Systems, Properties, and Engineering Models

Introduction

Signals and systems analysis provides a disciplined way to describe how physical devices and mathematical operators transform information. A signal may represent voltage, current, sound pressure, temperature, or a sampled data sequence. A system accepts one or more input signals and produces one or more output signals according to a rule. In electronics and communication engineering, this rule may model an amplifier, filter, transmission channel, sensor, controller, modulator, or software algorithm.

The central engineering task is not merely to calculate an output. It is to determine what a system can do, under what assumptions it operates, and whether its behavior is suitable for a particular application. Properties such as linearity, time invariance, causality, memory, invertibility, and stability allow engineers to classify systems before attempting detailed solutions.

A system is represented abstractly by

for continuous time, or

for discrete time. Here, x is the input, y is the output, and 𝒯 denotes the system operation. The notation does not assume that the system is linear, time invariant, or physically realizable.

Learning Objectives

After studying this chapter, you should be able to:

1. Model physical and communication systems using input-output descriptions and block diagrams.
2. Distinguish continuous-time and discrete-time system models.
3. Test systems for linearity, time invariance, causality, memory, invertibility, and stability.
4. Analyze the effects of series, parallel, and feedback interconnections.
5. Determine input-output behavior for common engineering models.
6. Recognize assumptions, limitations, and practical implications of idealized models.

1. System Models in Engineering

1.1 Mathematical and physical viewpoints

A physical system consists of interacting components governed by physical laws. A mathematical model is a simplified representation of that system. For example, an RC low-pass circuit can be modeled, under ideal component assumptions, by the differential equation

where x(t) is the applied voltage, y(t) is the capacitor voltage, R is resistance in ohms, and C is capacitance in farads. The product RC has units of seconds, so both terms have voltage units. This model neglects parasitic effects, component tolerances, noise, and nonlinear behavior.

A system model can be expressed in several equivalent ways:

• An input-output equation, such as y(t) = 2x(t) + 1.
• A differential or difference equation.
• An impulse response.
• A transfer function, when transform methods and suitable initial conditions apply.
• A block diagram showing functional interconnections.
• An algorithm or program implementing the input-output rule.

The choice depends on the purpose. A circuit designer may prefer a differential equation or transfer function, while a digital signal-processing engineer may use a difference equation or software implementation.

1.2 Continuous-time and discrete-time systems

A continuous-time signal x(t) is defined for every real value of t. A discrete-time signal x[n] is defined only at integer index n, usually after sampling. The independent variable t has units such as seconds; n is dimensionless and counts samples.

Sampling a continuous-time signal at sampling period T_s gives

where T_s is measured in seconds. This representation is useful only when the sampling process is adequately modeled and the sampling rate is appropriate for the signal bandwidth. In practical communication systems, insufficient sampling can cause aliasing, in which distinct continuous-time signals produce the same samples.

1.3 Systems with multiple inputs and outputs

Many systems have several inputs and outputs. A two-input adder may be written as

A multiple-input multiple-output system is more generally described by

where m is the number of inputs and p is the number of outputs. Wireless receivers, sensor arrays, and multiple-antenna communication systems commonly require this description.

Figure: Abstract input-output representation of a system.

2. Fundamental System Properties

The properties below are independent classifications. A system can be linear but time varying, causal but unstable, or memoryless but nonlinear. No single property determines all the others.

2.1 Linearity

A system is linear if it satisfies both superposition and homogeneity. For arbitrary inputs x₁ and x₂ and constants a and b,

This single equation combines additivity and scaling. It must hold for all permitted inputs and scalars, not merely for one example.

For instance, y(t) = 3x(t) is linear because

𝒯{a x₁ + b x₂} = 3(a x₁ + b x₂) = a(3x₁) + b(3x₂).

In contrast, y(t) = x²(t) is nonlinear. Choosing x₁ = x₂ = 1 and a = b = 1 gives an output of 4 for the combined input, whereas the sum of individual outputs is 1 + 1 = 2.

