- 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
Properties of Continuous-Time Fourier Transform (CTFT)
Fourier Transform
The Fourier transform of a continuous-time function $x(t)$ is defined as,
$$\mathrm{X(\omega) \:=\: \int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt}$$
Inverse Fourier Transform
The inverse Fourier transform of a continuous-time function is defined as,
$$\mathrm{x(t) \:=\: \frac{1}{2\pi}\int_{-\infty}^{\infty}X(\omega)e^{j\omega t}d\omega}$$
Properties of Fourier Transform
The continuous-time Fourier transform (CTFT) has a number of important properties. These properties are useful for driving Fourier transform pairs and also for deducing general frequency domain relationships. These properties also help to find the effect of various time domain operations on the frequency domain. Some of the important properties of continuous time Fourier transform are given in the table as −
| Property of CTFT | Time Domain x(t) | Frequency Domain X(ω) |
|---|---|---|
| Linearity Property | $ax_{1}(t)\:+\:bx_{2}(t)$ | $aX_{1}(\omega)+bX_{2}(\omega)$ |
| Time Shifting Property | $x(t \: \pm \: t_{0})$ | $e^{\pm j\:\omega\: t_{0}}X(\omega)$ |
| Frequency Shifting Property | $ e^{\pm j \:\omega_{0}\: t}x(t)$ | $X(\omega \: \pm \: \omega_{0})$ |
| Time Reversal Property | x(-t) | $x(-\omega)$ |
| Time Scaling Property | x(at) | $\frac{1}{|a|} X(\frac{\omega}{a})$ |
| Time Differentiation Property | $\frac{d}{dt} x(t)$ | $j \omega X(\omega)$ |
| Frequency Derivative Property | $t.x(t)$ | $j\frac{d}{d\omega}X(\omega)$ |
| Time Integration Property | $\int_{-\infty}^{\infty} x(t) d \tau$ | $\frac{X(\omega)}{j\omega}$ |
| Convolution Property | $x_{1}(t)*x_{2}(t)$ | $X_{1}(\omega)X_{2}(\omega)$ |
| Multiplication Property | $x_{1}(t)x_{2}(t)$ | $\frac{1}{2\pi}[X_{1}(\omega)*X_{2}(\omega)]$ |
| Duality or Symmetry Property | X(t) | $2\pi x(-\omega)$ |
| Modulation Property | $x(t)\:cos\:\omega_{0}t$ | $\frac{1}{2}[X(\omega-\omega_{0})+X(\omega+\omega_{0})]$ |
| $x(t)\:sin\:\omega_{0}t$ | $\frac{1}{2j}[X(\omega-\omega_{0})-X(\omega+\omega_{0})]$ | |
| Conjugation Property | x*(t) | $x*(-\omega)$ |
| Autocorrelation Property | R(τ) | $|X(-\omega)|^{2}$ |
| Parseval’s Theorem | $\int_{−\infty}^{\infty} x_{1}(t)x_{2}^*(t)dt$ | $\frac{1}{2\pi}\int_{−\infty}^{\infty}X_{1}(\omega)x_{2}^*(\omega)d\omega$ |
| Parseval’s Identity | $\int_{−\infty}^{\infty}|x(t)|^{2} dt$ | $\frac{1}{2\pi}\int_{−\infty}^{\infty}|X(\omega)|^{2}d\omega$ |
| Area Under the Curve Property | $\int_{−\infty}^{\infty}x(t)dt$ | $\frac{1}{2\pi}X(0)$ |
| x(0) | $\int_{−\infty}^{\infty}X(\omega)d\omega$ |
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