EE313 Linear Systems and Signals Fall 2010 Initial conversion of content to PowerPoint by Dr. Wade C. Schwartzkopf Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin Discrete-Time Signals and Systems

6 - 2 Signals A function, e.g. sin(t) in continuous-time or sin(2  n / 10) in discrete-time, useful in analysis A sequence of numbers, e.g. {1,2,3,2,1} which is a sampled triangle function, useful in simulation A collection of properties, e.g. even, causal, and stable, useful in reasoning about behavior A piecewise representation, e.g. A functional, e.g.  (t)

6 - 3 Kronecker Impulse (Function) Let  [n] be a discrete-time impulse function, a.k.a. the Kronecker delta function: Impulse response h[n]: response of a discrete- time LTI system to a discrete impulse function n [n][n] 1

6 - 4 T{} y(t)y(t)x(t)x(t) y[n]y[n]x[n]x[n] Systems Systems operate on signals to produce new signals or new signal representations Single-input one-dimensional continuous-time systems are commonly represented in two ways As operators As block diagrams

6 - 5 System Properties Let x[n], x 1 [n], and x 2 [n] be inputs to a linear system and let y[n], y 1 [n], and y 2 [n] be their corresponding outputs A linear system satisfies Additivity: x 1 [n] + x 2 [n]  y 1 [n] + y 2 [n] Homogeneity:  x[n]   y[n] for any constant  Let x[n] be the input to time-invariant system and y[n] be its corresponding output. Then, x[n - m]  y[n - m], for any integer m

6 - 6 Sampling Many signals originate as continuous-time signals, e.g. conventional music or voice By sampling a continuous-time signal at isolated, equally-spaced points in time, we obtain a sequence of numbers n  {…, -2, -1, 0, 1, 2,…} T s is the sampling period. Sampled analog waveform s(t)s(t) t TsTs TsTs

6 - 7 Optical Disk Writer Optical Disk Reader D/AA/D x(t)x(t)x(t)x(t) CDv[n]v[n]v[n]v[n] Recording StudioStereo System / PC F s = 44.1 kHz T s = ms F s = 44.1 kHz T s = ms Sampling Consider audio compact discs (CDs) Analog-to-digital (A/D) conversion consists of filtering, sampling, and quantization Digital-to-analog (D/A) conversion consists of interpolation and filtering

6 - 8 Generating Discrete-Time Signals Uniformly sampling a continuous-time signal –Obtain x[n] = x(n T s ) for -  < n < . –How to choose T s ? Using a formula –x[n] = n 2 – 5n + 3, for n  0 would give the samples {3, -1, -3, -3, -1, 3,...} –We really do not know what the sequence looks like in continuous time because we do not have a sampling period associated with it n stem plot