1. Overview of Sampling Theory
Lecture 4 Sampling
Overview of Sampling Theory

2. 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

3. 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

4. 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

5. 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…

6. 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

7. DAC Output Stair-step output
DAC needs filtering to reduce excess high frequency information

8. 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

9. 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

10. Operations on sequences
Addition:
Scaling:
Modulation:
Windowing is a type of modulation
Time-Shift:
Up-sampling:
Down-sampling:

11. Up-sampling

12. Down-sampling (Decimation)

13. Resampling (Integer Case)
Suppose we have x[n] sampled at T1 but want xR[n] sampled at T2=L T1

14. 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

15. Aliasing (Sinusoids)

16. 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