What is 1/2 e^{jθ}*1/2 e^{-jθ} ?

The solution to 1/2 e^{jθ}*1/2 e^{-jθ} is 1/4

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simplify 1/2 e^{jθ}*1/2 e^{-jθ}

simplify

\frac{1}{2} e^{j θ} \cdot \frac{1}{2} e^{- j θ}

\frac{1}{4}

Decimal Notation

0.25

Solution steps

\frac{1}{2} e^{j θ} \frac{1}{2} e^{- j θ}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

e^{j θ} e^{- j θ} = e^{j θ - j θ}

= \frac{1}{2} \cdot \frac{1}{2} e^{j θ - j θ}

Add similar elements:

j θ - j θ = 0

= \frac{1}{2} \cdot \frac{1}{2} e^{0}

Apply rule

a^{0} = 1, a \neq = 0

e^{0} = 1

= 1 \cdot \frac{1}{2} \cdot \frac{1}{2}

Multiply fractions:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= 1 \cdot \frac{1 \cdot 1}{2 \cdot 2}

\frac{1 \cdot 1}{2 \cdot 2} = \frac{1}{4}

= 1 \cdot \frac{1}{4}

Multiply:

1 \cdot \frac{1}{4} = \frac{1}{4}

= \frac{1}{4}

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