7.2 Laplace Transforms

7.2 Laplace Transforms

1. Definition and Properties

1.1 Definition of Laplace Transform

1. Bilateral (Two-sided) Laplace Transform: $X(s) = \mathcal{L}\{x(t)\} = \int_{-\infty}^{\infty} x(t) e^{-st} dt$ where $s = \sigma + j\omega$ is a complex variable.

2. Unilateral (One-sided) Laplace Transform: $X(s) = \mathcal{L}\{x(t)\} = \int_{0^-}^{\infty} x(t) e^{-st} dt$

   • More common in engineering applications.

   • Lower limit $0^-$ includes any impulses at $t=0$.

   • Assumes $x(t)=0$ for $t<0$.

3. Region of Convergence (ROC):

   • Set of $s$ values for which the integral converges.

   • Must be specified along with $X(s)$ for complete representation.

   • For causal signals: ROC is right of a vertical line in s-plane.

1.2 Existence Conditions

For $x(t)$ to have a Laplace transform, it must:

1. Be piecewise continuous on $0 \leq t \leq A$ for any finite $A>0$.

2. Be of exponential order: $|x(t)| \leq Me^{at}$ for $t>T$.

3. ROC must exist (not empty).

1.3 Key Properties of Laplace Transform

1.3.1 Linearity

$\mathcal{L}\{a_1x_1(t) + a_2x_2(t)\} = a_1X_1(s) + a_2X_2(s)$

1.3.2 Time Shifting

$\mathcal{L}\{x(t-t_0)u(t-t_0)\} = e^{-st_0}X(s)$

1.3.3 Frequency Shifting (s-Shifting)

$\mathcal{L}\{e^{at}x(t)\} = X(s-a)$

1.3.4 Time Scaling

$\mathcal{L}\{x(at)\} = \frac{1}{|a|}X\left(\frac{s}{a}\right)$

1.3.5 Time Differentiation

1. First derivative: $\mathcal{L}\left\{\frac{dx(t)}{dt}\right\} = sX(s) - x(0^-)$

2. Second derivative: $\mathcal{L}\left\{\frac{d^2x(t)}{dt^2}\right\} = s^2X(s) - sx(0^-) - x'(0^-)$

3. n-th derivative: $\mathcal{L}\left\{\frac{d^nx(t)}{dt^n}\right\} = s^nX(s) - \sum_{k=1}^{n} s^{n-k}x^{(k-1)}(0^-)$

1.3.6 Time Integration

$\mathcal{L}\left\{\int_{0^-}^{t} x(\tau)d\tau\right\} = \frac{1}{s}X(s)$

1.3.7 Frequency Differentiation

$\mathcal{L}\{t^n x(t)\} = (-1)^n \frac{d^n}{ds^n}X(s)$

1.3.8 Frequency Integration

$\mathcal{L}\left\{\frac{x(t)}{t}\right\} = \int_{s}^{\infty} X(u) du$

1.3.9 Convolution in Time

$\mathcal{L}\{x_1(t) * x_2(t)\} = X_1(s)X_2(s)$

1.3.10 Initial Value Theorem

$x(0^+) = \lim_{s \to \infty} sX(s)$

1.3.11 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 left half-plane.

1.3.12 Time Periodicity

If $x(t) = x(t+T)$ for $t \geq 0$: $\mathcal{L}\{x(t)\} = \frac{X_1(s)}{1 - e^{-sT}}$ where $X_1(s) = \mathcal{L}\{x(t)\}$ for $0 \leq t \leq T$.

2. Transforms of Common Functions

2.1 Elementary Functions

1. Unit Impulse: $\mathcal{L}\{\delta(t)\} = 1$ ROC: All $s$.

2. Unit Step: $\mathcal{L}\{u(t)\} = \frac{1}{s}$ ROC: Re($s$) > 0.

3. Ramp Function: $\mathcal{L}\{t u(t)\} = \frac{1}{s^2}$ ROC: Re($s$) > 0.

4. n-th Power: $\mathcal{L}\{t^n u(t)\} = \frac{n!}{s^{n+1}}$ ROC: Re($s$) > 0.

