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- 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
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- Wave Symmetry
- Even Symmetry
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- 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
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- Conjugation and Autocorrelation Property of Fourier Transform
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- 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
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- Sampling
- Signals Sampling Theorem
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- Laplace Transform
- Laplace Transforms
- Common Laplace Transform Pairs
- Laplace Transform of Unit Impulse Function and Unit Step Function
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- 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
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- 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 Discrete-Time Fourier Transform
Discrete Time Fourier Transform
The discrete time Fourier transform is a mathematical tool which is used to convert a discrete time sequence into the frequency domain. Therefore, the Fourier transform of a discrete time signal or sequence is called the discrete time Fourier transform (DTFT).
Mathematically, if x(n) is a discrete time sequence, then the discrete time Fourier transform of the sequence is defined as â
$$\mathrm{F[x(n)] \:=\: X(\omega) \:=\: \sum_{n=-\infty}^{\infty}\: x(n) e^{-j \omega n}}$$
Properties of Discrete-Time Fourier Transform
Following table gives the important properties of the discrete-time Fourier transform −
| Property | Discrete-Time Sequence | DTFT |
|---|---|---|
| Notation | $\mathrm{x(n)}$ | $\mathrm{X(\omega)}$ |
| $\mathrm{x_1(n)}$ | $\mathrm{X_1(\omega)}$ | |
| $\mathrm{x_2(n)}$ | $\mathrm{X_2(\omega)}$ | |
| Linearity | $\mathrm{ax_1(n)\:+\:bx_2(n)}$ | $\mathrm{aX_1(\omega) \:+\: bX_2(\omega)}$ |
| Time Shifting | $\mathrm{x(n\:â\:k)}$ | $\mathrm{e^{âjÏk}X(\omega)}$ |
| Frequency Shifting | $\mathrm{x(n)e^{j\omega_0 n}}$ | $\mathrm{X(\omega \:â\:\omega_0)}$ |
| Time Reversal | $\mathrm{x(ân)}$ | $\mathrm{X(â\omega)}$ |
| Frequency Differentiation | $\mathrm{nx(n)}$ | $\mathrm{j\frac{d}{d\omega}X(\omega)}$ |
| Time Convolution | $\mathrm{x_1(n)\:\cdot\:x_2(n)}$ | $\mathrm{X_1(\omega)X_2(\omega)}$ |
| Frequency Convolution (Multiplication in time domain) | $\mathrm{x_1(n)x_2(n)}$ | $\mathrm{X_1(\omega)\:\cdot\:X_2(\omega)}$ |
| Correlation | $\mathrm{R_{x_1x_2}(l)}$ | $\mathrm{X_1(\omega)X_2(â\omega)}$ |
| Modulation Property | $\mathrm{x(n)cos\omega_0n}$ | $\mathrm{\frac{1}{2}[X(\omega\:+\:\omega_0)\:+\:X(\omega\:â\:\omega_0)]}$ |
| Parsevalâs Relation | $\mathrm{\sum_{n=-\infty}^{\infty}\: |x(n)|^2}$ | $\mathrm{\frac{1}{2\pi} \int_{-\pi}^{\pi}\: |X(\omega)|^2 \: d\omega}$ |
| Conjugation | $\mathrm{x^*(n)}$ | $\mathrm{X(â\omega)}$ |
| $\mathrm{x^*(ân)}$ | $\mathrm{X^*(\omega)}$ | |
| Symmetry Properties | $\mathrm{x_R(n)}$ | $\mathrm{X_e(\omega)}$ |
| $\mathrm{jx_I(n)}$ | $\mathrm{X_0(\omega)}$ | |
| $\mathrm{x_e(n)}$ | $\mathrm{X_R(\omega)}$ | |
| $\mathrm{x_0(n)}$ | $\mathrm{jX_I(Ï)}$ |
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