Laplace & Z-Transform for Signals & Systems: GATE Preparation Notes

Laplace Transform

Laplace Transform - Definition

Bilateral Laplace Transform

\[\begin{aligned} X(s) = \mathcal{L}\{x(t)\} = \int_{-\infty}^{\infty} x(t) e^{-st} dt \end{aligned}\]
where \(s = \sigma + j\omega\) is the complex frequency.

Unilateral Laplace Transform

\[\begin{aligned} X(s) = \mathcal{L}\{x(t)\} = \int_{0^-}^{\infty} x(t) e^{-st} dt \end{aligned}\]
Most commonly used in engineering applications.

Inverse Laplace Transform

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

Region of Convergence (ROC)

Definition

The set of values of \(s\) for which the Laplace transform converges.

ROC Properties

  • ROC is a vertical strip in the s-plane

  • ROC cannot contain poles

  • For right-sided signals: \(\Re(s) > \sigma_0\)

  • For left-sided signals: \(\Re(s) < \sigma_0\)

  • For two-sided signals: \(\sigma_1 < \Re(s) < \sigma_2\)

  • For finite-duration signals: entire s-plane (except possibly \(s = 0\) or \(s = \infty\))

Stability Condition

A system is stable if and only if the ROC includes the imaginary axis (\(\sigma = 0\)).

Common Laplace Transform Pairs

Signal Laplace Transform ROC
\(\delta(t)\) \(1\) All \(s\)
\(u(t)\) \(\frac{1}{s}\) \(\Re(s) > 0\)
\(t^n u(t)\) \(\frac{n!}{s^{n+1}}\) \(\Re(s) > 0\)
\(e^{-at}u(t)\) \(\frac{1}{s+a}\) \(\Re(s) > -a\)
\(t e^{-at}u(t)\) \(\frac{1}{(s+a)^2}\) \(\Re(s) > -a\)
\(\sin(\omega_0 t)u(t)\) \(\frac{\omega_0}{s^2 + \omega_0^2}\) \(\Re(s) > 0\)
\(\cos(\omega_0 t)u(t)\) \(\frac{s}{s^2 + \omega_0^2}\) \(\Re(s) > 0\)
\(e^{-at}\sin(\omega_0 t)u(t)\) \(\frac{\omega_0}{(s+a)^2 + \omega_0^2}\) \(\Re(s) > -a\)
\(e^{-at}\cos(\omega_0 t)u(t)\) \(\frac{s+a}{(s+a)^2 + \omega_0^2}\) \(\Re(s) > -a\)

Properties of Laplace Transform

Linearity

\[\begin{aligned} ax_1(t) + bx_2(t) \leftrightarrow aX_1(s) + bX_2(s) \end{aligned}\]

Time Shifting

\[\begin{aligned} x(t-t_0)u(t-t_0) \leftrightarrow e^{-st_0}X(s) \end{aligned}\]

Frequency Shifting

\[\begin{aligned} e^{s_0 t}x(t) \leftrightarrow X(s-s_0) \end{aligned}\]

Time Scaling

\[\begin{aligned} x(at) \leftrightarrow \frac{1}{|a|}X\left(\frac{s}{a}\right) \end{aligned}\]

Differentiation and Integration Properties

Time Differentiation

\[\begin{aligned} \frac{dx(t)}{dt} &\leftrightarrow sX(s) - x(0^-) \\ \frac{d^2x(t)}{dt^2} &\leftrightarrow s^2X(s) - sx(0^-) - x'(0^-) \\ \frac{d^nx(t)}{dt^n} &\leftrightarrow s^nX(s) - \sum_{k=0}^{n-1} s^{n-1-k}x^{(k)}(0^-) \end{aligned}\]

Time Integration

\[\begin{aligned} \int_{0^-}^{t} x(\tau) d\tau \leftrightarrow \frac{X(s)}{s} \end{aligned}\]

Frequency Differentiation

\[\begin{aligned} tx(t) \leftrightarrow -\frac{dX(s)}{ds} \end{aligned}\]

Convolution and Final Value Theorems

Convolution

\[\begin{aligned} x_1(t) * x_2(t) \leftrightarrow X_1(s)X_2(s) \end{aligned}\]

Initial Value Theorem

\[\begin{aligned} \lim_{t \to 0^+} x(t) = \lim_{s \to \infty} sX(s) \end{aligned}\]
provided the limit exists.

