Chapter 2. Signals and Linear Systems Text Book: Essentials of Communication Systems Engineering - John G. Proakis and Masoud Salehi Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

2.0 Pre-requisites: 2.0.1 Sine & Cosine Functions Definisions: Y y r θ X O x Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

2.0 Pre-requisites: 2.0.1 Sine & Cosine Functions Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

2.0 Pre-requisites: 2.0.2 Complex Numbers Notations: Y y r θ X O x Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

2.0 Pre-requisites: Exercises Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

2.1 Basic Concepts: 2.1.1 Basic Operations on Signals x(t) : A signal Time shifting or delaying y(t) : Time shifted version of x(t) : , t0 : Time shift When t0 > 0, x(t) is shifted to the right : time delay When t0 < 0, x(t) is shifted to the left : time advance Time reversal, or flipping y(t) : Signal by replacing time t by –t : Reflected version of x(t) about t = 0 Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

Basic Operations on Signals Time scaling Expands (stretches) or contracts (compresses) a signal along the time axis y(t) : Signal by scaling the independent variable, time t, by a factor a If a>1, the signal y(t) is a compressed version of x(t) If 0<a<1, the signal y(t) is an expanded version of x(t) Combination of these operation x(-2t) = Combination of flipping the signal then contracting it by a factor of 2 x(2t-3) : x[2(t-1.5)] = Contract the signal by a factor of 2 and then shifting it to the right by 1.5 Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

2.1.2 Classification of Signals: Real and Complex Signals Real signals Takes its values in the set of real numbers x(t)  R, t  R Complex signals Takes its values in the set of complex numbers x(t)  C , t  R In communications Used to model signals Convey amplitude and phase information Represented by two real signals Real and imaginary parts Absolute value (or modulus or magnitude) and phase Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

Representation of Complex signals – Example 2.1.3 The signal : Complex signals From Euler’s relation : Its real part : Its imaginary part : Absolute value of x(t) : Its phase : Figure 2.8 Representation of complex signals Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

Periodic and Nonperiodic Signals The signal x(t) is periodic if and if only (iff) there exists a T0 >0 such that T0 : period of the signals (positive real number) the minimum value satisfying the condion A signal that does not satisfy the conditions of periodicity is called non-periodic Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

Even and Odd Signals Even Signals Odd Signals A signal x(t) is said to be an even signals if Odd Signals A signal x(t) is said to be an odd signals if In general, any signal x(t) can be written as the sum of its even and odd part as Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

Hermitian Symmetry for Complex Signals Another form of symmetry for complex signals A complex signal x(t) is called Hermitian if its real part is even and its imaginary part is odd Its magnitude is even and its phase is odd Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

Energy-Type and Power-Type Signals Energy of the signal For any signal x(t), the energy of the signal is defined by Power of the signal For any signal x(t), the power of the signal is defined by For real signal, is replaced by A signal is an energy-type signal if and only if Ex is finite A signal is an power-type signal if and only if Px satisfies Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

Example 2.1.9 The energy of Therefore, this signal is an energy-type signal. Hence, x(t) is a power-type signal and its power is Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

Example 2.1.10 The energy content of Therefore, this signal is not an energy-type signal However, the power of this signal is Hence, x(t) is a power-type signal and its power is Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

Example 2.1.11 For any periodic signal with period T0, the energy is Therefore, periodic signals are not typically energy type The power of any periodic signal is This means that the power content of a periodic signal is equal to the average power in one period Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

2.1.3 Some Important Signals & Their properties Sinusoidal Signal & Complex Exponential Signal Sinusoidal signals Definition : Real Part of complex exponential signal A : Amplitude f0 : Frequency  : Phase Period : T0 = 1/f0 Figure 2.6 Complex exponential signal Figure 2.8 Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

Unit Step, Rectangular & Triangular Signal Unit step signal Definition The unit step multiplied by any signal produces a “causal version” of the signal Note that for positive a, we have Figure 2.9 Rectangular pulse, Square signal, П signal Figure 2.13 Triangular Signal Figure 2.15 Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

Example 2.1.12, 13, 14 Plot the signal . Figure 2.12 Represent x(t) in Figure 2.14 using rectangular pulses. Plot the signal . Figure 2.16 Therefore, periodic signals are not typically energy type The power content of any periodic signal is This means that the power content of a periodic signal is equal to the average power in one period Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

