GATE ECE 2008 | Discrete Fourier Transform and Fast Fourier Transform Question 14 | Signals and Systems

{x(n)} is a real-valued periodic sequence with a period N. x(n) and X(k) form N-point. Discrete Fourier Transform (DFT) pairs. The DFT Y(k) of the sequence
y (n) = $${1 \over N}\,\sum\limits_{r = 0}^{N - 1} x \,\left( r \right)x\,(n + r\,)$$ is

$${\left| {X(k)} \right|^2}$$

$${1 \over N}\,\sum\limits_{r = 0}^{N - 1} X \,\left( r \right){X^*}\,(k + r\,)$$

$${1 \over N}\,\,\sum\limits_{r = 0}^{N - 1} X \,(r\,)X(k + r)$$

GATE ECE Subjects

Signals and Systems

Representation of Continuous Time Signal Fourier Series Discrete Time Signal Fourier Series Fourier Transform Discrete Time Signal Z Transform Continuous Time Linear Invariant System Transmission of Signal Through Continuous Time LTI Systems Discrete Time Linear Time Invariant Systems Sampling Continuous Time Signal Laplace Transform Discrete Fourier Transform and Fast Fourier Transform Transmission of Signal Through Discrete Time Lti Systems Miscellaneous Fourier Transform

Network Theory

Network Elements Network Theorems Transient Response Sinusoidal Steady State Response Two Port Networks Network Graphs State Equations For Networks Miscellaneous

Control Systems

Basic of Control Systems Signal Flow Graph and Block Diagram Time Response Analysis Stability Root Locus Diagram Frequency Response Analysis Compensators State Space Analysis

Digital Circuits

Logic Gates Boolean Algebra Analog to Digital and Digital to Analog Converters Sequential Circuits Combinational Circuits Logic Families Number System and Code Convertions Semiconductor Memories

General Aptitude

Numerical Ability Verbal Ability

Electronic Devices and VLSI

Semiconductor Physics PN Junction BJT and FET IC Basics and MOSFET

Analog Circuits

Bipolar Junction Transistor FET and MOSFET Oscillators Operational Amplifier 555 Timer Frequency Response Diodes Feedback Amplifier Power Amplifier

Engineering Mathematics

Linear Algebra Calculus Vector Calculus Probability and Statistics Differential Equations Complex Variable Numerical Methods Transform Theory

Microprocessors

Pin Details of 8085 and Interfacing with 8085 Instruction Set and Programming with 8085

Communications

Analog Communication Systems Digital Communication Systems Random Signals and Noise Fundamentals of Information Theory Noise In Digital Communication

Electromagnetics

Maxwell Equations Transmission Lines Uniform Plane Waves Waveguides Antennas Miscellaneous