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- Properties of Even and Odd Signals
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- Properties of Discrete Time Unit Impulse Signal
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- Energy and Power Signals
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- Signals Analysis
- Types of Systems
- What is a Linear System?
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- Linear Time-Invariant Systems
- Transfer Function of LTI System
- Properties of LTI Systems
- Response of LTI System
- Fourier Series
- Fourier Series
- Fourier Series Representation of Periodic Signals
- Fourier Series Types
- Trigonometric Fourier Series Coefficients
- Exponential Fourier Series Coefficients
- Complex Exponential Fourier Series
- Relation between Trigonometric & Exponential Fourier Series
- Fourier Series Properties
- Properties of Continuous-Time Fourier Series
- Time Differentiation and Integration Properties of Continuous-Time Fourier Series
- Time Shifting, Time Reversal, and Time Scaling Properties of Continuous-Time Fourier Series
- Linearity and Conjugation Property of Continuous-Time Fourier Series
- Multiplication or Modulation Property of Continuous-Time Fourier Series
- Convolution Property of Continuous-Time Fourier Series
- Convolution Property of Fourier Transform
- Parseval’s Theorem in Continuous Time Fourier Series
- Average Power Calculations of Periodic Functions Using Fourier Series
- GIBBS Phenomenon for Fourier Series
- Fourier Cosine Series
- Trigonometric Fourier Series
- Derivation of Fourier Transform from Fourier Series
- Difference between Fourier Series and Fourier Transform
- Wave Symmetry
- Even Symmetry
- Odd Symmetry
- Half Wave Symmetry
- Quarter Wave Symmetry
- Wave Symmetry
- Fourier Transforms
- Fourier Transforms Properties
- Fourier Transform – Representation and Condition for Existence
- Properties of Continuous-Time Fourier Transform
- Table of Fourier Transform Pairs
- Linearity and Frequency Shifting Property of Fourier Transform
- Modulation Property of Fourier Transform
- Time-Shifting Property of Fourier Transform
- Time-Reversal Property of Fourier Transform
- Time Scaling Property of Fourier Transform
- Time Differentiation Property of Fourier Transform
- Time Integration Property of Fourier Transform
- Frequency Derivative Property of Fourier Transform
- Parseval’s Theorem & Parseval’s Identity of Fourier Transform
- Fourier Transform of Complex and Real Functions
- Fourier Transform of a Gaussian Signal
- Fourier Transform of a Triangular Pulse
- Fourier Transform of Rectangular Function
- Fourier Transform of Signum Function
- Fourier Transform of Unit Impulse Function
- Fourier Transform of Unit Step Function
- Fourier Transform of Single-Sided Real Exponential Functions
- Fourier Transform of Two-Sided Real Exponential Functions
- Fourier Transform of the Sine and Cosine Functions
- Fourier Transform of Periodic Signals
- Conjugation and Autocorrelation Property of Fourier Transform
- Duality Property of Fourier Transform
- Analysis of LTI System with Fourier Transform
- Relation between Discrete-Time Fourier Transform and Z Transform
- Convolution and Correlation
- Convolution in Signals and Systems
- Convolution and Correlation
- Correlation in Signals and Systems
- System Bandwidth vs Signal Bandwidth
- Time Convolution Theorem
- Frequency Convolution Theorem
- Energy Spectral Density and Autocorrelation Function
- Autocorrelation Function of a Signal
- Cross Correlation Function and its Properties
- Detection of Periodic Signals in the Presence of Noise (by Autocorrelation)
- Detection of Periodic Signals in the Presence of Noise (by Cross-Correlation)
- Autocorrelation Function and its Properties
- PSD and Autocorrelation Function
- Sampling
- Signals Sampling Theorem
- Nyquist Rate and Nyquist Interval
- Signals Sampling Techniques
- Effects of Undersampling (Aliasing) and Anti Aliasing Filter
- Different Types of Sampling Techniques
- Laplace Transform
- Laplace Transforms
- Common Laplace Transform Pairs
- Laplace Transform of Unit Impulse Function and Unit Step Function
- Laplace Transform of Sine and Cosine Functions
- Laplace Transform of Real Exponential and Complex Exponential Functions
- Laplace Transform of Ramp Function and Parabolic Function
- Laplace Transform of Damped Sine and Cosine Functions
- Laplace Transform of Damped Hyperbolic Sine and Cosine Functions
