1. ΚΕΦΑΛΑΙΟ: Sampling And reconstruction
ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ
(22Δ802)
Β΄ ΕΞΑΜΗΝΟ
Καθηγητής Πέτρος Π. Γρουμπός 
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Τμήμα ΗΜ&ΤΥ
ΚΕΦΑΛΑΙΟ: Sampling And reconstruction

2. Continuous–Time Sinusoidal Signals
It is periodic for every fixed value of F, I.e. xa(t+Tp)=xa(t), where Tp=1/F
For distinct (different) frequencies they are themselves distinct
Increasing F results in an increase in the rate of oscillation

3. SAMPLING Discrete–Time Sinusoidal Signals
It is periodic only if f is a rational number
Discrete-Time sinusoids whose frequencies are separated by an integer multiple of 2π are identical
The highest rate of oscillation is attained when ω=π (or ω=-π) or f=1/2 (or f=-1/2)

4. Sampling the continuous-time (analog) sinusoid signal at a frequency of Fs=1/T, we get the discrete-time signal x(n):
i.e. or

5. ALIASING
Continuous-Time
Sampling
Discrete-Time
where
where

6. Proof:
Frequencies Fk=F0+kFs cannot be distinguished from F0 after sampling. In other words, they are aliases of F0.
This phenomenon is called aliasing or spectral overlap.

7. Aliasing is higher frequency impersonating lower frequencies due to the sampling rate not satisfying the Nyquist sampling criteria.

8. If Fk > Fs/2 then the actual frequency obtained is given by
Aliased frequencies
If Fk > Fs/2 then the actual frequency obtained is given by
where k is any integer such that

9. Aliasing example
Proof

10. Aliasing example

11. Aliasing example

12. Aliasing example
Similar to one-dimensional discrete-time signals, images can also suffer from aliasing if the sampling resolution or pixel density, is inadequate.
(Moiré pattern)

13. Aliasing demo

14. SAMPLING AND RECONSTRUCTION

15. S&H

16. S&H

17. S&H

18. S&H

19. S&H

20. S&H

21. S&H

28. Sampling Theorem or Nyquist Criteria or Shannon Theorem
If a signal contains no frequency components above a frequency F0 the signal can be uniquely represented by equally spaced samples if the sampling frequency Fs is greater than twice F0, i.e. Fs>2F0

29. Aliasing in the Frequency Domain

30. Aliasing in the Frequency Domain
SAMPLING

31. Sampling and the Frequency Domain

32. Aliasing in the Frequency Domain

33. Analog Anti-Aliasing Filter (Lowpass Filter)
Analog signals must be band-limited to proper frequency before sampling, because:
Input signal is time-limited and therefore cannot be band-limited
Even if the signal is “naturally” band-limited, additive noise has a much broader spectrum than the signal.

34. Hold

35. Hold

36. ZOH

37. ZOH

38. s --- z

39. s --- z

40. s --- z

41. s --- z

42. s --- z

43. s --- z

44. s --- z

45. s --- z

46. s --- z

47. ΕΥΧΑΡΙΣΤΩ ΓΙΑ ΤΗΝ ΠΡΟΣΟΧΗ ΣΑΣ
Καθ.Γρουμπός Π. Πέτρος .