Sampling and Reconstruction of Analog Signals

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Sampling • Sample x a (t) at sampling interval T s sec apart to obtain the discretetime signal x(n) x ( n ) = x a ( nT s ) • Let X(e jω ) be discrete-time Fourier transform of x(n). • X(e jω ) is a countable sum of amplitude-scaled, frequency-scaled, and translated version of the Fourier transform X a (jΩ) X ( e ⎡ ⎢ ⎣ jω 1 + ∞ ω 2 ) = ∑ X ⎜ a j − Ts l= −∞ Ts Ts ⎛ ⎝ π ⎞⎤ l ⎟⎥ ⎠⎦ This is known as the aliasing formula
The analog and digital frequencies w = ΩT F s = 1 T s s F s : the sampling frequency, sam/sec –Amplitude scaled factor: 1/T s ; –Frequency-scaled factor: ω=ΩT s (ω=0~2π) –Frequency-translated factor: 2πk/T s ;
Page 1 and 2: Sampling and Reconstruction of Anal
Page 3: Sampling Continuous-time Fourier tr
Page 7 and 8: X ( e ⎡ ⎢ ⎣ jω 1 + ∞ ω 2
Page 9 and 10: Sampling Principle • A band-limit
Page 11 and 12: Example 1 • Let x a (t)=e -1000|t
Page 13 and 14: Example 2a • Sample x a (t) in ex
Page 15 and 16: Reconstruction Impulse train conver
Page 17 and 18: Practical D/A converters • Zero-o
Page 19 and 20: Cubic-order-hold (COH) interpolatio
Page 21 and 22: Example 3 • From the sample x(n)
Page 23 and 24: Example 5a • Reconstruct signal f
Page 25 and 26: Example 6a • Reconstruct the samp

Sampling and Reconstruction of Analog Signals

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