1. The sampling of continuous-time signals is an important topic
It is required by many important technologies such as:
Digital Communication Systems ( Wireless Mobile Phones, Digital TV (Coming) ,
Digital Radio etc )
CD and DVD
Digital Photos

2. Switch close and open
Periodically with period Ts
Analog or continues level
Discrete Level
If you have levels you will need 3 bits
If you have 16 levels you will need 4 bits

3. Commercial type ADC or A/D

4. Some application of ADC or A/D

5. Recall Fourier Transform of periodical signal

6. Fourier Transform of periodical signal
Generating function

7. Generating function

8. Since

9. Let
Ideal Sampling in time
Maximum Frequency in F(w)

10. Ideal Sampling in time
Question : How to recover the original signal f(t) ↔F(w) from the sampled Fs(w)

11. If I use ideal low pass filter I will be able to extract this
Question : How to recover the original continuous signal f(t) ↔F(w) from the sampled fs(t) ↔ Fs(w)
Ideal Sampling in time
If I can recover this
in frequency
Ideal Sampling in time
With a constant correction A/Ts → A I can Fourier inverse back to recover the original continuous signal f(t)
If I use ideal low pass filter I will be able to extract this

12. If the bandwidth of the ideal low pass filter is greater than
We will get distorted shape
Therefore the ideal low pass filter bandwidth should be
When we inverse back we will not get the original signal f(t)

13. Now what will happened if you lowered the sampling frequency
Ideal Sampling in time
Now what will happened if you lowered the sampling frequency
The frequencies from adjacent part of the spectrum will interfere with each other
مستعار
We get distortion
Aliasing

15. Therefore to avoid aliasing and recover the original signal the sampling should be such that
Therefore Nyquist proposed the following
The Sampling Theorem
The sampling rate (ws) must be at least twice the highest frequency (wB) component present in the sample in order to reconstruct the original signal.

16. Practical non ideal sampling
If we sampled using train of pulses rather than impulses
Discrete Time
Discrete Level
We still can recover the original continuous signal
f(t) ↔F(w) from the sampled fs(t) ↔ Fs(w)

17. Next we develop the mathematics for discrete signals
Switch close and open
Periodically with period Ts
Analog or continues level
Discrete Level
We know have the following definition
Therefore we will have a sequence of numbers
Next we develop the mathematics for discrete signals

18. The shifted unit impulse function is defined by
Note that the discrete-time impulse function is well behaved mathematically and presents none of the problems of the continuous time impulse function

19. The discrete-time unit impulse function can be expressed as the difference of two step functions

21. 10 Discrete-Time Linear Time-Invariant Systems
Recall from the continuous case

23. Recall from the continuous case

24. Recall from the continuous case
An equation relating the output of a discrete LTI system to its input will now be developed
Recall from the continuous case
Impulse response
Impulse Input
Linear –Time Invariant
Shifted Impulse Input
Shifted Impulse Response
Multiply by the response by the same constant
Multiply by constant

25. Now relating the output of a discrete LTI system to its input will now be developed
Multiply each side by x[k]

26. (Discrete Convolution)
Let the input to a discrete-time system and the unit impulse response

34. Consider the system

36. Recall from the continuous case
Recall that a memory less (static) system is one whose current value of output depends on only the current value of input. A system with memory is called a dynamic system
Recall from the continuous case

37. A discrete-time LTI system is causal if the current value of the output depends on only the current value and past values of the input
Recall from the continuous case

40. ( )

41. The coefficient x[n] denotes the sample values and z-n denote the
The unilateral Z-Transform of the Sequence samples x(nT) ≡ x[n] defined by
The coefficient x[n] denotes the sample values and z-n denote the
sample occurs n sample periods after t = 0
However similar to Laplace we will not evaluate this complex integration we will Z-Transform of known tabulated sequences

42. Define the unite impulse sequence by ,
Note : the unit impulse here (the discrete) is different from the impulse d(t)
Laplace Z

43. Define the unite step by the sample values

44. Z- Transform Properties
(1) Linearity
Z- Transform is Linear operator
Then
Proof

45. Proof

46. Define the unite step by the sample values

47. Proof : see the book page 549

49. We want the z-Transform of cos(bn)
Entry on the Table 11.2
Similarly

50. Let the input to a discrete-time system and the unit impulse response
Using the same procedures we used in Fourier Transform and
Laplace Transform we get
The transfer function

52. Using partial fraction
One problem occurs in the use of the partial-fraction expansion procedure of Appendix F
The numerator is constant
However Z-transform for the exponential
Has z variable in the numerator
To solve this problem we expand (partial fraction ) Y(z)/z

54. Using the Z-Transform Table
Find x(n) ?
Since the degree of the Numerator equal the degree of the denominator
Polynomial division
This form is not available on the table

55. Now the degree of the Numerator less than the degree of the denominator
Using partial fraction , we have
Using Table 11-1

56. x(nT) will be the coefficients of the polynomial of X(z)
Inverse Z-Transform
Since
Therefore , if we can put X(z) into the form shown above,
Then we can determine x(nT) by inspection
x(nT) will be the coefficients of the polynomial of X(z)

57. Using polynomial division, we get
Therefore
The disadvantage of this method is that , we do not get x(nT) in closed form