Material presented here is from the course 6.003, Signals & Systems offered by MIT faculty member, Prof. Alan Willsky, Copyright c 2003.

Transcription

1 EE-295 Image Processing, Spring 2008 Lecture 1 Material presented here is from the course 6.003, Signals & Systems offered by MIT faculty member, Prof. Alan Willsky, Copyright c This material is subject to the MIT OpenCourseWare Creative Commons license, 1

2 Signals and Systems Fall 2003 Lecture #1 Prof. Alan S. Willsky 4 September ) Administrative details 2) Signals 3) Systems 4) For examples... Figures and images used in these lecture notes by permission, copyright 1997 by Alan V. Oppenheim and Alan S. Willsky 1

3 SIGNALS Signals are functions of independent variables that carry information. For example: Electrical signals --- voltages and currents in a circuit Acoustic signals --- audio or speech signals (analog or digital) Video signals --- intensity variations in an image (e.g. a CAT scan) Biological signals --- sequence of bases in a gene... 2

4 THE INDEPENDENT VARIABLES Can be continuous Trajectory of a space shuttle Mass density in a cross-section of a brain Can be discrete DNA base sequence Digital image pixels Can be 1-D, 2-D, N-D For this course: Focus on a single (1-D) independent variable which we call time. Continuous-Time (CT) signals: x(t), t continuous values Discrete-Time (DT) signals: x[n], n integer values only 3

5 CT Signals Most of the signals in the physical world are CT signals E.g. voltage & current, pressure, temperature, velocity, etc. 4

6 DT Signals x[n], n integer, time varies discretely Examples of DT signals in nature: DNA base sequence Population of the nth generation of certain species 5

7 Many human-made DT Signals Ex.#1 Weekly Dow-Jones industrial average Ex.#2 digital image Courtesy of Jason Oppenheim. Used with permission. Why DT? Can be processed by modern digital computers and digital signal processors (DSPs). 6

8 SYSTEMS For the most part, our view of systems will be from an input-output perspective: A system responds to applied input signals, and its response is described in terms of one or more output signals x(t) CT System y(t) x[n] DT System y[n] 7

9 An RLC circuit EXAMPLES OF SYSTEMS Dynamics of an aircraft or space vehicle An algorithm for analyzing financial and economic factors to predict bond prices An algorithm for post-flight analysis of a space launch An edge detection algorithm for medical images 8

10 SYSTEM INTERCONNECTIOINS An important concept is that of interconnecting systems To build more complex systems by interconnecting simpler subsystems To modify response of a system Signal flow (Block) diagram Cascade Parallel + Feedback + 9

11 Signals and Systems Fall 2003 Lecture #10 7 October Examples of the DT Fourier Transform 2. Properties of the DT Fourier Transform 3. The Convolution Property and its Implications and Uses

12 DT Fourier Transform Pair Analysis Equation FT Synthesis Equation Inverse FT

13 Convergence Issues Synthesis Equation: Analysis Equation: None, since integrating over a finite interval Need conditions analogous to CTFT, e.g. Finite energy Absolutely summable

14 Examples Parallel with the CT examples in Lecture #8

15 More Examples Infinite sum formula

16 Still More 4) DT Rectangular pulse (Drawn for N 1 = 2)

17 5)

18 DTFTs of Sums of Complex Exponentials Recall CT result: What about DT: a) We expect an impulse (of area 2π) at ω = ω o b) But X(e jω ) must be periodic with period 2π In fact Note: The integration in the synthesis equation is over 2π period, only need X(e jω ) in one 2π period. Thus,

19 DTFT of Periodic Signals DTFS synthesis eq. Linearity of DTFT

20 Example #1: DT sine function

21 Example #2: DT periodic impulse train Also periodic impulse train in the frequency domain!

22 Properties of the DT Fourier Transform Different from CTFT

23 More Properties Example Important implications in DT because of periodicity

24 Still More Properties

25 7) Time Expansion Recall CT property: But in DT: Yet Still More Properties x[n/2] makes no sense x[2n] misses odd values of x[n] Time scale in CT is infinitely fine But we can slow a DT signal down by inserting zeros: k an integer 1 x (k) [n] insert (k - 1) zeros between successive values Insert two zeros in this example (k=3)

26 Time Expansion (continued) Stretched by a factor of k in time domain -compressed by a factor of k in frequency domain

27 Is There No End to These Properties? 8) Differentiation in Frequency Multiplication by n 9) Parseval s Relation Differentiation in frequency Total energy in time domain Total energy in frequency domain

28 Example #1: The Convolution Property

29 Example #2: Ideal Lowpass Filter

30 Example #3:

31 Signals and Systems Fall 2003 Lecture #11 9 October DTFT Properties and Examples 2. Duality in FS & FT 3. Magnitude/Phase of Transforms and Frequency Responses

32 Convolution Property Example

33 DT LTI System Described by LCCDE s Rational function of e -jω, use PFE to get h[n]

34 Example: First-order recursive system with the condition of initial rest causal

35 DTFT Multiplication Property

36 Calculating Periodic Convolutions

37 Example:

38 Duality in Fourier Analysis Fourier Transform is highly symmetric CTFT: Both time and frequency are continuous and in general aperiodic Same except for these differences Suppose f( ) and g( ) are two functions related by Then

