1. Signal and System Foundations

Introduction and Context

Signals and systems provide the conceptual language for describing physical measurements, communication processes, control actions, biological activity, and computational operations. A voltage waveform from a sensor, a sampled audio recording, a radar echo, and a temperature history can all be treated as signals. A circuit, filter, communication channel, control plant, or algorithm that transforms one signal into another is a system.

At postgraduate level, the study of signals and systems is not limited to identifying waveforms or manipulating formulas. It requires the ability to select an appropriate representation, determine the mathematical class of a signal or system, interpret the physical meaning of that classification, and recognize the assumptions under which a model is valid. These foundations support later work in Fourier analysis, Laplace and z-transforms, convolution, sampling, digital signal processing, communications, estimation, and control.

The central discipline is abstraction. A real process is represented by variables defined over time, space, or another independent variable. The representation may be continuous or discrete, deterministic or random, periodic or aperiodic, and finite-energy or finite-power. A system is then studied through properties such as linearity, time invariance, causality, stability, and memory.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Represent continuous-time and discrete-time signals mathematically and graphically.

2. Classify signals according to time, amplitude, predictability, periodicity, symmetry, energy, and power.

3. Perform time shifting, scaling, reversal, amplitude scaling, and combinations of elementary operations.

4. Explain standard signal models, including impulses, steps, ramps, exponentials, sinusoids, and rectangular pulses.

5. Determine whether a system is linear, time invariant, causal, memoryless, invertible, and stable.

6. Apply these concepts to engineering analysis and interpret their practical implications.

Signals as Mathematical and Physical Objects

A signal is a function that conveys information about a phenomenon. In a continuous-time signal, the independent variable is continuous and the signal is written as x(t), where t commonly denotes time. In a discrete-time signal, the independent variable takes integer values and the signal is written as x[n], with n belonging to the integers. A discrete-time signal may arise by sampling a continuous waveform, but discreteness can also be intrinsic, as in a sequence of daily measurements.

The distinction between continuous time and discrete time is independent of whether amplitude is continuous. A continuous-time signal may be quantized in amplitude, and a discrete-time signal may have real-valued amplitude. When both time and amplitude are discrete, the result is a digital signal in the strict engineering sense. In many practical discussions, however, “digital signal” refers broadly to a digitally represented sequence.

A signal can be represented in several ways. A formula is useful for derivation; a graph emphasizes waveform geometry; a table is convenient for finite sequences; and a decomposition into standard signals reveals how a system will process it. The choice of representation is therefore part of engineering reasoning rather than a merely cosmetic decision.

Deterministic and Random Signals

A deterministic signal is completely specified by a mathematical rule or known record. For example,

x(t) = 3 cos(100πt + π/4)

can be evaluated at every time. A random or stochastic signal cannot be predicted exactly at every instant, although its statistical properties may be known. Thermal noise and communication-channel noise are typical examples. Deterministic signal analysis is generally developed first because it establishes the structural ideas later used in stochastic processes.

Even and Odd Signals

A signal is even if

x(t) = x(-t),

or, in discrete time, x[n] = x[-n]. It is symmetric about the vertical axis. A signal is odd if

x(t) = -x(-t),

with the corresponding discrete-time condition x[n] = -x[-n]. An odd signal must satisfy x(0)=0 when its value at zero is defined.

Every signal can be uniquely decomposed into even and odd parts:

x_e(t) = [x(t)+x(-t)]/2,

x_o(t) = [x(t)-x(-t)]/2,

so that x(t)=x_e(t)+x_o(t). The same formulas apply with t replaced by n. This decomposition is useful in Fourier analysis and in simplifying integrations involving symmetric limits.

Periodic and Aperiodic Signals

A continuous-time signal is periodic if there exists a positive number T such that

x(t+T)=x(t)

for every t. The smallest positive such T is the fundamental period T0. A discrete-time signal is periodic if there is a positive integer N satisfying x[n+N]=x[n] for all n; the smallest such integer is the fundamental period N0.

A continuous-time sinusoid cos(ω0t+φ) has period T0=2π/|ω0| for nonzero ω0. For a discrete-time sinusoid cos(ω0n+φ), periodicity requires ω0/(2π) to be rational. If ω0/(2π)=p/q in lowest terms, the fundamental period is q. Thus a discrete-time sinusoid with frequency 0.2π is periodic with period 10, whereas one with frequency √2π is not periodic.

