filters, and solve complex differential equations.

12. Z-Transforms: Analyzing Discrete-Time Systems

Z-transform is a very powerful mathematical technique for analysis of discrete-time systems.

Discrete-time counterpart. The Z-transform is the discrete-time equivalent of the Laplace transform, providing a powerful tool for analyzing and designing discrete-time systems. It transforms a discrete-time signal into the frequency domain, allowing for easier analysis and manipulation.

Key concepts in Z-transforms:

• Region of Convergence (ROC): The range of complex values for which the Z-transform converges.
• Transfer function: The ratio of the Z-transform of the output to the Z-transform of the input.
• Impulse response: The system's output when the input is a unit impulse sequence.

Applications of Z-transforms. Z-transforms are widely used in digital signal processing, control systems, and communication systems to analyze system stability, design digital filters, and solve difference equations.

Last updated: March 13, 2025