The system y(t) = x(t) + 5 is also nonlinear under the usual definition, because the zero input produces y(t) = 5 rather than zero. A constant bias can sometimes be handled by defining an affine model, but affine and linear are not identical.

2.2 Time invariance

A system is time invariant if delaying the input delays the output by the same amount, without any other change. For a continuous-time system, if

then time invariance requires

for every allowed shift t₀. For discrete time, the corresponding condition is

A quick test is to calculate the output for x(t), then replace the input by x(t − t₀), and compare the result with y(t − t₀).

Example: y(t) = t x(t) is time varying. With a shifted input, the output is t x(t − t₀), while shifting the original output gives (t − t₀)x(t − t₀). These are generally unequal.

The system y[n] = x[n] + x[n − 1] is time invariant because the delays are fixed relative to the input index.

2.3 Causality

A system is causal if its output at a given time depends only on present and past input values, not on future input values. For a continuous-time system, y(t₀) may depend on x(t) for t ≤ t₀, but not on x(t) for t > t₀. For discrete time, y[n₀] may depend on x[n] for n ≤ n₀.

Examples:

• y(t) = x(t) + x(t − 2) is causal.
• y[n] = x[n + 1] is noncausal because it requires a future sample.
• y(t) = ∫_−∞^t x(τ)dτ is causal, assuming the integral is physically initialized or well defined.

Causality is a physical realizability condition for real-time systems. A noncausal operation may still be implemented offline when the entire signal has already been recorded. For example, zero-phase filtering commonly uses future samples and is therefore noncausal in real time, although useful in post-processing.

2.4 Memory and memorylessness

A system is memoryless if its output at each instant depends only on the input at that same instant. Thus,

or

for some function f. Examples include y(t) = 4x(t) and y[n] = |x[n]|.

A delay, derivative, integral, or moving average generally has memory. The system

stores or uses a previous sample and therefore has memory. Physical energy storage in capacitors and inductors is a common source of memory.

Memorylessness and causality should not be confused. Every memoryless system is causal, but a causal system need not be memoryless.

2.5 Invertibility

A system is invertible if distinct inputs cannot produce the same output. Equivalently, there must exist an inverse system 𝒯⁻¹ such that

for every signal in the specified input class.

The system y(t) = 5x(t) is invertible, with x(t) = y(t)/5. The system y(t) = x²(t) is not invertible over all real-valued signals because x(t) and −x(t) produce the same output. It becomes invertible if the input is restricted to nonnegative values, in which case x(t) = √y(t).

A delay y[n] = x[n − 3] is invertible if the signal is available over the required index range; its inverse is an advance, x[n] = y[n + 3]. The inverse may be mathematically valid but noncausal, which matters in real-time implementation.

2.6 Stability

The standard input-output notion is bounded-input bounded-output (BIBO) stability. A system is BIBO stable if every bounded input produces a bounded output. If there is a finite constant B_x such that

for all t, then stability requires a finite B_y such that

for all t. The same definition applies to sequences, with t replaced by n.

The system y(t) = 3x(t) is stable: if |x(t)| ≤ B_x, then |y(t)| ≤ 3B_x. The system y(t) = t x(t) is not BIBO stable, since the bounded input x(t) = 1 produces the unbounded output y(t) = t.

For a continuous-time linear time-invariant (LTI) system with impulse response h(t), BIBO stability requires

For a discrete-time LTI system, the corresponding condition is

These conditions assume the usual convolution model and finite-valued signals. They are not conditions for every nonlinear or time-varying system.

3. LTI Systems and Input-Output Behavior

Linear time-invariant systems are especially important because their response is determined by the impulse response. For continuous time,

For discrete time,

Here, * denotes convolution, x is the input, h is the impulse response, and y is the output. The equations assume an LTI system and convergence of the integral or sum.

The reason convolution applies is that an input can be represented as a weighted collection of shifted impulses. Linearity determines how the system responds to the weights, and time invariance determines how it responds to shifts. The impulse response therefore contains the complete input-output description of an LTI system.

