ECE 700 Lecture Notes (Phil Schniter)

ECE 700 Lecture Notes (Phil Schniter) - ECE-700 Review Phil...

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Unformatted text preview: ECE-700 Review Phil Schniter January 5, 2007 1 Transforms Using x c ( t ) to denote a continuous-time signal at time t R , Laplace Transform: X c ( s ) = integraldisplay - x c ( t ) e- st dt, s C Continuous-Time Fourier Transform (CTFT): X c ( j ) = integraldisplay - x c ( t ) e- j t dt, R x c ( t ) = 1 2 integraldisplay - X c ( j ) e j t d , t R Note that: X c ( j ) is the Laplace transform evaluated at s = j . x c ( t ) R implies X c ( j ) = X * c ( j ), i.e., conjugate symmetry. x c ( t ) is bandlimited to if X c ( j ) = 0 for all | | > . Using x [ n ] to denote a discrete-time signal at index n Z , z-transform: X ( z ) = summationdisplay n =- x [ n ] z- n , z C Denoting a transform pair by x [ n ] X ( z ), some useful properties are x [ n ] X ( z- 1 ) ( 1) n x [ n ] X ( z ) c circlecopyrt P. Schniter, 2002 1 Discrete-Time Fourier Transform (DTFT): X ( e j ) = summationdisplay n =- x [ n ] e- jn , R x [ n ] = 1 2 integraldisplay - X ( e j ) e jn d, n Z Note that: X ( e j ) is the z-transform evaluated on the unit circle in C-plane: z = e j . X ( e j ) is 2 -periodic in . x [ n ] R implies X ( e j ) = X * ( e- j ), i.e., conjugate symmetry. Other DTFT properties are: x [ n ] X ( e- j ) x * [ n ] X * ( e- j ) x [ n ] X ( e j ) e- j x [ n ] e j n X ( e j ( - ) ) x [ n ] y [ n ] 1 2 integraltext - X ( e j ) Y ( e j ( - ) ) d x [ n ] y [ n ] X ( e j ) Y ( e j ) n | x [ n ] | 2 1 2 integraltext - | X ( e j ) | 2 d where denotes linear convolution: x [ n ] y [ n ] = m =- x [ m ] y [ n m ]. 2 Uniform Sampling Say that x [ n ] is sampled from x c ( t ) with uniform sampling interval T : x [ n ] = x c ( nT ) , n Z Let us define the continuous-time impulse train p ( t ) = summationdisplay n =- ( t nT ) where ( t ) denotes the Dirac delta function, defined by the properties integraltext - ( t ) dt = 1 unit area integraltext - f ( t ) ( t ) dt = f ( ) sifting property Using Fourier series, it can be shown that p ( t ) = 1 T summationdisplay k =- e j 2 T kt c circlecopyrt P. Schniter, 2002 2 x c ( t ) x s ( t ) x [ n ] t t n Figure 1: Signals used in sampling theorem. Multiplying x c ( t ) by the impulse train yields the continuous-time sampled signal x s ( t ) which will help us to derive the sampling theorem. (See Fig. 1.) x s ( t ) = x c ( t ) summationdisplay n =- ( t nT ) = x c ( t ) 1 T summationdisplay k =- e j 2 T kt Taking the CTFT of x s ( t ), X s ( j ) = integraldisplay - parenleftBigg x c ( t ) 1 T summationdisplay k =- e j 2 T kt parenrightBigg e- j t dt = 1 T...
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ECE 700 Lecture Notes (Phil Schniter) - ECE-700 Review Phil...

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