Question
Download Solution PDFδ(ω - ω0) का व्युत्क्रम फॉरियर रूपांतरण__________है।
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFसंकल्पना:
फॉरियर रूपांतरण के लिए समय-स्थानांतरण गुण:
यदि X(ω), x(t) का फॉरियर रूपांतरण है,
फॉरियर रूपांतरण का द्विविधता गुण:
यदि X(ω), x(t) का फॉरियर रूपांतरण है,
गणना:
δ(t) का फॉरियर रूपांतरण = 1
अब व्युत्क्रम फॉरियर रूपांतरण की परिभाषा से
अब समीकरण (1) से समय-स्थानांतरण गुण का प्रयोग करने पर
अतः विकल्प (3) सही उत्तर है।
Important Points
इकाई आवेग फलन:
एक निरंतर-समय वाले इकाई आवेश फलन δ(t), जिसे डिराक डेल्टा फलन भी कहा जाता है, को निम्न रूप में परिभाषित किया गया है:
δ (t) = ∞ , t = 0
= 0, हालाँकि
इकाई आवेग फलन को ‘1’ की दृढ़ता के साथ एक तीर के निशान द्वारा दर्शाया गया है जो इसके क्षेत्रफल को दर्शाता है।
डेल्टा फलन का गुण:
शल्कन गुण:
गुणन गुण:
X(t).δ(t – t0) = x(t0)δ(t – t0)
प्रतिचयन गुण:
महत्वपूर्ण समीकरण:
X(t).δ(t) = x(0)δ(t)
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