# AdvDSP_lecture1 | Discrete Fourier Transform | Convolution

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Advanced DSP Lecture 1 Review Signals & Systems The Delta Function ã Dirac Delta (generalized) Function ã Sifting Property: Any continuous function can be expressed in terms of the delta function as: ¹ ´ ¦ = · = = 0 , 0 , 0 ) ( t t t o ) (t o ) ( ) ( ) ( 0 0 t dt t t t | o | } · · ÷ = ÷ · For any general function: } · · ÷ ÷ · = t t o t d t x t x ) ( ) ( ) ( ¹ ´ ¦ = · = = 0 0 0 , , 0 ) ( t t t t t o t The Delta Function Discrete Dirac D
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Advanced DSP Lecture 1Review  Signals & Systems  The Delta Function ã Dirac Delta (generalized) Function ã Sifting Property: Any continuous function can be expressedin terms of the delta function as:  0 ,0 ,0 )( t t t    )( t    )()()( 00 t dt t t t          For any general function:         d t  xt  x )()()(  000  , ,0)( t t t t t     t  The Delta Function  Discrete Dirac Delta Function  Sifting Property: Any sequence can be expressed in termsof the delta function as:  0 ,1 0 ,0 ][ nnn    k nk nk n  ,1 ,0][   ][ n    n1    k  k nk  xn x ][][][   ][]0[][][ n xnn x      For any general function:
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