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Lecture 4 Sampling Overview of Sampling Theory Sampling Continuous Signals Sample Period is T, Frequency is 1/T x[n] = xa(n) = x(t)|t=nT Samples of x(t) from an infinite discrete sequence Continuous-time Sampling Delta function d(t) Zero everywhere except t=0 Integral of d(t) over any interval including t=0 is 1 (Not a function – but the limit of functions) Sifting f (t )d(t t )dt f (t ) 0 0 Continuous-time Sampling Defining the sequence by multiple sifts: xa (t ) Equivalently: xa (t ) x(t )d(t nT ) n x(nT )d(t nT ) n Note: xa(t) is not defined at t=nT and is zero for other t Reconstruction Given a train of samples – how to rebuild a continuous-time signal? In general, Convolve some impluse function with the samples: x(t ) x(nT )im p(t nT ) n Imp(t) can be any function with unit integral… Example Linear interpolation: 1 t 1 0 t 2 im p(t ) else 0 Integral (0,2) of imp(t) = 1 Imp(t) = 0 at t=0,2 Reconstucted function is piecewise-linear interpolation of sample values DAC Output Stair-step output 1 0 t 1 imp(t ) else 0 DAC needs filtering to reduce excess high frequency information Sinc(x) – ‘Perfect Reconstruction’ Is there an impulse function which needs no filtering? im p(t ) sin( t T t T ) Why? – Remember that Sin(t)/t is Fourier Transform of a unit impulse Perfect Reconstruction II Note – Sinc(t) is non-zero for all t Implies that all samples (including negative time) are needed (t nT ) sin T x(t ) x[n] (t nT ) n T Note that x(t) is defined for all t since Sinc(0)=1 Operations on sequences Addition: y[n] x[n] w[n] Scaling: y[n] A x[n] Modulation: y[n] x[n] w[n] Windowing is a type of modulation Time-Shift: y[n] x[n 1] x[n / L], n 0, L, 2L, Up-sampling: x [n] 0, else Down-sampling: xd [n] x[nM ] u Up-sampling x[n] x [n] 1 1 0.5 Amplitude Amplitude 0.5 0 -0.5 -1 u up-sampled by 3 Output sequence Input Sequence 0 -0.5 0 10 20 30 Time index n 40 50 -1 0 xu [n] x[n / 3] 10 20 30 Time index n 40 50 Down-sampling (Decimation) x[n] x [n] Input Sequence 1 1 0.5 Amplitude Amplitude 0.5 0 -0.5 -1 d down-sampled by 3 Output sequence 0 -0.5 0 10 20 30 Time index n 40 50 -1 0 xd [n] x[3n] 10 20 30 Time index n 40 50 Resampling (Integer Case) Suppose we have x[n] sampled at T1 but want xR[n] sampled at T2=L T1 x(t ) x(nT1 )im pulse(t nT1 ) n xR [ k ] x(nT )im pulse(t nT ) 1 n 1 x(nT )im pulse(kT n 1 2 t kT2 nT1 ) x(nLT )im pulse((k nL)T ) n xR [ k ] 2 2 x[n]im pulse[k n] n Sampling Theorem Perfect Reconstruction of a continuoustime signal with Bandlimit f requires samples no longer than 1/2f Bandlimit is not Bandwidth – but limit of maximum frequency Any signal beyond f aliases the samples Aliasing (Sinusoids) Alaising For Sinusoid signals (natural bandlimit): For Cos(wn), w=2k+w0 Samples for all k are the same! Unambiguous if 0<w< Thus One-half cycle per sample So if sampling at T, frequencies of f=e+1/2T will map to frequency e