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Complex numbers

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Fourier transform The Fourier transform of a continuous-time signal may be defined as The discrete version of this is Both have a j term so we need a basic understanding of complex numbers

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Complex numbers There are 2 parts to a complex number, a real part and an imaginary part z=a±jb

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Complex number application

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But what about this one?

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Complex arithmetic Add: (a+jb) + (c+jd) = (a+c) + j(b+d) –e.g. (4+j5) + (3-j2) = (7+j3) Subtract: (a+jb) – (c+jd) = (a-c) + j(b-d) –e.g. (4+j7) – (2-j5) = (2+j12) Multiply: (a+jb)(c+jd) = ac+jad+jbc+j 2 bd –e.g. (3+j4)(2+j5) = 6+j15+j8+j 2 20 = -14+j23 Note that all results are complex

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Using complex arithmetic, check that the previous quadratic has the solutions:

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Complex conjugate Complex conjugate: flip the sign of j –e.g. the complex conjugate of (5+j8) is (5-j8) Multiplication of a complex number with its conjugate produces a real number –e.g. (5+j8)(5-j8) = 25-j40+j40-j 2 64 = 25+64=89 Now consider division of complex numbers But what about

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Complex division Division: make the denominator real by multiplying top and bottom by the denominators complex conjugate

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Complex numbers as vectors Complex numbers can not be enumerated but they can be represented diagrammatically A vector (line with magnitude and direction) of a number pointing at an angle of zero can be represented as a line on the +ve x axis Multiply the vector by -1 and it points the other way i.e. a 180° shift As -1 = j 2 then j lies between them i.e. a 90° shift

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Complex plane or argand diagram Real Imaginary j j 2 = -1 j 3 =-j a+jb -c+jd -e-jf g-jh

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Polar form of a complex number The number may be represented by its vector magnitude and angle

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Or using trig for X and Y

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Polar manipulation It is easy to multiply and divide complex numbers in this form even if using degrees

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Remember the series form?

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Summary It is easy to add and subtract in Cartesian form It is easy to multiply and divide in polar form The exponential form is useful when dealing with sine and cosine waveforms

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One final note, using even-odd trig identities (or looking at waveforms), because the Fourier transform uses e -θ :

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