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Signals and Systems Dr Reza Danesfahani Faculty of Engineering University of Kashan Kashan, Iran

28th January 2006

Signals and Systems

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Part I Preamble, Introduction, and Overview

Signals and Systems

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Preamble

I Slide 3

Syllabus 1

2

3

4

5

6

7

Continuous-time and discrete-time signals: classification and properties. Basic properties of systems: linearity, time-invariance, causality, stability. Linear time-invariant (LTI) systems: convolution, characterization, impulse response. Fourier series and Fourier transform: definition, properties, frequency response of LTI Introduction to filtering. Sampling: impulse train sampling, sampling theorem, reconstruction of signals, effects of under-sampling. Z-transforms: definitions, properties, analysis of LTI systems, transfer functions. Communication systems: Amplitude modulation (AM), demodulation

Grading Signals and Systems

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Preamble

II Slide 4

1 2

Midterm Exam (20%) Final Exam(80%)

Type of Exam: Closed-book. Students may bring one A4 sheet of formulae to exam. References A.V. Oppenheim, A.S. Willsky and S.H. Nawab, Signals & Systems, Prentice-Hall International, second edition, 1997. G.E. Carlson, Signal and Linear System Analysis., John Wiley & Sons, second edition, 1998. R.E. Ziemer, W.H. Tranter and D.R. Fannin, Signals & Systems, Continuous and Discrete, Prentice Hall, fourth edition, 1998. Signals and Systems

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Preamble

III Slide 5

S. Haykin and B.V. Veen, Signals and Systems, John Wiley & Sons, 1999. C.L. Phillips and J.M. Parr, Signals, Systems, and Transforms, Prentice Hall, second edition, 1999. F.J. Taylor, Principles of Signals and Systems, McGraw-Hill, 1994. H.P. Hsu, Theory and Problems of Signals and Systems, McGraw Hill, 1995.

Signals and Systems

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Preamble

IV Slide 6

J.R. Buck, M.M. Daniel and A.C. Singer, Computer Explorations in Signals and Systems Using M ATLAB, Prentice Hall, 1997. L. Balmer, Signals and Systems, An Introduction, Prentice Hall, 1998. E.W. Kamen and B.S. Heck, Fundamentals of Signals and Systems using M ATLAB, Prentice Hall, 1998. L.B. Jackson, Signals, Systems, and Transforms, Addison-Wesley, 1991. Signals and Systems

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Preamble

V Slide 7

Z.Z. Karu, Signals and Systems Made Rediculously Simple, ZiZi Press, 2001. S.T. Karris, Signals and Systems with Matlab Applications, Orchard Publications, second edition, 2003. B.P. Lathi, Signal Processing & Linear Systems, Oxford University Press, 1998. S. Haykin, Communication Systems, John Wiley and Sons, fourth edition, 2001.

Signals and Systems

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Part II Signals and Systems

Signals and Systems

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Signals and Systems

I Slide 9

Definition A signal is a function representing a physical quantity or variable, and typically it contains information about the behavior or nature or phenomenon. For instance, in a RC circuit the signal may represent the voltage across the capacitor or the current flowing in the resistor. Mathematically, a signal is represented as a function of an independent variable t. Thus, a signal is denoted by x(t).

Signals and Systems

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Signals and Systems

II Slide 10

Definition An energy signal is a signal whose total energy is finite, i.e. those signals for which E∞ < ∞. Such a signal must have zero average power, since in the continuous time case, for example, E∞ =0 T →∞ 2T

P∞ = lim Example

An example of a finite-energy signal is a signal that takes on the value 1 for 0 ≤ t ≤ 1 and 0 otherwise. In this case E∞ = 1 and P∞ = 0.

Signals and Systems

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Signals and Systems

III Slide 11

Definition A power signal is a signal whose average power is finite, i.e. those signals for which P∞ < ∞. Such a signal must have infinite total energy. Example The constant signal x[n] = 4 is a power signal. It has infinite energy, but average power P∞ = 16. There are signals for which neither P∞ nor E∞ are finite. A simple example is the signal x(t) = t. Transformation of the independent variable 1

Time shift: A signal and its time shifted version are identical in shape, but are displaced or shifted relative to each other. Signals and Systems

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Signals and Systems

IV Slide 12

2

3

Time reversal: The time reversed version of the signal x(t) is obtained by a reflection about t = 0. Time scale: the independent variable is scaled.

