Signal and systems with matlab

[...]... of signals and systems We also discuss the basic concepts on signal and system analysis such as linearity, time-invariance, causality, stability, impulse response, and system function (transfer function) W.Y Yang et al., Signals and Systems with MATLAB R , DOI 10.1007/978-3-540-92954-3 1, C Springer-Verlag Berlin Heidelberg 2009 1 2 1 Signals and Systems 1.1 Signals 1.1.1 Various Types of Signal A signal, ... sin(Ω1 n) (1.1.10b) 6 1 Signals and Systems 1.1.3 Analog Frequency and Digital Frequency A continuous-time signal x(t) is periodic with period P if P is generally the smallest positive value such that x(t + P) = x(t) Let us consider a continuous-time periodic signal described by x(t) = e jω1 t (1.1.11) The analog or continuous-time (angular) frequency1 of this signal is ω1 [rad/s] and its period is P=... slower as Ω1 increases from π to 2π (see Fig 1.5(d)–(h)) Particularly with Ω1 = 2π (Fig 1.5(h)) or 2mπ , it is not distinguishable from a DC signal with Ω1 = 0 The discrete-time periodic signal is seen to oscillate faster as Ω1 increases from 2π to 3π (Fig 1.5(h) and (i)) and slower again as Ω1 increases from 3π to 4π 8 1 Signals and Systems 1 1 1 0 0 0 –1 –1 0 0.5 1 1.5 (a) cos(πnT), T = 0.25 –1 0 2... analog signal to a digital one usually performs the operations of sampling -and- hold, quantization, and encoding However, throughout this book, we ignore the quantization effect and use “discrete-time signal/ system” and “digital signal/ system” interchangeably Discrete-time Continuous-time Continuous-time Continuous-time continuous-amplitude continuous-amplitude discrete-amplitude discrete-amplitude signal. .. 1 Signals and Systems Contents 1.1 1.2 1.3 1.4 1.5 Signals 1.1.1 Various Types of Signal 1.1.2 Continuous/Discrete-Time Signals 1.1.3 Analog Frequency and Digital Frequency 1.1.4 Properties of the Unit Impulse Function and. .. example, a speech signal may be represented as an amplitude function of time and a picture as a brightness function of two spatial variables Depending on whether the independent variables and the values of a signal are continuous or discrete, the signal can be classified as follows (see Fig 1.1 for examples): - Continuous-time signal - Discrete-time signal (sequence) - Continuous-amplitude(value) signal - Discrete-amplitude(value)... steady-state 20 1 Signals and Systems 1 0.5 ^ :y (t ) = 2 : x(t ) 0.4 ^ : x(t ) 3 1 0.2 : y (t ) 0 0.5 1 1.5 2 0 ^ (a1) The input x (t ) and its approximation x (t ) with T = 0.5 1 : x(t ) ^ 3 : x(t ) 0.5 0 0 0 ^ :y (t ) = 0.4 2 0.2 0 0.5 1 1.5 2 ^ (a2) The input x (t ) and its approximation x (t ) with T = 0.25 0 0 1 4 + 2 + 3 + : y (t ) T 1 3 1 2 3 ^ (b1) The outputs to x (t ) and x (t ) 4 5 +2+... discrete-amplitude discrete-amplitude signal signal sampled signal signal xd [n] x (t ) x∗(t ) xd (t ) A/D conversion D/A conversion hold sampling at t = nT T : sample period Continuous-time continuous-amplitude signal x (t ) T (a) (b) (c) (d) (e) Fig 1.1 Various types of signal 1.1.2 Continuous/Discrete-Time Signals In this section, we introduce several elementary signals which not only occur frequently... periodic signals with increasing analog/digital frequency This implies that the frequency characteristic of a discrete-time signal is periodic with period 2π in the digital frequency Ω This is because e jΩ1 n is also periodic with period 2π in Ω1 , i.e., e j(Ω1 +2mπ )n = e jΩ1 n e j2mnπ = e jΩ1 n for any integer m (4) Note that if a discrete-time signal obtained from sampling a continuous-time periodic signal. .. continuous-time/discrete-time system if its input and output are both continuous-time/discrete-time signals Continuous-time/discrete-time systems with the input and output are often described by the following equations and the block diagrams depicted in Fig 1.7(a)/(b) Discrete-time system Continuous-time system G{} G{} x(t) → y(t); y(t) = G{x(t)} x[n] → y[n]; y[n] = G{x[n]} 1.2.1 Linear System and Superposition Principle