2.2 Exponential Functions

1. Real Exponential: $\mathcal{L}\{e^{at}u(t)\} = \frac{1}{s-a}$ ROC: Re($s$) > $a$.

2. Complex Exponential: $\mathcal{L}\{e^{j\omega_0 t}u(t)\} = \frac{1}{s-j\omega_0}$ ROC: Re($s$) > 0.

2.3 Trigonometric Functions

1. Cosine: $\mathcal{L}\{\cos(\omega_0 t)u(t)\} = \frac{s}{s^2 + \omega_0^2}$ ROC: Re($s$) > 0.

2. Sine: $\mathcal{L}\{\sin(\omega_0 t)u(t)\} = \frac{\omega_0}{s^2 + \omega_0^2}$ ROC: Re($s$) > 0.

3. Damped Cosine: $\mathcal{L}\{e^{at}\cos(\omega_0 t)u(t)\} = \frac{s-a}{(s-a)^2 + \omega_0^2}$ ROC: Re($s$) > $a$.

4. Damped Sine: $\mathcal{L}\{e^{at}\sin(\omega_0 t)u(t)\} = \frac{\omega_0}{(s-a)^2 + \omega_0^2}$ ROC: Re($s$) > $a$.

2.4 Hyperbolic Functions

1. Hyperbolic Cosine: $\mathcal{L}\{\cosh(at)u(t)\} = \frac{s}{s^2 - a^2}$ ROC: Re($s$) > |$a$|.

2. Hyperbolic Sine: $\mathcal{L}\{\sinh(at)u(t)\} = \frac{a}{s^2 - a^2}$ ROC: Re($s$) > |$a$|.

2.5 Summary Table of Common Transforms

Time Function $x(t)$	Laplace Transform $X(s)$	ROC
$\delta(t)$	1	All $s$
$u(t)$	$\frac{1}{s}$	Re($s$) > 0
$t u(t)$	$\frac{1}{s^2}$	Re($s$) > 0
$t^n u(t)$	$\frac{n!}{s^{n+1}}$	Re($s$) > 0
$e^{at}u(t)$	$\frac{1}{s-a}$	Re($s$) > $a$
$te^{at}u(t)$	$\frac{1}{(s-a)^2}$	Re($s$) > $a$
$\cos(\omega t)u(t)$	$\frac{s}{s^2+\omega^2}$	Re($s$) > 0
$\sin(\omega t)u(t)$	$\frac{\omega}{s^2+\omega^2}$	Re($s$) > 0

3. Inverse Laplace Transform Techniques

3.1 Definition of Inverse Laplace Transform

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

• Integration along vertical line in ROC.

• In practice, use partial fraction expansion and transform tables.

3.2 Partial Fraction Expansion (PFE)

3.2.1 Case 1: Distinct Real Poles

If $X(s) = \frac{N(s)}{(s-p_1)(s-p_2)\cdots(s-p_n)}$: $X(s) = \frac{A_1}{s-p_1} + \frac{A_2}{s-p_2} + \cdots + \frac{A_n}{s-p_n}$ where $A_k = (s-p_k)X(s)\big|_{s=p_k}$

3.2.2 Case 2: Repeated Real Poles

If $X(s) = \frac{N(s)}{(s-p)^r}$: $X(s) = \frac{A_1}{s-p} + \frac{A_2}{(s-p)^2} + \cdots + \frac{A_r}{(s-p)^r}$ where $A_k = \frac{1}{(r-k)!} \frac{d^{r-k}}{ds^{r-k}}[(s-p)^r X(s)]\big|_{s=p}$

3.2.3 Case 3: Complex Conjugate Poles

If $X(s) = \frac{N(s)}{(s-\alpha-j\beta)(s-\alpha+j\beta)}$: Write as: $X(s) = \frac{A}{s-(\alpha+j\beta)} + \frac{A^*}{s-(\alpha-j\beta)}$ where $A$ and $A^*$ are complex conjugates.

3.3 Heaviside's Cover-up Method (For Distinct Poles)

For $X(s) = \frac{N(s)}{(s-p_1)(s-p_2)\cdots(s-p_n)}$:

1. To find $A_k$, "cover up" $(s-p_k)$ in denominator.

2. Evaluate remaining expression at $s=p_k$.

3. $A_k = \frac{N(p_k)}{\prod_{i\neq k}(p_k-p_i)}$

3.4 Complete Procedure for Inverse Transform

1. Ensure proper fraction: If degree of numerator ≥ degree of denominator, perform long division first.

2. Factor denominator completely.

3. Set up partial fractions based on pole types.

4. Solve for coefficients using:

   • Heaviside's method for distinct poles.

