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Linear System Signal Analysis

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1. Signals, systems, and core models

Signals

A signal is a function that carries information. In engineering, signals usually represent voltage, current, force, pressure, sound, position, or sensor output.

Common classifications:

TypeExampleNotes
Continuous-time$x(t)$Defined for every real $t$
Discrete-time$x[n]$Defined only at integer index $n$
AnalogMicrophone voltageContinuous amplitude
DigitalQuantized samplesFinite resolution and sampled in time
Deterministic$e^{-at}$Exactly described by an equation
RandomThermal noiseDescribed statistically

Useful elementary signals:

$$ u(t) = \begin{cases} 0, & t < 0 \\ 1, & t \ge 0 \end{cases} $$
$$ \delta(t) \text{ is an impulse with } \int_{-\infty}^{\infty} \delta(t)\,dt = 1 $$

In discrete time:

$$ u[n] = \begin{cases} 0, & n < 0 \\ 1, & n \ge 0 \end{cases} \qquad \delta[n] = \begin{cases} 1, & n = 0 \\ 0, & n \ne 0 \end{cases} $$

Systems

A system transforms an input signal into an output signal.

Examples:

  • Electrical filter

  • Mechanical mass-spring-damper

  • Communication channel

  • Digital controller

The most important system class in this subject is the linear time-invariant system, or LTI system.

2. Basic signal operations

Time shifting

Delaying a signal by $t_0$:

$$ x(t - t_0) $$

Advancing a signal:

$$ x(t + t_0) $$

Time scaling

Stretching or compressing time:

$$ x(at) $$

If $|a| > 1$, the signal compresses in time. If $0 < |a| < 1$, it stretches.

Time reversal

$$ x(-t) $$

Amplitude scaling

$$ Ax(t) $$

This changes amplitude without changing the time axis.

Decomposition into even and odd parts

Any signal can be decomposed as:

$$ x(t) = x_e(t) + x_o(t) $$

where

$$ x_e(t) = \frac{x(t) + x(-t)}{2} $$
$$ x_o(t) = \frac{x(t) - x(-t)}{2} $$

The same formulas hold in discrete time by replacing $t$ with $n$.

3. System properties

Linearity

A system is linear if it satisfies superposition:

$$ \mathcal{T}\{a x_1 + b x_2\} = a\mathcal{T}\{x_1\} + b\mathcal{T}\{x_2\} $$

Two parts:

  • Additivity

  • Homogeneity

Quick test

  1. Send in $x_1$ and get $y_1$.

  2. Send in $x_2$ and get $y_2$.

  3. Check whether $a x_1 + b x_2$ produces $a y_1 + b y_2$.

Nonlinear examples:

  • Squaring: $y = x^2$

  • Saturation and clipping

  • Multiplication of two signals

Time invariance

A system is time invariant if shifting the input shifts the output by the same amount.

If

$$ x(t) \to y(t) $$

then for any shift $t_0$,

$$ x(t - t_0) \to y(t - t_0) $$

If the system changes behavior over time, it is time-varying.

Causality

A system is causal if the output at time $t$ depends only on present and past inputs, not future inputs.

For discrete time, a causal system can depend on $x[n], x[n-1], x[n-2], \dots$

Stability

In the bounded-input bounded-output sense, a system is stable if every bounded input produces a bounded output.

If

$$ |x(t)| \le M_x $$

then stability requires

$$ |y(t)| \le M_y $$

for some finite $M_y$.

Memory

A system is memoryless if output at time $t$ depends only on the input at the same time.

Examples:

  • Memoryless: $y(t) = 3x(t)$

  • Has memory: $y(t) = x(t) + x(t-1)$

Invertibility

A system is invertible if the input can be uniquely recovered from the output.

Example:

$$ y(t) = 2x(t) $$

is invertible, with inverse

$$ x(t) = \frac{y(t)}{2} $$

4. Linear time-invariant systems

LTI systems are central because they are completely characterized by their response to an impulse.

Impulse response

For a continuous-time LTI system, the impulse response is

$$ h(t) = \mathcal{T}\{\delta(t)\} $$

For discrete time:

$$ h[n] = \mathcal{T}\{\delta[n]\} $$

Once $h$ is known, the output for any input is determined by convolution.

Why LTI systems matter

LTI systems are useful because:

  • Superposition applies.

  • Shift invariance simplifies analysis.

  • Frequency-domain methods become available.

  • Differential or difference equations often reduce to algebra.

