ENGR 451 Lecture 3 Discrete-time Fourier Transform - Chapter 4 Discrete-time Fourier Transform 4.1 Introduction 4-3 Complex exponential sequences.4-3

# ENGR 451 Lecture 3 Discrete-time Fourier Transform -...

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Discrete-time Fourier Transform October 22, 2015 ©2015 T. Holton ([email protected]) 4-1 Chapter 4 Discrete-time Fourier Transform 4.1 Introduction ................................................................................................................................................... 4-3 Complex exponential sequences ........................................................................................................................... 4-3 Complex exponentials are eigenfunctions of linear time-invariant systems. ................................................ 4-3 Response of linear time-invariant systems to sinusoids ................................................................................ 4-5 Complex exponential sequences are orthogonal ........................................................................................... 4-7 4.2 Discrete-time Fourier transform .................................................................................................................... 4-9 Definition and derivation ...................................................................................................................................... 4-9 Alternate derivation of the DTFT ......................................................................................................................... 4-10 Notation of the DTFT ........................................................................................................................................... 4-10 Existence of the DTFT .......................................................................................................................................... 4-11 The system function (again) ................................................................................................................................ 4-11 Periodicity of the DTFT ........................................................................................................................................ 4-12 DTFT of finite-length sequences .......................................................................................................................... 4-12 DTFT of infinite-length sequences ....................................................................................................................... 4-17 4.3 Magnitude and phase description of the DTFT ............................................................................................. 4-19 Magnitude and phase of the DTFT of an impulse ............................................................................................... 4-19 Essential phase discontinuities ............................................................................................................................ 4-21 4.4 Important sequences and their transforms .................................................................................................. 4-25 4.5 Symmetry properties of the DTFT ................................................................................................................ 4-25 Time reversal ....................................................................................................................................................... 4-26 Even and odd symmetry ...................................................................................................................................... 4-26 Even sequences ............................................................................................................................................ 4-26 Odd sequences ............................................................................................................................................. 4-26 Conjugate symmetry and antisymmetry ............................................................................................................. 4-27 Real sequences ............................................................................................................................................. 4-27 Imaginary sequences .................................................................................................................................... 4-28 Conjugate symmetric and antisymmetric sequences ................................................................................... 4-29 Consequences of symmetry ................................................................................................................................ 4-29 Real and even sequences ............................................................................................................................. 4-29 Real and odd sequences ............................................................................................................................... 4-29 Imaginary sequences .................................................................................................................................... 4-30 Putting it all together ................................................................................................................................... 4-30 Additional properties .......................................................................................................................................... 4-34 DC and π -value property .............................................................................................................................. 4-34 Symmetry summary ............................................................................................................................................ 4-34 4.6 Response of a system to sinusoidal input ..................................................................................................... 4-35 4.7 Linear phase ................................................................................................................................................. 4-37 Causal symmetrical sequences ..................................................................................................................... 4-37 Causal antisymmetrical sequences .............................................................................................................. 4-38 4.8 The inverse discrete-time Fourier transform ................................................................................................ 4-40