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Chapter 1 Fundamental Concepts

Chapter 1 Fundamental Concepts. Signals. A signal is a pattern of variation of a physical quantity, often as a function of time (but also space, distance, position, etc). These quantities are usually the independent variables of the function defining the signal.

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Chapter 1 Fundamental Concepts

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  1. Chapter 1Fundamental Concepts

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

  3. Normal ECG Signal Voltage Time

  4. Radar Signal North East

  5. “Hello” Voltage Time

  6. Signal Processing • Signal 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

  7. Characterization and Classification of Signals • The type of signal depends on the nature of the independent variables and on the value of the function defining the signal • For example, the independent variables can be continuous or discrete • Likewise, the signal can be a continuous or discrete function of the independent variables

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

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

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

  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)

  12. Systems • A system is any device that can process signals for analysis, synthesis, enhancement, format conversion, recording, transmission, etc. • A 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

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

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

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

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

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

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

  19. The Rectangular Pulse Function

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

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

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

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

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

  25. Example: The Sinusoid

  26. Time-Shifted Signals

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

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

  29. Example of Continuous Signal: The Triangular Pulse Function

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

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

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

  33. Derivative of a Continuous-Time Signal • A 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 • If the limit exists, has a derivative at

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

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

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

  37. Another Example of Generalized Derivative: Cont’d

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

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

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

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

  42. Basic System Properties: Causality • A system is said to be causal if, for any time t1, the output response at time t1 resulting from input x(t) does not depend on values of the input for t > t1. • A system is said to be noncausal if it is not causal

  43. Example: The Ideal Predictor

  44. Example: The Ideal Delay

  45. Memoryless Systems and Systems with Memory • A causal system is memoryless or static if, for any time t1, the value of the output at time t1 depends only on the value of the input at time t1 • A causal system that is not memoryless is said to have memory. A system has memory if the output at time t1 depends in general on the past values of the input x(t) for some range of values of t up to t = t1

  46. Examples • Ideal Amplifier/Attenuator • RC Circuit

  47. Basic System Properties: Additive Systems • A system is said to be additive if, for any two inputs x1(t) and x2(t), the response to the sum of inputs x1(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) system

  48. Basic System Properties: Homogeneous Systems • A 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 system

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

  50. Example of Nonlinear System: Circuit with a Diode

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