Transform Methods in Mathematics

4.

Discrete Transform Methods

4.1.

4.1.1.

Definition and Motivation

4.1.1.1.

Discrete-Time Signals

4.1.1.2.

Relationship to Difference Equations

4.1.2.

Bilateral and Unilateral Z-Transform

4.1.3.

Region of Convergence

4.1.3.1.

4.1.3.2.

Causality and Stability

4.1.3.3.

ROC for Different Sequence Types

4.1.4.

4.1.4.1.

4.1.4.2.

4.1.4.3.

Scaling in z-Domain

4.1.4.4.

4.1.4.5.

Differentiation in z-Domain

4.1.4.6.

Convolution Property

4.1.4.7.

Initial and Final Value Theorems

4.1.5.

Inverse Z-Transform

4.1.5.1.

Power Series Method

4.1.5.2.

Partial Fraction Method

4.1.5.3.

Contour Integration Method

4.1.6.

4.1.6.1.

Difference Equations

4.1.6.2.

Digital Filter Analysis

4.1.6.3.

System Function Representation

4.2.

Discrete-Time Fourier Transform

4.2.1.

Definition and Properties

4.2.1.1.

Periodicity in Frequency

4.2.1.2.

Convergence Conditions

4.2.2.

Relationship to Z-Transform

4.2.2.1.

Unit Circle Evaluation

4.2.3.

4.2.3.1.

4.2.3.2.

Time and Frequency Shifting

4.2.3.3.

4.2.3.4.

Parseval's Relation

4.3.

Discrete Fourier Transform

4.3.1.

Definition for Finite Sequences

4.3.1.1.

Circular Nature

4.3.1.2.

Matrix Representation

4.3.2.

4.3.2.1.

4.3.2.2.

Circular Shifting

4.3.2.3.

Circular Convolution

4.3.2.4.

Symmetry Properties

4.3.3.

Relationship to Other Transforms

4.3.3.1.

Connection to DTFT

4.3.3.2.

Sampling in Frequency Domain

4.4.

Fast Fourier Transform

4.4.1.

Computational Efficiency

4.4.1.1.

Complexity Analysis

4.4.1.2.

Comparison with Direct DFT

4.4.2.

Decimation-in-Time Algorithm

4.4.2.1.

4.4.2.2.

Butterfly Computations

4.4.2.3.

Signal Flow Graphs

4.4.3.

Decimation-in-Frequency Algorithm

4.4.3.1.

Alternative Decomposition

4.4.3.2.

Implementation Considerations

4.4.4.

Practical Applications

4.4.4.1.

Spectral Analysis

4.4.4.2.

Digital Filtering

4.4.4.3.

Convolution Implementation

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3. Fourier Analysis

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5. Specialized Integral Transforms