tailieunhanh - Understanding digital signal processing - Appendix

Tài liệu tham khảo giáo trình tìm hiểu xử lý tín hiệu số bằng tiếng anh ( Understanding digital signal processing ) Phụ lục | APPENDIX A The Arithmetic of Complex Numbers To understand digital signal processing we have to get comfortable using complex numbers. The first step toward this goal is learning to manipulate complex numbers arithmetically. Fortunately we can take advantage of our knowledge of real numbers to make this job easier. Although the physical significance of complex numbers is discussed in Appendix c the following discussion provides the arithmetic rules governing complex numbers. Graphical Representation of Real and Complex Numbers To get started real numbers are those positive or negative numbers we re used to thinking about in our daily lives. Examples of real numbers are etc. Keeping this in mind we see how a real number can be represented by a point on a one-dimensional axis called the real axis as shown in Figure A-l. We can in fact consider that all real numbers correspond to all of the points on the real axis line on a one-to-one basis. This point represents the real -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 Real axis Figure A-l The representation of a real number as a point on the onedimensional real axis. 443 444 Appendix A The Arithmetic of Complex Numbers Figure A-2 The phasor representation of the complex number c R jl on the complex plane. A complex number unlike a real number has two parts a real part and an imaginary part. Just as a real number can be considered to be a point on the one-dimensional real axis a complex number can be treated as a point on a complex plane as shown in Figure A-2. We ll use this geometrical concept to help US understand the arithmetic of complex numbers Arithmetic Representation of Complex Numbers A complex number c is represented in a number of different ways in the literature such as Rectangular form - C R jI Trigonometric form - c M cos o sinfa A-l J Exponential form C Me B A-l Magnitude and angle form c Al 0 . A-T Equations A-l and A-l remind US that the complex number c can also be considered .