1 Introduction CHAPTER 1.1 What is a signal? A signal is formally defined as a function of one or more variables that conveys information on the nature of a physical phenomenon. 1.2 What is a system? Figure 1.1 (p. 2) Block diagram representation of a system. A system is formally defined as an entity that manipulates one or more signals to accomplish a function, thereby yielding new signals. 1.3 Overview of Specific Systems ★ Communication systems 1. Analog communication system: modulator + channel + demodulator Elements of a communication system Fig. 1.2

2 Introduction CHAPTER Figure 1.2 (p. 3) Elements of a communication system. The transmitter changes the message signal into a form suitable for transmission over the channel. The receiver processes the channel output (i.e., the received signal) to produce an estimate of the message signal. ◆ Modulation: 2. Digital communication system: sampling + quantization + coding  transmitter  channel  receiver sampling + quantization + coding  transmitter  channel  receiver ◆ Two basic modes of communication: 1.Broadcasting 2.Point-to-point communication Radio, television Telephone, deep-space communication Fig. 1.3

3 Introduction CHAPTER Figure 1.3 (p. 5) (a) Snapshot of Pathfinder exploring the surface of Mars. (b) The 70-meter (230-foot) diameter antenna located at Canberra, Australia. The surface of the 70-meter reflector must remain accurate within a fraction of the signal’s wavelength. (Courtesy of Jet Propulsion Laboratory.)

4 Introduction CHAPTER ★ Control systems Figure 1.4 (p. 7) Block diagram of a feedback control system. The controller drives the plant, whose disturbed output drives the sensor(s). The resulting feedback signal is subtracted from the reference input to produce an error signal e(t), which, in turn, drives the controller. The feedback loop is thereby closed. ◆ Reasons for using control system: 1. Response, 2. Robustness ◆ Closed-loop control system: Fig Single-input, single-output (SISO) system 2.Multiple-input, multiple-output (MIMO) system Controller: digital computer Fig. 1.5 (Fig. 1.5.)

5 Introduction CHAPTER Figure 1.5 (p. 8) NASA space shuttle launch. (Courtesy of NASA.)

6 Introduction CHAPTER Analog Versus Digital Signal Processing Digital approach has two advantages over analog approach: 1.Flexibility 2.Repeatability 1.4 Classification of Signals 1. Continuous-time and discrete-time signals Continuous-time signals: x(t) Discrete-time signals: (1.1) Fig Fig Parentheses ( ‧ ) Brackets [ ‧ ] where t = nT s

7 Introduction CHAPTER Figure 1.11 (p. 17) Continuous-time signal. Figure 1.12 (p. 17) (a) Continuous-time signal x(t). (b) Representation of x(t) as a discrete-time signal x[n].

8 Symmetric about vertical axis Introduction CHAPTER 2. Even and odd signals Even signals: (1.2) Odd signals: (1.3) Antisymmetric about origin Example 1.1 Consider the signal Is the signal x(t) an even or an odd function of time? <Sol.> odd function

9 Introduction CHAPTER ◆ Even-odd decomposition of x(t): (1.4) where (1.5) Example 1.2 Find the even and odd components of the signal <Sol.> Even component: Odd component:

10 Introduction CHAPTER ◆ Conjugate symmetric: A complex-valued signal x(t) is said to be conjugate symmetric if (1.6) Let 3. Periodic and nonperiodic signals (Continuous-Time Case) Periodic signals: (1.7) and Fundamental frequency: (1.8) Angular frequency: (1.9) Refer to Fig Problem 1-2 Figure 1.13 (p. 20) (a) One example of continuous- time signal. (b) Another example of a continuous-time signal.

11 Introduction CHAPTER Fig ◆ Example of periodic and nonperiodic signals: Fig Figure 1.14 (p. 21) (a) Square wave with amplitude A = 1 and period T = 0.2s. (b) Rectangular pulse of amplitude A and duration T 1. ◆ Periodic and nonperiodic signals (Discrete-Time Case) (1.10) N = positive integer Fundamental frequency of x[n]: (1.11)

12 Introduction CHAPTER Figure 1.15 (p. 21) Triangular wave alternative between –1 and +1 for Problem 1.3. Figure 1.16 (p. 22) Discrete-time square wave alternative between –1 and +1. ◆ Example of periodic and nonperiodic signals: Fig and Fig Fig and Fig

13 Introduction CHAPTER Figure 1.17 (p. 22) Aperiodic discrete-time signal consisting of three nonzero samples. 4. Deterministic signals and random signals A deterministic signal is a signal about which there is no uncertainty with respect to its value at any time. Figure 1.13 ~ Figure 1.17 A random signal is a signal about which there is uncertainty before it occurs. Figure Energy signals and power signals Instantaneous power: (1.12) (1.13) If R = 1  and x(t) represents a current or a voltage, then the instantaneous power is (1.14)

