Laplace Transform
Laplace Transform - Definition
Bilateral Laplace Transform
Unilateral Laplace Transform
Inverse Laplace Transform
Region of Convergence (ROC)
Definition
The set of values of \(s\) for which the Laplace transform converges.
ROC Properties
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ROC is a vertical strip in the s-plane
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ROC cannot contain poles
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For right-sided signals: \(\Re(s) > \sigma_0\)
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For left-sided signals: \(\Re(s) < \sigma_0\)
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For two-sided signals: \(\sigma_1 < \Re(s) < \sigma_2\)
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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
Time Shifting
Frequency Shifting
Time Scaling
Differentiation and Integration Properties
Time Differentiation
Time Integration
Frequency Differentiation
Convolution and Final Value Theorems
Convolution
Initial Value Theorem
Final Value Theorem
System Analysis Using Laplace Transform
Transfer Function
Definition
For a linear time-invariant system:
Relationship to Impulse Response
System Response
Poles, Zeros, and Stability
Transfer Function Form
Definitions
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Zeros: Values of \(s\) where \(H(s) = 0\)
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Poles: Values of \(s\) where \(H(s) = \infty\)
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Order: \(n\) = number of poles, \(m\) = number of zeros
Stability Conditions
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BIBO Stable: All poles in left half-plane (\(\Re(p_i) < 0\))
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Marginally Stable: Simple poles on imaginary axis, rest in LHP
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Unstable: At least one pole in right half-plane or repeated poles on imaginary axis
Partial Fraction Expansion
Distinct Poles
Repeated Poles
For a pole \(p_1\) of multiplicity \(r\):
Complex Conjugate Poles
Combine conjugate terms to get real coefficients in time domain.
Z-Transform
Z-Transform - Definition
Bilateral Z-Transform
Unilateral Z-Transform
Inverse Z-Transform
Z-Transform ROC
ROC Properties
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ROC is an annular region in the z-plane
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ROC cannot contain poles
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For right-sided sequences: \(|z| > R_1\)
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For left-sided sequences: \(|z| < R_2\)
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For two-sided sequences: \(R_1 < |z| < R_2\)
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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
Time Shifting
Time Scaling
Time Reversal
More Z-Transform Properties
Differentiation in z-domain
Convolution
Initial Value Theorem
Final Value Theorem
Discrete-Time System Analysis
Discrete-Time Transfer Function
Definition
Difference Equation
Impulse Response
System Response
Stability Analysis
BIBO Stability
A discrete-time LTI system is BIBO stable if and only if:
Pole-Zero Analysis
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Stable: All poles inside unit circle (\(|p_i| < 1\))
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Marginally Stable: Simple poles on unit circle, rest inside
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Unstable: At least one pole outside unit circle or repeated poles on unit circle
Causal vs Anti-causal
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Causal: Right-sided ROC (\(|z| > R\))
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Anti-causal: Left-sided ROC (\(|z| < R\))
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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
3. Convolution Method
If \(X(s) = X_1(s)X_2(s)\), then:
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
3. Residue Method
4. Long Division
Direct division of numerator by denominator.
Relationship Between Transforms
Laplace to Fourier Transform
Relationship
Condition
The imaginary axis (\(s = j\omega\)) must be in the ROC of \(X(s)\).
Z-Transform to DTFT
Relationship
Condition
The unit circle (\(|z| = 1\)) must be in the ROC of \(X(z)\).
Laplace to Z-Transform
Sampling Relationship
For a sampled signal \(x_s(t) = \sum_{n=-\infty}^{\infty} x(nT)\delta(t-nT)\):
Mapping
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Left half s-plane → Inside unit circle in z-plane
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Right half s-plane → Outside unit circle in z-plane
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Imaginary axis → Unit circle
Applications and Examples
Circuit Analysis Example
RC Circuit
For an RC circuit with transfer function:
Step Response
Digital Filter Example
First-Order Digital Filter
Unit Step Response
Important Formulas Summary
Key Formulas - Laplace Transform
Essential Relations
Key Formulas - Z-Transform
Essential Relations
Common GATE Problems
Typical Question Types
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Finding transfer functions from difference/differential equations
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Partial fraction expansion and inverse transforms
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Stability analysis using pole locations
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System response to standard inputs
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ROC determination and significance
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Initial and final value theorem applications
Problem-Solving Tips
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Always identify the ROC for uniqueness
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Check stability using pole locations
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Use properties to simplify complex transforms
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Remember initial conditions in differentiation property
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Verify final value theorem applicability