brPage 9br 6 ft 6 2 4 0 0 0 0 0 0 w 0 0 2p Figure 4 sinct and its ourier transform An imp ortan oin is that signal that is bandlimited is not timelimited while signal that is timelimited is not bandlimited ID: 23811 Download Pdf

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brPage 9br 6 ft 6 2 4 0 0 0 0 0 0 w 0 0 2p Figure 4 sinct and its ourier transform An imp ortan oin is that signal that is bandlimited is not timelimited while signal that is timelimited is not bandlimited

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Some Example Con tin uous ourier transforms )) )) 1 dt Giv en that 1 dt (0) 1 dt (0) 1 dt Therefore ))

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Linearit of the ourier transform )) ), where is constan Using the dualit prop ert and the linearit prop ert of the ourier transform A ourier transform of at (see Figure 1)

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at dt 1/a p/2 p/2 Magnitude Phase −at Figure 1: The exp onen tial function and its ourier transform

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ourier transform of the unit step function The ourier transform of the unit step function can obtained only in the

limit )) lim at ourier transform of (see Figure 2)

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|a|t 2/a Magnitude F( ) Figure 2: j and its ourier transform

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at at dt 1 dt ourier transform of the rectangular function A; other ise

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Ae dt AT sin sinc The rectangular function ect and its ourier transform are sho wn in Figure

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T AT 2 4 F( ) Figure 3: rect(t) and its ourier transform ourier transform of the sinc function Using the dualit prop ert the ourier transform of the sinc function can determined (see Figure 4).

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6 ......... ......... f(t) 6

2 4 0 0 0 0 0 0 -w 0 0 2p Figure 4: sinc(t) and its ourier transform An imp ortan oin is that signal that is \bandlimited" is not \time-limited" while signal that is \time-limited" is not \bandlimited"

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