GATE EE

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

Lecture Notes

SEC 01

Laplace Transform

SEC 02

Laplace Transform - Definition

1Laplace Transform - Definition
1Bilateral 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.
1Unilateral 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.
1Inverse Laplace Transform
\[\begin{aligned} x(t) = \mathcal{L}^{-1}\{X(s)\} = \frac{1}{2\pi j} \int_{\sigma-j\infty}^{\sigma+j\infty} X(s) e^{st} ds \end{aligned}\]
SEC 03

Region of Convergence (ROC)

1Region of Convergence (ROC)
1Definition

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

1ROC 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 \(s = \infty\))

1Stability Condition

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

SEC 04

Common Laplace Transform Pairs

1Common Laplace Transform Pairs
Signal Laplace Transform ROC
\(\delta(t)\) \(1\) All \(s\)
\(u(t)\) \(\frac{1}{s}\) \(\Re(s) > 0\)
\(t^n u(t)\) \(\frac{n!}{s^{n+1}}\) \(\Re(s) > 0\)
\(e^{-at}u(t)\) \(\frac{1}{s+a}\) \(\Re(s) > -a\)
\(t e^{-at}u(t)\) \(\frac{1}{(s+a)^2}\) \(\Re(s) > -a\)
\(\sin(\omega_0 t)u(t)\) \(\frac{\omega_0}{s^2 + \omega_0^2}\) \(\Re(s) > 0\)
\(\cos(\omega_0 t)u(t)\) \(\frac{s}{s^2 + \omega_0^2}\) \(\Re(s) > 0\)
\(e^{-at}\sin(\omega_0 t)u(t)\) \(\frac{\omega_0}{(s+a)^2 + \omega_0^2}\) \(\Re(s) > -a\)
\(e^{-at}\cos(\omega_0 t)u(t)\) \(\frac{s+a}{(s+a)^2 + \omega_0^2}\) \(\Re(s) > -a\)
SEC 05

Properties of Laplace Transform

1Properties of Laplace Transform
1Linearity
\[\begin{aligned} ax_1(t) + bx_2(t) \leftrightarrow aX_1(s) + bX_2(s) \end{aligned}\]
1Time Shifting
\[\begin{aligned} x(t-t_0)u(t-t_0) \leftrightarrow e^{-st_0}X(s) \end{aligned}\]
1Frequency Shifting
\[\begin{aligned} e^{s_0 t}x(t) \leftrightarrow X(s-s_0) \end{aligned}\]
1Time Scaling
\[\begin{aligned} x(at) \leftrightarrow \frac{1}{|a|}X\left(\frac{s}{a}\right) \end{aligned}\]
SEC 06

Differentiation and Integration Properties

1Differentiation and Integration Properties
1Time Differentiation
\[\begin{aligned} \frac{dx(t)}{dt} &\leftrightarrow sX(s) - x(0^-) \\ \frac{d^2x(t)}{dt^2} &\leftrightarrow s^2X(s) - sx(0^-) - x'(0^-) \\ \frac{d^nx(t)}{dt^n} &\leftrightarrow s^nX(s) - \sum_{k=0}^{n-1} s^{n-1-k}x^{(k)}(0^-) \end{aligned}\]
1Time Integration
\[\begin{aligned} \int_{0^-}^{t} x(\tau) d\tau \leftrightarrow \frac{X(s)}{s} \end{aligned}\]
1Frequency Differentiation
\[\begin{aligned} tx(t) \leftrightarrow -\frac{dX(s)}{ds} \end{aligned}\]
SEC 07

Convolution and Final Value Theorems

1Convolution and Final Value Theorems
1Convolution
\[\begin{aligned} x_1(t) * x_2(t) \leftrightarrow X_1(s)X_2(s) \end{aligned}\]
1Initial Value Theorem
\[\begin{aligned} \lim_{t \to 0^+} x(t) = \lim_{s \to \infty} sX(s) \end{aligned}\]
provided the limit exists.
1Final 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.
SEC 08

System Analysis Using Laplace Transform

SEC 09

Transfer Function

1Transfer Function
1Definition

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 \(X(s) = \mathcal{L}\{x(t)\}\).
1Relationship to Impulse Response
\[\begin{aligned} H(s) = \mathcal{L}\{h(t)\} \end{aligned}\]
1System Response
\[\begin{aligned} Y(s) &= H(s)X(s) \\ y(t) &= h(t) * x(t) \end{aligned}\]
SEC 10

Poles, Zeros, and Stability

1Poles, Zeros, and Stability
1Transfer Function Form
\[\begin{aligned} H(s) = K \frac{N(s)}{D(s)} = K \frac{(s-z_1)(s-z_2)\cdots(s-z_m)}{(s-p_1)(s-p_2)\cdots(s-p_n)} \end{aligned}\]
1Definitions

  • 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

1Stability 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

SEC 11

Partial Fraction Expansion

1Partial Fraction Expansion
1Distinct 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)
1Repeated 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 \(A_{1,k} = \frac{1}{(r-k)!} \lim_{s \to p_1} \frac{d^{r-k}}{ds^{r-k}}[(s-p_1)^r H(s)]\)
1Complex Conjugate Poles

Combine conjugate terms to get real coefficients in time domain.

