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Simulink ModelingPage 7−318 The Fourier Transform 8−18.1 Definition and Special Forms ................................................................................ 8−18.2 Special Forms of the Fourier Transform ................................................................ 8−28.2.1 Real Time Functions.................................................................................. 8−38.2.2 Imaginary Time Functions ......................................................................... 8−68.3 Properties and Theorems of the Fourier Transform .............................................. 8−98.3.1 Linearity...................................................................................................... 8−98.3.2 Symmetry.................................................................................................... 8−98.3.3 Time Scaling............................................................................................. 8−108.3.4 Time Shifting............................................................................................ 8−118.3.5 Frequency Shifting ................................................................................... 8−118.3.6 Time Differentiation ................................................................................ 8−128.3.7 Frequency Differentiation ........................................................................ 8−138.3.8 Time Integration ...................................................................................... 8−138.3.9 Conjugate Time and Frequency Functions.............................................. 8−138.3.10 Time Convolution.................................................................................... 8−148.3.11 Frequency Convolution............................................................................ 8−158.3.12 Area Under ft ()........................................................................................ 8−158.3.13 Area Under F( ω)...................................................................................... 8−158.3.14 Parseval’s Theorem................................................................................... 8−168.4 Fourier Transform Pairs of Common Functions.................................................. 8−188.4.1 The Delta Function Pair .......................................................................... 8−188.4.2 The Constant Function Pair .................................................................... 8−188.4.3 The Cosine Function Pair........................................................................ 8−198.4.4 The Sine Function Pair............................................................................. 8−208.4.5 The Signum Function Pair........................................................................ 8−208.4.6 The Unit Step Function Pair .................................................................... 8−22e – jω 0 t u 0 t8.4.7 The () Function Pair.................................................................... 8−248.4.8 The ( cosω 0 t) ( u 0 t)Function Pair ............................................................... 8−248.4.9 The ( sin t) ( u 0 t)Function Pair................................................................ 8−25ω 08.5 Derivation of the Fourier Transform from the Laplace Transform .................... 8−258.6 Fourier Transforms of Common Waveforms ...................................................... 8−278.6.1 The Transform of ft () = A[ u 0 ( t + T) – u 0 ( t – T)] ....................................... 8−278.6.2 The Transform of ft () = A[ u 0 () t – u 0 ( t – 2T)] ........................................... 8−288.6.3 The Transform of ft () = A[ u 0 ( t + T) + u 0() t – u 0 ( t – T)– u 0 ( t – 2T)] ........... 8−29Signals and Systems with MATLAB ® Computing and Simulink ® Modeling, Fourth EditionCopyright © Orchard Publicationsv