39 Example of CTFT duality Square pulse in either time or frequency domain

40 DTFS Duality in DTFS Then

41 Duality between CTFS and DTFT CTFS DTFT

42 CTFS-DTFT Duality

43 Magnitude and Phase of FT, and Parseval Relation CT: Parseval Relation: Energy density in ω DT: Parseval Relation:

44 Effects of Phase Not on signal energy distribution as a function of frequency Can have dramatic effect on signal shape/character Constructive/Destructive interference Is that important? Depends on the signal and the context

45 Demo: 1) Effect of phase on Fourier Series 2) Effect of phase on image processing

46 Log-Magnitude and Phase Easy to add

47 Plotting Log-Magnitude and Phase a) For real-valued signals and systems Plot for ω 0, often with a logarithmic scale for frequency in CT b) In DT, need only plot for 0 ω π (with linear scale) c) For historical reasons, log-magnitude is usually plotted in units of decibels (db): power magnitude So 20 db or 2 bels: = 10 amplitude gain = 100 power gain

48 A Typical Bode plot for a second-order CT system 20 log H(jω) and H(jω) vs. log ω 40 db/decade Changes by -π

49 A typical plot of the magnitude and phase of a secondorder DT frequency response 20log H(e jω ) and H(e jω ) vs. ω For real signals, 0 to π is enough

50 Signals and Systems Fall 2003 Lecture #12 16 October Linear and Nonlinear Phase 2. Ideal and Nonideal Frequency-Selective Filters 3. CT & DT Rational Frequency Responses 4. DT First- and Second-Order Systems

51 Linear Phase CT Result: Linear phase simply a rigid shift in time, no distortion Nonlinear phase distortion as well as shift DT Question:

52 All-Pass Systems CT DT

53 Demo: Impulse response and output of an all-pass system with nonlinear phase

54 φ How do we think about signal delay when the phase is nonlinear? Group Delay

55 Ideal Lowpass Filter CT Noncausal h(t <0) 0 Oscillatory Response e.g. step response Overshoot by 9%, Gibbs phenomenon

56 Nonideal Lowpass Filter Sometimes we don t want a sharp cutoff, e.g. Often have specifications in time and frequency domain Trade-offs Freq. Response Step response

57 CT Rational Frequency Responses CT: If the system is described by LCCDEs, then Prototypical Systems First-order system, has only one energy storing element, e.g. L or C Second-order system, has two energy storing elements, e.g. L and C

58 DT Rational Frequency Responses If the system is described by LCCDE s (Linear-Constant-Coefficient Difference Equations), then

59 DT First-Order Systems

60 Demo: Unit-sample, unit-step, and frequency response of DT first-order systems

61 DT Second-Order System decaying oscillations

62 Demo: Unit-sample, unit-step, and frequency response of DT second-order systems

63 Signals and Systems Fall The Concept and Representation of Periodic Sampling of a CT Signal 2. Analysis of Sampling in the Frequency Domain 3. The Sampling Theorem the Nyquist Rate 4. In the Time Domain: Interpolation 5. Undersampling and Aliasing Lecture #13 21 October 2003

64 SAMPLING We live in a continuous-time world: most of the signals we encounter are CT signals, e.g. x(t). How do we convert them into DT signals x[n]? Sampling, taking snap shots of x(t) every T seconds. T sampling period x[n] x(nt), n =..., -1, 0, 1, 2,... regularly spaced samples Applications and Examples Digital Processing of Signals Strobe Images in Newspapers Sampling Oscilloscope How do we perform sampling?

65 Why/When Would a Set of Samples Be Adequate? Observation: Lots of signals have the same samples By sampling we throw out lots of information all values of x(t) between sampling points are lost. Key Question for Sampling: Under what conditions can we reconstruct the original CT signal x(t) from its samples?

66 Impulse Sampling Multiplying x(t) by the sampling function

67 Analysis of Sampling in the Frequency Domain Important to note: ω s 1/T

68 Illustration of sampling in the frequency-domain for a band-limited (X(jω)=0 for ω > ω M ) signal No overlap between shifted spectra

69 Reconstruction of x(t) from sampled signals If there is no overlap between shifted spectra, a LPF can reproduce x(t) from x p (t)

70 The Sampling Theorem Suppose x(t) is bandlimited, so that Then x(t) is uniquely determined by its samples {x(nt)} if

71 (1) In practice, we obviously don t sample with impulses or implement ideal lowpass filters. One practical example: The Zero-Order Hold Observations on Sampling

72 Observations (Continued) (2) Sampling is fundamentally a time-varying operation, since we multiply x(t) with a time-varying function p(t). However, is the identity system (which is TI) for bandlimited x(t) satisfying the sampling theorem (ω s > 2ω M ). (3) What if ω s 2ω M? Something different: more later.

