Chapter 1. -Signals and Systems Media Network Lab.

Signals and Systems Why signals and systems? 1. 1 Signal? Signal processing : from signal to signal (communication, control, multimedia) Pattern recognition : from signal to information Synthesis : from information to signal 1. 1 Signal? f(t): 1D f(x, y): 2D f(x, y, z) or f(x, y, t): 3D 4D? Signals and systems Cont.

1. 4 Classification of Signals Continuous-time vs discrete-time Even vs odd signal Conjugate symmetric signal Periodic signal Deterministic and random signal Power/energy signal Communication system Cont.

1. 4 Classification of Signals a. Continuous-time -> sampling -> discrete-time b. Even and odd signals Odd signal (1.3) Example 1.1 Even and Odd Signal Fig 1.12 (a) Continuous-time signal x(t). (b) discrete-time signal x[n]. -> (1.5절 먼저하고 올 것) Cont.

1.4.b Odd signals and even signals Theorem: All signals can be represented by sum of odd signal and even signal. Problem 1.1 Odd signal + even signal (a) (b) (c) (d) Cont.

1.4.c symmetric 1.4.d periodic c. Conjugate symmetric signal (1. 6) d. Periodic vs aperiodic signal Definition : A signal x(t) is periodic if and only if there exists non-zero T which satisfies Frequency: (1. 8) angular frequency: (1.9) Cont.

1.4.d Periodic signals Periodic discrete signal (1. 11) Cont. Figure1.14 (a) Square wave with amplitude A = 1 and period T = 0.2s. (b) Rectangular pulse of amplitude A and duration T1. Figure 1.16 Discrete-time square wave alternative between –1 and +1. Figure 1.17 Aperiodic discrete-time signal consisting of three nonzero samples. Cont.

1.4.d periodic 1.4.e Deterministic Problem 1.5 periodic or aperiodic? (a) (b) (c) (d) (e) (f) (g) How about e. Deterministic and random signals random signals => 확률 및 랜덤변수 Cont.

1.4.f Energy signal and power signals Power [Watt or J/sec] 가 유한하면, power signal. (ex. 집에 들어오는 전력) Energy [J or W∙sec]가 유한하면, energy signal. (ex. ‘아’하는 소리) (1. 12) (1. 13) normalized power (R=1Ω) -> (1.14)  (1. 15) (1. 16) (1. 18) (1. 19) Cont.

1.4.f Energy signal and power signals Problem 1.9 Power or energy signal (a) (b) (c) 

1.5 Basic operations 1.5.1 basic operations amplitude scaling (노래방 볼륨) (1. 21) addition (1. 22), (노래방 마이크 2개) multiplication (modulation, sampling) differentiation  (1. 24) Cont.

1.5.2.a time scaling a. time scaling Prob 1.10 Let -> Cont. Up-sampling and down-sampling (decimation) Cont.

1.5.2.b Reflection b. reflection (reverse) : Michael Jackson, 서태지 ‘피가 모자라’ Example 1.13 Problem1. 11 The discrete-time signal -> Find Cont.

1.5.2.c Time shifting c. Time shifting Problem 1. 13 The discrete-time signal -> Find Cont.

1.5.3 Shifting and Time Scaling Example 1.5 Precedence Rule incorrect! Cont.

1.5.3 Shifting and Time Scaling Problem 1. 14 (a) (b) (c) (d) (e) (f) Answer: Cont.

1.5.3 Shifting and Time Scaling Ex 1.16 Precedence Rule for Discrete-Time Signal -> Find Problem 1.15 consider a discrete-time signal Answer: Cont.

1.6 Elementary Signals 1.6.1 Exponential signals Exponential Sinusoidal Complex exponential Step function and impulse => ramp signals 1.6.1 Exponential signals (1. 31)  (1. 32) Figure 1.28 (a) a<0, B>0. (b) a>0, B>0 Cont.

1.6.2 Sinusoidal Signals 1.6.2 Sinusoidal Signals (1. 35) Periodic with period of Periodic? -> Figure 1.30 (a) a<0, B>0. (b) a>0, B>0 Cont.

