Digital Signal Processing - A Modern Introduction

Ashok Ambardar

Language: English

Pages: 608

ISBN: 0534405096

Format: PDF / Kindle (mobi) / ePub

This book provides a modern and self-contained introduction to digital signal processing (DSP). It is supplemented by a vast number of end-of-chapter problems such as worked examples, drill exercises, and application oriented problems that require the use of computational resources such as MATLAB. Also, many figures have been included to help grasp and visualize critical concepts. Results are tabulated and summarized for easy reference and access. The text also provides a broader perspective to the content by introducing useful applications and additional special topics in each chapter. These form the background for more advanced graduate courses.

system equation has a form different from y[n + α] = Kx[n + α]. EXAMPLE 3.4 (Causal and Dynamic Systems) (a) y[n] = x[n + 2] is noncausal (to find y[0], we need x[2]) and dynamic (y[n0 ] does not depend on x[n0 ] but on x[n0 + 2]). (b) y[n + 4] + y[n + 3] = x[n + 2]. By time invariance, y[n] + y[n − 1] = x[n − 2]. So it is causal and dynamic. (c) y[n] = 2x[αn] is causal and instantaneous for α = 1, causal and dynamic for α < 1, and noncausal and dynamic for α > 1. It is also time varying if α =

September 1, 2003 3.4 The Natural and Forced Response 57 input. Table 3.1 summarizes these forms for various types of inputs. The constants in the forced response can be found uniquely and independently of the natural response or initial conditions simply by satisfying the given differential equation. The characteristic equation is defined by the polynomial equation 1 + A1 z −1 + A2 z −2 + · · · + AN z −N = z N + A1 z N −1 + · · · + AN = 0 (3.15) This equation has N roots, z1 , z2 , . . . ,

change the frequency (or period) of the signal (to double the frequency for example, we would cycle over alternate samples) or for storing a new signal. EXAMPLE 3.18 (Generating Periodic Signals Using Recursive Filters) Suppose we wish to generate the periodic signal x[n] described by x[n] = {2, 3, 1, 6, 2, 3, 1, 6, 2, 3, 1, 6, 2, 3, 1, 6, . . .} The impulse response sequence of a nonrecursive filter that generates this signal is simply h[n] = {2, 3, 1, 6, 2, 3, 1, 6, 2, 3, 1, 6, 2, 3, 1, 6, . .

1] + u[n] (b) x[n] = 2 + cos(0.5nπ) [Hints and Suggestions: For part (a), find Y (z) and its inverse transform by partial fractions. For part (b), add the steady state response for each component of the input.] 4.68 (System Response) The signal x[n] = (0.5)n u[n] is applied to a digital filter, and the response is y[n]. Find the filter transfer function and state whether it is an IIR or FIR filter and whether it is a linear-phase filter if the system output y[n] is the following: (a) y[n] =

Find two different recursive difference equations (with different orders) for each of the following filters. ⇓ (a) h[n] = {1, 2, 1} z 2 − 2z + 1 (b) H(z) = z2 (c) y[n] = x[n] − x[n − 2] [Hints and Suggestions: Set up H(z) and multiply both numerator and denominator by identical polynomials in z (linear, quadratic etc). Use this to find the recursive difference equation.] COMPUTATION AND DESIGN 4.76 (System Response in Symbolic Form) The Matlab based routine sysresp2 (on the author’s website)