73 Time-Domain Interpretation of Reconstruction of Sampled Signals Band-Limited Interpolation The lowpass filter interpolates the samples assuming x(t) contains no energy at frequencies ω c

74 Original CT signal Graphic Illustration of Time-Domain Interpolation h(t) After sampling T After passing the LPF

75 Interpolation Methods Bandlimited Interpolation Zero-Order Hold First-Order Hold Linear interpolation

76 When ω s 2 ω M Undersampling Undersampling and Aliasing

77 Undersampling and Aliasing (continued) X r (jω) X(jω) Distortion because of aliasing Higher frequencies of x(t) are folded back and take on the aliases of lower frequencies Note that at the sample times, x r (nt) = x(nt)

78 A Simple Example Picture would be Modified Demo: Sampling and reconstruction of cosω o t

79 Signals and Systems Fall 2003 Lecture #14 23 October Review/Examples of Sampling/Aliasing 2. DT Processing of CT Signals

80 Sampling Review Demo: Effect of aliasing on music.

81 Strobe Demo > 0, strobed image moves forward, but at a slower pace = 0, strobed image still < 0, strobed image moves backward. Applications of the strobe effect (aliasing can be useful sometimes): E.g., Sampling oscilloscope

82 DT Processing of Band-Limited CT Signals Why do this? Inexpensive, versatile, and higher noise margin. How do we analyze this system? We will need to do it in the frequency domain in both CT and DT In order to avoid confusion about notations, specify ω CT frequency variable Ω DT frequency variable (Ω = ωτ) Step 1: Find the relation between x c (t) and x d [n], or X c (jω) and X d (e jω )

83 Note: Not full analog/digital (A/D) conversion not quantizing the x[n] values Time-Domain Interpretation of C/D Conversion

84 Frequency-Domain Interpretation of C/D Conversion Note: ω s 2π CT DT

85 Illustration of C/D Conversion in the Frequency-Domain X d (e jω ) X d (e jω ) Ω = ωt 1 Ω = ωt2

86 D/C Conversion y d [n] y c (t) Reverse of the process of C/D conversion

87 Now the whole picture Overall system is time-varying if sampling theorem is not satisfied It is LTI if the sampling theorem is satisfied, i.e. for bandlimited inputs x c (t), with ω M < ω s 2 When the input x c (t) is band-limited (X(jω) = 0 at ω > ω Μ ) and the sampling theorem is satisfied (ω s > 2ω M ), then DT omege needs to changed

88 Frequency-Domain Illustration of DT Processing of CT Signals DT filter Sampling DT freq CT freq CT freq DT freq Interpolate (LPF) equivalent CT filter

89 Assuming No Aliasing In practice, first specify the desired H c (jω), then design H d (e jω ).

90 Example: Digital Differentiator Applications: Edge Enhancement Courtesy of Jason Oppenheim. Used with permission. Courtesy of Jason Oppenheim. Used with permission.

91 Bandlimited Differentiator Construction of Digital Differentiator

92 Band-Limited Digital Differentiator (continued) CT DT

93 Signals and Systems Fall 2003 Lecture #2 9 September ) Some examples of systems 2) System properties and examples a) Causality b) Linearity c) Time invariance

94 SYSTEM EXAMPLES x(t) CT System y(t) x[n] DT System y[n] Ex. #1 RLC circuit

95 Ex. #2 Mechanical system Force Balance: Observation: Very different physical systems may be modeled mathematically in very similar ways.

96 Ex. #3 Thermal system Cooling Fin in Steady State

97 Ex. #3 (Continued) Observations Independent variable can be something other than time, such as space. Such systems may, more naturally, have boundary conditions, rather than initial conditions.

98 Ex. #4 Financial system Fluctuations in the price of zero-coupon bonds t = 0 Time of purchase at price y 0 t = T Time of maturity at value y T y(t) = Values of bond at time t x(t) = Influence of external factors on fluctuations in bond price Observation: Even if the independent variable is time, there are interesting and important systems which have boundary conditions.

99 Ex. #5 A rudimentary edge detector This system detects changes in signal slope

100 Observations 1) A very rich class of systems (but by no means all systems of interest to us) are described by differential and difference equations. 2) Such an equation, by itself, does not completely describe the input-output behavior of a system: we need auxiliary conditions (initial conditions, boundary conditions). 3) In some cases the system of interest has time as the natural independent variable and is causal. However, that is not always the case. 4) Very different physical systems may have very similar mathematical descriptions.

101 WHY? SYSTEM PROPERTIES (Causality, Linearity, Time-invariance, etc.) Important practical/physical implications They provide us with insight and structure that we can exploit both to analyze and understand systems more deeply.

102 CAUSALITY A system is causal if the output does not anticipate future values of the input, i.e., if the output at any time depends only on values of the input up to that time. Allreal-time physical systems are causal, because time only moves forward. Effect occurs after cause. (Imagine if you own a noncausal system whose output depends on tomorrow s stock price.) Causality does not apply to spatially varying signals. (We can move both left and right, up and down.) Causality does not apply to systems processing recorded signals, e.g. taped sports games vs. live broadcast.

103 CAUSALITY (continued) Mathematically (in CT): A system x(t) y(t) is causal if when and x 1 (t) y 1 (t) x 2 (t) y 2 (t) x 1 (t) = x 2 (t) for all t t o Then y 1 (t) = y 2 (t) for all t t o

104 CAUSAL OR NONCAUSAL

105 TIME-INVARIANCE (TI) Informally, a system is time-invariant (TI) if its behavior does not depend on what time it is. Mathematically (in DT): A system x[n] y[n] is TI if for any input x[n] and any time shift n 0, If x[n] y[n] then x[n - n 0 ] y[n - n 0 ]. Similarly for a CT time-invariant system, If x(t) y(t) then x(t - t o ) y(t - t o ).

106 TIME-INVARIANT OR TIME-VARYING? TI Time-varying (NOT time-invariant)

107 NOW WE CAN DEDUCE SOMETHING! Fact: If the input to a TI System is periodic, then the output is periodic with the same period. Proof : Suppose x(t + T) = x(t) and x(t) y(t) Then by TI x(t + T) y(t + T). These are the same input! So these must be the same output, i.e., y(t) = y(t + T).