A sum of continuous-time periodic signals is periodic if the ratio of every pair of component periods is rational. For discrete-time sinusoids, a common integer period must exist. These conditions prevent a frequent error: assuming that every sinusoidal expression is periodic in discrete time.

Energy, Power, and Signal Magnitude

The energy of a continuous-time signal is

E = ∫ from −∞ to ∞ |x(t)|² dt,

and for a discrete-time signal,

E = Σ from n=−∞ to ∞ |x[n]|².

The average power of a continuous-time signal is

P = lim as T→∞ of (1/2T) ∫ from −T to T |x(t)|² dt,

while for a discrete-time signal,

P = lim as N→∞ of (1/(2N+1)) Σ from n=−N to N |x[n]|².

A nonzero finite-energy signal has E finite and P=0 and is called an energy signal. A signal with finite, nonzero average power generally has infinite energy and is called a power signal. Periodic signals of finite amplitude are normally power signals, while decaying pulses are commonly energy signals. The zero signal belongs to both categories by convention.

Energy and power describe different engineering situations. Energy is appropriate for a transient event, such as a transmitted pulse of finite duration. Power is appropriate for a sustained waveform, such as a carrier or a periodic supply voltage. Neither classification should be inferred solely from whether a signal is continuous or discrete.

For complex signals, the magnitude squared |x|², rather than x², is used. The absolute value also ensures that energy and power are nonnegative.

Elementary Operations on Signals

Signal operations alter the independent or dependent variable. Their correct order matters.

Amplitude scaling produces y(t)=A x(t), changing magnitude and possibly sign. Time shifting produces y(t)=x(t−t0). A positive t0 delays the signal, while a negative t0 advances it. Time reversal produces y(t)=x(−t), reflecting the waveform about t=0. Time scaling produces y(t)=x(at). For |a|>1 the waveform is compressed in time; for 0<|a|<1 it is expanded. A negative a combines scaling with reversal.

In discrete time, shifts must preserve integer indexing: x[n−n0] is a delay by n0 samples when n0 is positive. Expressions such as x[0.5n] require special interpretation and are not generally valid as ordinary discrete-time sequences unless an indexing convention is specified. This difference is fundamental: continuous time permits arbitrary time scaling, whereas discrete time permits only operations that yield defined integer-indexed samples.

For composite transformations, rewrite the argument carefully. For example, y(t)=x(2t−4)=x(2(t−2)) means that the original signal is first compressed by a factor of 2 and then delayed by 2 time units when interpreted through the final waveform. Graphical analysis should begin by locating significant points, transforming their coordinates, and checking the result algebraically.

Standard Signal Models

Several elementary signals act as building blocks for more complicated descriptions.

The unit step is

u(t)=1 for t>0 and 0 for t<0,

with the value at t=0 convention-dependent. In discrete time, u[n]=1 for n≥0 and 0 otherwise. The unit impulse δ(t) is a generalized signal characterized by zero value away from the origin and unit area, ∫δ(t)dt=1. It has the sampling property

∫ x(t)δ(t−t0)dt = x(t0).

The discrete-time impulse δ[n] equals 1 at n=0 and 0 elsewhere, and satisfies Σx[n]δ[n−n0]=x[n0]. Unlike δ(t), it is an ordinary sequence rather than a generalized function.

The continuous-time ramp is r(t)=t u(t), and the discrete-time ramp is r[n]=n u[n]. In distributional analysis, the derivative of the unit step is the impulse, while the derivative of the ramp is the step, subject to the usual generalized-function interpretation.

Exponential signals include x(t)=Ae^{st}, where s may be real or complex. If s=σ+jω, then e^{st}=e^{σt}e^{jωt}; σ controls growth or decay and ω controls oscillation. Discrete-time exponentials have the form Az^n. Sinusoids are central because complex exponentials are eigenfunctions of linear time-invariant systems: an input e^{jωt} produces an output H(jω)e^{jωt}, when the system is characterized by frequency response H(jω).

Finite-duration pulses, rectangular signals, triangular signals, and sinc-type waveforms are also widely used. They model switching events, communication symbols, sampled spectra, and filter responses. Piecewise descriptions and combinations of shifted steps often provide efficient mathematical representations.

Systems and Their Structural Properties

A system maps an input signal to an output signal. We write y=T{x}, where T denotes the transformation. A system model may be a differential equation, difference equation, integral relation, transfer function, algorithm, or physical block diagram.