Final Value Theorem

\[\begin{aligned} \lim_{t \to \infty} x(t) = \lim_{s \to 0} sX(s) \end{aligned}\]
provided \(x(t)\) has a final value and all poles of \(sX(s)\) are in the left half-plane.

System Analysis Using Laplace Transform

Transfer Function

Definition

For a linear time-invariant system:

\[\begin{aligned} H(s) = \frac{Y(s)}{X(s)} \end{aligned}\]
where \(Y(s) = \mathcal{L}\{y(t)\}\) and \(X(s) = \mathcal{L}\{x(t)\}\).

Relationship to Impulse Response

\[\begin{aligned} H(s) = \mathcal{L}\{h(t)\} \end{aligned}\]

System Response

\[\begin{aligned} Y(s) &= H(s)X(s) \\ y(t) &= h(t) * x(t) \end{aligned}\]

Poles, Zeros, and Stability

Transfer Function Form

\[\begin{aligned} H(s) = K \frac{N(s)}{D(s)} = K \frac{(s-z_1)(s-z_2)\cdots(s-z_m)}{(s-p_1)(s-p_2)\cdots(s-p_n)} \end{aligned}\]

Definitions

  • Zeros: Values of \(s\) where \(H(s) = 0\)

  • Poles: Values of \(s\) where \(H(s) = \infty\)

  • Order: \(n\) = number of poles, \(m\) = number of zeros

Stability Conditions

  • BIBO Stable: All poles in left half-plane (\(\Re(p_i) < 0\))

  • Marginally Stable: Simple poles on imaginary axis, rest in LHP

  • Unstable: At least one pole in right half-plane or repeated poles on imaginary axis

Partial Fraction Expansion

Distinct Poles

\[\begin{aligned} H(s) = \frac{N(s)}{(s-p_1)(s-p_2)\cdots(s-p_n)} = \sum_{i=1}^{n} \frac{A_i}{s-p_i} \end{aligned}\]
where \(A_i = \lim_{s \to p_i} (s-p_i)H(s)\) (Residue method)

Repeated Poles

For a pole \(p_1\) of multiplicity \(r\):

\[\begin{aligned} H(s) = \frac{A_{1,r}}{s-p_1} + \frac{A_{1,r-1}}{(s-p_1)^2} + \cdots + \frac{A_{1,1}}{(s-p_1)^r} + \cdots \end{aligned}\]
where \(A_{1,k} = \frac{1}{(r-k)!} \lim_{s \to p_1} \frac{d^{r-k}}{ds^{r-k}}[(s-p_1)^r H(s)]\)

Complex Conjugate Poles

Combine conjugate terms to get real coefficients in time domain.

Z-Transform

Z-Transform - Definition

Bilateral Z-Transform

\[\begin{aligned} X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n] z^{-n} \end{aligned}\]

Unilateral Z-Transform

\[\begin{aligned} X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=0}^{\infty} x[n] z^{-n} \end{aligned}\]

Inverse Z-Transform

\[\begin{aligned} x[n] = \mathcal{Z}^{-1}\{X(z)\} = \frac{1}{2\pi j} \oint_C X(z) z^{n-1} dz \end{aligned}\]
where \(C\) is a counterclockwise contour in the ROC.

Z-Transform ROC

ROC Properties

  • ROC is an annular region in the z-plane

  • ROC cannot contain poles

  • For right-sided sequences: \(|z| > R_1\)

  • For left-sided sequences: \(|z| < R_2\)

  • For two-sided sequences: \(R_1 < |z| < R_2\)

  • For finite-duration sequences: entire z-plane (except possibly \(z = 0\) or \(z = \infty\))

Stability Condition

A discrete-time system is stable if and only if the ROC includes the unit circle (\(|z| = 1\)).

Common Z-Transform Pairs

Sequence Z-Transform ROC
\(\delta[n]\) \(1\) All \(z\)
\(u[n]\) \(\frac{z}{z-1}\) \(|z| > 1\)
\(n u[n]\) \(\frac{z}{(z-1)^2}\) \(|z| > 1\)
\(a^n u[n]\) \(\frac{z}{z-a}\) \(|z| > |a|\)
\(n a^n u[n]\) \(\frac{az}{(z-a)^2}\) \(|z| > |a|\)
\(-a^n u[-n-1]\) \(\frac{z}{z-a}\) \(|z| < |a|\)
\(\cos(\omega_0 n)u[n]\) \(\frac{z(z-\cos\omega_0)}{z^2-2z\cos\omega_0+1}\) \(|z| > 1\)
\(\sin(\omega_0 n)u[n]\) \(\frac{z\sin\omega_0}{z^2-2z\cos\omega_0+1}\) \(|z| > 1\)
\(a^n\cos(\omega_0 n)u[n]\) \(\frac{z(z-a\cos\omega_0)}{z^2-2az\cos\omega_0+a^2}\) \(|z| > |a|\)

