- 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
Even and Odd Components of a Signal
Even Signal
A signal is said to be an even signal if it is symmetrical about the vertical axis or time origin, i.e.,
$$\mathrm{x(t) = x(-t);\: \text{ for all t ..... continuous time signal}}$$
$$\mathrm{x(n) = x(-n);\: \text{ for all n ..... discrete time signal}}$$
Odd Signal
A signal is said to be an odd signal if it is anti-symmetrical about the vertical axis, i.e.,
$$\mathrm{x(-t) \:=\: -x(t);\: \text{ for all t ..... continuous time signal}}$$
$$\mathrm{x(-n) \:=\: -x(n);\: \text{ for all n ..... discrete time signal}}$$
Determination of Even and Odd Components of a Signal
Continuous-time Case
Every signal need not be either purely even signal or purely odd signal, but the signal can be expressed as the sum of even and odd components, i.e.,
$$\mathrm{x(t) \:=\: x_e (t) \:+\: x_o \:(t) \:\: \dotso \: (1)}$$
Where,
- xe (t) is the even component of the signal, and
- xo (t) is the odd component of the signal.
By the definition of even and odd signals, we have,
$$\mathrm{x(-t) \:=\: x_e (-t) \:+\: x_o (-t)}$$
$$\mathrm{\Rightarrow \: x(-t) \:=\: x_e (t) \:-\: x_o (t) \:\: \dotso \: (2)}$$
Adding eqns. (1) & (2), we get,
$$\mathrm{x(t) \:+\: x(-t) \:=\: x_e (t) \:+\: x_o (t) \:+\: x_e (t) \:-\: x_o (t) \:=\: 2x_e (t)}$$
$$\mathrm{\therefore \: x_e (t) \:=\: 12[x(t) \:+\: x(-t)] \:\: \dotso \: (3)}$$
$$\mathrm{\therefore \: x_{e}(t) \:=\: \frac{1}{2}\left[ x(t) \:+\: x(-t) \right] \:\: \dotso \: (3)}$$
Again, subtracting eqn. (2) from eqn. (1), we get,
$$\mathrm{x(t) \:-\: x(-t) \:=\: [x_e (t) \:+\: x_o (t)] \:-\: [x_e (t) \:-\: x_o (t)]}$$
$$\mathrm{\Rightarrow \: x(t) \:-\: x(-t) \:=\: x_e (t) \:+\: x_o (t) \:-\: x_e (t) \:+\: x_o (t) \:=\: 2x_o (t)}$$
$$\mathrm{\therefore \: x_{0}(t) \:=\: \frac{1}{2}\left[x(t) \:+\: x(-t) \right]\:\: \dotso \: (4)}$$
Thus, the equations (3) and (4) gives the even and odd components of a continuous-time signal respectively.
Discrete-time Case
The even and odd components of a discrete-time signal x(n) are given by,
$$\mathrm{\therefore \: x_{e}(n) \:=\: \frac{1}{2}\left[ x(n)\:+\:x(-n) \right ]} \:\: \dotso \:(5)$$
$$\mathrm{\therefore \: x_{0}(n) \:=\: \frac{1}{2}\left[ x(n)\:-\:x(-n) \right ]} \:\: \dotso \:(6)$$
Numerical Example 1
Find the even and odd components of the continuous-time signal x(t) = ej4t.
Solution
Given signal is,
$$\mathrm{x(t) \:=\: e^{j4t}}$$
$$\mathrm{\therefore \: x(-t) = e^{-j4t}}$$
The even component of the signal is,
$$\mathrm{\therefore \: x_{e}(t) \:=\: \frac{1}{2}\left[ x(t)\:+\:x(-t) \right]\:=\:\frac{1}{2}\left( e^{j4t}\:+\:e^{-j4t} \right ) \:=\: \cos 4t}$$
And, the odd component of the signal is,
$$\mathrm{\therefore \: x_{0}(t) \:=\: \frac{1}{2}\left[ x(t) \:-\: x(-t) \right] \:=\: \frac{1}{2}\left( e^{j4t} \:- \: e^{-j4t} \right) \:=\: j\:\sin 4t }$$
Numerical Example 2
Find the even and odd components of the discrete-time signal x(n), where,
$$\mathrm{x(n) \:=\: \begin{Bmatrix} 5,6,3,4,1\ \uparrow \ \end{Bmatrix}}$$
Solution
The given discrete time sequence is,
$$\mathrm{x(n) \:=\: \begin{Bmatrix} 5,6,3,4,1\ \uparrow \ \end{Bmatrix}}$$
Here,
$$\mathrm{n \:=\: 0,\: 1,\: 2,\: 3,\: 4}$$
$$\mathrm{\therefore \: x(-n) \:=\: \begin{Bmatrix} 1, 4, 3, 6, 5\ \uparrow \ \end{Bmatrix}}$$
Hence, the even component of the sequence is,
$$\mathrm{x_{e}(n) \:=\: \frac{1}{2}\left[ x(n) \:+\: x(-n) \right ] \:=\: \frac{1}{2}\left[5,\: 6,\: 3,\: 4,\: 1 + 1,\: 4,\: 3,\: 6,\: 5 \right ]}$$
$$\mathrm{\Rightarrow\: x_{e}(n)\:=\: \frac{1}{2}\left[5 + 1,\: 6 + 4,\: 3 + 3,\: 4 + 6,\: 1 + 5 \right ]\:= \:\frac{1}{2} \left [6,\: 10,\: 6,\: 10,\: 6 \right ]}$$
$$\mathrm{\therefore \: x_{e}(n) \:=\: \begin{Bmatrix} 3, 5, 3, 5, 3\ \uparrow \ \end{Bmatrix}}$$
And the odd component of the sequence is,
$$\mathrm{ x_{0}(n) \:=\: \frac{1}{2}\left[ x(n) \:-\: x(-n) \right ] \:=\: \frac{1}{2}\left[5,\: 6,\: 3,\: 4,\: 1 - 1,\: 4,\: 3,\: 6,\: 5 \right ]}$$
$$\mathrm{\Rightarrow \: x_{0}(n) \:=\: \frac{1}{2}\left [5-1,\:6-4,\:3-3,\:4-6,\:1-5\right ] \:=\: \frac{1}{2}\left [4,\:2,\:0,\:-2,\:-4\right ]}$$
$$\mathrm{\therefore \: x_{0}(n) \:=\: \begin{Bmatrix} 2, 1, 0, -1, -4\ \uparrow \ \end{Bmatrix}}$$