Linearity

A system is linear if it satisfies superposition:

T{a x1+b x2}=aT{x1}+bT{x2}

for arbitrary signals x1, x2 and scalars a, b. This requires both homogeneity and additivity. The system y(t)=3x(t)+2dx(t)/dt is linear, whereas y(t)=x²(t) and y(t)=x(t)+5 are nonlinear. A nonzero constant offset violates homogeneity even if the remaining operation is linear.

Time Invariance

A system is time invariant if a shift in the input produces the same shift in the output. If T{x(t)}=y(t), then the system must satisfy T{x(t−t0)}=y(t−t0) for every t0. The system y(t)=t x(t) is time varying because the explicit factor t changes under a shift. A system involving fixed delays, such as y[n]=x[n−2], is time invariant.

Memory and Memorylessness

A memoryless system has an output at an instant determined solely by the input at that same instant: y(t)=f(x(t)). The system y(t)=x(t)+x(t−1) has memory because it requires a past value. Differentiation and integration also have memory-related distinctions: differentiation uses behavior in a neighborhood, while integration generally depends on an interval of values.

Causality

A causal system cannot respond before the input is applied. For a system initially considered over all time, causality means that y(t0) depends only on x(t) for t≤t0. A delay is causal; an advance y(t)=x(t+1) is noncausal. Noncausal systems can be implemented in offline data processing when the complete record is available, but they cannot generally operate in real time without prediction or stored future data.

Invertibility

A system is invertible if distinct inputs always produce distinct outputs, allowing an inverse system to recover the input. The system y(t)=2x(t) is invertible, with inverse x(t)=y(t)/2. The squaring system y(t)=x²(t) is not invertible over unrestricted real-valued signals because x and −x produce the same output.

Stability

The bounded-input bounded-output (BIBO) criterion states that a system is stable if every bounded input produces a bounded output. If |x(t)|≤M_x, then stability requires the existence of finite M_y such that |y(t)|≤M_y. The system y(t)=5x(t) is stable; an accumulator y[n]=Σ from k=−∞ to n x[k] is not BIBO stable, since a bounded constant input can produce an unbounded output.

For a continuous-time linear time-invariant system with impulse response h(t), BIBO stability requires ∫|h(t)|dt to be finite. In discrete time, the corresponding condition is Σ|h[n]|<∞. These results connect an abstract property to a practical test based on the system’s response to an impulse.

A Worked Classification Example

Consider the discrete-time system

y[n]=0.5x[n]+x[n−1].

It is linear because the output is a weighted sum of input samples. It is time invariant because shifting x by n0 shifts both terms by n0. It has memory because x[n−1] is required. It is causal because only the present and past inputs occur. It is BIBO stable because if |x[n]|≤M, then |y[n]|≤1.5M. It is not memoryless, although it is simple to implement in real time. This example illustrates why system properties must be tested independently; causality does not imply memorylessness, and linearity does not imply stability.

An Illustrative Applied Case: Sensor-Based Motor Protection

Background

Consider an industrial motor whose vibration sensor produces a continuous voltage x(t). A data-acquisition unit samples the voltage at regular intervals, producing x[n]. The protection controller must detect excessive vibration and issue a shutdown command.

Situation and Problem

The engineering team initially proposes a moving-average algorithm,

y[n]=(1/3)[x[n]+x[n−1]+x[n−2]],

followed by a threshold detector. The team must determine whether the model is suitable for real-time protection and how its signal properties affect the design.

Analysis

The moving average is linear and time invariant, has memory, and is causal. It is also BIBO stable: if the sensor signal is bounded by M, then |y[n]|≤M. The averaging suppresses rapid fluctuations and sensor noise but introduces delay. A sudden change in vibration is distributed across three output samples, so the threshold crossing may occur later than it would with the raw signal. If the sampling interval is Δ seconds, the oldest sample used is two intervals old, and the effective response is delayed in a manner dependent on the signal shape.

The sampled sequence is discrete time, but amplitude may still be represented with finite precision in the actual controller. Quantization and sensor saturation can introduce nonlinear effects not present in the ideal mathematical model. The team must also distinguish a finite transient vibration event, which may be analyzed through energy, from continuing periodic vibration, for which average power is more meaningful.