Properties of Z-Transform

Linearity

\[\begin{aligned} ax_1[n] + bx_2[n] \leftrightarrow aX_1(z) + bX_2(z) \end{aligned}\]

Time Shifting

\[\begin{aligned} x[n-n_0] &\leftrightarrow z^{-n_0}X(z) \\ x[n+n_0] &\leftrightarrow z^{n_0}X(z) - \sum_{k=0}^{n_0-1} x[k]z^{n_0-k} \end{aligned}\]

Time Scaling

\[\begin{aligned} a^n x[n] \leftrightarrow X(z/a) \end{aligned}\]

Time Reversal

\[\begin{aligned} x[-n] \leftrightarrow X(z^{-1}) \end{aligned}\]

More Z-Transform Properties

Differentiation in z-domain

\[\begin{aligned} nx[n] \leftrightarrow -z\frac{dX(z)}{dz} \end{aligned}\]

Convolution

\[\begin{aligned} x_1[n] * x_2[n] \leftrightarrow X_1(z)X_2(z) \end{aligned}\]

Initial Value Theorem

\[\begin{aligned} x[0] = \lim_{z \to \infty} X(z) \end{aligned}\]

Final Value Theorem

\[\begin{aligned} \lim_{n \to \infty} x[n] = \lim_{z \to 1} (z-1)X(z) \end{aligned}\]
provided the limit exists and all poles of \((z-1)X(z)\) are inside the unit circle.

Discrete-Time System Analysis

Discrete-Time Transfer Function

Definition

\[\begin{aligned} H(z) = \frac{Y(z)}{X(z)} = \frac{\sum_{k=0}^{M} b_k z^{-k}}{\sum_{k=0}^{N} a_k z^{-k}} \end{aligned}\]

Difference Equation

\[\begin{aligned} \sum_{k=0}^{N} a_k y[n-k] = \sum_{k=0}^{M} b_k x[n-k] \end{aligned}\]

Impulse Response

\[\begin{aligned} H(z) = \mathcal{Z}\{h[n]\} \end{aligned}\]

System Response

\[\begin{aligned} Y(z) &= H(z)X(z) \\ y[n] &= h[n] * x[n] \end{aligned}\]

Stability Analysis

BIBO Stability

A discrete-time LTI system is BIBO stable if and only if:

\[\begin{aligned} \sum_{n=-\infty}^{\infty} |h[n]| < \infty \end{aligned}\]

Pole-Zero Analysis

  • Stable: All poles inside unit circle (\(|p_i| < 1\))

  • Marginally Stable: Simple poles on unit circle, rest inside

  • Unstable: At least one pole outside unit circle or repeated poles on unit circle

Causal vs Anti-causal

  • Causal: Right-sided ROC (\(|z| > R\))

  • Anti-causal: Left-sided ROC (\(|z| < R\))

  • Two-sided: Annular ROC (\(R_1 < |z| < R_2\))

Inverse Transform Methods

Inverse Laplace Transform Methods

1. Partial Fraction Expansion

Most common method for rational functions.

2. Residue Method

\[\begin{aligned} x(t) = \sum_{i} \text{Res}[X(s)e^{st}, s_i] \quad \text{for } t > 0 \end{aligned}\]

3. Convolution Method

If \(X(s) = X_1(s)X_2(s)\), then:

\[\begin{aligned} x(t) = x_1(t) * x_2(t) \end{aligned}\]

4. Long Division

For improper rational functions (numerator degree \(\geq\) denominator degree).

Inverse Z-Transform Methods

1. Partial Fraction Expansion

Convert to partial fractions and use known pairs.

2. Power Series Expansion

\[\begin{aligned} X(z) = \sum_{n=0}^{\infty} x[n] z^{-n} \end{aligned}\]
Coefficient of \(z^{-n}\) gives \(x[n]\).

3. Residue Method

\[\begin{aligned} x[n] = \sum_{i} \text{Res}[X(z)z^{n-1}, z_i] \quad \text{for } n \geq 0 \end{aligned}\]

4. Long Division

Direct division of numerator by denominator.

