1. 1 Chapter 1 Fundamental Concepts

2. 2 signalpattern of variation of a physical quantity,A signal is a pattern of variation of a physical quantity, often as a function of time (but also space, distance, position, etc). independent variablesThese quantities are usually the independent variables of the function defining the signal. informationA signal encodes information, which is the variation itself. Signals

3. 3 Normal ECG Signal Time Voltage

4. 4 Radar Signal East North

5. 5 “Hello” Time Voltage

6. 6 extracting, analyzing, and manipulating the informationSignal processing is the discipline concerned with extracting, analyzing, and manipulating the information carried by signals The processing method depends on the type of signal and on the nature of the information carried by the signal Signal Processing

7. 7 type of signalThe type of signal depends on the nature of the independent variables and on the value of the function defining the signal continuous or discreteFor example, the independent variables can be continuous or discrete continuous or discrete functionLikewise, the signal can be a continuous or discrete function of the independent variables Characterization and Classification of Signals

8. 8 real- valued functioncomplex-valued functionMoreover, the signal can be either a real- valued function or a complex-valued function scalar or one-dimensional (1-D) signalA signal consisting of a single component is called a scalar or one-dimensional (1-D) signal Characterization and Classification of Signals – Cont’d

9. 9 Examples: CT vs. DT Signals stem(n,x)plot(t,x)

10. 10 Discrete-time signals are often obtained by sampling continuous-time signals Sampling..

11. Code for MATLAB demo >> syms t >> syms x >> x=exp(-.3*t)*sin(2*t/3); >> ezplot(x) >> ezplot(x, 0, 30) >> grid on >> x=exp(-.1*t)*sin(2*t/3); >> ezplot(x, 0, 30) >> grid on >> n=1:2:100; >> y=n.^1.5; >> stem(n,y) >> n=-10:10; >> y=n.^2; >> stem(n,y) 11

12. 12 systemprocess signalsA system is any device that can process signals for analysis, synthesis, enhancement, format conversion, recording, transmission, etc. I/O characterizationA system is usually mathematically defined by the equation(s) relating input to output signals (I/O characterization) A system may have single or multiple inputs and single or multiple outputs Systems

13. 13 Block Diagram Representation of Single-Input Single-Output (SISO) CT Systems input signal output signal

14. 14 Differential equation Convolution model Transfer function representation (Fourier transform, Laplace transform) Types of input/output representations considered

15. 15 Examples of 1-D, Real-Valued, CT Signals: Temporal Evolution of Currents and Voltages in Electrical Circuits

16. 16 Examples of 1-D, Real-Valued, CT Signals: Temporal Evolution of Some Physical Quantities in Mechanical Systems

17. 17 Unit-step functionUnit-step function Unit-ramp functionUnit-ramp function Continuous-Time (CT) Signals

18. 18 Unit-Ramp and Unit-Step Functions: Some Properties (with exception of )

19. 19 The Rectangular Pulse Function

20. 20 delta functionDirac distributionA.k.a. the delta function or Dirac distribution It is defined by:It is defined by: The value is not defined, in particularThe value is not defined, in particular The Unit Impulse

21. 21 The Unit Impulse: Graphical Interpretation A is a very large number

22. 22 is the impulse with area, i.e., The Scaled Impulse K(t)

23. 23 Properties of the Delta Function except 1) 2) sifting property (sifting property)

24. 24 Definition: a signal is said to be periodic with period, if Notice that is also periodic with period where is any positive integer fundamental period is called the fundamental period Periodic Signals

25. 25 Example: The Sinusoid

26. 26 Time-Shifted Signals

27. 27 A continuous-time signal is said to be discontinuous at a point if where and, being a small positive number Points of Discontinuity

28. 28 A signal is continuous at the point if continuous signalIf a signal is continuous at all points t, is said to be a continuous signal Continuous Signals

29. 29 Example of Continuous Signal: The Triangular Pulse Function

30. 30 A signal is said to be piecewise continuous if it is continuous at all except a finite or countably infinite collection of points Piecewise-Continuous Signals

31. 31 Example of Piecewise-Continuous Signal: The Rectangular Pulse Function

32. 32 Another Example of Piecewise- Continuous Signal: The Pulse Train Function

33. 33 differentiableA signal is said to be differentiable at a point if the quantity has limit as independent of whether approaches 0 from above or from below derivativeIf the limit exists, has a derivative at Derivative of a Continuous-Time Signal

34. 34 However, piecewise-continuous signals may have a derivative in a generalized sense Suppose that is differentiable at all except generalized derivativeThe generalized derivative of is defined to be Generalized Derivative ordinary derivative of at all except

35. 35 Define The ordinary derivative of is 0 at all points except Therefore, the generalized derivative of is Example: Generalized Derivative of the Step Function

36. 36 Consider the function defined as Another Example of Generalized Derivative

37. 37 Another Example of Generalized Derivative: Cont’d

38. 38 Example of CT System: An RC Circuit Kirchhoff’s current law:

39. 39 The v-i law for the capacitor is Whereas for the resistor it is RC Circuit: Cont’d

40. 40 Constant-coefficient linear differential equationConstant-coefficient linear differential equation describing the I/O relationship if the circuit RC Circuit: Cont’d

41. 41 Step response when R=C=1 RC Circuit: Cont’d

42. 42 causalA system is said to be causal if, for any time t 1, the output response at time t 1 resulting from input x(t) does not depend on values of the input for t > t 1. noncausalA system is said to be noncausal if it is not causal Basic System Properties: Causality

43. 43 Example: The Ideal Predictor

44. 44 Example: The Ideal Delay

45. 45 memorylessstaticA causal system is memoryless or static if, for any time t 1, the value of the output at time t 1 depends only on the value of the input at time t 1 memoryA causal system that is not memoryless is said to have memory. A system has memory if the output at time t 1 depends in general on the past values of the input x(t) for some range of values of t up to t = t 1 Memoryless Systems and Systems with Memory

46. 46 Ideal Amplifier/AttenuatorIdeal Amplifier/Attenuator RC CircuitRC Circuit Examples

47. 47 additiveA system is said to be additive if, for any two inputs x 1 (t) and x 2 (t), the response to the sum of inputs x 1 (t) + x 2 (t) is equal to the sum of the responses to the inputs (assuming no initial energy before the application of the inputs) Basic System Properties: Additive Systems system

48. 48 homogeneousA system is said to be homogeneous if, for any input x(t) and any scalar a, the response to the input ax(t) is equal to a times the response to x(t), assuming no energy before the application of the input Basic System Properties: Homogeneous Systems system

49. 49 linearA system is said to be linear if it is both additive and homogeneous nonlinearA system that is not linear is said to be nonlinear Basic System Properties: Linearity system

50. 50 Example of Nonlinear System: Circuit with a Diode

51. 51 Example of Nonlinear System: Square-Law Device

52. 52 Example of Linear System: The Ideal Amplifier

53. 53 Example of Nonlinear System: A Real Amplifier

54. 54 time invariantA system is said to be time invariant if, for any input x(t) and any time t 1, the response to the shifted input x(t – t 1 ) is equal to y(t – t 1 ) where y(t) is the response to x(t) with zero initial energy time varyingtime variantA system that is not time invariant is said to be time varying or time variant Basic System Properties: Time Invariance system

55. 55 Amplifier with Time-Varying GainAmplifier with Time-Varying Gain First-Order SystemFirst-Order System Examples of Time Varying Systems

56. 56 Basic System Properties: CT Linear Finite-Dimensional Systems If the N-th derivative of a CT system can be written in the form then the system is both linear and finite dimensional To be time-invariant