Unit 5 - Notes

ECE220

Unit 5: The Laplace Transform

1. Introduction

The Laplace Transform is a powerful mathematical tool used for the analysis of continuous-time signals and systems. While the Fourier Transform decomposes a signal into sinusoidal components (basis functions $e^{j\omega t}$ ), it is strictly defined only for stable signals where the integral converges (absolute integrability).

The Laplace Transform generalizes the Fourier Transform by using complex exponentials $e^{st}$ where $s = \sigma + j\omega$ . This introduction of a real part ( $\sigma$ ) allows the transform to converge for a broader class of signals (including unstable ones) and provides a mechanism to analyze transient responses and stability in Linear Time-Invariant (LTI) systems.

Key differences:

Fourier Transform: Physical frequency domain ( $j\omega$ ). Used primarily for steady-state analysis and signal processing.
Laplace Transform: Complex frequency domain ( $s$ -plane). Used for stability analysis, transient response, and solving differential equations.

2. The Laplace Transform

2.1 The Complex Variable $s$

The Laplace variable is defined as a complex number:

s = \sigma + j\omega

where:

$\sigma$ (Sigma) = Neper frequency (Real part, attenuation/growth factor).
$\omega$ (Omega) = Radial frequency (Imaginary part, oscillation factor).

2.2 The Bilateral (Two-Sided) Laplace Transform

Mathematically, the bilateral Laplace transform of a continuous-time signal $x(t)$ is defined as:

X(s) \triangleq \mathcal{L}\{x(t)\} = \int_{-\infty}^{\infty} x(t) e^{-st} dt

2.3 The Unilateral (One-Sided) Laplace Transform

Used primarily for solving differential equations with initial conditions and causal systems ( $t \ge 0$ ).

X(s) = \int_{0^-}^{\infty} x(t) e^{-st} dt

2.4 Relationship with Fourier Transform

Substituting $s = \sigma + j\omega$ into the definition:

X(\sigma + j\omega) = \int_{-\infty}^{\infty} [x(t)e^{-\sigma t}] e^{-j\omega t} dt

Thus, the Laplace Transform of

x(t)

is essentially the Fourier Transform of the signal

x(t)

weighted by an exponential

e^{-\sigma t}

. If

\sigma = 0

(meaning

s = j\omega

), the Laplace Transform reduces to the Fourier Transform, provided the region of convergence includes the imaginary axis.

3. The Region of Convergence (ROC)

The Region of Convergence is the set of values of $s$ (points in the complex plane) for which the integral $X(s)$ converges to a finite value. The ROC is crucial because different time-domain signals can result in the same algebraic expression for $X(s)$ ; the ROC makes the transform unique.

3.1 Properties of the ROC

Strip Shape: The ROC consists of strips parallel to the $j\omega$ -axis in the $s$ -plane.
No Poles: The ROC cannot contain any poles. (A pole is a value of $s$ where $X(s) \to \infty$ ).
Finite Duration Signals: If $x(t)$ is of finite duration and absolutely integrable, the ROC is the entire $s$ -plane (possibly excluding $s=0$ or $s=\infty$ ).
Right-Sided Signals: If $x(t)$ is right-sided (e.g., $x(t) = 0$ for $t < T_1$ ), the ROC is of the form $\text{Re}\{s\} > \sigma_{\max}$ , where $\sigma_{\max}$ is the real part of the rightmost pole.
Left-Sided Signals: If $x(t)$ is left-sided (e.g., $x(t) = 0$ for $t > T_2$ ), the ROC is of the form $\text{Re}\{s\} < \sigma_{\min}$ , where $\sigma_{\min}$ is the real part of the leftmost pole.
Two-Sided Signals: If $x(t)$ is two-sided (infinite duration in both directions), the ROC is a vertical strip bounded by two poles: $\sigma_1 < \text{Re}\{s\} < \sigma_2$ .
Stability: For a system to be stable, the ROC must include the $j\omega$ -axis ( $\text{Re}\{s\} = 0$ ).

4. The Inverse Laplace Transform

The Inverse Laplace Transform recovers $x(t)$ from $X(s)$ .

4.1 Formal Definition (Bromwich Integral)

x(t) = \frac{1}{2\pi j} \int_{\sigma - j\infty}^{\sigma + j\infty} X(s) e^{st} ds

This involves contour integration in the complex plane and is rarely used for standard engineering problems.

4.2 Partial Fraction Expansion (Practical Method)

For rational transforms (ratio of polynomials), we use Partial Fraction Expansion (PFE).

Form of X(s):

X(s) = \frac{N(s)}{D(s)} = \frac{b_M s^M + \dots + b_0}{a_N s^N + \dots + a_0}

Roots of $N(s)$ : Zeros (where $X(s) = 0$ ).
Roots of $D(s)$ : Poles (where $X(s) \to \infty$ ).

Procedure:

Factor the denominator $D(s)$ .
Express $X(s)$ as a sum of simpler fractions:
$X(s) = \frac{C_1}{s - p_1} + \frac{C_2}{s - p_2} + \dots + \frac{C_N}{s - p_N}$
Determine the coefficients ( $C_k$ ) using the residue method (cover-up method).
Apply the inverse transform to each term using standard pairs (e.g., $\frac{1}{s+a} \leftrightarrow e^{-at}u(t)$ for right-sided ROC).

