1. Echivalarea sistemelor analogice cu sisteme digitale

2. An analogic system is composed from circuits
A digital system is a computer algorithm

3. (continuous time) signal
An analogic
(continuous time) signal
Digital signal, a numerical sequence, obtained by sampling and AD conversion

4. Constant signal with digital frequency 0 cycles/sample
The maximum digital frequency is 0.5 cycles/sample

5. There are some “problems” with sampling
There are some “problems” with sampling. The signal having a 0 digital frequency can be obtained from many analogic signals
The signal with 0.5 digital frequency can be also obtained from many analogic signals

6. Sampling Sampling – time-domain discretization The Sampling Theorem
Ideal Sampling.

7. A sample of x(t) is obtained multiplying the continuous-time signal x(t) with the rectangular impulse uΔ(t ):
Another sample can be obtained using the same rectangular impulse shifted by kTs.

8. sampling process : small values of Δ:
ideal sampling of the signal x(t) :

9. Ideal Sampling
Mathematical model
System model

10. Spectrum of Ideal Sampled Signal
the spectrum of the ideal sampled signal :

11. The Fourier transform of a product is convolution of the Fourier transforms. The Fourier transform of the periodic Dirac’s distribution is also a periodic Dirac’s distribution.
The effect of the convolution of a specified function with the periodic Dirac’s distribution is the periodic repetition of the considered function.

12. The spectrum of an ideal sampled signal is the periodic repetition of the spectrum of the original signal. The period is inverse proportional with the sampling step Ts.

13. Spectrum of original signal
Spectrum of periodic Dirac distribution
Spectrum of ideal sampled signal
Aliasing error

14. Sampling Band-limited Signals
x(t)-band-limited

15. successive replica of the original signal spectrum are not superimposed and the spectrum of the original signal can be recovered by low-pass ideal filtering for

16. Ideal Low-pass Filter

17. the aliasing error can be avoided.
For perfect reconstruction :
Sampling freq.
Cutoff freq for the low pass filter

18. The frequency response of the reconstruction filter is:
Its response :
with the spectrum:

19. When the condition: is not verified, the aliasing error appears.

20. Reconstruction

21. WKS (Whittaker, Kotelnikov, Shannon) Sampling Theorem
If the finite energy signal x(t) is band limited at ωM , ( X(ω)=0 for |ω | > ωM), it is uniquely determined by its samples if the sampling frequency is higher or equal than twice the maximum frequency of the signal:
the original signal can be reconstructed from its samples a.e.w:
if the cut-off frequency of ideal low-pass reconstruction filter :

22. The link between the two frequency axes of spectra of the discrete signal and of the corresponding continuous signal is:
 spectrum periodic
The maximum frequencies are also related:

23. Ideal Low-pass Filtering Reconstruction
The signal reconstructed from curves of type sin x / x.
Interpolation

25. Linear interpolation
approximates the signal using straight lines that unify points determined by the samples
Reconstruction filter is triangular.
errors

27. Reconstruction by Zero Order Extrapolation

28. The frequency response of the reconstruction filter is not perfectly flat in the pass band.

33. The Fourier Transform of Distributions
1) The spectrum of the Dirac’s distribution
for any test function (t):
the Dirac’s distribution is even.
Hence, we have obtained:

34. 2) The spectrum of the constant 1(t)
duality

35. Approximation of continuous-time systems with discrete-time systems
The continuous-time systems are replaced by discrete-time systems even for the processing of continuous-time signals.
Impulse invariance method
Step invariance method
Finite Difference Approximation (FDA)
Bilinear Transform

36. General Method for Quality Evaluation of Approximation for Band-limited Systems
AD converter – Analog to Digital Converter
DA converter –Digital to Analog Converter

38. Identification of the impulse response of the system h(t)

39. Identification of the impulse response of a band-limited system using the cardinal sine.
Frequency response of the band-limited system that must be identified
Spectrum of the input signal (cardinal sine)
Spectrum of the output signal

41. 1. Impulse Invariance Method

43. Change the input signal : changes the digital system.
No perfect approximation.
Approximation using “standard” signals:
Unit step (t)
Ramp signal t (t).

44. Approximation by sum of unit step signals, shifted and weighted

45. Approximation by sum of ramp signals, shifted and weighted

46. Impulse invariance method for digital systems equivalent with band-limited c.t. systems

48. Relation between s and z planes for the Impulse Invariance Method
Left half plane <  interior of unit disc
|z|=r<1
Imaginary  unit disc |z|=1
axis =0
Right half plane >  exterior of unit disc |z|=r>1
segment [-/T, /T) on imaginary axis  one wrapping on the unit disc

49. Avoid aliasing errors from the frequency response of the digital system obtained:
frequency response of the analog system: completely included in freq. band -π/T , π/T.
band-limited analog system
Sampling freq.

50. Impulse invariance method: link between frequency responses of equivalent systems
Transfer functions
Freq. responses
The frequency response of the digital system is the same with the frequency response of the analog system of limited band for frequency less than half of sampling frequency

51. frequency response of the digital system ~ comb

52. Example: RC circuit
non band limited system  aliasing errors

54. Unit step response

57. Frequency response

60. 2. Step Invariance Method
Input signal: unit step
Step response of the analog signal
Digital input signal:

61. Digital system:

62. Example: RC circuit

64. 3. Finite difference approximation
First derivative approximation
Transfer function of the digital system:

65. General case

67. Relation Between s and z planes Finite Difference Approximation

69. s-plane | z-plane

70. Stable analog system => stable digital system
Analog system and digital system: identical freq. response if imaginary axis on s-plane = unit circle on the z-plane.
Not true for this method!!!

72. Example: first order LPF

73. Finite difference approximation

74. Finite difference approximation

75. Finite difference approximation
higher accuracy !

76. 4. Bilinear Transform Input signal
Area An (ABCD)~ integral In: numerical methods (trapezoidal rule)

77. Bilinear transform:

78. Relation of the s and z planes for the bilinear transformation method

80. Relation between frequency responses for the bilinear transform

82. Distorted digital freq
Distorted digital freq.response due to non-linear relation between frequencies!!!