Manipulating an expression into alternate form

Question

I'm trying to get $1-1.4e^{-j\theta}+.81e^{-2j\theta}$ into the form $(1-d_ke^{-j\theta})$. I'm not sure which rules I could apply to get it into that form. May I have a hint at it or even if it is possible.

Answer 1

This is not possible. If we take $u = ae^{-j\theta} + be^{-2j\theta}$ and $v = ce^{-j\theta}$, then we have that

$$\frac{du}{d\theta} = -jae^{-j\theta} -2jbe^{-2j\theta} = -ju -jbe^{-2j\theta}$$ $$\frac{dv}{d\theta} = -jce^{-j\theta} = -jv$$

The conclusion is that even if we choose constants $a,b,c$ such that $u(\theta_0) = v(\theta_0)$ for some specific value of $\theta_0$, then their derivatives will not be equal there unless $b=0$. In your case, $a = 1.4$ and $b=-.81 \neq 0$, so we cannot rewrite this complex number in this form.