Signals and Systems

Signals and systems is the study of how information, energy, and physical measurements vary with an independent variable, usually time, and how devices or mathematical rules transform those variations. In Oppenheim, Willsky, and Nawab's treatment, the main theme is representation: a complicated waveform becomes easier to understand after it is decomposed into impulses, shifted copies, exponentials, sinusoids, or transform-domain components.

The notes in this section follow the usual second-edition course arc: begin with signal and system models, build linear time-invariant systems through convolution, move into Fourier series and Fourier transforms, then use sampling, modulation, Laplace transforms, and $z$-transforms to analyze frequency behavior, stability, and realizations. The pages are working study notes rather than a replacement for the textbook. They emphasize definitions, formulas, example calculations, and the decision process for choosing the right domain.

Definitions

A signal is a function of one or more independent variables. In this section the independent variable is time, so continuous-time signals are written as $x(t)$ and discrete-time signals are written as $x[n]$. Continuous time uses real-valued time $t\in\mathbb{R}$; discrete time uses integer sample index $n\in\mathbb{Z}$. This difference determines whether formulas use integrals or sums, whether frequency is unique or periodic, and whether time shifts can be arbitrary real numbers or must be integer sample shifts.

A system is a mapping from an input signal to an output signal:

$$y=T\{x\}.$$

The most important system class in these notes is the linear time-invariant system. Linearity means superposition holds. Time invariance means a time shift at the input produces the same time shift at the output. These two properties imply that the impulse response determines the entire input-output behavior through convolution.

The chapter list used by these notes is:

Order	Topic area	Wiki page
1	Signal models, independent variables, transformations of time	Signals and Time Transformations
2	Periodicity, energy, power, and signal classes	Periodicity, Energy, and Power
3	System properties: linearity, time invariance, causality, stability	System Properties
4	Continuous- and discrete-time LTI systems	LTI Systems and Convolution
5	Periodic signal expansions	Fourier Series for Periodic Signals
6	Aperiodic continuous-time spectra	Continuous-Time Fourier Transform
7	Aperiodic discrete-time spectra	Discrete-Time Fourier Transform
8	Sampling, aliasing, and reconstruction	Sampling, Aliasing, and Reconstruction
9	Laplace-domain analysis and ROC	Laplace Transform and ROC
10	$z$-domain analysis and ROC	Z-Transform and ROC
11	Frequency response and filters	Frequency Response and Filtering
12	Modulation and communication models	Modulation and Communication Systems
13	State variables and first-order vector models	State-Space Introduction

Key results

The first organizing result is impulse decomposition. In discrete time,

$$x[n]=\sum_{k=-\infty}^{\infty}x[k]\delta[n-k].$$

In continuous time,

$$x(t)=\int_{-\infty}^{\infty}x(\tau)\delta(t-\tau)\,d\tau.$$

These formulas say that a signal can be assembled from shifted impulses. For LTI systems, each shifted impulse produces a shifted impulse response, so the weighted sum or integral of those responses gives convolution:

$$y(t)=x(t)*h(t), \qquad y[n]=x[n]*h[n].$$

The second organizing result is the complex exponential eigenfunction property. For an LTI system,

$$e^{j\omega t}\to H(j\omega)e^{j\omega t}$$

in continuous time, and

$$e^{j\Omega n}\to H(e^{j\Omega})e^{j\Omega n}$$

in discrete time. The frequency is preserved; only amplitude and phase change. Fourier series and Fourier transforms exploit this fact by decomposing signals into exponential components.

The third organizing result is that transform algebra must be paired with convergence information. A Laplace transform $X(s)$ or a $z$-transform $X(z)$ is incomplete without its region of convergence. The ROC determines sidedness, whether the Fourier transform exists, and whether a causal rational LTI system is stable.

Use this decision path when solving problems:

If the problem gives...	Usually start with...	Reason
a graph or shifted expression	signal transformations	locate support and landmarks
system property claims	property tests	avoid assuming LTI
LTI impulse response	convolution or frequency response	$h$ determines the system
periodic signal	Fourier series	spectrum is harmonic lines
aperiodic CT signal	CTFT or Laplace	continuous spectrum or ROC needed
aperiodic DT sequence	DTFT or $z$-transform	periodic spectrum or ROC needed
sampling rate and bandwidth	sampling theorem	check aliasing before reconstruction

Another useful habit is to separate representation questions from system questions. A representation question asks, "How can this signal be decomposed?" A system question asks, "How does the operator act on each component?" For example, Fourier series is a representation method for periodic signals, while frequency response is a system method for LTI systems. They work together only after the system has been identified as LTI.

