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Question 8.9: Use of the s-domain shifting property If X1(s) = 1/s+5, σ &g......

Use of the s-domain shifting property

If X_1(s) = \frac{1}{s+5}, σ > -5 and X_2 (s)=X_1(s-j4)+X_1(s+j4), σ > -5 find x_2 (t).

e^{-5t}u(t) \stackrel{\mathcal{L}}{\longleftrightarrow} \frac{1}{s+5}, σ > -5
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Using the s-domain shifting property

e^{-(5-j 4) t} \mathrm{u}(t) \stackrel{\mathcal{L}}{\longleftrightarrow} \frac{1}{s-j 4+5}, \sigma>-5 \text { and } e^{-(5+j 4) t} \mathrm{u}(t) \stackrel{\mathcal{L}}{\longleftrightarrow} \frac{1}{s+j 4+5}, \sigma>-5 \text {. }

Therefore

\mathrm{x}_2(t)=e^{-(5-j 4) t} \mathrm{u}(t)+e^{-(5+j 4) t} \mathrm{u}(t)=e^{-5 t}\left(e^{j 4 t}+e^{-j 4 t}\right) \mathrm{u}(t)=2 e^{-5 t} \cos (4 t) \mathrm{u}(t)

The effect of shifting equal amounts in opposite directions parallel to the axis in the s domain and adding corresponds to multiplication by a causal cosine in the time domain. The overall effect is double-sideband suppressed carrier modulation, which will be discussed in Chapter 12.

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