Worked Example 1: Discrete convolution

Let x[n] be nonzero only at n = 0 and n = 1, with x[0] = 1 and x[1] = 2. Let h[n] be nonzero only at n = 0 and n = 1, with h[0] = 3 and h[1] = 1. The convolution is

The nonzero output samples are:

• y[0] = x[0]h[0] = 1 × 3 = 3.
• y[1] = x[0]h[1] + x[1]h[0] = 1 × 1 + 2 × 3 = 7.
• y[2] = x[1]h[1] = 2 × 1 = 2.

Thus y[0] = 3, y[1] = 7, y[2] = 2, with all other samples zero.

Differential and difference equation models

A continuous-time LTI system may be represented by a constant-coefficient differential equation:

The coefficients a_i and b_i must be dimensionally consistent. Initial conditions are required to determine a unique response. A discrete-time counterpart is

If the current output depends on past outputs, the system is recursive. If it depends only on present and past inputs, it is nonrecursive or finite impulse response (FIR) when the input delay structure is finite.

4. Interconnection of Systems

4.1 Series and cascade connection

In a cascade, the output of one system becomes the input of the next:

For LTI systems with impulse responses h₁ and h₂, the equivalent impulse response is

Because convolution is associative and commutative under appropriate convergence conditions, the order of ideal LTI blocks does not affect the overall input-output response. In practical systems, however, quantization, saturation, noise, and finite precision can make the physical order important.

4.2 Parallel connection

For parallel systems whose outputs are added,

For LTI blocks,

This structure is used to combine low-pass and high-pass paths, construct filter banks, or represent direct and interference paths.

4.3 Feedback connection

A negative-feedback system can be represented by a forward system G and feedback system H. Under zero initial conditions and transform-domain assumptions, the closed-loop transfer function is

For positive feedback, the denominator becomes 1 − GH. The formula assumes compatible signal definitions, a well-defined transform, and algebraic feedback without unmodeled delays. Feedback can improve robustness and reduce sensitivity to parameter variations, but excessive loop gain or delay can cause instability.

5. Worked Classification Examples

Example 2: Classifying a system

Consider the discrete-time system

Linearity: The system is linear because it is a weighted sum of input samples; the coefficient n does not depend on the input amplitude.

Time invariance: It is time varying. For a shifted input x[n − n₀], the output is n x[n − n₀] + x[n − n₀ − 1]. Shifting the original output gives (n − n₀)x[n − n₀] + x[n − n₀ − 1]. These expressions differ generally.

Causality: It is causal because y[n] uses x[n] and x[n − 1], never a future sample.

Memory: It has memory because it uses x[n − 1].

BIBO stability: It is not stable. The bounded input x[n] = 1 produces y[n] = n + 1, which is unbounded.

Invertibility: Invertibility cannot be concluded merely from the presence of a current and delayed term; it depends on the signal domain and whether the resulting recurrence can be uniquely solved with appropriate boundary conditions. This illustrates why properties should be tested independently.

Example 3: Stability of an LTI impulse response

Let h[n] = (1/2)^n u[n], where u[n] is the unit-step sequence. Since h[n] is nonzero for n ≥ 0,

The sum is finite, so the system is BIBO stable. The result is dimensionless when h is a gain sequence. If the system includes physical units, the gain and signal units must be interpreted consistently.

6. Engineering Case Study: Digital Noise Reduction in a Sensor Node

This is an illustrative case.

Background

A battery-powered sensor node measures temperature once every second. The raw measurement contains random short-term fluctuations caused by sensor noise and electromagnetic interference. The processor applies a two-point moving-average filter:

Problem

The design team must determine whether the filter is suitable for real-time use and whether it introduces delay, memory, or instability.