- Laplace Transform of Periodic Functions
- Laplace Transform of Rectifier Function
- Laplace Transforms Properties
- Linearity Property of Laplace Transform
- Time Shifting Property of Laplace Transform
- Time Scaling and Frequency Shifting Properties of Laplace Transform
- Time Differentiation Property of Laplace Transform
- Time Integration Property of Laplace Transform
- Time Convolution and Multiplication Properties of Laplace Transform
- Initial Value Theorem of Laplace Transform
- Final Value Theorem of Laplace Transform
- Parseval's Theorem for Laplace Transform
- Laplace Transform and Region of Convergence for right sided and left sided signals
- Laplace Transform and Region of Convergence of Two Sided and Finite Duration Signals
- Circuit Analysis with Laplace Transform
- Step Response and Impulse Response of Series RL Circuit using Laplace Transform
- Step Response and Impulse Response of Series RC Circuit using Laplace Transform
- Step Response of Series RLC Circuit using Laplace Transform
- Solving Differential Equations with Laplace Transform
- Difference between Laplace Transform and Fourier Transform
- Difference between Z Transform and Laplace Transform
- Relation between Laplace Transform and Z-Transform
- Relation between Laplace Transform and Fourier Transform
- Laplace Transform – Time Reversal, Conjugation, and Conjugate Symmetry Properties
- Laplace Transform – Differentiation in s Domain
- Laplace Transform – Conditions for Existence, Region of Convergence, Merits & Demerits
- Z Transform
- Z-Transforms (ZT)
- Common Z-Transform Pairs
- Z-Transform of Unit Impulse, Unit Step, and Unit Ramp Functions
- Z-Transform of Sine and Cosine Signals
- Z-Transform of Exponential Functions
- Z-Transforms Properties
- Properties of ROC of the Z-Transform
- Z-Transform and ROC of Finite Duration Sequences
- Conjugation and Accumulation Properties of Z-Transform
- Time Shifting Property of Z Transform
- Time Reversal Property of Z Transform
- Time Expansion Property of Z Transform
- Differentiation in z Domain Property of Z Transform
- Initial Value Theorem of Z-Transform
- Final Value Theorem of Z Transform
- Solution of Difference Equations Using Z Transform
- Long Division Method to Find Inverse Z Transform
- Partial Fraction Expansion Method for Inverse Z-Transform
- What is Inverse Z Transform?
- Inverse Z-Transform by Convolution Method
- Transform Analysis of LTI Systems using Z-Transform
- Convolution Property of Z Transform
- Correlation Property of Z Transform
- Multiplication by Exponential Sequence Property of Z Transform
- Multiplication Property of Z Transform
- Residue Method to Calculate Inverse Z Transform
- System Realization
- Cascade Form Realization of Continuous-Time Systems
- Direct Form-I Realization of Continuous-Time Systems
- Direct Form-II Realization of Continuous-Time Systems
- Parallel Form Realization of Continuous-Time Systems
- Causality and Paley Wiener Criterion for Physical Realization
- Discrete Fourier Transform
- Discrete-Time Fourier Transform
- Properties of Discrete Time Fourier Transform
- Linearity, Periodicity, and Symmetry Properties of Discrete-Time Fourier Transform
- Time Shifting and Frequency Shifting Properties of Discrete Time Fourier Transform
- Inverse Discrete-Time Fourier Transform
- Time Convolution and Frequency Convolution Properties of Discrete-Time Fourier Transform
- Differentiation in Frequency Domain Property of Discrete Time Fourier Transform
- Parseval’s Power Theorem
- Miscellaneous Concepts
- What is Mean Square Error?
- What is Fourier Spectrum?
- Region of Convergence
- Hilbert Transform
- Properties of Hilbert Transform
- Symmetric Impulse Response of Linear-Phase System
- Filter Characteristics of Linear Systems
- Characteristics of an Ideal Filter (LPF, HPF, BPF, and BRF)
- Zero Order Hold and its Transfer Function
- What is Ideal Reconstruction Filter?
- What is the Frequency Response of Discrete Time Systems?
- Basic Elements to Construct the Block Diagram of Continuous Time Systems
- BIBO Stability Criterion
- BIBO Stability of Discrete-Time Systems
- Distortion Less Transmission
- Distortionless Transmission through a System
- Rayleigh’s Energy Theorem
Common Laplace Transform Pairs
Laplace Transform
The linear time invariant (LTI) system is described by differential equations. The Laplace transform is a mathematical tool which converts the differential equations in time domain into algebraic equations in the frequency domain (or s-domain).
