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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+j2bd e.g. (3+j4)(2+j5) = 6+j15+j8+j220 = -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-j264 = 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 denominator’s 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 = j2 then j lies between them i.e. a 90° shift

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**Complex plane or argand diagram**

j -c+jd Imaginary a+jb j2= -1 Real g-jh -e-jf j3=-j

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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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STROUD Worked examples and exercises are in the text Programme 1: Complex numbers 1 COMPLEX NUMBERS 1 PROGRAMME 1.

STROUD Worked examples and exercises are in the text Programme 1: Complex numbers 1 COMPLEX NUMBERS 1 PROGRAMME 1.

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