Analysis

The filter is linear because it is a weighted sum of input samples. It is time invariant because the coefficients and delays do not change with n. It is causal because it uses the current and previous samples. It has memory because x[n − 1] is required. It is BIBO stable because its impulse response is h[0] = 0.5 and h[1] = 0.5, and

The filter is not perfectly invertible for arbitrary sequences: some rapid variations are attenuated and information may be lost. For a constant input, however, the output equals that constant, so the filter has unity gain at zero frequency.

Approach and outcome

The processor stores one previous sample, computes the average, and transmits y[n]. The first output requires an initialization policy, such as setting x[−1] = x[0] or declaring the first output invalid. The filter reduces high-frequency fluctuations but also smooths genuine rapid temperature changes. Its computational cost is low, making it appropriate for a resource-constrained node.

Learning points

1. A simple equation can reveal implementation requirements before coding.
2. Stability does not guarantee that a filter preserves all information.
3. Boundary and initialization conditions are part of a practical system specification.
4. A mathematically causal model supports real-time implementation.
5. Engineering decisions require balancing noise reduction, responsiveness, memory, and energy consumption.

7. Model Limitations and Professional Practice

Ideal system classifications depend on the chosen model. An amplifier may be treated as linear for small signals but become nonlinear near saturation. A communication channel may be considered time invariant over a short interval but time varying as a user moves. A circuit may be modeled as memoryless at one time scale while parasitic capacitance becomes important at another.

Good modeling practice therefore requires:

• Clearly identifying input and output variables and their units.
• Stating the operating range and initial conditions.
• Distinguishing ideal, nominal, and measured behavior.
• Checking causality and stability before deployment.
• Testing models against laboratory or field data.
• Documenting approximations, tolerances, delays, saturation, noise, and quantization.
• Considering safety and regulatory constraints in systems connected to medical, automotive, aerospace, or public communication infrastructure.

AI-based signal-processing systems do not eliminate these requirements. A learned model still has inputs, outputs, memory, latency, and operating limits. Engineers must evaluate robustness to distribution shifts, adversarial or corrupted signals, privacy risks, and explainability requirements where decisions affect people or safety.

Chapter Summary

A system maps input signals to output signals and may be modeled by equations, block diagrams, impulse responses, differential equations, difference equations, or algorithms. The main system properties are linearity, time invariance, causality, memory, invertibility, and BIBO stability. Linearity is tested by superposition; time invariance by comparing shifted inputs and shifted outputs; causality by checking for future-input dependence; memorylessness by checking whether only the present input is used; invertibility by determining whether distinct inputs can share an output; and stability by testing whether bounded inputs produce bounded outputs.

LTI systems are especially useful because their behavior is completely characterized by an impulse response and convolution. Series and parallel interconnections have simple equivalent forms for LTI systems, while feedback changes the overall transfer relationship and can affect stability. Practical models must include assumptions, initialization, finite precision, delays, nonlinearities, and operating limits. System classification is therefore both a mathematical activity and an engineering judgment.

Key Terms

System: A rule or physical arrangement that maps input signals to output signals.

Model: A simplified mathematical or computational representation of a physical system.

Continuous-time signal: A signal defined for every value of a continuous independent variable.

Discrete-time signal: A sequence defined at discrete index values.

Linearity: The property of satisfying superposition and homogeneity.

Time invariance: The property that an input shift produces the same shift in output.

Causality: Dependence only on present and past inputs.

Memoryless system: A system whose output depends only on the input at the same time or index.

Memory: Dependence on input values at other times or indices, or on stored internal state.

Invertibility: The ability to recover the input uniquely from the output.

BIBO stability: The property that every bounded input produces a bounded output.

Impulse response: The output of an LTI system produced by a unit impulse.

Convolution: The operation that determines the output of an LTI system from input and impulse response.

LTI system: A system that is both linear and time invariant.

Cascade: A series interconnection in which one system feeds another.

Parallel interconnection: Systems receiving a common input whose outputs are combined.

Feedback: An interconnection in which part of the output is returned to the input path.

FIR system: A discrete-time system with a finite-duration impulse response.

Affine system: A system consisting of a linear transformation plus a nonzero constant offset.