If x(t) is a time function, then the Laplace transform of the function is defined as −
$$\mathrm{L[x(t)] \:=\: X(s) \:=\: \int_{-\infty}^{\infty}\: x(t) e^{-st} \: dt\:\:\dotso\:(1)}$$
Where, s is a complex variable and it is given by,
$$\mathrm{s \:=\: \sigma \:+\: j\omega}$$
Inverse Laplace Transform
The inverse Laplace transform is defined as −
$$\mathrm{L^{-1}[X(s)] \:=\: x(t)\:=\: \frac{1}{2\pi j} \int_{\sigma - j\infty}^{\sigma + j\infty}\: X(s) e^{st} \: ds\:\:\dotso\:(2)}$$
The equations (1) and (2) constitute the Laplace transform pair, and it may be represented as,
$$\mathrm{x(t) \:\overset{LT}\longleftrightarrow\: X(s)}$$
Common Laplace Transform Pairs
Following table provides a number of Laplace transforms. The table also specifies the region of convergence (ROC) −
| Function $\mathrm{\{x(t) \:=\: L^{−1}[x(t)]\}}$ | Laplace Transform $\mathrm{\{L[x(t)] \:=\: X(s)\}}$ | Region of Convergence (ROC) |
|---|---|---|
| $\mathrm{\delta(t)}$ | $\mathrm{1}$ | $\mathrm{All s}$ |
| $\mathrm{\delta(t\:−\:a)}$ | $\mathrm{e^{−as}}$ | $\mathrm{All s}$ |
| $\mathrm{u(t)}$ | $\mathrm{\frac{1}{s}}$ | $\mathrm{Re(s) \:\gt\: 0}$ |
| $\mathrm{u(t\:−\:a)}$ | $\mathrm{\frac{e^{−as}}{s}}$ | $\mathrm{Re(s) \:\gt\: 0}$ |
| $\mathrm{u(−t)}$ | $\mathrm{−\frac{1}{s}}$ | $\mathrm{Re(s) \:\lt\: 0}$ |
| $\mathrm{tu(t)}$ | $\mathrm{\frac{1}{s^2}}$ | $\mathrm{Re(s)\:\gt\:0}$ |
| $\mathrm{t^2u(t)}$ | $\mathrm{\frac{2!}{s^3}}$ | $\mathrm{Re(s) \:\gt\: 0}$ |
| $\mathrm{t^nu(t)}$ | $\mathrm{\frac{n!}{s^{(n+1)}}}$ | $\mathrm{Re(s) \:\gt\: 0}$ |
| $\mathrm{e^{−at}u(t)}$ | $\mathrm{\frac{1}{(s\:+\:a)}}$ | $\mathrm{Re(s) \:\gt\: −a}$ |
| $\mathrm{e^{at}u(t)}$ | $\mathrm{\frac{1}{(s\:−\:a)}}$ | $\mathrm{Re(s) \:\gt\: a}$ |
| $\mathrm{te^{−at}u(t)}$ | $\mathrm{\frac{1}{(s\:+\:a)^2}}$ | $\mathrm{Re(s) \:\gt\: −a}$ |
| $\mathrm{t^ne^{−at}u(t)}$ | $\mathrm{\frac{n!}{(s+a)^{n+1}}}$ | $\mathrm{Re(s) \:\gt\: −a}$ |
| $\mathrm{sin\:\omega t\:u(t)}$ | $\mathrm{\frac{\omega}{(s^2\:+\:\omega^2)}}$ | $\mathrm{Re(s) \:\gt\: 0}$ |
| $\mathrm{cos\:\omega t\:u(t)}$ | $\mathrm{\frac{s}{(s^2\:+\:\omega^2)}}$ | $\mathrm{Re(s) \:\gt\: 0}$ |
| $\mathrm{e^{−at}sin\:\omega t\:u(t)}$ | $\mathrm{\frac{\omega}{(s\:+\:a)^2\:+\:\omega^2}}$ | $\mathrm{Re(s) \:\gt\: −a}$ |
| $\mathrm{e^{−at}cos\:\omega\:t\:u(t)}$ | $\mathrm{\frac{(s\:+\:a)}{(s\:+\:a)^2\:+\:\omega^2}}$ | $\mathrm{Re(s) \:\gt\: −a}$ |
| $\mathrm{sin(\omega t \:+\: \theta)}$ | $\mathrm{\frac{s\:sin\theta \:+\: \omega\:cos\theta}{(s^2 \:+\: \omega^2 )}}$ | $\mathrm{Re(s) \:\gt\: 0}$ |
| $\mathrm{cos(\omega t \:+\: \theta)}$ | $\mathrm{\frac{s\:cos\theta \:+\: \omega\:sin\theta}{(s^2 \:+\: \omega^2 )}}$ | $\mathrm{Re(s) \:\gt\: 0}$ |
| $\mathrm{t\:sin\:\omega t\:u(t)}$ | $\mathrm{\frac{2\omega s}{(s^2 \:+\: \omega^2)^2}}$ | $\mathrm{Re(s) \:\gt\: 0}$ |
| $\mathrm{t\:cos\:\omega t\:u(t)}$ | $\mathrm{\frac{(s^2 \:−\: \omega^2)}{(s^2 \:+\:\omega^2)^2}}$ | $\mathrm{Re(s) \:\gt\: 0}$ |
| $\mathrm{sin\:h\:\omega t\:u(t)}$ | $\mathrm{\frac{\omega}{(s^2 \:−\: \omega^2)}}$ | $\mathrm{Re(s) \:\gt\: \omega}$ |
| $\mathrm{cos\:h\:\omega t\:u(t)}$ | $\mathrm{\frac{s}{(s^2 \:−\: \omega^2)}}$ | $\mathrm{Re(s) \:\gt\: \omega}$ |
| $\mathrm{e^{−at}sin\:h\:\omega t\:u(t)}$ | $\mathrm{\frac{\omega}{(s\:+\:a)^2 \:−\: \omega^2}}$ | $\mathrm{Re(s) \:\gt\: (\omega \:−\: a)}$ |
| $\mathrm{e^{−at}cos\:h\:\omega t\:u(t)}$ | $\mathrm{\frac{s\:+\:a}{(s\:+\:a)^2 \:−\: \omega^2}}$ | $\mathrm{Re(s) \:\gt\: (\omega − a)}$ |
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