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lecture6_small

# lecture6_small - Sampling Model of the sampling process r(t...

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Unformatted text preview: Sampling Model of the sampling process r(t) time: t time: t Sampling impulse train: v(t) Continuous-time signal: T 2T 3T 4T 5T 6T 7T time: t 1 T 2T 3T 4T 5T 6T 7T Sampled signal: r(t)v(t) Model sampling as time-domain multiplication by a train of impulses. r ( t ) = r ( t ) v ( t ) = summationdisplay k = r ( t ) ( t kT ) Taking the Laplace transform, L{ r ( t ) } = integraldisplay r ( t )e st dt = integraldisplay summationdisplay k = r ( t ) ( t kT )e st dt = summationdisplay k = r ( kT )e skT =: R ( s ) Roy Smith: ECE 147b 6 : 2 Sampling What is really happening with sampling? a27 a64 a100 r ( t ) r ( t ) T Recall that sampling maps strips of the s-plane onto the z-plane. Imaginary Imaginary Real Real 1-1 /T- /T Mapping via sampling s- plane z- plane What does this look like from a Fourier Transform point of view? Roy Smith: ECE 147b 6 : 1 Sampling Impulse trains-2 s-3T time: t Sampling impulse train (time-domain)-T-2T T 2T 3T 1 frequency: - 2 /T Sampling impulse train (frequency domain) s s 2 s Fourier series representation: v ( t ) = summationdisplay k = ( t kT ) = 1 T summationdisplay n = e jn ( 2 T ) t = 1 T summationdisplay n = e jn s t Fourier Transform: v ( s ) = 2 T summationdisplay n = ( n s ) (note: s = 2 /T ). A train of impulses is equivalent to an infinite sum of (equal) sinusoids....
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lecture6_small - Sampling Model of the sampling process r(t...

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