Example Suppose that we would like to determine the effect of transforming the independent variable of a given signal, x(t), to obtain a signal of the form x(αt + β), where α and β are given numbers. A systematic approach to doing this is to first delay or advance x(t) in accordance with the value of β, and then to perform time scaling and/or time time reversal on the resulting signal in accordance with the value of α. The delayed or advanced signal is linearly stretched if |α| < 1, linearly compressed if |α| > 1, and reversed in time if α < 0.

Signals and Systems

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Signals and Systems

V Slide 13

Definition If a continuous-time signal x(t) can take on any value in the continuous interval (a, b), where a may be −∞ and b may be +∞, then the continuous-time signal x(t) is called an analog signal. If a discrete-time signal x[n] can take on only a finite number of distinct values, then we call this signal a digital signal.

Signals and Systems

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Signals and Systems

VI Slide 14

Definition A signal x(t) is a real signal if its value is a real number, and a signal x(t) is a complex signal if its value is a complex number. A general complex signal x(t) is a function of the form x(t) = x1 (t) + jx2 (t) where x1 (t) and x2 (t) are real signals and j =

√

−1.

Definition Deterministic signals are those signals whose values are completely specified for any given time. Random signals are those signals that take random values at any given time and must be characterized statistically.

Signals and Systems

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Signals and Systems

VII Slide 15

Definition A signal x(t) or x[n] is referred to as an even signal if x(−t) = x(t) x[−n] = x[n] A signal x(t) or x[n] is referred to as an odd signal if x(−t) = −x(t) x[−n] = −x[n]

Signals and Systems

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Signals and Systems

VIII Slide 16

The even part of the signal x(t) is E{x(t)} =

1 [x(t) + x(−t)] 2

The odd part of the signal x(t) is O{x(t)} =

1 [x(t) − x(−t)] 2

Similarly, 1 [x[n] + x[−n]] 2 The odd part of the signal x[n] is E{x[n]} =

O{x[n]} = Signals and Systems

1 [x[n] − x[−n]] 2 www.danesfahani.com

Signals and Systems

IX Slide 17

Example The odd and even components of x(t) = ejt are 1 jt (e + e−jt ) 2 1 xo (t) = (ejt − e−jt ) 2 xe (t) =

Signals and Systems

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Signals and Systems

X Slide 18

Definition A continuous-time signal x(t) is said to be periodic with period T if there is a positive nonzero value of T for which x(t) = x(t + T )

all t

Any sequence which is not periodic is called a nonperiodic (or aperiodic) sequence. The fundamental period T0 of x(t) is the smallest positive value of T for which the above equation holds.

Signals and Systems

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Signals and Systems

XI Slide 19

Definition A discrete-time signal x[n] is periodic with period N, where N is a positive integer, if it is unchanged by a time shift of N, i.e. if x[n] = x[n + N] for all values of n. The fundamental period N0 is the smallest positive value of N for which the above equation holds. Definition The unit step function is defined as  1 t >0 u(t) ∆ 0 t 0}

