Anonymous
Anonymous asked in Science & MathematicsMathematics · 9 years ago

What is the magnitude of this function?

I'm asked to find the magnitude of a complex function R(jw) = 1 + exp(-jw) + exp(-j2w) + exp(-j3w) + exp(-j4w)

where w is the angular frequency, j is the imaginary number (j = sqrt(-1)), and exp(-jnw) is a complex sinusoid.

So what I did was: |R(jw)| = |1 + exp(-jw) + exp(-j2w) + exp(-j3w) + exp(-j4w)| and I don't know how to proceed from here. Do we have to do it like this:

= |1| + |exp(-jw)| + |exp(-j2w)| + |exp(-j3w)| + |exp(-j4w)|

= 1 + 1 + 1 + 1 + 1

= 5

?

2 Answers

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  • 9 years ago
    Favorite Answer

    Re-quired magnitude = +sq rt[R(jw)*R(-jw)} = [{1-exp(-j4w)}/{1-exp(-jw}]*[{1-exp(+j4w)}/{1-exp(+jw}]

    = [-{exp(+j4w) + exp(-j4w)}/-{exp(+jw) + exp(-jw)}] = cos (4w)/cos(w)

    = (1/cos w)*[2 cos^2(2w) -1] = (1/cos w)*[(2cos^2(w) -1)^2 -1]

    (1/cos w)*[(4cos^4(w) +4cos^2(w)+1 -1] = 4 cos^3 w + 4 cos w

  • 9 years ago

    Review your complex numbers in exponential form. You have something missing.

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