Relationship Between Transforms

Laplace to Fourier Transform

Relationship

\[\begin{aligned} X(\omega) = X(s)\Big|_{s=j\omega} \end{aligned}\]
provided the Fourier transform exists.

Condition

The imaginary axis (\(s = j\omega\)) must be in the ROC of \(X(s)\).

Z-Transform to DTFT

Relationship

\[\begin{aligned} X(\omega) = X(z)\Big|_{z=e^{j\omega}} \end{aligned}\]
provided the DTFT exists.

Condition

The unit circle (\(|z| = 1\)) must be in the ROC of \(X(z)\).

Laplace to Z-Transform

Sampling Relationship

For a sampled signal \(x_s(t) = \sum_{n=-\infty}^{\infty} x(nT)\delta(t-nT)\):

\[\begin{aligned} X(z) = X_s(s)\Big|_{s=\frac{1}{T}\ln z} \end{aligned}\]
or equivalently:
\[\begin{aligned} z = e^{sT} \end{aligned}\]

Mapping

  • Left half s-plane → Inside unit circle in z-plane

  • Right half s-plane → Outside unit circle in z-plane

  • Imaginary axis → Unit circle

Applications and Examples

Circuit Analysis Example

RC Circuit

For an RC circuit with transfer function:

\[\begin{aligned} H(s) = \frac{1}{RCs + 1} \end{aligned}\]

Step Response

\[\begin{aligned} X(s) &= \frac{1}{s} \quad \text{(unit step)} \\ Y(s) &= H(s)X(s) = \frac{1}{s(RCs + 1)} \\ &= \frac{1}{s} - \frac{RC}{RCs + 1} \\ y(t) &= \left(1 - e^{-t/RC}\right)u(t) \end{aligned}\]

Digital Filter Example

First-Order Digital Filter

\[\begin{aligned} H(z) = \frac{1 - a}{1 - az^{-1}} = \frac{z(1-a)}{z-a} \end{aligned}\]

Unit Step Response

\[\begin{aligned} X(z) &= \frac{z}{z-1} \quad \text{(unit step)} \\ Y(z) &= H(z)X(z) = \frac{z(1-a)}{(z-a)(z-1)} \\ &= \frac{z}{z-1} - \frac{za}{z-a} \\ y[n] &= \left(1 - a^{n+1}\right)u[n] \end{aligned}\]

Important Formulas Summary

Key Formulas - Laplace Transform

Essential Relations

\[\begin{aligned} \text{Definition: } & X(s) = \int_{0^-}^{\infty} x(t) e^{-st} dt \\ \text{Differentiation: } & \frac{dx}{dt} \leftrightarrow sX(s) - x(0^-) \\ \text{Integration: } & \int_{0^-}^{t} x(\tau) d\tau \leftrightarrow \frac{X(s)}{s} \\ \text{Final Value: } & \lim_{t \to \infty} x(t) = \lim_{s \to 0} sX(s) \\ \text{Convolution: } & x_1(t) * x_2(t) \leftrightarrow X_1(s)X_2(s) \\ \text{Stability: } & \text{All poles in LHP for BIBO stability} \end{aligned}\]

Key Formulas - Z-Transform

Essential Relations

\[\begin{aligned} \text{Definition: } & X(z) = \sum_{n=0}^{\infty} x[n] z^{-n} \\ \text{Time Shift: } & x[n-k] \leftrightarrow z^{-k}X(z) \\ \text{Final Value: } & \lim_{n \to \infty} x[n] = \lim_{z \to 1} (z-1)X(z) \\ \text{Convolution: } & x_1[n] * x_2[n] \leftrightarrow X_1(z)X_2(z) \\ \text{Stability: } & \text{All poles inside unit circle} \\ \text{Causality: } & \text{ROC: } |z| > R \end{aligned}\]

Common GATE Problems

Typical Question Types

  1. Finding transfer functions from difference/differential equations

  2. Partial fraction expansion and inverse transforms

  3. Stability analysis using pole locations

  4. System response to standard inputs

  5. ROC determination and significance

  6. Initial and final value theorem applications

Problem-Solving Tips

  • Always identify the ROC for uniqueness

  • Check stability using pole locations

  • Use properties to simplify complex transforms

  • Remember initial conditions in differentiation property

  • Verify final value theorem applicability