By John G. Proakis, Dimitris K. Manolakis
Compatible for a one- or two-semester undergraduate-level electric engineering, laptop engineering, and desktop technology direction in Discrete structures and electronic sign Processing. Assumes a few past wisdom of complicated calculus, linear platforms for continuous-time signs, and Fourier sequence and transforms. Giving scholars a legitimate stability of thought and functional software, this no-nonsense textual content provides the elemental recommendations and strategies of contemporary electronic sign processing with similar algorithms and functions. masking either time-domain and frequency- area tools for the research of linear, discrete-time platforms, the e-book bargains state of the art insurance on such subject matters as sampling, electronic clear out layout, filter out realizations, deconvolution, interpolation, decimation, state-space tools, spectrum research, and extra. Rigorous and not easy, it additional prepares scholars with a variety of examples, workouts, and experiments emphasizing software program implementation of electronic sign processing algorithms built-in all through.
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Extra info for Digital Signal Processing: Principles, Algorithms and Applications (3rd Edition)
A rational expression is the quotient of two polynomials. For example, 3 , 5 2 , xϪ3 and x2 Ϫ 4 x Ϫ 4x Ϫ 5 2 are rational expressions. , publishing as Pearson Addison-Wesley 99. 3͑z Ϫ 2͒ ͑z 2 ϩ 2z ϩ 4͒ 100. 4͑t ϩ 3͒ ͑t 2 Ϫ 3t ϩ 9͒ 101. 2ab͑2a2 ϩ 3b 2 ͒ ͑4a4 Ϫ 6a2b 2 ϩ 9b 4 ͒ 102. 3a2x͑2x Ϫ 5a2 ͒ ͑4x 2 ϩ 10a2x ϩ 25a4 ͒ 116. a͑5 Ϫ 2a͒ ͑25 ϩ 10a ϩ 4a 2 ͒ 128. 5 • Rational Expressions 31 The Domain of a Rational Expression The domain of an algebraic expression is the set of all real numbers for which the expression is defined.
The factorization is ͑x ϩ 2͒ ͑x ϩ 3͒. We have x 2 ϩ 5x ϩ 6 ͑x ϩ 2͒ ͑x ϩ 3͒. We can check this by multiplying: ͑x ϩ 2͒ ͑x ϩ 3͒ x 2 ϩ 3x ϩ 2x ϩ 6 x 2 ϩ 5x ϩ 6. EXAMPLE 4 Factor: 2y 2 Ϫ 14y ϩ 24. Solution First, we look for a common factor. Each term has a factor of 2, so we factor it out first: 2y 2 Ϫ 14y ϩ 24 2͑ y 2 Ϫ 7y ϩ 12͒. Now we consider the trinomial y 2 Ϫ 7y ϩ 12. We look for two numbers whose product is 12 and whose sum is Ϫ7. Since the constant term, 12, is positive and the coefficient of the middle term, Ϫ7, is negative, we look for a factorization of 12 in which both factors are negative.
180 5 16. ͙250 3 3 19. ͙128c 2d 4 18. ͙ 135 8͙2 ͉ c ͉ d 2 4 45 3 6 m12n24 64 m2n4 2 ͙ 40m 2 3 ͙ 5m 3 3 3͙ 5 4 2 4 3͉ m ͉n ͙ mn2 23. ͙x 2 Ϫ 4x ϩ 4 ͉x Ϫ 2͉ 24. ͙x 2 ϩ 16x ϩ 64 Simplify. Assume that no radicands were formed by raising negative quantities to even powers. 25. ͙10 ͙30 10͙3 26. ͙28 ͙14 14͙2 ͙ 3x 2 39. 3 ͙ 24x 5 ͱ ͱ 41. 43. 3 5 3 2͑x ϩ 4͙͒ ͑x ϩ 4͒2 3 ͱ 5͙10 21. ͙ 48x 6y 4 2͉ x ͉ ͉ y ͉͙ 3x 2 22. ͙ 243m5n 10 5 32. ͙ 8x 3y 4 ͙ 4x 4y 34. ͙ 4͑x ϩ 1͒2 ͙ 18͑x ϩ 1͒2 35. 5͙21 30. ͙3y 4z ͙20z 33.