04Discrete_FT

04Discrete_FT - Discrete vs Continuous Fourier Transform...

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Discrete vs Continuous Fourier Transform Discrete F.T. Continous F.T. pseudo-random function non-periodic, random th = sampling time interval dt N n t n t n ,..... , 2 , 1 ; = = time t T = Nh (finite) period infinite f TN == 11 h f loop frequency, Hz fd () k N Nh = 1 1 frequency operator −∞ df n N h = 1 time operator dt XX k f hx j kn N k n n N = =− = exp 2 1 π kN = 123 , , ,. ..... , Fourier Transform Xf x te d t jf t ( ) = 2 π xx n t fX j kn N n k k N = = = exp 2 1 n N = , Inverse Fourier Transform xt X f e d f t ( ) = 2 π −∞< <∞ t discrete spectrum continuous Properties (conjugate even) : nN n + = (periodic) for all n XXX X kkk N −+ * , k 2 for all k X X Nk k N k N k * (/ ) ) * , 2 for k N = 2 , k 1 2 3 N 2 -1 N 2 N 2 +1 N-3 N-2 N-1 N Re { X k } ar br cr gr Nyquist gr cr br ar mean Im { X k } ai bi ci gi 0 -gi -ci -bi -ai 0
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MAE 591 RANDOM DATA Test for Invariance of PSD : Random Signal PSD: 2 ,.... , 3 , 2 , 1 , 2 exp ) ( 2 ) , ( 2 ) ( 2 1 2 N k N kn j nh x N h T T f X E f N n k k xx = = = = π G System: Butterworth filter; 4th, low = 20 Hz, high =50 Hz; Input: Normal distributed normalized random; Sampling interval h = 1 ms;
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04Discrete_FT - Discrete vs Continuous Fourier Transform...

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