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- Signals & Systems Overview
- Introduction
- Signals Basic Types
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- 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
Table of Fourier Transform Pairs
Fourier Transform
Fourier transform is a transformation technique that transforms signals from the continuous-time domain to the corresponding frequency domain and vice-versa.
The Fourier transform of a continuous-time function $x(t)$ is defined as,
$$\mathrm{X(\omega) \:=\: \int_{-\infty}^{\infty} x(t)e^{-j\omega t}dt\:\:\dotso\:(1)}$$
Inverse Fourier Transform
The inverse Fourier transform of a continuous-time function is defined as,
$$\mathrm{x(t) \:=\: \frac{1}{2\pi}\int_{-\infty}^{\infty}X(\omega)\:e^{j\omega t}d\omega\:\:\dotso\:(2)}$$
Equations (1) and (2) for $X(\omega)$ and $x(t)$ are known as Fourier transform pair and can be represented as −
$$\mathrm{X(\omega) \:=\: F[x(t)]}$$
And
$$\mathrm{x(t) \:=\: F^{-1}[X(\omega)]}$$
Table of Fourier Transform Pairs
| Function,x(t) | Fourier Transform, X(ω) |
|---|---|
| $\mathrm{\delta(t)}$ | 1 |
| $\mathrm{\delta(t \:-\: t_{0})}$ | $\mathrm{e^{-j\: \omega\: t_{0}}}$ |
| 1 | $\mathrm{2\pi \delta(\omega)}$ |
| u(t) | $\mathrm{\pi\delta(\omega)\:+\:\frac{1}{j\omega}}$ |
| $\mathrm{\sum_{n= -\infty}^{\infty}\delta(t \:-\: nT)}$ | $\mathrm{\omega_{0}\sum_{n= -\infty}^{\infty}\:\delta(\omega \:-\: n\omega_{0});\:\:\left(\omega_{0} \:=\: \frac{2\pi}{T} \right)}$ |
| sgn(t) | $\mathrm{\frac{2}{j\omega}}$ |
| $\mathrm{e^{j\omega_{0}t}}$ | $\mathrm{2\pi\delta(\omega \:-\: \omega_{0})}$ |
| $\mathrm{cos\:\omega_{0}t}$ | $\mathrm{\pi[\delta(\omega \:-\: \omega_{0}) \:+\: \delta(\omega \:+\: \omega_{0})]}$ |
| $\mathrm{sin\:\omega_{0}t}$ | $\mathrm{-j\pi[\delta(\omega \:-\: \omega_{0}) \:-\: \delta(\omega \:+\: \omega_{0})]}$ |
| $\mathrm{e^{-at}u(t);\:\: a \:\gt \:0}$ | $\mathrm{\frac{1}{a \:+\: j\omega}}$ |
| $\mathrm{t\:e^{at}u(t);\:\:a \: \gt \:0}$ | $\mathrm{\frac{1}{(a \:+\: j\omega)^{2}}}$ |
| $\mathrm{e^{-|at|};\:\:a \:\gt\:0}$ | $\mathrm{\frac{2a}{a^{2} \:+\: \omega^{2}}}$ |
| $\mathrm{e^{-|t|}}$ | $\mathrm{\frac{2}{1 \:+\: \omega^{2}}}$ |
| $\mathrm{\frac{1}{\pi t}}$ | $\mathrm{-j\:sgn(\omega)}$ |
| $\mathrm{\frac{1}{a^{2} \:+\: t^{2}}}$ | $\mathrm{\frac{\pi}{a}e^{-a|\omega|}}$ |
| $\mathrm{\Pi \left(\frac{t}{\tau}\right)}$ | $\mathrm{\tau\:sin c\left(\frac{\omega \tau}{2}\right)}$ |
| $\mathrm{\Delta\left(\frac{t}{\tau}\right)}$ | $\mathrm{\frac{\tau}{2}sin C^{2}\left(\frac{\omega \tau}{4}\right)}$ |
| $\mathrm{\frac{sin\:at}{\pi t}}$ | $\mathrm{P_{a}(\omega) \:=\: \begin{cases} 1 \:\: for\:|\omega|\:\lt\: a \\\\0 \:\: for\:|\omega|\: \gt\: a \end{cases}}$ |
| $\mathrm{cos\:\omega_{0}t\:u(t)}$ | $\mathrm{\frac{\pi}{2}[\delta(\omega \:-\: \omega_{0}) \:+\: \delta(\omega \:+\: \omega_{0})] \:+\:\left[ \frac{j\omega}{(j\omega)^{2} \:+\: \omega_{0}^{2}} \right]}$ |
| $\mathrm{sin\:\omega_{0}t\:u(t)}$ | $\mathrm{-j\frac{\pi}{2}[\delta(\omega \:-\: \omega_{0}) \:-\: \delta(\omega \:+\: \omega_{0})] \:+\: \left [\frac{\omega_{0}}{(j\omega)^{2} \:+\: \omega_{0}^{2}} \right ]}$ |
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