2 o w p329 f cos 0 t 0 0 f sin

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Unformatted text preview: x[n] and y [n], respectively. � ⎧−j 2, k = −4 ⎧ ⎧ ⎧ j�/4 ⎧e ⎧ , k = −1 ⎧ ⎧ ⎧ ⎧2, ⎧ k=0 � bk jk�0 jk � −j�/4 � H (e )= � H (e 4 ) = e , k=1 ⎧ ak ⎧ ⎧j 2, k=4 ⎧ ⎧ ⎧ ⎧0, ⎧ k = ±, 2 ⎧ ⎧ ⎧ �?, otherwise 12 Problem 5 Compute the Fourier transform of each of the following signals: (a) x(t) = e−|t| cos 2t Trying to compute the Fourier transform of x(t) using the analysis equation (O & W, p.288) might require going through a lengthy integration. Instead we will use the Fourier transform properties. Let x(t) = e−|t| cos 2t = s(t)p(t), where s(t) = e−|t| and p(t) = cos 2t. From the Multiplication property (O & W, Section 4.5, p.322) we have: �� 1 1 X (j� ) = S (jβ )P (j (� − β ))dβ = S (j� ) ⇒ P (j� ) 2ω −� 2ω Now, we need to find the Fourier transforms of s(t) and p(t) and plug them in the expression above. F From Example 4.2 (O & W, p.291): e−a|t| �� F � s(t) = e−|t| �� S (j� ) = From Table 4.2 (O & W, p.329): a2 2a + �2 2 1 + �2 F cos �0 t �� ω [α (� − �0 ) + α (� + �0 )] F � p(t) = cos 2t �� P (j� ) = ω [α (� − 2) + α (� + 2)] �� 1 X (j� )...
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