108 LINEAR AND NONLINEAR SYSTEMS Many systems are nonlinear. For example: many circuit elements (e.g., diodes), dynamics of aircraft, econometric models, However, in we focus exclusively on linear systems. Why? Linear models represent accurate representations of behavior of many systems (e.g., linear resistors, capacitors, other examples given previously, ) Can often linearize models to examine small signal perturbations around operating points Linear systems are analytically tractable, providing basis for important tools and considerable insight

109 LINEARITY A (CT) system is linear if it has the superposition property: If x 1 (t) y 1 (t) and x 2 (t) y 2 (t) then ax 1 (t) + bx 2 (t) ay 1 (t) + by 2 (t) y[n] = x 2 [n] y(t) = x(2t) Nonlinear, TI, Causal Linear, notti, Noncausal Can you find systems with other combinations? - e.g. Linear, TI, Noncausal Linear, not TI, Causal

110 PROPERTIES OF LINEAR SYSTEMS Superposition If Then For linear systems, zero input zero output "Proof" 0 = 0 x[n] 0 y[n] = 0

111 Properties of Linear Systems (Continued) A linear system is causal if and only if it satisfies the condition of initial rest: Proof a) Suppose system is causal. Show that (*) holds. b) Suppose (*) holds. Show that the system is causal.

112 LINEAR TIME-INVARIANT (LTI) SYSTEMS Focus of most of this course - Practical importance (Eg. #1-3 earlier this lecture are all LTI systems.) - The powerful analysis tools associated with LTI systems A basic fact: If we know the response of an LTI system to some inputs, we actually know the response to many inputs

113 Example: DT LTI System

114 Signals and Systems Fall 2003 Lecture #21 25 November Feedback a) Root Locus b) Tracking c) Disturbance Rejection d) The Inverted Pendulum 2. Introduction to the Z-Transform

115 The Concept of a Root Locus C(s),G(s) Designed with one or more free parameters Question: How do the closed-loop poles move as we vary these parameters? Root locus of 1+ C(s)G(s)H(s)

116 The Classical Root Locus Problem C(s) = K a simple linear amplifier Closed-loop poles are the same.

117 A Simple Example In either case, pole is at s o = -2 - K Sketch where pole moves as K increases... Becomes more stable Becomes less stable

118 What Happens More Generally? For simplicity, suppose there is no pole-zero cancellation in G(s)H(s) Closed-loop poles are the solutions of That is Difficult to solve explicitly for solutions given any specific value of K, unless G(s)H(s) is second-order or lower. Much easier to plot the root locus, the values of s that are solutions for some value of K, because: 1) It is easier to find the roots in the limiting cases for K = 0, ±. 2) There are rules on how to connect between these limiting points.

119 Rules for Plotting Root Locus End points At K = 0, G(s o )H(s o ) = s o are poles of the open-loop system function G(s)H(s). At K =, G(s o )H(s o ) = 0 s o are zeros of the open-loop system function G(s)H(s). Thus: Rule #1: A root locus starts (at K = 0) from a pole of G(s)H(s) and ends (at K = ) at a zero of G(s)H(s). Question: Answer: What if the number of poles the number of zeros? Start or end at ±.

120 Rule #2: Angle criterion of the root locus Thus, s 0 is a pole for some positive value of K if: In this case, s 0 is a pole if K = 1/ G(s 0 ) H(s 0 ). Similarly s 0 is a pole for some negative value of K if: In this case, s 0 is a pole if K = -1/ G(s 0 ) H(s 0 ).

121 Example of Root Locus. One zero at -2, two poles at 0, -1.

122 In addition to stability, we may want good tracking behavior, i.e. for at least some set of input signals. Tracking + = ) ( ) ( ) ( 1 1 ) ( s X s H s C s E ) ( ) ( ) ( 1 1 ) ( ω ω ω ω j X j H j C j E + = We want to be large in frequency bands in which we want good tracking ) ( ) ( ω ω j P j C

123 Tracking (continued) Using the final-value theorem Basic example: Tracking error for a step input

124 Disturbance Rejection There may be other objectives in feedback controls due to unavoidable disturbances. Clearly, sensitivities to the disturbances D 1 (s) and D 2 (s) are much reduced when the amplitude of the loop gain

125 Internal Instabilities Due to Pole-Zero Cancellation H w(t) ) ( ) ( ) ( ) ( 1 ) ( ) ( ) ( 2 ) ( 1) ( 1 ) ( Stable 2 s X s s s X s H s C s H s C s Y s s s H, s s s C + + = + = + = + = However ) ( 3) 3 ( 2 ) ( ) ( ) ( 1 ) ( ) ( Unstable 2 s X s s s s s X s H s C s C s W = + =

126 Inverted Pendulum Unstable!

127 Feedback System to Stabilize the Pendulum a PI feedback stabilizes θ Subtle problem: internal instability in x(t)! Additional PD feedback around motor / amplifier centers the pendulum

128 Root Locus & the Inverted Pendulum Attempt #1: Negative feedback driving the motor Root locus of M(s)G(s) Remains unstable! after K. Lundberg

129 Root Locus & the Inverted Pendulum Attempt #2: Proportional/Integral Compensator Root locus of K(s)M(s)G(s) Stable for large enough K after K. Lundberg

