Lecture 1: Discrete Time Signal Processing

Discrete Time Signal Processing ChuSong Chen （陳祝嵩） songiis.sinica.edu.tw Institute of Information Science Academia Sinica 中央研究院 資訊科學研究所Textbook „ Main Textbook „ Alan V. Oppenheim and Ronald W. Schafer, DiscreteTime Signal Processing, Second Edition, PrenticeHall, 1999. （全華代理） „ Reference „ James. H. McClellan, Ronald W. Schafer, and Mark. A. Yoder, Signal Processing First, Prentice Hall, 2004. （開發代理）Activities „ Homework – about three times. „ Tests: twice „ First test: October 17 „ Second test: to be announced „ Term projectTeach Assistant „ 鄭文皇 wiselycmlab.csie.ntu.edu.tw 通訊與多媒體實驗室 http:www.cmlab.csie.ntu.edu.twwisleyContents „ Discretetime signals and systems „ The ztransform „ Sample of continuoustime signals „ Transform analysis of linear time invariant systems „ Structure for discretetime systemsContents (continue) „ Filter design techniques „ The discrete Fourier transform „ Computation of the discrete Fourier transform „ Fourier analysis of signals using the discrete Fourier transformSignals „ Something that conveys information „ Generally convey information about the state or behavior of a physical system. „ Signal representation „ represented mathematically as functions of one or more independent variables.Signal Examples „ Speech signal: represented as a function over time. 1D signal „ Image signal: represented as a brightness function of two spatial variables. 2D signal „ Ultra sound data or image sequence – 3D signalSignal Types „ Continuoustime signal „ defined along a continuum of times and thus are represented by a continuous independent variable. „ also referred to as analog signal „ Discretetime signal „ defined at discrete times, and thus, the independent variable has discrete values: „ i.e., a discretetime signal is represented as a sequence of numbersSignal Types (continue) „ Digital Signals „ those for which both time and amplitude are discrete „ Signal Processing System: map an input signal to an output signal „ Continuoustime systems „ Systems for which both input and output are continuoustime signals „ Discretetime system „ Both input and output are discretetime signals „ Digital system „ Both input and output are digital signalsExample of Discretetime Signal „ Discretetime signal x = {xn}, ∞ < n < ∞ „ Discretetime Signal where n is an integerGeneration of Discretetime Signal „ In practice, such sequences can often arise from periodic sampling of an analog signal. x = x nT ∞ < n < ∞ a ,Signal Operations „ Multiplication and addition „ The product and sum of two sequences xn and yn are defined as the samplebysample product and sum, respectively. „ Multiplication by a number a is defined as multiplication of each sample value by a. „ Shift operation: yn is a delayed or shifted version of xn where n 0 is an integer. yn= xn − n0A Particular Signal „ Unit sample sequence „ Unit impulse function, Dirac delta function, impulse ⎧⎨⎩ ≠ = = 1 0 0 0 n n δ nSignal Representation „ An arbitrary sequence can be represented as a sum of scaled, delayed impulses. ∑ ∞ =−∞ = − k x n x k δ n kSome Signal Examples „ Unit step sequence ⎧⎨⎩ ≥ < = 0 0 1 0 n n u nSome Signal Examples (cont.) „ Real exponential sequence xn= Aα n ⎧⎨⎩ ≥ < = 0 0 0 n n A y n n α „ y n can be represented as yn= Aα nunSome Signal Examples (cont.) „ Sinusoidal sequence xn= Acos(w0n +φ)Complex Exponential Sequence „ Consider an exponential sequence xn = Aαn, where α is a complex number having real and imaginary parts ( ) α ( ) α ( ) ( ) ( ) φ φ α α φ φ = = + + + = = + A e A w n j w n x n A Ae e n n j w n n j n j w n 0 0 0 0 cos sin „ The sequence oscillates with an exponentialy growing envelope if |α|>1, or with an exponentially decaying envelope if |α|