Approach and Solution

The moving average is retained for noise reduction, but a parallel fast threshold path is added to detect severe instantaneous vibration. The sampling rate is selected sufficiently above the dominant vibration frequency, and anti-alias filtering is placed before sampling. Thresholds are tested against calibration data, and fail-safe behavior is defined for sensor disconnection or implausible values.

Outcome

The resulting design combines a stable smoothing system with a rapid protective path. The mathematical classification clarifies expected delay, real-time implementability, noise behavior, and failure modes before hardware deployment.

Learning Points

A system model is useful only when its assumptions are connected to the physical task. Causality matters for protection, stability matters for bounded operation, memory explains delay, and sampling determines whether the discrete representation preserves the relevant phenomenon. A mathematically elegant filter is not automatically an adequate safety function.

Common Errors and Professional Considerations

A frequent error is confusing a shifted argument with a shift in the wrong direction. The expression x(t−3) is delayed, not advanced. Another is treating a discrete-time sinusoid as periodic without checking whether its normalized frequency is rational. Students also commonly classify every bounded signal as an energy signal; boundedness alone does not determine energy, since a nonzero periodic signal usually has infinite energy.

System properties should not be inferred from one example. Testing linearity with only one input is insufficient; the superposition condition must hold for arbitrary inputs. Similarly, a system may be causal but unstable, stable but noncausal, or linear but time varying.

In professional systems, model boundaries must be explicit. Sensor saturation, quantization, finite word length, delays, missing data, and computational overflow can invalidate an ideal model. Safety-critical applications require traceable requirements, validation under abnormal conditions, fail-safe behavior, and documented assumptions. Signal processing should not conceal meaningful abnormalities merely to improve visual smoothness.

Modern computational tools can plot signals, verify transformations symbolically, estimate energy and power numerically, and simulate system responses. Machine-learning models may classify or predict signals, but their outputs still require signal-quality assessment, temporal validation, robustness testing, and responsible deployment. Automation does not remove the need to understand causality, sampling, stability, or uncertainty.

Current Directions

Contemporary signal and system practice increasingly combines classical models with embedded computation, distributed sensors, adaptive filtering, and data-driven inference. Edge devices perform real-time analysis under constraints of energy, memory, and latency. Cyber-physical systems integrate sensors, communication networks, and control algorithms, making timing and reliability central design issues. Hybrid models that combine physical equations with learned components are becoming common, but they must be evaluated for stability, interpretability, and behavior outside the training data.

Best Practices for Analysis

Begin by stating whether the independent variable is continuous or discrete and define the domain clearly. Draw or tabulate the signal before manipulating it. Identify symmetry and periodicity using definitions rather than visual guesses. For energy and power, write the appropriate integral or sum and examine convergence. When classifying a system, test linearity, time invariance, memory, causality, invertibility, and stability separately. Finally, relate every mathematical conclusion to implementation requirements such as latency, sampling, boundedness, and available information.

Chapter Summary

Signals are mathematical representations of physical or informational quantities, while systems transform signals. Continuous-time signals use a continuous independent variable; discrete-time signals use integer indices. Signals may be deterministic or random, even or odd, periodic or aperiodic, and classified as energy or power signals. Every signal has unique even and odd components. Fundamental operations include amplitude scaling, shifting, reversal, and time scaling, with important differences between continuous and discrete time.

Unit steps, impulses, ramps, exponentials, sinusoids, and pulses form standard models for analysis. Systems are evaluated through linearity, time invariance, memory, causality, invertibility, and BIBO stability. These properties determine whether a system can be superposed, shifted, implemented in real time, reversed, or safely operated with bounded signals. Applied analysis must also account for sampling, quantization, delay, noise, saturation, and safety requirements. These foundations prepare the reader for transform methods, convolution, frequency response, sampling theory, and control-oriented system analysis.

Key Terms

Signal: A function representing a physical quantity or information.

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

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

Deterministic signal: A signal completely specified by a rule or known record.

Random signal: A signal described through probabilistic or statistical properties.

Even signal: A signal satisfying x(t)=x(−t) or x[n]=x[−n].

Odd signal: A signal satisfying x(t)=−x(−t) or x[n]=−x[−n].

Periodic signal: A signal that repeats after a positive period.

Fundamental period: The smallest positive period of a periodic signal.

Energy signal: A signal with finite total squared magnitude and normally zero average power.

Power signal: A signal with finite, nonzero average power and generally infinite energy.