Sampling: Conversion of a continuous-time signal into a sequence of values at specified instants.

Review Questions

Short Answer Questions

1. State the superposition condition for a linear system.
2. What distinguishes a causal system from a noncausal system?
3. Define BIBO stability.
4. Why does a moving-average filter have memory?
5. What is the role of an impulse response in an LTI system?

Long Answer Questions

1. Explain how linearity and time invariance are tested, and illustrate each test with an example.
2. Discuss causality, memory, and invertibility as separate system properties. Explain why one property does not automatically determine the others.
3. Derive the convolution representation for an LTI system conceptually from shifted impulses.
4. Compare series, parallel, and feedback interconnections, including their practical engineering uses.
5. Explain why physical models require assumptions, operating ranges, initialization conditions, and validation.

Analytical / Application Questions

1. Classify y[n] = x[n] − 2x[n − 2] with respect to linearity, time invariance, causality, memory, and BIBO stability. Justify each conclusion.
2. A sensor processor uses y[n] = 0.25x[n] + 0.5x[n − 1] + 0.25x[n − 2]. Determine its impulse response, assess BIBO stability, and discuss one benefit and one limitation for real-time sensing.
3. A communication receiver has a forward gain G and feedback gain H. Explain how negative feedback changes the closed-loop model and identify practical factors that could invalidate an ideal algebraic analysis.

Multiple Choice Questions with Answers

Question 1

Which condition defines linearity for a system 𝒯?

A. 𝒯{x(t − t₀)} = y(t − t₀)

B. 𝒯{a x₁ + b x₂} = a𝒯{x₁} + b𝒯{x₂}

C. Every bounded input produces a bounded output

D. The output depends only on the present input

Correct Answer: B. 𝒯{a x₁ + b x₂} = a𝒯{x₁} + b𝒯{x₂}

Question 2

The system y[n] = x[n + 1] is noncausal because it

A. uses a past sample

B. squares the input

C. uses a future sample

D. has no memory

Correct Answer: C. uses a future sample

Question 3

Which system is time varying?

A. y[n] = x[n] + x[n − 1]

B. y(t) = 3x(t)

C. y(t) = t x(t)

D. y[n] = |x[n]|

Correct Answer: C. y(t) = t x(t)

Question 4

For an LTI system, the output is obtained from the input and impulse response by

A. differentiation only

B. convolution

C. sampling only

D. feedback cancellation

Correct Answer: B. convolution

Question 5

Which system is memoryless?

A. y[n] = x[n − 2]

B. y(t) = ∫_−∞^tx(τ)dτ

C. y[n] = 0.5x[n] + 0.5x[n − 1]

D. y(t) = 4x(t)

Correct Answer: D. y(t) = 4x(t)

Question 6

For an LTI discrete-time system, which condition guarantees BIBO stability?

A. ∑_n |h[n]| is finite

B. h[n] is nonzero for infinitely many n

C. The input is always sinusoidal

D. The system has memory

Correct Answer: A. ∑_n |h[n]| is finite

Question 7

The system y(t) = x²(t) is not invertible over all real-valued inputs because

A. it is time invariant

B. x(t) and −x(t) produce the same output

C. it has no output

D. it uses a future input

Correct Answer: B. x(t) and −x(t) produce the same output

Question 8

A two-point moving average has impulse response values h[0] = 0.5 and h[1] = 0.5. What is the sum of absolute impulse-response values?

A. 0

B. 0.5

C. 1

D. 2

Correct Answer: C. 1

Question 9

For two cascaded LTI systems with impulse responses h₁ and h₂, the equivalent impulse response is

A. h₁ + h₂

B. h₁ − h₂

C. h₁ × h₂ pointwise only

D. h₁ * h₂

Correct Answer: D. h₁ * h₂

Question 10

Under the stated ideal negative-feedback assumptions, the closed-loop transfer function is

A. G/(1 + GH)

B. G/(1 − GH)

C. H/(1 + G)

D. G + H

Correct Answer: A. G/(1 + GH)