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Complex numbers

Complex numbers. 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. Complex numbers.

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Complex numbers

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  1. Complex numbers

  2. 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

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

  4. Complex number application

  5. But what about this one?

  6. 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

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

  8. 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

  9. Complex division • Division: make the denominator real by multiplying top and bottom by the denominator’s complex conjugate

  10. 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

  11. Complex plane or argand diagram j -c+jd Imaginary a+jb j2= -1 Real g-jh -e-jf j3=-j

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

  13. Or using trig for X and Y

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

  15. Remember the series form?

  16. 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

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

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