discrete time signals and systems

1. Discrete-Time signals: sequences Discreet-Time signals are represented mathematically as sequences of numbers T he sequence is denoted , and it is written formally as where n is an integer number In practice sequences arises from the periodic sampling of an analog signal   1

2. Discrete-Time signals: sequences In this case the numeric value of the n th number in the sequence is equal to the value of the analog signal, , at time   2

3. Examples of sequences 3

4. Basic sequences and sequence operation The product and sum of two sequences x[n] and are defined as the sample by sample product and sum Multiplication of a sequence by a number is defined as the multiplication of each sample value by A sample is said to be delayed or shifted version of if   4

5. MATLAB exercise Record a voice signal using the audiorecorder function for 5 seconds with the following specifications sampling frequency of 44100 Number of quantization bits 16 Number of channels = 1 for mono Try to multiply the recorded samples by a scaling factor of then by Play the signal and hear the voice   5

6. Special sequences Unit sample sequence Unit sample sequence is defined as the sequence One of the important aspects of the impulse sequence is that an arbitrary sequence can be presented as a sum of scaled, delayed impulses as shown in the next slide 6

7. Special sequences Unit sample sequence In general any sequence can be written as   7

8. Special sequences Unit step sequence The unit step sequence is given by 8

9. Special sequences Unit step sequence The unit step sequence is given by 9

10. Special sequences Unit step sequence The unit step sequence in terms of delayed impulses can be written as Note that the impulse sequence can be expressed as the first backward difference of the unit step sequence   10

11. Special sequences exponential sequences Exponential sequence are important in representing and analyzing linear time invariant systems The general form of an exponential sequence is given by If and are real then the sequence is real If and is positive then the sequence values are positive and decreasing with increasing   11

12. Special sequences exponential sequences Graphical representation of exponential sequence 12

13. Special sequences sinusoidal sequences The general form of sinusoidal sequence is given by as shown   13

14. Special sequences sinusoidal and complex exponential sequence The exponential sequence with complex has a real and imaginary parts that are exponentially weighted sinusoids If and then the sequence can be expressed in either one of the following forms   14

15. Notes about sequences When discussing either complex exponential signals of the form or real sinusoidal signal of the form we need only to consider frequencies in an interval of length of only because   15

16. Periodic sequence A periodic sequence is a sequence that satisfies the following equation , Where is an integer number If this condition is tested for the discrete time sinusoids, then Which requires   16

17. Periodic sequence Where is an integer A similar statement holds for the complex exponential Where is an integer number Again   17

18. Example Determine if the following sequences are periodic or not. If the sequence is periodic find its period   18

19. solution For the first sequence we have or since is an integer value the sequence is periodic For the second sequence or since is not an integer value for the sequence is aperiodic if   19

20. 2.2 Discrete time systems A discrete-time system is a system that maps an input sequence with an output sequence   20

21. Discrete time system examples There are many systems will be investigated through out this course Examples of these systems are The ideal delay system which is described mathematically by Moving average system which is described mathematically by   21

22. Discrete time system classifications Systems can be classifieds into one of the following categories Memoryless Systems. A system is classified into memoryless system if the output at every value of depends only on the input of at the same value of . An example of a memoryless system is the squarer system described by   22

23. Discrete time system classifications Linear systems. Any system satisfies the superposition and the scaling property is classifieds as a linear system. As an example of a linear system is the accumulator system described by Time-invariant system is a system for which a time shift or delay of the input sequence causes a corresponding shift in the output sequence   23

24. Discrete time system classifications Example show that the accumulator system is a time invariant system solution Assume that the input to the accumulator is , then its output is Let This means that   24

25. Discrete time system classifications Causality, a system is causal if the output sequence value at the index depends only on the input sequence values for For example the forward difference system described by is not causal because the current value of the output depends on future value of the input Another example is the backward difference system is a causal system since the output depends only on the present and past values of the input   25

26. Discrete time system classifications Stability, a system is stable if and only if every bounded input sequence produces a bounded output sequence Such a system is called BIBO in equation form In general any sequence that has the form is stable system   26

27. Linear time-invariant system The linear time-invariant system is an important system since many of the system we deal with in signal processing are of this type The output sequence in response to the input sequence applied to the input of the linear time-invariant system is given by the convolutional sum   27

28. Linear time-invariant system In order to compute the convolution we draw both and sequences as shown below   28

29. Linear time-invariant system From the Figure, we have The next sequence interval is shown by the next graph that is   29

30. Linear time-invariant system The output sequence for this interval is given by This equation can be solved analytically by using the geometric series expansion   30

31. Linear time-invariant system The output sequence for this interval is given by This equation can be solved analytically by using the geometric series expansion   31

32. Convolution example Which yields the following result We consider the next interval when The output sequence is given by   32

