Linear Systems and Signals Prof. Shun-Pin Hsu ( 許舜斌 ) Dept. of Electrical Engineering National Chung-Hsin University Course 2303 Spring 2008 Lecture 1 : An Introduction

1-2 Audio CD Samples at 44.1 kHz Human hearing is from about 20 Hz to 20 kHz Sampling theorem (covered last week of class): sample analog signal at a rate of more than twice the highest frequency in the analog signal –Apply a filter to pass frequencies up to 20 kHz (called a lowpass filter) and reject high frequencies, e.g. a coffee filter passes water through but not coffee grounds –Lowpass filter needs 10% of cutoff frequency to roll off to zero (filter can reject frequencies above 22 kHz) –Sampling at 44.1 kHz captures analog frequencies of up to but not including kHz.

1-3 Coverage of Course Analysis of linear subsystems within control, communication, and signal processing systems Examples of electronic control systems? –Antilock brakes –Engine control –Chemical processing plant Emerging trend: brake by wire Examples of signal processing systems?

1-4 Signal Processing Systems Speech and audio –Speech synthesis and recognition –Audio CD players –Audio compression: AC3, MPEG 1 layer 3 audio (MP3) Image and video compression –Image compression: JPEG, JPEG 2000 –Video CDs: MPEG 1 –DVD, digital cable, HDTV: MPEG 2 –Wireless video: MPEG 4/H.263, MPEG 4 Advanced Video Coding/H.264 Examples of communication systems? Moving Picture Experts Group (MPEG) Joint Picture Experts Group (JPEG)

1-5 Communication Systems Voiceband modems (56k) Digital subscriber line (DSL) ISDN: 144 kilobits per second (kbps) Symmetric: High-speed DSL Asymmetric: Asymmetric DSL and Very High-Speed DSL Cable modems Cellular phones First generation (1G): AMPS Second generation (2G): Global System for Mobile (GSM) and Interim Standard-95 (Code Division Multiple Access) Third generation (3G): cdma2000, Wideband CDMA Integrated Digital Services Network (ISDN)

1-6 Wireline Communications HDSL High bitrate Mbps in both directions ADSL Asymmetric 1-15 Mbps down, Mbps up VDSL Very high bitrate, 22 Mbps down, 3 Mbps up Customer Premises downstream upstream Voice Switch Central Office DSLAM ADSL modem Lowpass Filter PSTNPSTN Interne t

1-7 Wireless Communications Time-frequency (Fourier) analysis Digital communication increases SNR/capacity Antenna array adds further increase in SNR & capacity by using spatial diversity 2.5G and 3G systems: transmit voice & data Picture by Prof. Murat Torlak, UT Dallas

1-8 Related Technical Areas Communication/networking Real-Time DSP Lab Digital Comm. Comm. Systems Intro to Telecom. Networks Info. Theory Net. Eng. Lab. Wireless Comm.Lab Signal/image processing Real-Time DSP Lab DSP Neural Nets Digital Image and Video Processing Related Technical Areas Electromagnetics Embedded Systems System Software VLSI Design

1-9 Signals Continuous-time signals are functions of a real argument x(t) where t can take any real value x(t) may be 0 for a given range of values of t Discrete-time signals are functions of an argument that takes values from a discrete set x[n] where n  {...-3,-2,-1,0,1,2,3...} We sometimes use “index” instead of “time” when discussing discrete-time signals Values for x may be real or complex

Analog vs. Digital The amplitude of an analog signal can take any real or complex value at each time/sample Amplitude of a digital signal takes values from a discrete set

1-11 Systems A system is a transformation from one signal (called the input) to another signal (called the output or the response). Continuous-time systems with input signal x and output signal y (a.k.a. the response): y(t) = x(t) + x(t-1) y(t) = x 2 (t) Discrete-time system examples y[n] = x[n] + x[n-1] y[n] = x 2 [n] x(t)x(t) y(t)y(t) x[n]x[n] y[n]y[n]

1-12 Linearity Linear systems –Output is linear transformation of input Linear transformation T{·} satisfies both –Homogeneity T[k x(t)] = k T[x(t)] k is a scalar constant and x(t) is an input –Additivity T[x 1 (t) + x 2 (t)] = T[x 1 (t)] + T[x 2 (t)] x 1 (t) and x 2 (t) are two inputs x 1 (t) + x 2 (t) is a superposition (addition) of inputs x(t)x(t) y(t)y(t)

1-13 Time-Invariance A shift in the input produces a shift in the output by the same amount –If T[x(t)] = y(t), then T[x(t -  )] = y(t -  ) for all real- valued time shifts  Is the following system time-invariant? y(t) = x(t) + x(t -1)

1-14 Causality Output depends only on the current and past inputs and past outputs y(t) = y(t -1) + x(t) + 2 x(t -1) is causal y(t) = y(t +1) + x(t +1) - 2 x(t -100) is NOT causal For systems that involve functions of time (e.g audio signals), we must use causal systems if we must process signals in real-time For systems that involve functions of spatial coordinates (images), this is not of concern when entire image is available for processing

1-15 Course Outline Time domain analysis (lectures 1-10) –Signals and systems in continuous and discrete time –Convolution: finding system response in time domain Generalized frequency domain analysis (lectures 11-18) –Laplace and z transforms of signals –Transfer functions of linear time-invariant systems –Tests for system stability Frequency domain analysis (lectures 19-25) –Fourier series –Fourier transform of continuous-time signals –Frequency responses of systems

1-16 Philosophy Pillars of electrical engineering (related) –Fourier analysis –Random processes Pillars of computer engineering (related) –System state –Complexity Finite-state machines for digital input/output –Finite number of states –Models all possible input- output combinations –Can two outputs be true at the same time? Given output observation, work backwards to inputs to see if output is possible This is called observability