Convolution of Two Signals Definition Example: p35 Example: p39, property 10 Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

Sinc & Sign or Signum Signal Sinc signal Definition The sinc signal achieves its maximum of 1 at t = 0. The zeros of the sinc signal are at t = 1, 2, 3,  Figure 2.17 Sign or Signum signal Definition : Sign of the independent variable t Can be expressed as the limit of the signal xn(t) when n   Figure 2.18 Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/ 21

Impulse, Delta, Sampling Signal 일반 함수가 아님, 특이함수임 defined in terms of its effect on another function (usually called the “test function”) under the integral sign. Definition of the impulse distribution (or signal) by the relation Helpful to visualize (t) as the limit of certain known signals Figure 2.19 Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

Impulse Signal – Properties Figure 2.20 : Schematic representation of the impulse signal (t) = 0 for all t  0 and (0) =  x(t)(t-t0) = x(t0)(t-t0) For any (t) continuous at t0, → 다른 책에서는 (t) 의 정의식임, Sampling Signal For any (t) continuous at t0, For all a  0, Let all a = -1, , even signal Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

Impulse Signal – Properties The result of the convolution of any signal with the impulse signal is the signal it self : The unit step signal is the integral of the impulse signal, and the impulse signal is the generalized derivative of the unit step signal Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

Impulse Signal – Properties Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

Impulse Signal – Exercises b) c) Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

2.1.4 Classification of Systems An interconnection of various elements or devices that behave as a whole. Communication point of view A entity that is excited by an input signal and , as a result of this excitation, produces an output signal Communication engineer’s point of view A law that assigns output signals to various input signals Its output must be uniquely defined for ant legitimate input x(t) : Input y(t) : Output  : Operation performed by the system Characteristics The operation that describes the system : Use the operator  to denote the operation The set of legitimate input signals : Use the operator H  Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

Linear and Nonlinear Systems Systems for which the superposition property is satisfied The system response to a linear combination of the inputs is the combination of the response to the corresponding inputs A system  is linear if and only if, for any two input signals x1(t) and x2(t) and for any two scalars  and , we have A system that does not satisfy this relationship is called nonlinear Linearity : Additive + homogeneous Output of a linear system Decompose the input into a linear combination of some fundamental signals Linear combination of the corresponding outputs L : Operation of linear systems Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

Time-Invariant and Time-Varying Systems Time-invariant system A system is called time invariant if its input-output relationship does not change with time A delayed version of an input results in a delayed version of the output A system is time-invariant if and only if, for all x(t) and all values of t0, its response to x(t-t0) is y(t-t0) , y(t) is the response of the system to x(t) Linear time-invariant systems The response of these systems to inputs can be derived simply by finding the convolution of the input and the impulse response of the system Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

Example 2.1.19-22 Differentiating & integrating systems: → Linear & Time Invariant → RLC circuits: LTI Time delaying system: → Linear & Time Invariant : Nonlinear & Time Invariant : Linear & Time Variant Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

Analysis of LTI Systems in the Time Domain Impulse response The impulse response h(t) of a system is the response of the system to a unit impulse input (t) The convolution integral: Output y(t) of an LTI system to any input signal x(t) Pf) Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

Example 2.1.25 Let a linear time-invariant system have the impulse response h(t) Assume this system has a complex exponential signal as input, The response to the input The response of an LTI system to the complex exponential with frequency f0 is a complex exponential with the same frequency Amplitude of the response : Multiplying the amplitude of the input by Phase response : Adding to the input phase : A function of the impulse response and the input frequency Complex exponential Eigenfunctions of the class of linear time-invariant systems Eigenfunctions of a system : Set of inputs for which the output is a scaling of the input Simple to find the response of LTI system to the class of complex exponential signals Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/

HW & QZ #1 Textbook Problems from p105 2.1.1, 2.1.3, 2.1.4, 2.1.6, 2.1.9 2.5.1, 2.5.2 2.6.2, 2.6.4, 2.6.6 2.7.1, 2.7.2, 2.7.3, 2.7.4 2.9.4 2.13.9, 2.13.12, 2.13.15 2.16.1, 2.16.3, 2.16.5, 2.16.7, 2.16.9 2.24.1, 2.24.3, 2.24.5, 2.24.7, 2.24.9 2.34.1, 2.34.4, 2.34.6 Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea: http://dasan.sejong.ac.kr/~ojkwon/