33. Convolution example Which yields the following result The final answer for the output sequence for these three intervals is given by 33

34. Convolution example 34

35. Convolution in Matlab Convolution can be accomplished easily in matlab by using the function conv ( u,v ) The above example can be solved easily in matalb by using the following code in matlab n=1:10; h=ones(1,5); x=0.4.^n; Y= conv ( x,h ); stem(y); 35

36. 2.4 Properties of linear time invariant system The output sequence of all LTI are described by the convolution sum Where is the impulse response of the LTI system This means that is a complete characterization of the properties of a specific LTI system   36

37. Properties of the convolution sum commutative Distribution over addition Associative   37

38. Graphical representation of combined LTI systems 38 Cascaded systems can be presented by a single system whose impulse response is given by . Cascaded systems satisfy the convolution commutative property   Systems connected in parallel can be replaced by a single system whose .  

39. Stability and causality in terms of   LTI are stable if and only if there impulse response is absolutely summable i.e. LTI is causal if Causality means that the difference equations describing the system can be solved recursively   39

40. FIR systems – reflected in the h[n] Ideal delay Forward difference Backward difference Finite-duration impulse response (FIR) system are characterized by an impulse response has that has only a finite number of nonzero samples   40

41. IIR systems – reflected in the   Accumulator Infinite duration impulse response (IIR) system has whose duration extends to infinity Stability FIR systems always are stable, if each value of values is finite in magnitude IIR systems can be stable, e.g .   41

42. Cascading system examples Determine if the following system is causal or not Solution Since the impulse response of the cascaded system satisfy the resulting cascaded system is stable Any FIR system can be made causal by cascading it with a sufficiently long delay   42

43. Cascading system examples Determine the impulse response of the following cascaded systems An inverse system is given by 43

44. Linear constant-coefficient difference equations The N th order linear constant coefficient equations are a subclass of linear time invariant systems The general form of these equations is 44

45. Example of difference equations Write the accumulator system in terms of difference equations Solution The accumulator equation is given by The output for can be written as   45

46. Example of difference equations Now the output sequence can be written as Or alternatively it can be written as If we compare the last equation with we find that   46

47. Example of difference equations The difference equations gives a better understanding of how we can be implement the accumulator system in this example 47

48. Solving the Linear constant coefficient difference equations Difference equations are similar to differential equations in continuous systems The solution for the difference equations is composed from the homogeneous and particular solutions as described mathematically by   48

49. Solving the Linear constant coefficient difference equations The homogeneous solution is obtained with This means that the difference equation reduces to Since has undetermined coefficients, a set of auxiliary conditions is required for the unique specification of for a given   49

50. Solving the Linear constant coefficient difference equations These auxiliary conditions might consist of specifying fixed values of at specific values of , such as The above step results in a set of linear equations for the undetermined coefficients, which can be solved to produce the required coefficients   50

51. Recursive solution of the difference equations The output samples for can be computed recursively by rearranging the difference equation as shown below If the input , together with a set of auxiliary values is specified then the output can be computed   51

52. Recursive solution of the difference equations With available can be computed To generate values of for , we can rearrange the linear constant coefficient difference equation as shown below   52

53. Recursive computation example Example: solve the following difference equation recursively Assume that the input is and   53

54. Recursive computation example When , we can use recursive computation as follows Let then Since , then   54

55. Recursive computation example Next we do the same procedure when To determine the output for , we express the difference equations in the form   55

56. Recursive computation example If we use the auxiliary conditions , we can compute for as follows By combining the solutions for and , we got the following solution   56

57. 2.6 Frequency-domain representation of discrete time signals and systems The frequency response of a given system with impulse response of is defined by The output of any system characterized by its frequency response is given by   57

58. Frequency response of the ideal delay system Example determine the frequency response of an ideal delay system described by the following equation Solution To find the frequency response we first find the impulse response of the system which can be found by substituting   58

59. Frequency response of the ideal delay system This means that Now the frequency response is given by can be written in rectangular form as illustrated below , from Euler identity   59

60. 2.7 Representation of sequences by Fourier transforms In order to represent a given sequence by its Fourier transform we can use the following equation However the inverse Fourier transform is given by   60

61. Representation of sequences by Fourier transforms For the discrete time signals, the value of is restricted to an interval of The low frequency component of discrete time signals are located around The high frequency component are located around   61

62. Convergence of the Fourier transform In general not all the signals have Fourier transform Only the absolutely summable signals have their Fourier transform exits Absolutely summable signals are signals satisfying the following condition   62

63. Example Determine if has a Fourier transform or not. If the Fourier transform exist, find the value of Solution The summation If and only if this means that the discrete Fourier transform exists only for   63

64. Example The summation If and only if this means that the discrete Fourier transform exists only for   64