10

Accumulation n X

x[k ] ↔ (1 − z −1 )−1 X (z), ROC: R ∩ {|z| > 1}

k =−∞ 11

Initial value theorem. If x[n] = 0 for n < 0, then x[0] = lim X (z) z→∞

Some common z-transform pairs are as follows: 1

δ[n] ↔ 1, ROC: All z

Signals and Systems

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The z-Transform

XVIII Slide 103

2

u[n] ↔

1 , ROC: |z| > 1 1 − z −1

3

−u[−n − 1] ↔

1 , ROC: |z| < 1 1 − z −1

4

δ[n−m] ↔ z −m , ROC: All z except 0 (if m > 0) or ∞ (if m < 0) 5

αn u[n] ↔

1 , ROC: |z| > |α| 1 − αz −1

6

−αn u[−n − 1] ↔

Signals and Systems

1 , ROC: |z| < |α| 1 − αz −1 www.danesfahani.com

The z-Transform

XIX Slide 104

7

nαn u[n] ↔

αz −1 , ROC: |z| > |α| (1 − αz −1 )2

8

−nαn u[−n − 1] ↔

αz −1 , ROC: |z| < |α| (1 − αz −1 )2

9

[cos ω0 n]u[n] ↔

1 − [cos ω0 ]z −1 , ROC: |z| > 1 1 − [2 cos ω0 ]z −1 + z −2

[sin ω0 n]u[n] ↔

[sin ω0 ]z −1 , ROC: |z| > 1 1 − [2 cos ω0 ]z −1 + z −2

10

Signals and Systems

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The z-Transform

XX Slide 105

11

[r n cos ω0 n]u[n] ↔

1 − [r cos ω0 ]z −1 , ROC: |z| > r 1 − [2r cos ω0 ]z −1 + r 2 z −2

[r n sin ω0 n]u[n] ↔

[r sin ω0 ]z −1 , ROC: |z| > r 1 − [2r cos ω0 ]z −1 + r 2 z −2

12

Definition A discrete-time LTI system is causal if and only if the ROC of its system function is the exterior of a circle, including infinity.

Signals and Systems

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The z-Transform

XXI Slide 106

Definition A discrete-time LTI system with rational system function H(z) is causal if and only if (a) the ROC is the exterior of a circle outside the outermost pole; and (b) with H(z) expressed as a ratio of polynomials in z, the order of the numerator cannot be greater than the order of the denominator. Example The system with the following system function is not causal, because the numerator of H(z) is of higher order than the denominator. z 3 − 2z 2 + z H(z) = 1 1 z2 + z + 4 8 Signals and Systems

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The z-Transform

XXII Slide 107

Example Consider a system with system function H(z) =

1 1 + , 1 −1 1 − 2z −1 1− z 2

|z| > 2

The impulse response of this system is   n 1 n + 2 u[n] h[n] = 2 Since h[n] = 0 for n < 0, we conclude that the system is causal.

Signals and Systems

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The z-Transform

XXIII Slide 108

Definition An LTI system is stable if and only if the ROC of its system function H(z) includes the unit circle, |z| = 1.

Signals and Systems

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The z-Transform

XXIV Slide 109

Example Consider a system with system function H(z) =

1 1 + , 1 −1 1 − 2z −1 1− z 2

1/2 < |z| < 2

The impulse response of this system is  n 1 h[n] = u[n] − 2n u[−n − 1] 2 Since h[n] is absolutely summable, we conclude that the system is stable. Further, it is seen that the system is noncausal.

Signals and Systems

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The z-Transform

XXV Slide 110

Example Consider a system with system function H(z) =

1 1 + , 1 −1 1 − 2z −1 1− z 2

|z| < 1/2

The impulse response of this system is   n 1 n + 2 u[−n − 1] h[n] = − 2 Since h[n] is neither stable nor causal.

Signals and Systems

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The z-Transform

XXVI Slide 111

Definition A causal LTI system with rational system function H(z) is stable if and only if all of the poles of H(z) lie inside the unit circle – i.e., they must all have magnitude smaller than 1. Example Consider a causal system with system function H(z) =

1 1 − az −1

which has a pole at z = a. For this system to be stable, its pole must be inside the unit circle, i.e., we must have |a| < 1.

Signals and Systems

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The z-Transform

XXVII Slide 112

For system characterized by linear constant-coefficient difference equation, the properties of the z-transform provide a particularly convenient procedure for obtaining the system function, frequency response, or time-domain response of the system.

Signals and Systems

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The z-Transform

XXVIII Slide 113

Example Consider an LTI system for which the input x[n] and output y [n] satisfy the linear constant-coefficient difference equation 1 1 y [n] − y [n − 1] = x[n] + x[n − 1] 2 3 Applying the z-trasform 1+ Y (z) = H(z) = X (z) 1−

Signals and Systems

1 −1 z 3 1 −1 z 2

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Part X Laplace Transform

Signals and Systems

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Laplace Tranform

I Slide 115

The basic inverse Laplace transform equation is 1 x(t) = 2πj

ˆ

σ+jω

X (s)est ds

σ−jω

This equation states that x(t) can be represented as a weighted integral of complex exponentials. The contour of integration in this equation is the straight line in the s-plane corresponding to all points s satisfying