The same separation helps with transform domains. The CTFT and DTFT describe spectra directly on frequency axes. The Laplace and $z$ transforms add convergence regions so that exponential growth, decay, sidedness, and stability can be represented. When a problem includes words such as causal, stable, right-sided, left-sided, pole, zero, or ROC, it is usually asking for Laplace or $z$ reasoning, not only Fourier algebra.

Finally, always track units. Continuous-time frequency $\omega$ is in radians per second, ordinary frequency $f$ is in cycles per second, and discrete-time frequency $\Omega$ is in radians per sample. The conversion $\Omega=\omega T$ is central in sampling problems and is a common place where otherwise correct solutions lose a factor of $2\pi$.

Visual

Worked example 1: choosing the right representation

Problem: A signal is described as a nonzero periodic square wave with period $T_0=4$, and it is passed through an LTI system with known frequency response $H(j\omega)$. Which representation should be used to compute the steady output?

Method:

1. The input is continuous-time because its period is given as a real time $T_0=4$ rather than an integer sample period.

2. The input is periodic and nonzero. A nonzero periodic signal usually has infinite total energy but finite average power, so a Fourier transform as an ordinary function is not the most direct representation.

3. Use the continuous-time Fourier series:

$$x(t)=\sum_{k=-\infty}^{\infty}a_k e^{jk\omega_0t}.$$

4. Compute the fundamental frequency:

$$\omega_0=\frac{2\pi}{T_0}=\frac{2\pi}{4}=\frac{\pi}{2}.$$

5. Because the system is LTI, each harmonic is scaled by the frequency response at that harmonic:

$$y(t)=\sum_{k=-\infty}^{\infty}a_k H\left(jk\frac{\pi}{2}\right)e^{jk(\pi/2)t}.$$

Checked answer: Use CT Fourier series coefficients, not a finite-energy CTFT calculation. The LTI system acts by multiplying the $k$th harmonic coefficient by $H(jk\pi/2)$.

Worked example 2: identifying when ROC is required

Problem: Two sequences have the same algebraic $z$-transform expression,

$$X(z)=\frac{1}{1-0.8z^{-1}}.$$

One has ROC $\vert z\vert \gt 0.8$, and the other has ROC $\vert z\vert \lt 0.8$. Are they the same sequence?

Method:

1. The expression has a pole at $z=0.8$.

2. ROC $\vert z\vert \gt 0.8$ corresponds to the right-sided geometric sequence

$$x_1[n]=(0.8)^n u[n].$$

3. ROC $\vert z\vert \lt 0.8$ corresponds to the left-sided sequence

$$x_2[n]=-(0.8)^n u[-n-1].$$

4. Compare a sample. At $n=0$,

$$x_1[0]=1,\qquad x_2[0]=0.$$

5. Therefore the sequences are different even though the rational expression is identical.

Checked answer: They are not the same sequence. The ROC is part of the transform specification.

Code

import numpy as np

def choose_tool(is_periodic, is_discrete, is_lti, has_roc_question):
    if has_roc_question:
        return "z-transform" if is_discrete else "Laplace transform"
    if is_periodic:
        return "DT Fourier series" if is_discrete else "CT Fourier series"
    if is_lti:
        return "DTFT/frequency response" if is_discrete else "CTFT/frequency response"
    return "start with definitions and system-property tests"

cases = [
    (True, False, True, False),
    (False, True, True, True),
    (False, False, False, False),
]

for case in cases:
    print(case, "->", choose_tool(*case))

Common pitfalls

• Treating continuous-time and discrete-time variables as interchangeable. Use $t$ for CT and $n$ for DT unless a page explicitly says otherwise.
• Forgetting the region of convergence for Laplace and $z$ transforms. The same rational expression can describe different signals.
• Assuming that every bounded-looking formula has finite energy. Periodic nonzero signals have infinite energy but finite average power.
• Calling every frequency-domain plot a Fourier transform. Periodic signals often use Fourier series coefficients, while aperiodic signals use transform densities.
• Applying ideal sampling conclusions without checking bandlimiting. Sampling below Nyquist creates aliasing that cannot be removed by an ideal reconstruction filter.
• Using LTI shortcuts before proving the system is LTI. Convolution and frequency response are not valid for arbitrary nonlinear or time-varying systems.

Connections

• Signals and Time Transformations
• Periodicity, Energy, and Power
• System Properties
• LTI Systems and Convolution
• Continuous-Time Fourier Transform
• Sampling, Aliasing, and Reconstruction
• Laplace Transform and ROC
• Z-Transform and ROC