1.6.2 Sinusoidal Signals  Periodic if  Angular frequency Cont. Figure 1.31 (a) A cos(ωt + Φ) with phase Φ = +/6 . (b) A sin (ωt + Φ) with phase Φ = +/6 Figure 1.33 Discrete-time sinusoidal signal. Cont.

1.6.2 Sinusoidal Signals Example 1.17 Discrete-Time sinusoidal signals  Problem 1.16 The fundamental period? Cont.

1.6.3 Complex Exponential signals Problem 1.17 Consider the following sinusoidal signals: (a) (b) (c) (d) 1.6.3 Complex Exponential signals -> -> -> Cont.

1.6.4 Exponential Damped Sinusoidal signals (1. 48) Problem 1.20 Complex-valued exponential signal Evaluate and for the following cases: (a) real, (b) imaginary, (c) complex, Figure 1.35 Exponentially damped sinusoidal signal Ae-at sin(ωt), with A = 60 and α = 6. Cont.

1.6.5 Step Function Example 1.8 Rectangular Pulse Problem 1.22 Represent x[n] by using step functions -> 

1.6.6 Impulse Function Continuous time impulse functions (1. 61) (1. 68)

1.6.6 Impulse Function Discrete time impulse function Cont.

Continuous time: δ(t)  integration  u(t) 1.6.8 Ramp Function Continuous time: δ(t)  integration  u(t) δ(t)  differentiation  u(t) Discrete time: δ[n]  summation  u[n] δ[n]  difference  u[n] Cont.

Signals and Systems 1.7 Systems viewed as interconnections of operations Example 1.12 Moving average system 1.8 Properties of Systems (p. 55) stable memory causal inverse time invariance linear Cont.

BIBO (bounded input, bounded output) 1.8.1 STABILITY  BIBO BIBO (bounded input, bounded output) For BI for all t if BO for all t, then Stable! Ex 1.13 Stable? Sol) Let  stable

1.8.1 STABILITY  BIBO Fig 1.52 Collapse of the Tacoma Narrows suspension bridge on November 7, 1940. Photograph showing the twisting motion of the bridge’s center span just before failure. A few minutes after the first piece of concrete fell, this second photograph shows a 600-ft section of the bridge breaking out of the suspension span and turning upside down as it crashed in Puget Sound, Washington. Note the car in the top right-hand corner of the photograph. (Courtesy of the Smithsonian Institution.) Cont.

1.8.1 STABILITY  BIBO Ex 1.1.4 , where r>1. Stable? Sol) for all n Feedback (recursive) system  (1. 117) The feedback coefficient ρ determines the stability of the filter. Stable if |ρ|<1 …………………………

1.8.2 Memory or Memory-less 1.8.3 Causal 1.8.3 CAUSALITY 출력이 현재 또는 과거의 입력에 의해 결정된다.  no memory

1.8.4 INVERTIBILITY (P.59) Ex) 2 2 Cont.

1.8.4 INVERTIBILITY (P.59) Example 1.16 Invertible? Multipath Communication Channels Fig. 1.67 Example of multiple paths in a wireless communication environment.   time-invariant. Cont.

1.8.5 TIME INVARIANCE (P.60) 언제나 같은 대답 (always the same response) 입력을 time-shift하면 출력도 time-shift (Time shifted input -> time shifted output) Problem 1.33 Let z[n]=x[n-2], then not T. I. Cont.

1.8.6 LINEARITY (P.63) The same amplitude scaling, addition in both input and output signals. Definition) For and , linear if where Superposition of the input -> superposition of output Cont.

1.8.6 LINEARITY (P.63) Example 1.19 Linear Discrete-Time System (P.64) linear? Solution) Let where  Linear! Yes!! Cont.

1.8.6 LINEARITY (P.63) Example 1.20 Linear? (p. 65) Solution) Let Problem 1.34 linear?  Linear!! Problem 1.35 Is it possible for a linear system to be non-causal? Problem 1.35 Linear? No But, Ex) y[n]=x[n]+1 Linear? -> Non-linear! non-linear. Cont.

Linear Time-Invariant Systems LTI System