130 Root Locus & the Inverted Pendulum BUT x(t) unstable: System subject to drift... Solution: add PD feedback around motor and compensator: after K. Lundberg

131 The z-transform Motivation: Analogous to Laplace Transform in CT We now do not restrict ourselves just to z = e jω The (Bilateral) z-transform

132 The ROC and the Relation Between zt and DTFT, r = z depends only on r = z, just like the ROC in s-plane only depends on Re(s) Unit circle (r = 1) in the ROC DTFT X(e jω ) exists

133 Example #1 This form for PFE and inverse z- transform = 1 1 az 1 = z z a That is, ROC z > a, outside a circle This form to find pole and zero locations

134 Example #2: Same X(z) as in Ex #1, but different ROC.

135 Rational z-transforms x[n] = linear combination of exponentials for n > 0 and for n < 0 Polynomials in z characterized (except for a gain) by its poles and zeros

136 Signals and Systems Fall 2003 Lecture #22 2 December Properties of the ROC of the z-transform 2. Inverse z-transform 3. Examples 4. Properties of the z-transform 5. System Functions of DT LTI Systems a. Causality b. Stability

137 The z-transform -depends only on r = z, just like the ROC in s-plane only depends on Re(s) Last time: Unit circle (r = 1) in the ROC DTFT X(e jω ) exists Rational transforms correspond to signals that are linear combinations of DT exponentials

138 Some Intuition on the Relation between zt and LT The (Bilateral) z-transform Can think of z-transform as DT version of Laplace transform with

139 More intuition on zt-lt, s-plane - z-plane relationship LHP in s-plane, Re(s) < 0 z = e st < 1, inside the z = 1 circle. Special case, Re(s) = - z = 0. RHP in s-plane, Re(s) > 0 z = e st > 1, outside the z = 1 circle. Special case, Re(s) = + z =. A vertical line in s-plane, Re(s) = constant e st = constant, a circle in z-plane.

140 Properties of the ROCs of z-transforms (1) The ROC of X(z) consists of a ring in the z-plane centered about the origin (equivalent to a vertical strip in the s-plane) (2) The ROC does not contain any poles (same as in LT).

141 More ROC Properties (3) If x[n] is of finite duration, then the ROC is the entire z-plane, except possibly at z = 0 and/or z =. Why? Examples: CT counterpart

142 ROC Properties Continued (4) If x[n] is a right-sided sequence, and if z = r o is in the ROC, then all finite values of z for which z > r o are also in the ROC.

143 Side by Side (5) If x[n] is a left-sided sequence, and if z = r o is in the ROC, then all finite values of z for which 0 < z < r o are also in the ROC. (6) If x[n] is two-sided, and if z = r o is in the ROC, then the ROC consists of a ring in the z-plane including the circle z = r o. What types of signals do the following ROC correspond to? right-sided left-sided two-sided

144 Example #1

145 Example #1 continued Clearly, ROC does not exist if b > 1 No z-transform for b n.

146 for fixed r: Inverse z-transforms

147 Example #2 Partial Fraction Expansion Algebra: A = 1, B = 2 Note, particular to z-transforms: 1) When finding poles and zeros, express X(z) as a function of z. 2) When doing inverse z-transform using PFE, express X(z) as a function of z -1.

148 ROC III: ROC II: ROC I:

149 Inversion by Identifying Coefficients in the Power Series Example #3: for all other n s A finite-duration DT sequence

150 Example #4: (a) (b)

151 Properties of z-transforms (1) Time Shifting The rationality of X(z) unchanged, different from LT. ROC unchanged except for the possible addition or deletion of the origin or infinity n o > 0 ROC z 0 (maybe) n o < 0 ROC z (maybe) (2) z-domain Differentiation same ROC Derivation:

152 Convolution Property and System Functions Y(z) = H(z)X(z), ROC at least the intersection of the ROCs of H(z) and X(z), can be bigger if there is pole/zero cancellation. e.g. H(z) + ROC tells us everything about system

153 CAUSALITY (1) h[n] right-sided ROC is the exterior of a circle possibly including z = : A DT LTI system with system function H(z) is causal the ROC of H(z) is the exterior of a circle including z =

154 Causality for Systems with Rational System Functions A DT LTI system with rational system function H(z) is causal (a) the ROC is the exterior of a circle outside the outermost pole; and (b) if we write H(z) as a ratio of polynomials then

155 Stability LTI System Stable ROC of H(z) includes the unit circle z = 1 Frequency Response H(e jω ) (DTFT of h[n]) exists. A causal LTI system with rational system function is stable all poles are inside the unit circle, i.e. have magnitudes < 1

156 Signals and Systems Fall 2003 Lecture #23 4 December Geometric Evaluation of z-transforms and DT Frequency Responses 2. First- and Second-Order Systems 3. System Function Algebra and Block Diagrams 4. Unilateral z-transforms

157 Geometric Evaluation of a Rational z-transform Example #1: Example #2: Example #3: All same as in s-plane

158 Geometric Evaluation of DT Frequency Responses First-Order System one real pole

159 Second-Order System Two poles that are a complex conjugate pair (z 1 = re jθ =z 2* ) Clearly, H peaks near ω = ±θ

160 Demo: DT pole-zero diagrams, frequency response, vector diagrams, and impulse- & step-responses

161 DT LTI Systems Described by LCCDEs Use the time-shift property Rational ROC: Depends on Boundary Conditions, left-, right-, or two-sided. For Causal Systems ROC is outside the outermost pole

