|
फलन |
रूपान्तर |
टिप्पणी
|
1 |
![{\displaystyle af(t)+bg(t)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60e906555fec88aa79baa114327b609cb83030bb) |
![{\displaystyle aF(\omega )+bG(\omega )\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2268c4e3ab05688cc0f83a4d407fdddd1adac2e2) |
रैखिकता
|
2 |
![{\displaystyle f(t-a)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91c4bb340c0a97ba84e97a69e44d2b2b0ac9bac5) |
![{\displaystyle e^{-i\omega a}F(\omega )\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f703f82494ddf9a2aeabe18c66006829faf2e7a) |
विलम्ब (delay)
|
3 |
![{\displaystyle e^{iat}f(t)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef45326f3a2cafa16f1fcc673b9a8fa30dfa9ed7) |
![{\displaystyle F(\omega -a)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d8c1aa5f53a92fffad8c81c5d028b174aac56c0) |
आवृत्ति शिफ्ट
|
4 |
![{\displaystyle f(at)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d68e32dabd7150ba1bfacd6c13c3fc600f03870) |
![{\displaystyle |a|^{-1}F\left({\frac {\omega }{a}}\right)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0000f3df1bf44cb79c94fdc7e5c72e83bd37663) |
यदि बहुत बड़ा हो तो 0 के आसपास केन्द्रित होगा और 'चपटा' हो जाएगा।
|
5 |
![{\displaystyle {\frac {d^{n}f(t)}{dt^{n}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52c839ecdcba871f1176a0534f676b1a6757580b) |
![{\displaystyle (i\omega )^{n}F(\omega )\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb803052ef099c524b10c00ad417accb97b0892) |
Свойство преобразования Фурье от
|
6 |
![{\displaystyle t^{n}f(t)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/388b4862f73e88738172a41b8c57949be936e8c6) |
![{\displaystyle i^{n}{\frac {d^{n}F(\omega )}{d\omega ^{n}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c98075eadc997ffc614dc760e4890c13c1f506de) |
|
7 |
![{\displaystyle (f*g)(t)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22637a8f010c81dafe6121061e6ed09e6154de50) |
![{\displaystyle {\sqrt {2\pi }}F(\omega )G(\omega )\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8d4cb93fe1e053ffe000cf31797240f01b6901b) |
फलन का अर्थ है - फलन और का कॉनवोलुशन.
|
8 |
![{\displaystyle f(t)g(t)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e54087b12611ecb6d565378f447aeb8e8a345186) |
![{\displaystyle {\frac {(F*G)(\omega )}{\sqrt {2\pi }}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/712dd0615b2a1b7b9b53b9adfeafbf652d9e14dc) |
|
9 |
![{\displaystyle \delta (t)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a6eb27873ffae4d91f1f3a174131b84c9141edb) |
![{\displaystyle {\frac {1}{\sqrt {2\pi }}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2adeddfedc8362c827aca3ae9efa7a82534c7acd) |
का अर्थ है - डिरैक डेल्टा फलन
|
10 |
![{\displaystyle 1\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfd1e7984fe6e1b79a26404a8138a6c6ee41a476) |
![{\displaystyle {\sqrt {2\pi }}\delta (\omega )\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/252cae36173adfd951b148c44b7b36b775967d89) |
|
11 |
![{\displaystyle t^{n}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb672701379c41c98a5c6a4d6be21f8652c1d288) |
![{\displaystyle i^{n}{\sqrt {2\pi }}\delta ^{(n)}(\omega )\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/445f7a2a7630766920d322750869a7c11050c94b) |
|
12 |
![{\displaystyle e^{iat}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2538d1ebdf70ba0cedc3bd574f23ab76b8e70bba) |
![{\displaystyle {\sqrt {2\pi }}\delta (\omega -a)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4af2a6db3aac758c1dcaf3f93ebbe5db99de5618) |
|
13 |
![{\displaystyle \cos(at)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c89d72e1e6bea14b393ac46483390c1a674f5d1c) |
![{\displaystyle {\sqrt {2\pi }}{\frac {\delta (\omega -a)+\delta (\omega +a)}{2}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/879f916c4fcb1513f9a19c2bee63140fd5732272) |
1 और 12 का उपसाध्य ; आयलर प्रमेय से -
|
14 |
![{\displaystyle \sin(at)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc1d893181c207dc354d47cdd3e50d45e72dc7fa) |
![{\displaystyle {\sqrt {2\pi }}{\frac {\delta (\omega -a)-\delta (\omega +a)}{2i}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8b9fe1978d4bfc1b6d5aa54400082c85fa17cce) |
|
15 |
![{\displaystyle \exp(-at^{2})\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd2c9536d6169110d88be8bfd23dd0484e6c72ac) |
![{\displaystyle {\frac {1}{\sqrt {2a}}}\exp \left({\frac {-\omega ^{2}}{4a}}\right)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c8cd6cf2918eab6ec52a2d83442512178b7dbcf) |
इससे स्पष्ट है कि गासियन फलन का फूर्ये रूपान्तर भी गासियन फलन ही होगा।
|
16 |
![{\displaystyle W{\sqrt {\frac {2}{\pi }}}\mathrm {sinc} (Wt)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a9d231eee148ecea58cd6115f49511a5c179d1d) |
![{\displaystyle \mathrm {rect} \left({\frac {\omega }{2W}}\right)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ad143384522d212f6f86751255171266ee4edc1) |
रेक्टैगुलर फलन अर्थात्, आदर्श लो-पास-फिल्टर
|
17 |
![{\displaystyle {\frac {1}{t}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15405d4483c111940a42d48f592db680e3b84512) |
![{\displaystyle -i{\sqrt {\frac {\pi }{2}}}\operatorname {sgn}(\omega )\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab9b23eb0f44cbffe5024b75f36c49cf0d69fec8) |
यहाँ — sgn फलन (चिह्न फलन) है।
|
18 |
![{\displaystyle {\frac {1}{t^{n}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b63d8f689b821c3489977ea76ef61558d82ddd22) |
![{\displaystyle -i{\sqrt {\frac {\pi }{2}}}{\frac {(-i\omega )^{n-1}}{(n-1)!}}\operatorname {sgn}(\omega )\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93449709999b65aaab1505cbfe04c7dca288dd3c) |
17 का सामान्यीकृत रूप
|
19 |
![{\displaystyle \operatorname {sgn}(t)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d568cf2831d83676198d17cac12520363c313e6e) |
![{\displaystyle {\sqrt {\frac {2}{\pi }}}(i\omega )^{-1}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe9a9df726a4c05d1a62c5fa62427b6391673937) |
17 का द्वैत
|
20 |
![{\displaystyle {\sqrt {2\pi }}\theta (t)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/754769c5d2094ed0977afd1345b0ad7b41fe60a9) |
![{\displaystyle {\frac {1}{i\omega }}+\pi \delta (\omega )\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd91ac8bceadb9a98778eadc50e042cb82119c96) |
यहाँ — हेविसाइड फलन है।
|