Differential and difference equation models

Continuous-time example:

$$ a_1 \frac{dy(t)}{dt} + a_0 y(t) = b_1 \frac{dx(t)}{dt} + b_0 x(t) $$

Discrete-time example:

$$ y[n] - 0.5y[n-1] = x[n] + x[n-1] $$

To test whether such a model is LTI, check linearity and whether the coefficients are constant over time.

5. Convolution

Convolution is the main tool for computing the output of an LTI system.

Continuous-time convolution

For input $x(t)$ and impulse response $h(t)$:

$$ y(t) = (x * h)(t) = \int_{-\infty}^{\infty} x(\tau) h(t - \tau)\,d\tau $$

Equivalent form:

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

Discrete-time convolution

$$ y[n] = (x * h)[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k] $$

Graphical convolution workflow

  1. Time-reverse one signal.

  2. Shift it by $t$ or $n$.

  3. Multiply pointwise with the other signal.

  4. Integrate or sum over overlap.

Key properties of convolution

$$ x * h = h * x $$
$$ x * (h_1 + h_2) = x * h_1 + x * h_2 $$
$$ x * \delta = x $$
$$ x * \delta(t - t_0) = x(t - t_0) $$

If one factor is a delayed impulse, convolution simply shifts the other signal.

Physical meaning

Convolution can be viewed as:

  • Decomposing the input into shifted impulses

  • Scaling each impulse by the local input value

  • Summing the shifted impulse responses

Example: moving average filter

For discrete time:

$$ h[n] = \frac{1}{3}\big(\delta[n] + \delta[n-1] + \delta[n-2]\big) $$

Then

$$ y[n] = \frac{1}{3}\big(x[n] + x[n-1] + x[n-2]\big) $$

This is a simple smoothing filter.

6. Frequency response and steady-state sinusoidal analysis

Sinusoids as test signals

For LTI systems, complex exponentials are eigenfunctions:

$$ x(t) = e^{j\omega t} \quad \Rightarrow \quad y(t) = H(j\omega)e^{j\omega t} $$

where $H(j\omega)$ is the frequency response.

This means the system changes only amplitude and phase, not frequency, for a sinusoidal input.

Frequency response

If the impulse response is $h(t)$, then

$$ H(j\omega) = \int_{-\infty}^{\infty} h(t)e^{-j\omega t}\,dt $$

For discrete time:

$$ H(e^{j\omega}) = \sum_{n=-\infty}^{\infty} h[n]e^{-j\omega n} $$

Amplitude and phase interpretation

Write

$$ H(j\omega) = |H(j\omega)|e^{j\angle H(j\omega)} $$

Then a sinusoidal input produces:

$$ y(t) = |H(j\omega)| \cos(\omega t + \angle H(j\omega)) $$

for a cosine input of unit amplitude.

Bode plots

A Bode plot shows:

  • Magnitude in dB:

$$ 20\log_{10}|H(j\omega)| $$

  • Phase in degrees or radians

Useful for identifying:

  • Low-pass behavior

  • High-pass behavior

  • Band-pass behavior

  • Resonance

Poles and zeros

For rational transfer functions, poles and zeros strongly shape the response.

  • Zeros can attenuate specific frequencies.

  • Poles can amplify or destabilize the response.

General rule:

  • Poles closer to the imaginary axis or unit circle create stronger effects.

7. Fourier series and Fourier transform

Fourier series for periodic signals

If $x(t)$ is periodic with period $T$, it can be written as:

$$ x(t) = \sum_{k=-\infty}^{\infty} c_k e^{jk\omega_0 t} $$

where

$$ \omega_0 = \frac{2\pi}{T} $$

and

$$ c_k = \frac{1}{T}\int_{T} x(t)e^{-jk\omega_0 t}\,dt $$

Fourier series decomposes a periodic signal into harmonics of the fundamental frequency.

Symmetry shortcuts

If $x(t)$ is real and even:

  • Coefficients are real

  • Sine terms vanish

If $x(t)$ is real and odd:

  • Cosine terms vanish

Use symmetry whenever possible to reduce algebra.

Fourier transform for aperiodic signals

The Fourier transform is

$$ X(j\omega) = \int_{-\infty}^{\infty} x(t)e^{-j\omega t}\,dt $$

with inverse

$$ x(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} X(j\omega)e^{j\omega t}\,d\omega $$

It expresses a signal as a continuum of frequencies.