162 Feedback System (causal systems) System Function Algebra and Block Diagrams negative feedback configuration Example #1: z -1 D Delay

163 Example #2: Cascade of two systems

164 Unilateral z-transform Note: (1) If x[n] = 0 for n < 0, then (2) UZT of x[n] = BZT of x[n]u[n] ROC always outside a circle and includes z = (3) For causal LTI systems,

165 Properties of Unilateral z-transform Many properties are analogous to properties of the BZT e.g. Convolution property (for x 1 [n<0] = x 2 [n<0] = 0) But there are important differences. For example, time-shift Derivation: Initial condition

166 Use of UZTs in Solving Difference Equations with Initial Conditions UZT of Difference Equation ZIR Output purely due to the initial conditions, ZSR Output purely due to the input.

167 Example (continued) β = 0 System is initially at rest: ZSR α = 0 Get response to initial conditions ZIR

168 Signals and Systems Fall 2003 Lecture #3 11 September ) Representation of DT signals in terms of shifted unit samples 2) Convolution sum representation of DT LTI systems 3) Examples 4) The unit sample response and properties of DT LTI systems

169 Exploiting Superposition and Time-Invariance Question: Are there sets of basic signals so that: a) We can represent rich classes of signals as linear combinations of these building block signals. b) The response of LTI Systems to these basic signals are both simple and insightful. Fact: For LTI Systems (CT or DT) there are two natural choices for these building blocks Focus for now: DT Shifted unit samples CT Shifted unit impulses

170 Representation of DT Signals Using Unit Samples

171 That is... Coefficients Basic Signals The Sifting Property of the Unit Sample

172 x[n] DT System y[n] Suppose the system is linear, and define h k [n] as the response to δ[n - k]: From superposition:

173 x[n] DT System y[n] Now suppose the system is LTI, and define the unit sample response h[n]: From TI: From LTI:

174 Convolution Sum Representation of Response of LTI Systems Interpretation n n n n

175 Visualizing the calculation of Choose value of n and consider it fixed View as functions of k with n fixed y[0] = prod of overlap for n = 0 y[1] = prod of overlap for n = 1

176 Calculating Successive Values: Shift, Multiply, Sum = = 2 (-1) (-1) = -2 (-1) (-1) + 1 (-1) = -3 (-1) (-1) + 0 (-1) = 1 (-1) (-1) = 1 4

177 Properties of Convolution and DT LTI Systems 1) A DT LTI System is completely characterized by its unit sample response

178 Unit Sample response

179 The Commutative Property Ex: Step response s[n] of an LTI system step input input Unit Sample response of accumulator

180 Interpretation The Distributive Property

181 The Associative Property Implication (Very special to LTI Systems)

182 Properties of LTI Systems 1) Causality 2) Stability

183 Signals and Systems Fall 2003 Lecture #4 16 September Representation of CT Signals in terms of shifted unit impulses 2. Convolution integral representation of CT LTI systems 3. Properties and Examples 4. The unit impulse as an idealized pulse that is short enough : The operational definition of δ(t)

184 Representation of CT Signals Approximate any input x(t) as a sum of shifted, scaled pulses

185 The Sifting Property of the Unit Impulse has unit area

186 Response of a CT LTI System LTI

187 Operation of CT Convolution Example: CT convolution

188

189 PROPERTIES AND EXAMPLES 1) Commutativity: 2) 3) An integrator: 4) Step response:

190 DISTRIBUTIVITY

191 ASSOCIATIVITY

192

193 The impulse as an idealized short pulse Consider response from initial rest to pulses of different shapes and durations, but with unit area. As the duration decreases, the responses become similar for different pulse shapes.

194 The Operational Definition of the Unit Impulse δ(t) δ(t) idealization of a unit-area pulse that is so short that, for any physical systems of interest to us, the system responds only to the area of the pulse and is insensitive to its duration Operationally: The unit impulse is the signal which when applied to any LTI system results in an output equal to the impulse response of the system. That is, δ(t) is defined by what it does under convolution.

195 The Unit Doublet Differentiator Impulse response = unit doublet The operational definition of the unit doublet:

196 Triplets and beyond! n is number of differentiations

197 Integrators -1 derivatives" = integral I.R. = unit step

198 Integrators (continued)

199 Notation Define Then E.g.

200 Sometimes Useful Tricks Differentiate first, then convolve, then integrate

201 Example

202 Example (continued)

203 Signals and Systems Fall 2003 Lecture #5 18 September Complex Exponentials as Eigenfunctions of LTI Systems 2. Fourier Series representation of CT periodic signals 3. How do we calculate the Fourier coefficients? 4. Convergence and Gibbs Phenomenon

204 Portrait of Jean Baptiste Joseph Fourier Image removed due to copyright considerations. Signals & Systems, 2nd ed. Upper Saddle River, N.J.: Prentice Hall, 1997, p. 179.

205 Desirable Characteristics of a Set of Basic Signals a. We can represent large and useful classes of signals using these building blocks b. The response of LTI systems to these basic signals is particularly simple, useful, and insightful Previous focus: Unit samples and impulses Focus now: Eigenfunctions of all LTI systems

206 The eigenfunctions φ k (t) and their properties (Focus on CT systems now, but results apply to DT systems as well.) eigenvalue eigenfunction Eigenfunction in same function out with a gain From the superposition property of LTI systems: Now the task of finding response of LTI systems is to determine λ k.

