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Unformatted text preview: ECE700 Review Phil Schniter January 5, 2007 1 Transforms Using x c ( t ) to denote a continuoustime signal at time t R , Laplace Transform: X c ( s ) = integraldisplay  x c ( t ) e st dt, s C ContinuousTime 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 discretetime signal at index n Z , ztransform: 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 DiscreteTime 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 ztransform evaluated on the unit circle in Cplane: 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 continuoustime 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 continuoustime 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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