   • System of equations for repeated/complex poles.

5. Use transform table to find inverse of each term.

6. Combine results to get $x(t)$.

3.5 Example of PFE

Find $\mathcal{L}^{-1}\left\{\frac{s+3}{s^2+3s+2}\right\}$:

1. Factor denominator: $s^2+3s+2 = (s+1)(s+2)$

2. Partial fractions: $\frac{s+3}{(s+1)(s+2)} = \frac{A}{s+1} + \frac{B}{s+2}$

3. Solve: $A = (s+1)X(s)\big|_{s=-1} = \frac{-1+3}{-1+2} = 2$ $B = (s+2)X(s)\big|_{s=-2} = \frac{-2+3}{-2+1} = -1$

4. Inverse transform: $x(t) = 2e^{-t}u(t) - e^{-2t}u(t)$

4. Application to Solving Linear Differential Equations

4.1 General Approach

1. Take Laplace transform of both sides of differential equation.

2. Use derivative properties to convert to algebraic equation in $s$.

3. Solve algebraic equation for $X(s)$.

4. Apply inverse Laplace transform to get $x(t)$.

4.2 Solving nth-Order LTI Differential Equations

Consider: $\sum_{k=0}^{n} a_k \frac{d^ky(t)}{dt^k} = \sum_{k=0}^{m} b_k \frac{d^kx(t)}{dt^k}$

1. Transform equation: $\sum_{k=0}^{n} a_k [s^kY(s) - \sum_{i=0}^{k-1} s^{k-1-i}y^{(i)}(0^-)] = \sum_{k=0}^{m} b_k [s^kX(s) - \sum_{i=0}^{k-1} s^{k-1-i}x^{(i)}(0^-)]$

2. Group terms: $Y(s)\sum_{k=0}^{n} a_k s^k = X(s)\sum_{k=0}^{m} b_k s^k + IC \text{ terms}$

3. Solve for $Y(s)$: $Y(s) = \frac{\sum_{k=0}^{m} b_k s^k}{\sum_{k=0}^{n} a_k s^k} X(s) + \frac{\text{Initial condition terms}}{\sum_{k=0}^{n} a_k s^k}$

4.3 Example: Solving Second-Order DE

Solve: $\frac{d^2y}{dt^2} + 5\frac{dy}{dt} + 6y = x(t)$ with $x(t)=u(t)$, $y(0)=1$, $y'(0)=2$.

1. Transform: $[s^2Y(s) - sy(0) - y'(0)] + 5[sY(s) - y(0)] + 6Y(s) = \frac{1}{s}$ $[s^2Y(s) - s - 2] + 5[sY(s) - 1] + 6Y(s) = \frac{1}{s}$

2. Solve for $Y(s)$: $(s^2 + 5s + 6)Y(s) = \frac{1}{s} + s + 2 + 5$ $Y(s) = \frac{s^2 + 7s + 1}{s(s^2 + 5s + 6)}$

3. Partial fractions: $Y(s) = \frac{1/6}{s} + \frac{1}{s+2} - \frac{7/6}{s+3}$

4. Inverse transform: $y(t) = \frac{1}{6}u(t) + e^{-2t}u(t) - \frac{7}{6}e^{-3t}u(t)$

4.4 Advantages of Laplace Transform Method

1. Handles initial conditions naturally.

2. Converts calculus to algebra.

3. Works for any forcing function with known transform.

4. Provides complete solution (transient + steady-state).

5. Transfer Function and Frequency Response

5.1 Transfer Function Definition

For LTI system described by differential equation: $H(s) = \frac{Y(s)}{X(s)} = \frac{\sum_{k=0}^{m} b_k s^k}{\sum_{k=0}^{n} a_k s^k}$

• Assumes zero initial conditions.

• Characterizes system completely in s-domain.

5.2 Properties of Transfer Function

1. Poles and Zeros:

   • Poles: Roots of denominator (characteristic equation).

