1 Chapter 1 Fundamental Concepts

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 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

4 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

5 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

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

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

8 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

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

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

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

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

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

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

15 The Rectangular Pulse Function

16 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

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

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

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

20 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

21 Example: The Sinusoid

22 Time-Shifted Signals

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

24 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

25 Example of Continuous Signal: The Triangular Pulse Function

26 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

27 Example of Piecewise-Continuous Signal: The Rectangular Pulse Function

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

29 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

30 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

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

32 Consider the function defined as Another Example of Generalized Derivative

33 Another Example of Generalized Derivative: Cont’d

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

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

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

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

38 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

39 Example: The Ideal Predictor

40 Example: The Ideal Delay

41 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

42 Ideal Amplifier/AttenuatorIdeal Amplifier/Attenuator RC CircuitRC Circuit Examples

43 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

44 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

45 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

46 Example of Nonlinear System: Circuit with a Diode

47 Example of Nonlinear System: Square-Law Device

48 Example of Linear System: The Ideal Amplifier

49 Example of Nonlinear System: A Real Amplifier

50 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

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

52 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