Unit impulse: A generalized continuous-time signal of unit area, or a discrete sequence equal to one at the origin.

Unit step: A signal that changes from zero to one at the origin.

Linearity: Satisfaction of additivity and homogeneity, jointly expressed as superposition.

Time invariance: The property that input shifts produce identical output shifts.

Causality: Dependence of the present output only on present and past inputs.

Memoryless system: A system whose output at an instant depends only on input at that instant.

Invertibility: The property that permits unique recovery of the input from the output.

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

Sampling: Representation of a continuous-time waveform by values at discrete time instants.

Review Questions

Short Answer Questions

1. Distinguish continuous-time, discrete-time, and digital signals.

2. State the conditions defining an even signal and an odd signal.

3. Define an energy signal and a power signal.

4. What is the sampling property of the unit impulse?

5. Why can a discrete-time sinusoid fail to be periodic?

Long Answer Questions

1. Explain the decomposition of an arbitrary signal into even and odd components and discuss its analytical value.

2. Derive and explain the energy and average-power expressions for continuous-time and discrete-time signals.

3. Describe amplitude scaling, time shifting, time reversal, and time scaling, including their effects on a waveform.

4. Explain linearity, time invariance, memory, causality, invertibility, and BIBO stability with suitable examples.

5. Discuss the standard signal models and explain how impulses, steps, ramps, exponentials, and sinusoids are used in system analysis.

Analytical / Application Questions

1. Classify the system y[n]=2x[n]−x[n−3] with respect to linearity, time invariance, memory, causality, and BIBO stability. Justify each conclusion.

2. Determine whether x[n]=cos(πn/3)+sin(πn/4) is periodic. If it is, find its fundamental period.

3. An industrial monitoring system uses a three-sample moving average before threshold detection. Analyze the advantages and disadvantages of this choice in terms of stability, causality, memory, delay, noise suppression, and protection response.

22. Multiple Choice Questions with Answers

Question 1

Which condition is necessary for a discrete-time sinusoid cos(ω0n+φ) to be periodic?

A. ω0 must be an integer

B. ω0/(2π) must be rational

C. φ must be zero

D. The amplitude must exceed one

Correct Answer: B. ω0/(2π) must be rational

Question 2

For x(t)=3cos(100πt+π/4), the fundamental period is

A. 0.01 s

B. 0.02 s

C. 100π s

D. 2π s

Correct Answer: B. 0.02 s

Question 3

Which signal is most appropriately classified as a power signal under ordinary conditions?

A. A finite-duration rectangular pulse

B. A decaying exponential extending over all time

C. A nonzero periodic sinusoid

D. An isolated impulse-like transient

Correct Answer: C. A nonzero periodic sinusoid

Question 4

The operation y(t)=x(t−4) represents

A. An advance by four time units

B. A reversal about the origin

C. A delay by four time units

D. Compression by a factor of four

Correct Answer: C. A delay by four time units

Question 5

Which system is linear and time invariant but not memoryless?

A. y(t)=x²(t)

B. y(t)=x(t)+x(t−2)

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

D. y(t)=x(t)+5

Correct Answer: B. y(t)=x(t)+x(t−2)

Question 6

The system y[n]=x[n+1] is generally

A. Causal because it uses one sample

B. Noncausal because it uses a future input sample

C. Memoryless because it uses only one input sample

D. Nonlinear because it shifts the input

Correct Answer: B. Noncausal because it uses a future input sample

Question 7

If a signal satisfies x(t)=−x(−t), it is

A. Even

B. Periodic

C. Odd

D. A power signal

Correct Answer: C. Odd

Question 8

Which condition guarantees BIBO stability for a discrete-time LTI system with impulse response h[n]?

A. Σ|h[n]| is finite

B. h[n] is periodic

C. h[n] is odd

D. h[n] has finite duration only when n is positive

Correct Answer: A. Σ|h[n]| is finite

Question 9

Why is the system y(t)=x²(t) not invertible for unrestricted real-valued inputs?

A. It is time varying

B. It has memory

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

D. It is not causal

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

Question 10

A three-sample moving-average filter is best described as

A. Nonlinear, noncausal, and unstable

B. Linear, causal, with memory, and BIBO stable

C. Memoryless and time varying

D. Noninvertible solely because it is causal

Correct Answer: B. Linear, causal, with memory, and BIBO stable