207 Complex Exponentials as the Eigenfunctions of any LTI Systems eigenvalue eigenfunction eigenvalue eigenfunction

208 DT:

209 What kinds of signals can we represent as sums of complex exponentials? For Now: Focus on restricted sets of complex exponentials CT: DT: Magnitude 1 CT & DT Fourier Series and Transforms Periodic Signals

210 Fourier Series Representation of CT Periodic Signals -smallest such T is the fundamental period - is the fundamental frequency ω o = 2π T - periodic with period T -{a k } are the Fourier (series) coefficients - k = 0 DC - k = ±1 first harmonic - k = ±2 second harmonic

211 Question #1: How do we find the Fourier coefficients? First, for simple periodic signals consisting of a few sinusoidal terms Euler's relation (memorize!) 0 no dc component 0 0

212 For real periodic signals, there are two other commonly used forms for CT Fourier series: Because of the eigenfunction property of e jωt, we will usually use the complex exponential form in A consequence of this is that we need to include terms for both positive and negative frequencies:

213 Now, the complete answer to Question #1

214

215 Ex: Periodic Square Wave DC component is just the average

216 Convergence of CT Fourier Series How can the Fourier series for the square wave possibly make sense? The key is: What do we mean by One useful notion for engineers: there is no energy in the difference (just need x(t) to have finite energy per period)

217 Under a different, but reasonable set of conditions (the Dirichlet conditions) Condition 1. x(t) is absolutely integrable over one period, i. e. And Condition 2. of Ex. In a finite time interval, x(t) has a finite number maxima and minima. An example that violates Condition 2. And Condition 3. Ex. In a finite time interval, x(t) has only a finite number of discontinuities. An example that violates Condition 3.

218 Dirichlet conditions are met for the signals we will encounter in the real world. Then - The Fourier series = x(t) at points where x(t) is continuous - The Fourier series = midpoint at points of discontinuity Still, convergence has some interesting characteristics: - As N, x N (t) exhibits Gibbs phenomenon at points of discontinuity Demo: Fourier Series for CT square wave (Gibbs phenomenon).

219 Signals and Systems Fall 2003 Lecture #6 23 September CT Fourier series reprise, properties, and examples 2. DT Fourier series 3. DT Fourier series examples and differences with CTFS

220 CT Fourier Series Pairs Skip it in future for shorthand

221 Another (important!) example: Periodic Impulse Train All components have: (1) the same amplitude, & (2) the same phase.

222 Linearity (A few of the) Properties of CT Fourier Series Conjugate Symmetry Time shift Introduces a linear phase shift t o

223 Example: Shift by half period

224 Parseval s Relation Energy is the same whether measured in the time-domain or the frequency-domain Multiplication Property

225 Periodic Convolution x(t), y(t) periodic with period T

226 Periodic Convolution (continued) Periodic convolution: Integrate over any one period (e.g. -T/2 to T/2)

227 Periodic Convolution (continued) Facts 1) z(t) is periodic with period T (why?) 2) Doesn t matter what period over which we choose to integrate: 3) Periodic convolution in time Multiplication in frequency!

228 Fourier Series Representation of DT Periodic Signals x[n] - periodic with fundamental period N, fundamental frequency Only e jω n which are periodic with period N will appear in the FS There are only N distinct signals of this form So we could just use However, it is often useful to allow the choice of N consecutive values of k to be arbitrary.

229 DT Fourier Series Representation k =<N > = Sum over any N consecutive values of k This is a finite series {a k } - Fourier (series) coefficients Questions: 1) What DT periodic signals have such a representation? 2) How do we find a k?

230 Answer to Question #1: Any DT periodic signal has a Fourier series representation

231 Finite geometric series A More Direct Way to Solve for a k

232 So, from

233 DT Fourier Series Pair Note: It is convenient to think of a k as being defined for all integers k. So: 1) a k+n = a k Special property of DT Fourier Coefficients. 2) We only use N consecutive values of a k in the synthesis equation. (Since x[n] is periodic, it is specified by N numbers, either in the time or frequency domain)

234 Example #1: Sum of a pair of sinusoids 0 1/2 1/2 e jπ/4 /2 e -jπ/4 /2 0 0 a = a -1 = 1/2 a = a 2 = e jπ/4 /2

235 Example #2: DT Square Wave Using n = m - N 1

236 Example #2: DT Square wave (continued)

237 Convergence Issues for DT Fourier Series: Not an issue, since all series are finite sums. Properties of DT Fourier Series: Lots, just as with CT Fourier Series Example:

238 Signals and Systems Fall 2003 Lecture #7 25 September Fourier Series and LTI Systems 2. Frequency Response and Filtering 3. Examples and Demos

239 The Eigenfunction Property of Complex Exponentials CT: CT "System Function" DT: DT "System Function"

240 Fourier Series: Periodic Signals and LTI Systems

241 The Frequency Response of an LTI System CT notation

242 Frequency Shaping and Filtering By choice of H(jω) (or H(e jω )) as a function of ω, we can shape the frequency composition of the output - Preferential amplification - Selective filtering of some frequencies Example #1: Audio System Adjustable Filter Equalizer Speaker Bass, Mid-range, Treble controls For audio signals, the amplitude is much more important than the phase.

