Discrete-Time Signals and Systems Linear Systems and Signals Lecture 7 Spring 2008

7 - 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)

7 - 3 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

7 - 4 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

7 - 5 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

7 - 6 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

7 - 7 Ideal Differentiator ContinuousDiscrete f(t)f(t)y(t)y(t)f[n]f[n]y[n]y[n]

7 - 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