SEC 12

Z-Transform

SEC 13

Z-Transform - Definition

1Z-Transform - Definition
1Bilateral Z-Transform
\[\begin{aligned} X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n] z^{-n} \end{aligned}\]
1Unilateral Z-Transform
\[\begin{aligned} X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=0}^{\infty} x[n] z^{-n} \end{aligned}\]
1Inverse 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.
SEC 14

Z-Transform ROC

1Z-Transform ROC
1ROC 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 \(z = \infty\))

1Stability Condition

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

SEC 15

Common Z-Transform Pairs

1Common Z-Transform Pairs
Sequence Z-Transform ROC
\(\delta[n]\) \(1\) All \(z\)
\(u[n]\) \(\frac{z}{z-1}\) \(|z| > 1\)
\(n u[n]\) \(\frac{z}{(z-1)^2}\) \(|z| > 1\)
\(a^n u[n]\) \(\frac{z}{z-a}\) \(|z| > |a|\)
\(n a^n u[n]\) \(\frac{az}{(z-a)^2}\) \(|z| > |a|\)
\(-a^n u[-n-1]\) \(\frac{z}{z-a}\) \(|z| < |a|\)
\(\cos(\omega_0 n)u[n]\) \(\frac{z(z-\cos\omega_0)}{z^2-2z\cos\omega_0+1}\) \(|z| > 1\)
\(\sin(\omega_0 n)u[n]\) \(\frac{z\sin\omega_0}{z^2-2z\cos\omega_0+1}\) \(|z| > 1\)
\(a^n\cos(\omega_0 n)u[n]\) \(\frac{z(z-a\cos\omega_0)}{z^2-2az\cos\omega_0+a^2}\) \(|z| > |a|\)
SEC 16

Properties of Z-Transform

1Properties of Z-Transform
1Linearity
\[\begin{aligned} ax_1[n] + bx_2[n] \leftrightarrow aX_1(z) + bX_2(z) \end{aligned}\]
1Time Shifting
\[\begin{aligned} x[n-n_0] &\leftrightarrow z^{-n_0}X(z) \\ x[n+n_0] &\leftrightarrow z^{n_0}X(z) - \sum_{k=0}^{n_0-1} x[k]z^{n_0-k} \end{aligned}\]
1Time Scaling
\[\begin{aligned} a^n x[n] \leftrightarrow X(z/a) \end{aligned}\]
1Time Reversal
\[\begin{aligned} x[-n] \leftrightarrow X(z^{-1}) \end{aligned}\]
SEC 17

More Z-Transform Properties

1More Z-Transform Properties
1Differentiation in z-domain
\[\begin{aligned} nx[n] \leftrightarrow -z\frac{dX(z)}{dz} \end{aligned}\]
1Convolution
\[\begin{aligned} x_1[n] * x_2[n] \leftrightarrow X_1(z)X_2(z) \end{aligned}\]
1Initial Value Theorem
\[\begin{aligned} x[0] = \lim_{z \to \infty} X(z) \end{aligned}\]
1Final 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.
SEC 18

Discrete-Time System Analysis

SEC 19

Discrete-Time Transfer Function

1Discrete-Time Transfer Function
1Definition
\[\begin{aligned} H(z) = \frac{Y(z)}{X(z)} = \frac{\sum_{k=0}^{M} b_k z^{-k}}{\sum_{k=0}^{N} a_k z^{-k}} \end{aligned}\]
1Difference Equation
\[\begin{aligned} \sum_{k=0}^{N} a_k y[n-k] = \sum_{k=0}^{M} b_k x[n-k] \end{aligned}\]
1Impulse Response
\[\begin{aligned} H(z) = \mathcal{Z}\{h[n]\} \end{aligned}\]
1System Response
\[\begin{aligned} Y(z) &= H(z)X(z) \\ y[n] &= h[n] * x[n] \end{aligned}\]
SEC 20

Stability Analysis

1Stability Analysis
1BIBO Stability

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

\[\begin{aligned} \sum_{n=-\infty}^{\infty} |h[n]| < \infty \end{aligned}\]
1Pole-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

1Causal vs Anti-causal

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

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

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

SEC 21

Inverse Transform Methods

SEC 22

Inverse Laplace Transform Methods

1Inverse Laplace Transform Methods
11. Partial Fraction Expansion

Most common method for rational functions.