243 Example #2: Frequency Selective Filters Filter out signals outside of the frequency range of interest Lowpass Filters: Only show amplitude here. low frequency low frequency

244 Highpass Filters Remember: high frequency high frequency

245 Bandpass Filters Demo: Filtering effects on audio signals

246 Idealized Filters CT ω c cutoff frequency DT Note: H = 1 and H = 0 for the ideal filters in the passbands, no need for the phase plot.

247 Highpass CT DT

248 Bandpass CT lower cut-off upper cut-off DT

249 Example #3: DT Averager/Smoother FIR (Finite Impulse Response) filters LPF

250 Example #4: Nonrecursive DT (FIR) filters Rolls off at lower ω as M+N+1 increases

251 Example #5: Simple DT Edge Detector DT 2-point differentiator Passes high-frequency components

252 Demo: DT filters, LP, HP, and BP applied to DJ Industrial average

253 Example #6: Edge enhancement using DT differentiator Courtesy of Jason Oppenheim. Used with permission. Courtesy of Jason Oppenheim. Used with permission.

254 Example #7: A Filter Bank

255 Demo: Apply different filters to two-dimensional image signals. Face of a monkey. HP Image removed do to copyright considerations LP LP BP HP BP Note: To really understand these examples, we need to understand frequency contents of aperiodic signals the Fourier Transform

256 Signals and Systems Fall 2003 Lecture #8 30 September Derivation of the CT Fourier Transform pair 2. Examples of Fourier Transforms 3. Fourier Transforms of Periodic Signals 4. Properties of the CT Fourier Transform

257 Fourier s Derivation of the CT Fourier Transform x(t) - an aperiodic signal - view it as the limit of a periodic signal as T For a periodic signal, the harmonic components are spaced ω 0 = 2π/T apart... As T, ω 0 0, and harmonic components are spaced closer and closer in frequency Fourier series Fourier integral

258 Motivating Example: Square wave increases kept fixed Discrete frequency points become denser in ω as T increases

259 So, on with the derivation... For simplicity, assume x(t) has a finite duration.

260 Derivation (continued)

261 Derivation (continued)

262 For what kinds of signals can we do this? (1) It works also even if x(t) is infinite duration, but satisfies: a) Finite energy In this case, there is zero energy in the error b) Dirichlet conditions c) By allowing impulses in x(t)or inx(jω), we can represent even more signals E.g. It allows us to consider FT for periodic signals

263 Example #1 (a) (b)

264 Example #2: Exponential function Even symmetry Odd symmetry

265 Example #3: A square pulse in the time-domain Note the inverse relation between the two widths Uncertainty principle Useful facts about CTFT s

266 Example #4: x(t) = e at 2 A Gaussian, important in probability, optics, etc. (Pulse width in t) (Pulse width in ω) t ω ~ (1/a 1/2 ) (a 1/2 ) = 1 Also a Gaussian! Uncertainty Principle! Cannot make both t and ω arbitrarily small.

267 CT Fourier Transforms of Periodic Signals periodic in t with frequency ω o All the energy is concentrated in one frequency ω o

268 Example #4: Line spectrum

269 Example #5: Sampling function Same function in the frequency-domain! Note: (period in t) T (period in ω) 2π/T Inverse relationship again!

270 1) Linearity Properties of the CT Fourier Transform 2) Time Shifting FT magnitude unchanged Linear change in FT phase

271 3) Conjugate Symmetry Properties (continued) Even Odd Even Odd

272 The Properties Keep on Coming... 4) Time-Scaling a) x(t) real and even b) x(t) real and odd c)

273 Signals and Systems Fall 2003 Lecture #9 2 October The Convolution Property of the CTFT 2. Frequency Response and LTI Systems Revisited 3. Multiplication Property and Parseval s Relation 4. The DT Fourier Transform

274 The CT Fourier Transform Pair (Analysis Equation) (Synthesis Equation) Last lecture: Today: some properties further exploration

275 Convolution Property A consequence of the eigenfunction property: Synthesis equation for y(t)

276 The Frequency Response Revisited impulse response The frequency response of a CT LTI system is simply the Fourier transform of its impulse response Example #1: frequency response

277 Example #2: A differentiator Differentiation property: 1) Amplifies high frequencies (enhances sharp edges) Larger at high ω o phase shift

278 Example #3: Impulse Response of an Ideal Lowpass Filter Questions: 1) Is this a causal system? 2) What is h(0)? No. 3) What is the steady-state value of the step response, i.e. s( )?

279 Example #4: Cascading filtering operations H(jω)

280 Example #5: Example #6: Gaussian Gaussian = Gaussian Gaussian Gaussian = Gaussian

281 Example #2 from last lecture

282 Example #7:

283 Example #8: LTI Systems Described by LCCDE s (Linear-constant-coefficient differential equations) Using the Differentiation Property 1) Rational, can use PFE to get h(t) 2) If X(jω) is rational e.g. then Y(jω) is also rational

284 Parseval s Relation FT is highly symmetric, Multiplication Property We already know that: Then it isn t a surprise that: Convolution in ω A consequence of Duality

285 Examples of the Multiplication Property For any s(t)...

286 Example (continued)

287 The Discrete-Time Fourier Transform

288 DTFT Derivation (Continued) DTFS synthesis eq. DTFS analysis eq. Define

289 DTFT Derivation (Home Stretch)