Important properties

PropertyTime domainFrequency domain
Linearity$ax_1 + bx_2$$aX_1 + bX_2$
Time shift$x(t-t_0)$$e^{-j\omega t_0}X(j\omega)$
Modulation$e^{j\omega_0 t}x(t)$$X(j(\omega-\omega_0))$
Convolution$x*h$$XH$
Multiplication$x h$Convolution in frequency

Parseval's relation

Energy is preserved between time and frequency domains:

$$ \int_{-\infty}^{\infty} |x(t)|^2\,dt = \frac{1}{2\pi}\int_{-\infty}^{\infty} |X(j\omega)|^2\,d\omega $$

This is useful for energy calculations and spectral interpretation.

Common transforms

SignalFourier transform
$\delta(t)$$1$
$u(t)$$\pi\delta(\omega) + \frac{1}{j\omega}$ in distribution sense
$e^{-at}u(t),\ a>0$$\frac{1}{a+j\omega}$
Rectangular pulseSinc-shaped spectrum

8. Laplace transform

The Laplace transform is the main tool for continuous-time linear systems with initial conditions.

Definition

$$ X(s) = \int_{-\infty}^{\infty} x(t)e^{-st}\,dt $$

where

$$ s = \sigma + j\omega $$

The unilateral Laplace transform, used heavily in engineering, is

$$ X(s) = \int_0^\infty x(t)e^{-st}\,dt $$

Why use Laplace transforms

  • Turns differential equations into algebraic equations

  • Handles initial conditions naturally

  • Provides a pole-zero view of system behavior

Transfer function

For an LTI system with zero initial conditions:

$$ H(s) = \frac{Y(s)}{X(s)} $$

For rational systems,

$$ H(s) = \frac{N(s)}{D(s)} $$

Poles are roots of $D(s)$ and zeros are roots of $N(s)$.

Stability and causality in the $s$-plane

For a causal continuous-time LTI system:

  • Causality generally requires the ROC to be to the right of the rightmost pole.

  • Stability requires the ROC to include the $j\omega$ axis.

For rational causal systems, stability typically means all poles lie in the left-half plane.

Solving differential equations

Example:

$$ \frac{dy(t)}{dt} + ay(t) = x(t) $$

Taking the Laplace transform with zero initial conditions:

$$ sY(s) + aY(s) = X(s) $$

So

$$ H(s) = \frac{Y(s)}{X(s)} = \frac{1}{s+a} $$

Partial fractions and inverse transform

Workflow:

  1. Compute $Y(s) = H(s)X(s)$.

  2. Factor the denominator.

  3. Use partial fractions.

  4. Match each term to a known inverse transform.

This is the standard route for transient response problems.

9. Z-transform and discrete-time analysis

The Z-transform is the discrete-time counterpart of the Laplace transform.

Definition

$$ X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n} $$

The unilateral form is

$$ X(z) = \sum_{n=0}^{\infty} x[n]z^{-n} $$

Transfer function

For a discrete-time LTI system:

$$ H(z) = \frac{Y(z)}{X(z)} $$

Difference equations

Example:

$$ y[n] - ay[n-1] = x[n] $$

Taking the Z-transform with zero initial conditions gives:

$$ Y(z)(1 - az^{-1}) = X(z) $$

So

$$ H(z) = \frac{1}{1 - az^{-1}} $$

Stability and causality in the $z$-plane

For a causal discrete-time LTI system:

  • The ROC is outside the outermost pole.

  • Stability requires the unit circle $|z| = 1$ to lie in the ROC.

For rational causal systems, stability typically means all poles lie inside the unit circle.

Relationship to Fourier analysis

If the unit circle is in the ROC, then the frequency response is found by evaluating:

$$ H(e^{j\omega}) = H(z)\big|_{z=e^{j\omega}} $$

This connects the Z-transform to steady-state sinusoidal behavior.

10. Sampling and reconstruction

Sampling converts a continuous-time signal into a discrete-time signal:

$$ x[n] = x(nT_s) $$

where $T_s$ is the sampling period and

$$ f_s = \frac{1}{T_s} $$

is the sampling frequency.

Nyquist criterion

To avoid aliasing for a band-limited signal with maximum frequency $f_{max}$:

$$ f_s > 2f_{max} $$

The quantity $2f_{max}$ is the Nyquist rate.

Aliasing

If sampling is too slow, high-frequency components fold into lower frequencies and distort the spectrum.

Common fixes:

  • Increase sampling rate

  • Use an anti-aliasing low-pass filter before sampling

Reconstruction

Ideal reconstruction uses sinc interpolation:

$$ x(t) = \sum_{n=-\infty}^{\infty} x[n]\operatorname{sinc}\left(\frac{t-nT_s}{T_s}\right) $$

This is idealized but useful for theory.

11. State-space viewpoint

State-space models represent systems using first-order vector equations.