14 CHAPTER Introduction Fig. 1.9

15 Introduction CHAPTER The total energy of the continuous-time signal x(t) is (1.15) Time-averaged, or average, power is (1.16) For periodic signal, the time-averaged power is (1.18) ◆ Discrete-time case: Total energy of x[n]: (1.17) Average power of x[n]: (1.19) (1.20) ★ Energy signal: If and only if the total energy of the signal satisfies the condition ★ Power signal: If and only if the average power of the signal satisfies the condition

16 Energy signal has zero time-average power (why?) Power signal has infinite energy (why?)  Energy signal and power signal are mutually exclusive Periodic signal and random signal are usually viewed as power signal Nonperiodic and deterministic are usually viewed as energy signal Introduction CHAPTER

17 Introduction CHAPTER Figure 1.18 (p. 26) Inductor with current i(t), inducing voltage v(t) across its terminals. 1.5 Basic Operations on Signals ★ Operations Performed on dependent Variables Amplitude scaling:x(t)x(t) (1.21) c = scaling factor Performed by amplifier Discrete-time case:x[n]x[n] Addition: (1.22) Discrete-time case: Multiplication: (1.23) Ex. AM modulation Differentiation: (1.24) Inductor: (1.25) Integration: (1.26)

18 Introduction CHAPTER Capacitor: (1.27) Figure 1.19 (p. 27) Capacitor with voltage v(t) across its terminals, inducing current i(t). ★ Operations Performed on independent Variables independent Variables Time scaling: a >1  compressed 0 < a < 1  expanded Fig Figure 1.20 (p. 27) Time-scaling operation; (a) continuous-time signal x(t), (b) version of x(t) compressed by a factor of 2, and (c) version of x(t) expanded by a factor of 2.

19 Introduction CHAPTER Discrete-time case:k = integer Some values lost! Figure 1.21 (p. 28) Effect of time scaling on a discrete-time signal: (a) discrete-time signal x[n] and (b) version of x[n] compressed by a factor of 2, with some values of the original x[n] lost as a result of the compression. Reflection: The signal y(t) represents a reflected version of x(t) about t = 0. Ex. 1-3 Consider the triangular pulse x(t) shown in Fig. 1-22(a). Find the reflected version of x(t) about the amplitude axis (i.e., the origin). Fig.1-22(b) Fig.1-22(b).

20 Introduction CHAPTER Figure 1.22 (p. 28) Operation of reflection: (a) continuous-time signal x(t) and (b) reflected version of x(t) about the origin. Time shifting: t 0 > 0  shift toward right t 0 < 0  shift toward left Ex. 1-4 Time Shifting: Fig Figure 1.23 (p. 29) Time-shifting operation: (a) continuous- time signal in the form of a rectangular pulse of amplitude 1.0 and duration 1.0, symmetric about the origin; and (b) time- shifted version of x(t) by 2 time shifts.

21 Introduction CHAPTER Discrete-time case:where m is a positive or negative integer ★ Precedence Rule for Time Shifting and Time Scaling 1. Combination of time shifting and time scaling: (1.28) (1.29) (1.30) 2. Operation order: To achieve Eq. (1.28), 1st step: time shifting 2nd step: time scaling Ex. 1-5 Precedence Rule for Continuous-Time Signal Fig. 1-24(a). Consider the rectangular pulse x(t) depicted in Fig. 1-24(a). Find y(t)=x(2t + 3). <Sol.> Case 1: Fig  Shifting first, then scaling Case 2: Fig  Scaling first, then shifting

22 Introduction CHAPTER Figure 1.24 (p. 31) The proper order in which the operations of time scaling and time shifting should be applied in the case of the continuous-time signal of Example 1.5. (a) Rectangular pulse x(t) of amplitude 1.0 and duration 2.0, symmetric about the origin. (b) Intermediate pulse v(t), representing a time-shifted version of x(t). (c) Desired signal y(t), resulting from the compression of v(t) by a factor of 2.

23 Introduction CHAPTER Figure 1.25 (p. 31) The incorrect way of applying the precedence rule. (a) Signal x(t). (b) Time-scaled signal v(t) = x(2t). (c) Signal y(t) obtained by shifting v(t) = x(2t) by 3 time units, which yields y(t) = x(2(t + 3)). Ex. 1-6 Precedence Rule for Discrete-Time Signal A discrete-time signal is defined by Find y[n] = x[2n + 3].

24 Introduction CHAPTER See Fig See Fig Figure 1.27 (p. 33) The proper order of applying the operations of time scaling and time shifting for the case of a discrete-time signal. (a) Discrete-time signal x[n], antisymmetric about the origin. (b) Intermediate signal v(n) obtained by shifting x[n] to the left by 3 samples. (c) Discrete-time signal y[n] resulting from the compression of v[n] by a factor of 2, as a result of which two samples of the original x[n], located at n = –2, +2, are lost.