5. Geometric Evaluation of Fourier Transform from Pole-Zero Plot

Since $H(j\omega) = H(s)|_{s=j\omega}$ , we can evaluate the magnitude and phase of the frequency response geometrically using the pole-zero plot in the $s$ -plane.

Given a transfer function in factored form:

H(s) = K \frac{\prod_{k=1}^M (s - z_k)}{\prod_{k=1}^N (s - p_k)}

To evaluate at a specific frequency $\omega_0$ , we set $s = j\omega_0$ .
We define vectors from each pole and zero to the point $j\omega_0$ on the imaginary axis.

Zero Vector: $V_{z_k} = (j\omega_0 - z_k)$
Pole Vector: $V_{p_k} = (j\omega_0 - p_k)$

5.1 Magnitude Evaluation

|H(j\omega_0)| = |K| \frac{\prod |j\omega_0 - z_k|}{\prod |j\omega_0 - p_k|} = |K| \frac{\text{Product of lengths of vectors from zeros to } j\omega_0}{\text{Product of lengths of vectors from poles to } j\omega_0}

5.2 Phase Evaluation

\angle H(j\omega_0) = \sum \angle(j\omega_0 - z_k) - \sum \angle(j\omega_0 - p_k)

\angle H(j\omega_0) = (\text{Sum of angles of zero vectors}) - (\text{Sum of angles of pole vectors})

6. Properties of the Laplace Transform

Let $x(t) \leftrightarrow X(s)$ with ROC $R$ .

6.1 Linearity

a x_1(t) + b x_2(t) \leftrightarrow a X_1(s) + b X_2(s)

ROC: At least $R_1 \cap R_2$ .

6.2 Time Shifting

x(t - t_0) \leftrightarrow e^{-st_0} X(s)

ROC: Same as $R$ .

6.3 Shifting in the s-Domain

e^{s_0 t} x(t) \leftrightarrow X(s - s_0)

ROC: $R + \text{Re}\{s_0\}$ (shifted region).

6.4 Time Scaling

x(at) \leftrightarrow \frac{1}{|a|} X\left(\frac{s}{a}\right)

ROC: Scaled version of $R$ .

6.5 Time Differentiation

Bilateral:

\frac{d x(t)}{dt} \leftrightarrow s X(s)

Unilateral (useful for circuits/initial conditions):

\frac{d x(t)}{dt} \leftrightarrow s X(s) - x(0^-)

6.6 Integration in Time

\int_{-\infty}^{t} x(\tau) d\tau \leftrightarrow \frac{1}{s} X(s)

ROC: $R \cap \{ \text{Re}\{s\} > 0 \}$ .

6.7 Convolution Property

x(t) * h(t) \leftrightarrow X(s) H(s)

This property converts the complex convolution operation in the time domain into simple algebraic multiplication in the

s

-domain.

6.8 Initial and Final Value Theorems (Unilateral)

Initial Value Theorem: $x(0^+) = \lim_{s \to \infty} sX(s)$ (strictly proper rational $X(s)$ ).
Final Value Theorem: $\lim_{t \to \infty} x(t) = \lim_{s \to 0} sX(s)$ (Valid only if all poles of $sX(s)$ are in the Left Half Plane).

7. Analysis and Characterization of LTI Systems using Laplace Transforms

7.1 The Transfer Function

For an LTI system with input $x(t)$ and output $y(t)$ :

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

H(s)

is the Laplace Transform of the impulse response

h(t)

7.2 Causality

A system is causal if $h(t) = 0$ for $t < 0$ .

Condition on $H(s)$ : The ROC of $H(s)$ must be a right-half plane (RHP) defined by $\text{Re}\{s\} > \sigma_{\max}$ (to the right of the rightmost pole).

7.3 Stability (BIBO Stability)

A system is Bounded-Input Bounded-Output (BIBO) stable if $\int_{-\infty}^{\infty} |h(t)| dt < \infty$ .

Condition on $H(s)$ : The ROC of $H(s)$ must include the imaginary axis ( $j\omega$ -axis).
Causal Stable Systems: All poles of $H(s)$ must lie strictly in the Left Half Plane (LHP) (i.e., $\text{Re}\{p_k\} < 0$ ).

7.4 Solving LTI Systems with Differential Equations

LTI systems are often described by Linear Constant Coefficient Differential Equations (LCCDE):

\sum_{k=0}^N a_k \frac{d^k y(t)}{dt^k} = \sum_{k=0}^M b_k \frac{d^k x(t)}{dt^k}

Applying the Laplace Transform:

H(s) = \frac{Y(s)}{X(s)} = \frac{\sum_{k=0}^M b_k s^k}{\sum_{k=0}^N a_k s^k}

This converts differential equations into algebraic equations.

8. Software Simulation of System Representation and Pole-Zero Analysis

Modern engineering relies on software tools like MATLAB or Python (SciPy/SymPy) to analyze systems.

8.1 Representation of Systems

Systems are represented by the numerator and denominator coefficient vectors.

Example (MATLAB):
$H(s) = \frac{2s + 1}{s^2 + 3s + 2}$

MATLAB

num = [2 1];      % Coefficients of numerator: 2s + 1
den = [1 3 2];    % Coefficients of denominator: 1s^2 + 3s + 2
sys = tf(num, den); % Create transfer function object

8.2 Pole-Zero Analysis

Software can automatically compute roots and visualize them.

MATLAB Code: