- Signals & Systems Home
- Signals & Systems Overview
- Introduction
- Signals Basic Types
- Signals Classification
- Signals Basic Operations
- Systems Classification
- Types of Signals
- Representation of a Discrete Time Signal
- Continuous-Time Vs Discrete-Time Sinusoidal Signal
- Even and Odd Signals
- Properties of Even and Odd Signals
- Periodic and Aperiodic Signals
- Unit Step Signal
- Unit Ramp Signal
- Unit Parabolic Signal
- Energy Spectral Density
- Unit Impulse Signal
- Power Spectral Density
- Properties of Discrete Time Unit Impulse Signal
- Real and Complex Exponential Signals
- Addition and Subtraction of Signals
- Amplitude Scaling of Signals
- Multiplication of Signals
- Time Scaling of Signals
- Time Shifting Operation on Signals
- Time Reversal Operation on Signals
- Even and Odd Components of a Signal
- Energy and Power Signals
- Power of an Energy Signal over Infinite Time
- Energy of a Power Signal over Infinite Time
- Causal, Non-Causal, and Anti-Causal Signals
- Rectangular, Triangular, Signum, Sinc, and Gaussian Functions
- Signals Analysis
- Types of Systems
- What is a Linear System?
- Time Variant and Time-Invariant Systems
- Linear and Non-Linear Systems
- Static and Dynamic System
- Causal and Non-Causal System
- Stable and Unstable System
- Invertible and Non-Invertible Systems
- 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 Z-Transform Pairs
Z-Transform
Z−transform is a mathematical tool which is used to convert the difference equations in time domain into the algebraic equations in the frequency domain.
Mathematically, if x(n) is a discrete−time sequence, then its Z−transform is defined as −
$$\mathrm{X(z) \:=\: \sum_{n=−\infty}^{\infty}\: x(n) z^{−n} \:\:\dotso\:(1)}$$
Where, z is a complex variable. The z−transform defined in eq. (1) is called bilateral or two−sided z−transform.
The unilateral or one−sided z−transform is defined as −
$$\mathrm{X(z) \:=\: \sum_{n\:=\:0}^{\infty}\: x(n) z^{−n} \:\:\dotso\:(2)}$$
Common Z-Transform Pairs
The following table gives a number of unilateral and bilateral z−transforms along with their region of convergence (ROC) −
| Discrete-Time Sequence, x(n) | Z-Transform, X(z) | ROC |
|---|---|---|
| $\mathrm{\delta(n)}$ | $\mathrm{1}$ | $\mathrm{All z}$ |
| $\mathrm{u(n)}$ | $\mathrm{\frac{z}{z−1}\:=\:\frac{1}{1−z^{−1}}}$ | $\mathrm{|z|\:\gt\:1}$ |
| $\mathrm{u(−n)}$ | $\mathrm{\frac{1}{1−z}}$ | $\mathrm{|z|\:\lt\:1}$ |
| $\mathrm{u(−n−1)}$ | $\mathrm{−\frac{z}{(z−1)}}$ | $\mathrm{|z| \:\lt\:1}$ |
| $\mathrm{u(−n−2)}$ | $\mathrm{−\frac{z^2}{(z−1)}}$ | $\mathrm{|z| \:\lt\: 1}$ |
| $\mathrm{u(−n−k)}$ | $\mathrm{−\frac{z^k}{(z−1)}}$ | $\mathrm{|z| \:\lt\: 1}$ |
| $\mathrm{\delta(n−k)}$ | $\mathrm{z^{−k}}$ |
If k > 0, all z except at z = 0 If k < 0, all z except at z = &infty; |
| $\mathrm{\frac{1}{n}\:;\:n \:\gt\:0 }$ | $\mathrm{−ln(1\:−\:z^{−1})}$ | $\mathrm{|z|\:\gt\:1}$ |
| $\mathrm{a^{|n|}\:;\:\:\text{for all n}}$ | $\mathrm{\frac{(1\:−\:a^2)}{[(1\:−\:az)(1\:−\:az^{−1})]}}$ | $\mathrm{|a| \:\lt\:|z|\:\lt\:|\frac{1}{a}|}$ |
| $\mathrm{a^{n}u(n)}$ | $\mathrm{\frac{z}{z\:−\:a}}$ | $\mathrm{|z| \:\gt\: |a|}$ |
| $\mathrm{−a^{n}u(−n)}$ | $\mathrm{\frac{a}{(z−a)}}$ | $\mathrm{|z| \:\lt\: |a|}$ |
| $\mathrm{−a^{n}u(−n−1)}$ | $\mathrm{\frac{z}{(z−a)}}$ | $\mathrm{|z| \:\lt\: |a|}$ |
| $\mathrm{nu(n)}$ | $\mathrm{\frac{z}{(z−1)^2}}$ | $\mathrm{|z| \:\gt\: 1}$ |
| $\mathrm{na^{n}u(n)}$ | $\mathrm{\frac{az}{(z−a)^2}}$ | $\mathrm{|z| \:\gt\: |a|}$ |
| $\mathrm{−nu(−n−1)}$ | $\mathrm{\frac{z}{(z−1)^2}}$ | $\mathrm{|z|\:\lt\:1}$ |
| $\mathrm{−na^n u(−n−1)}$ | $\mathrm{\frac{az}{(z−a)^2}}$ | $\mathrm{|z|\:\lt\:|a|}$ |
| $\mathrm{e^{−j\omega n}u(n)}$ | $\mathrm{\frac{z}{(z−e^{−j\omega})}}$ | $\mathrm{|z| \:\gt\: 1}$ |
| $\mathrm{cos\:\omega\:nu(n)}$ | $\mathrm{\frac{z(z\:−\:cos\omega)}{z^2\:−\:2z\:cos\omega\:+\:1}}$ | $\mathrm{|z| \:\gt\: 1}$ |
| $\mathrm{sin\:\omega\:nu(n)}$ | $\mathrm{\frac{z\:sin\:\omega}{z^2\:−\:2z\:cos\:\omega\:+\:1}}$ | $\mathrm{|z|\:\gt\:1}$ |
| $\mathrm{a^n\:cos\:\omega\:nu(n)}$ | $\mathrm{\frac{z(z\:−\:acos\:\omega)}{z^2\:−\:2az\:cos\omega\:+\:a^2}}$ | $\mathrm{|z|\:\gt\:|a|}$ |
| $\mathrm{a^n\:sin\omega\:nu(n)}$ | $\mathrm{\frac{az\:sin\omega}{z^2\:−\:2az\:cos\omega\:+\:a^2}}$ | $\mathrm{|z| \:\gt\: |a|}$ |
| $\mathrm{(n\:+\:1)a^nu(n)}$ | $\mathrm{\frac{Z^2}{(z−a)^2}}$ | $\mathrm{|z|\:\gt\:|a|}$ |
| $\mathrm{\frac{(n+1)(n+2)}{2!}\:a^n u(n)}$ | $\mathrm{\frac{z^3}{(z\:−\:a)^3}}$ | $\mathrm{|z| \:\gt\:|a|}$ |
| $\mathrm{\frac{n(n−1)}{2!}\:a^{(n−2)}u(n)}$ | $\mathrm{\frac{z}{(z−a)^3}}$ | $\mathrm{|z| \:>\: |a|}$ |
| $\mathrm{\frac{n(n−1)\:\cdots\:[n−(k−2)]}{(k−1)!}\:a^{(n−k+1)}u(n)}$ | $\mathrm{\frac{z}{(z−a)^k}}$ | $\mathrm{|z|\:\gt\:|a|}$ |
Advertisements