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[✓] Simplify InverseZTransform output?

GROUPS:

I am experiencing difficulty obtaining a Real result when performing the InverseZTransform function. As an example, I want to find a closed form expression for the output of a simple exponential smoothing filter to a sinusoidal input. I start by finding the z-transforms of the filter impulse response and the sinusoid:

ZTransform[a (1 - a)^n, n, z]

(a z)/(-1 + a + z)


ZTransform[Sin[w n], n, z]

(z Sin[w])/(1 + z^2 - 2 z Cos[w])

Then I take the inverse z-transform of their product:

InverseZTransform[((a z)/(-1 + a + z)) (z Sin[w])/( 1 + z^2 - 2 z Cos[w]), z, n,  Assumptions -> { a \[Element] Reals, w \[Element] Reals}] 

-((a E^(-I (-1 + n) w) (-1 + a + E^(I w) + (1 - a)^n E^(I n w) - (1 - a)^n a E^(I n w) + E^(2 I (1 + n) w) - a E^(2 I (1 + n) w) - 
E^(I (1 + 2 n) w) - (1 - a)^n E^(2 I w + I n w) + (1 - a)^n a E^(2 I w + I n w)) Sin[w] UnitStep[-1 + n])/((-1 + E^(I w)) (1 + E^(I w)) 
    (-1 + a + E^(I w)) (1 - E^(I w) + a E^(I w))))

Using FullSimplify[ComplexExpand[ ] ] to simplify this produces an expression that still contains complex exponentials:

(a E^(-I (-1 + n) w) Sin[w] (a (-1 + E^(2 I (1 + n) w)) - (-1 + E^(I w)) (1 + E^(I (w + 2 n w))) + 
     2 I (1 - a)^(1 + n) E^(I (1 + n) w) Sin[w]) UnitStep[-1 + n])/((-1 + a + E^(I w)) (1 + (-1 + a) E^(I w)) (-1 + E^(2 I w)))

This result is obtained even if I include the option {n [Element] Reals, a [Element] Reals, w [Element] Reals}

I know the result must be Real but I can't seem to nudge Mathematica to produce a result in that form. Any suggestions how to accomplish this will be most welcome.

KM

POSTED BY: K M
Answer
8 months ago
FullSimplify[
 ComplexExpand[
  Re[-((a E^(-I (-1 + n) w) (-1 + a + 
          E^(I w) + (1 - a)^n E^(I n w) - (1 - a)^n a E^(I n w) + 
          E^(2 I (1 + n) w) - a E^(2 I (1 + n) w) - 
          E^(I (1 + 2 n) w) - (1 - a)^n E^(2 I w + I n w) + (1 - a)^
            n a E^(2 I w + I n w)) Sin[
         w] UnitStep[-1 + n])/((-1 + E^(I w)) (1 + E^(I w)) (-1 + a + 
          E^(I w)) (1 - E^(I w) + a E^(I w))))]], {a ∈ Reals,
   w ∈ Reals, n ∈ Integers, n > 0}]

or

Assuming[{a ∈ Reals, w ∈ Reals, n ∈ Integers, n > 0}, FullSimplify[ComplexExpand[Re[-((a E^(-I (-1 + n) w) (-1 + a + 
E^(I w) + (1 - a)^n E^(I n w) - (1 - a)^n a E^(I n w) + E^(2 I (1 + n) w) - a E^(2 I (1 + n) w) - 
E^(I (1 + 2 n) w) - (1 - a)^n E^(2 I w + I n w) + (1 - a)^n a E^(2 I w + I n w)) 
Sin[w] UnitStep[-1 + n])/((-1 + E^(I w)) (1 + E^(I w)) (-1 + a +E^(I w)) (1 - E^(I w) + a E^(I w))))]]]]

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POSTED BY: Mariusz Iwaniuk
Answer
7 months ago

Thanks. Very helpful.

KM

POSTED BY: K M
Answer
7 months ago

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