Overview of Sampling Theory 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
Continuous-time Sampling Defining the sequence by multiple sifts: Equivalently: 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: Imp(t) can be any function with unit integral…
Example Linear interpolation: 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 DAC needs filtering to reduce excess high frequency information
Sinc(x) – ‘Perfect Reconstruction’ Is there an impulse function which needs no filtering? 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 Note that x(t) is defined for all t since Sinc(0)=1
Operations on sequences Addition: Scaling: Modulation: Windowing is a type of modulation Time-Shift: Up-sampling: Down-sampling:
Up-sampling
Down-sampling (Decimation)
Resampling (Integer Case) Suppose we have x[n] sampled at T1 but want xR[n] sampled at T2=L T1
Sampling Theorem Perfect Reconstruction of a continuous-time 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=2pk+w0 Samples for all k are the same! Unambiguous if 0<w<p Thus One-half cycle per sample So if sampling at T, frequencies of f=e+1/2T will map to frequency e