   • Zeros: Roots of numerator.

2. Order: Highest power of $s$ in denominator.

3. Proper/Strictly proper: Degree(num) ≤/＜ degree(denom).

5.3 Relationship with Impulse Response

$H(s) = \mathcal{L}\{h(t)\}$ $h(t) = \mathcal{L}^{-1}\{H(s)\}$

5.4 Frequency Response from Transfer Function

1. Definition: Frequency response is transfer function evaluated on imaginary axis. $H(j\omega) = H(s)\big|_{s=j\omega}$

2. Magnitude Response: $|H(j\omega)| = \sqrt{H(j\omega)H^*(j\omega)}$

3. Phase Response: $\angle H(j\omega) = \tan^{-1}\left(\frac{\text{Im}\{H(j\omega)\}}{\text{Re}\{H(j\omega)\}}\right)$

4. Bode Plot Construction:

   • Plot magnitude (dB) and phase vs. frequency (log scale).

   • Break frequencies at poles and zeros.

5.5 Stability Analysis from Transfer Function

1. BIBO Stability Criterion: System is stable if all poles of $H(s)$ have negative real parts.

   • All poles in left half of s-plane.

2. Routh-Hurwitz Criterion: Determines stability without finding poles explicitly.

5.6 Example: Finding Frequency Response

For $H(s) = \frac{1}{s+1}$:

1. Frequency response: $H(j\omega) = \frac{1}{j\omega + 1}$

2. Magnitude: $|H(j\omega)| = \frac{1}{\sqrt{\omega^2 + 1}}$

3. Phase: $\angle H(j\omega) = -\tan^{-1}(\omega)$

5.7 System Classification from Transfer Function

1. Low-pass filter: $H(0) \neq 0$, $H(\infty) = 0$

2. High-pass filter: $H(0) = 0$, $H(\infty) \neq 0$

3. Band-pass filter: $H(0) = 0$, $H(\infty) = 0$, peak at mid-frequency

4. Band-stop filter: $H(0) \neq 0$, $H(\infty) \neq 0$, notch at mid-frequency

6. Important Formulas Summary

6.1 Key Laplace Transform Properties

1. Differentiation: $\mathcal{L}\{f'(t)\} = sF(s) - f(0^-)$

2. Integration: $\mathcal{L}\left\{\int_{0^-}^{t} f(\tau)d\tau\right\} = \frac{1}{s}F(s)$

3. Convolution: $\mathcal{L}\{f(t)*g(t)\} = F(s)G(s)$

4. Initial Value: $f(0^+) = \lim_{s\to\infty} sF(s)$

5. Final Value: $f(\infty) = \lim_{s\to 0} sF(s)$

6.2 Common Inverse Transforms

1. $\mathcal{L}^{-1}\left\{\frac{1}{s-a}\right\} = e^{at}u(t)$

2. $\mathcal{L}^{-1}\left\{\frac{1}{(s-a)^n}\right\} = \frac{t^{n-1}e^{at}}{(n-1)!}u(t)$

3. $\mathcal{L}^{-1}\left\{\frac{s}{s^2+\omega^2}\right\} = \cos(\omega t)u(t)$

4. $\mathcal{L}^{-1}\left\{\frac{\omega}{s^2+\omega^2}\right\} = \sin(\omega t)u(t)$

6.3 Partial Fraction Expansion Rules

1. Distinct poles: $X(s) = \sum_{i=1}^{n} \frac{A_i}{s-p_i}$

2. Repeated poles: $X(s) = \sum_{i=1}^{r} \frac{A_i}{(s-p)^i} + \text{other terms}$

3. Complex poles: $X(s) = \frac{A}{s-(\alpha+j\beta)} + \frac{A^*}{s-(\alpha-j\beta)}$

6.4 Stability Conditions

1. Continuous-time system: All poles have Re($s$) < 0.

2. Transfer function stability: $H(s)$ analytic in right half-plane.

3. Frequency response exists if imaginary axis is in ROC.

6.5 Solving Differential Equations Steps

1. Transform DE to algebraic equation.

2. Incorporate initial conditions.

3. Solve for $Y(s)$.

4. Apply inverse transform.

5. Verify solution satisfies original DE and ICs.