EE200_Weber_11-13

EE200_Weber_11-13 - EE 200 The 4 Fourier Transforms There...

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8 EE 200 The 4 Fourier Transforms There are four variations on the Fourier expansion for different types of signals. Periodic signals Nonperiodic signals Continuous time Discrete time X m = 1 p x ( t ) e " im # 0 t dt 0 p $ X k = 1 p x ( m ) e " imk 0 m = 0 p " 1 $ X ( " ) = x ( t ) e # i t dt #$ $ % X ( ) = x ( m ) e # im m = #$ $ % Continuous-Time Fourier Series DFS (Discrete-Time Fourier Series) DTFT (Discrete-Time Fourier Transform) CTFT (Continuous-Time Fourier Transform)
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9 EE 200 Discrete-Time Fourier Series The DFS converts a periodic discrete-time signal into a periodic discrete-frequency signal. where ω 0 =2 π /p An alternate form called the “Discrete Fourier Transform” (DFT) also exists. X k = 1 p x ( m ) e " imk # 0 m = 0 p " 1 $ = 1 p x ( m ) e " i 2 % mk / p m = 0 p " 1 $ DFT p :DiscPeriodic p DiscPeriodic p ! " X k = x ( m ) e # imk $ 0 m = 0 p # 1 % x ( n ) = X k e ik " 0 n k = 0 p # 1 $ x ( n ) = 1 p " X k e ik 0 n k = 0 p $ 1 %
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10 EE 200 Discrete-Time Fourier Series For the DFT both the time domain and frequency domain signals are periodic, with the same period. " X k + p = x ( m ) e # im ( k + p ) $ 0 m = 0 p # 1 % = x ( m ) e # imk 0 e # imp 0 m = 0 p # 1 % = x ( m ) e # imk 0 e # im 2 m = 0 p # 1 % = x ( m ) e # imk 0 m = 0 p # 1 % = " X k
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11 EE 200 Discrete-Time Fourier Series The inverse DFT is the function InverseDFT p :DiscPeriodic p DiscPeriodic p ! x ( n ) = 1 p " X k e ik # 0 n k = 0 p $ 1 % x ( n ) = 1 p " X k e ik 0 n k = 0 p $ 1 % = 1 p x ( m ) e $ imk 0 m = 0 p $ 1 % ( ) * + e ik 0 n k = 0 p $ 1 % = 1 p x ( m ) m = 0 p $ 1 % e $ imk 0 e ik 0 n k = 0 p $ 1 % = 1 p x ( m ) m = 0 p $ 1 % e $ i ( m $ n ) k 0 k = 0 p $ 1 %
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12 EE 200 Discrete-Time Fourier Series We can separate the summation over m into just the value of n , plus the sum over all the others. The summation over
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EE200_Weber_11-13 - EE 200 The 4 Fourier Transforms There...

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