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

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

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

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

SEC 23

Inverse Z-Transform Methods

1Inverse Z-Transform Methods
11. Partial Fraction Expansion

Convert to partial fractions and use known pairs.

12. Power Series Expansion
\[\begin{aligned} X(z) = \sum_{n=0}^{\infty} x[n] z^{-n} \end{aligned}\]
Coefficient of \(z^{-n}\) gives \(x[n]\).
13. Residue Method
\[\begin{aligned} x[n] = \sum_{i} \text{Res}[X(z)z^{n-1}, z_i] \quad \text{for } n \geq 0 \end{aligned}\]
14. Long Division

Direct division of numerator by denominator.

SEC 24

Relationship Between Transforms

SEC 25

Laplace to Fourier Transform

1Laplace to Fourier Transform
1Relationship
\[\begin{aligned} X(\omega) = X(s)\Big|_{s=j\omega} \end{aligned}\]
provided the Fourier transform exists.
1Condition

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

SEC 26

Z-Transform to DTFT

1Z-Transform to DTFT
1Relationship
\[\begin{aligned} X(\omega) = X(z)\Big|_{z=e^{j\omega}} \end{aligned}\]
provided the DTFT exists.
1Condition

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

SEC 27

Laplace to Z-Transform

1Laplace to Z-Transform
1Sampling 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:
\[\begin{aligned} z = e^{sT} \end{aligned}\]
1Mapping

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

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

  • Imaginary axis → Unit circle

SEC 28

Applications and Examples

SEC 29

Circuit Analysis Example

1Circuit Analysis Example
1RC Circuit

For an RC circuit with transfer function:

\[\begin{aligned} H(s) = \frac{1}{RCs + 1} \end{aligned}\]
1Step Response
\[\begin{aligned} X(s) &= \frac{1}{s} \quad \text{(unit step)} \\ Y(s) &= H(s)X(s) = \frac{1}{s(RCs + 1)} \\ &= \frac{1}{s} - \frac{RC}{RCs + 1} \\ y(t) &= \left(1 - e^{-t/RC}\right)u(t) \end{aligned}\]
SEC 30

Digital Filter Example

1Digital Filter Example
1First-Order Digital Filter
\[\begin{aligned} H(z) = \frac{1 - a}{1 - az^{-1}} = \frac{z(1-a)}{z-a} \end{aligned}\]
1Unit Step Response
\[\begin{aligned} X(z) &= \frac{z}{z-1} \quad \text{(unit step)} \\ Y(z) &= H(z)X(z) = \frac{z(1-a)}{(z-a)(z-1)} \\ &= \frac{z}{z-1} - \frac{za}{z-a} \\ y[n] &= \left(1 - a^{n+1}\right)u[n] \end{aligned}\]
SEC 31

Important Formulas Summary

SEC 32

Key Formulas - Laplace Transform

1Key Formulas - Laplace Transform
1Essential Relations
\[\begin{aligned} \text{Definition: } & X(s) = \int_{0^-}^{\infty} x(t) e^{-st} dt \\ \text{Differentiation: } & \frac{dx}{dt} \leftrightarrow sX(s) - x(0^-) \\ \text{Integration: } & \int_{0^-}^{t} x(\tau) d\tau \leftrightarrow \frac{X(s)}{s} \\ \text{Final Value: } & \lim_{t \to \infty} x(t) = \lim_{s \to 0} sX(s) \\ \text{Convolution: } & x_1(t) * x_2(t) \leftrightarrow X_1(s)X_2(s) \\ \text{Stability: } & \text{All poles in LHP for BIBO stability} \end{aligned}\]
SEC 33

Key Formulas - Z-Transform

1Key Formulas - Z-Transform
1Essential Relations
\[\begin{aligned} \text{Definition: } & X(z) = \sum_{n=0}^{\infty} x[n] z^{-n} \\ \text{Time Shift: } & x[n-k] \leftrightarrow z^{-k}X(z) \\ \text{Final Value: } & \lim_{n \to \infty} x[n] = \lim_{z \to 1} (z-1)X(z) \\ \text{Convolution: } & x_1[n] * x_2[n] \leftrightarrow X_1(z)X_2(z) \\ \text{Stability: } & \text{All poles inside unit circle} \\ \text{Causality: } & \text{ROC: } |z| > R \end{aligned}\]
SEC 34

Common GATE Problems

1Common GATE Problems
1Typical 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

1Problem-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