Question
Download Solution PDFविविक्त-समय sinc फलन के लिए, चित्र में दिखाए गए फलन का व्युत्क्रम विविक्त-समय फूरियर रूपांतरण क्या है?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFसंप्रत्यय: व्युत्क्रम विविक्त समय फूरियर रूपांतरण सूत्र द्वारा दिया गया है
x(n) = \(\frac{1}{{2π }}\int\limits_{ - π }^π { \times \left( {{e^{jw}}} \right)} \,\,{e^{jwn}}\,dw\)
गणना: दिया गया DTFT
अर्थात x(ejn) = \(\left\{ {\begin{array}{*{20}{c}} 1&{\left| \Omega \right| < w} \\ 0&{w\, < \,\,\left| \Omega \right| \leqslant π } \end{array}} \right.\)
व्युत्क्रम DTFT के लिए सूत्र लागू करने पर हमें प्राप्त होता है
x[n] = \(\frac{1}{{2π }}\int\limits_{ - π }^π { \times \left( {{e^{j\Omega }}} \right)} \,\,{e^{j\Omega n}}\,d\Omega \)
= \(\frac{1}{{2π }}\int\limits_{ - w}^w {{e^{j\Omega n}}\,d\Omega } \,\,\)
= \(\frac{{{e^{j\Omega n}}}}{{2π jn}}\int_{ - w}^w {} = \,\frac{{{e^{jwn}} - {e^{ - jwn}}}}{{\left( {2j} \right)π n}}\,\, = \,\,\frac{{\sin \,(wn)}}{{π n}}\)
sinc फलन में परिवर्तित करना:
x[n] = \(\frac{sin(Wn)}{Wn} .\frac{Wn}{πn}\) = \(\frac{{\sin π \frac{{Wn}}{π }}}{{π \,.\frac{{Wn}}{π }}}\,\,.\,\,\frac{{π \frac{{Wn}}{π }}}{{π n}}\)
x [n] = \(\frac{w}{π} sinc(\frac{w}{π}n)\)
Last updated on May 28, 2025
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