Continuous-time form

$$ \dot{\mathbf{x}}(t) = A\mathbf{x}(t) + B\mathbf{u}(t) $$
$$ \mathbf{y}(t) = C\mathbf{x}(t) + D\mathbf{u}(t) $$

Discrete-time form

$$ \mathbf{x}[n+1] = A\mathbf{x}[n] + B\mathbf{u}[n] $$
$$ \mathbf{y}[n] = C\mathbf{x}[n] + D\mathbf{u}[n] $$

Why state-space is useful

  • Handles multiple inputs and outputs

  • Models internal dynamics directly

  • Supports modern control and estimation

Connection to transfer functions

For zero initial conditions:

$$ H(s) = C(sI - A)^{-1}B + D $$

and similarly in discrete time,

$$ H(z) = C(zI - A)^{-1}B + D $$

State-space and transfer-function descriptions are equivalent for linear systems, but each is more convenient in different contexts.

12. Problem-solving workflow

Decide the domain first

Ask:

  • Is the system continuous or discrete?

  • Is the analysis time-domain or frequency-domain?

  • Are initial conditions given?

  • Is the input periodic, impulsive, sinusoidal, or arbitrary?

This decides whether to use convolution, Fourier, Laplace, or Z-transform methods.

Standard workflow for LTI problems

  1. Identify whether the system is linear and time invariant.

  2. Write the model as a differential equation, difference equation, or impulse response.

  3. Choose the best domain:

    • Convolution for direct time-domain output

    • Fourier for steady-state frequency behavior

    • Laplace or Z-transform for transients and initial conditions

  4. Solve algebraically if possible.

  5. Check stability, causality, and physical units.

Common pitfalls

  • Confusing time shift with time reversal

  • Forgetting that convolution reverses one signal

  • Treating $H(s)$ or $H(z)$ as the same thing as a Fourier transform without checking the ROC

  • Ignoring initial conditions in differential equations

  • Mixing continuous-time frequency $\omega$ with discrete-time frequency $\omega$ without noting units

  • Using the wrong sign convention in the complex exponential

Sanity checks

  • Does the answer have the correct units?

  • Does the output remain bounded for a bounded input when the system is stable?

  • Does a shift in the input produce the same shift in the output for an LTI system?

  • Does the result reduce correctly for a simple test case such as a unit impulse or constant input?

13. Formula summary

Core definitions

$$ y(t) = (x * h)(t) = \int_{-\infty}^{\infty} x(\tau)h(t-\tau)\,d\tau $$
$$ y[n] = (x * h)[n] = \sum_{k=-\infty}^{\infty} x[k]h[n-k] $$
$$ H(j\omega) = \int_{-\infty}^{\infty} h(t)e^{-j\omega t}\,dt $$
$$ H(s) = \frac{Y(s)}{X(s)} $$
$$ H(z) = \frac{Y(z)}{X(z)} $$

Fourier transform pair

$$ X(j\omega) = \int_{-\infty}^{\infty} x(t)e^{-j\omega t}\,dt $$
$$ x(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} X(j\omega)e^{j\omega t}\,d\omega $$

Laplace transform pair

$$ X(s) = \int_0^\infty x(t)e^{-st}\,dt $$

Z-transform pair

$$ X(z) = \sum_{n=0}^{\infty} x[n]z^{-n} $$

Stability rules of thumb

  • Continuous-time causal rational system: stable if all poles are in the left-half plane.

  • Discrete-time causal rational system: stable if all poles are inside the unit circle.

Sampling rule

$$ f_s > 2f_{max} $$

Compact intuition

The subject is built around one idea:

An LTI system is fully described by its impulse response, and every other analysis tool is a different way of exploiting that description.

If you can move comfortably between the time domain, frequency domain, and transform domain, most signal-analysis problems become routine.

Sources

  • Engineering LibreTexts

  • Hibbeler, Engineering Mechanics

  • Nilsson and Riedel, Electric Circuits

  • Sedra and Smith, Microelectronic Circuits

  • Oppenheim and Willsky, Signals and Systems

  • Nise, Control Systems Engineering

  • Incropera et al., Fundamentals of Heat and Mass Transfer

  • Fox, McDonald, and Pritchard, Introduction to Fluid Mechanics

  • Groover, Fundamentals of Modern Manufacturing

  • Callister and Rethwisch, Materials Science and Engineering

  • Montgomery, Introduction to Statistical Quality Control

  • Kerzner, Project Management: A Systems Approach to Planning, Scheduling, and Controlling

  • Law, Simulation Modeling and Analysis

  • Fraden, Handbook of Modern Sensors

  • Leake and Borger, Engineering Design Graphics

  • Parell GitHub repository