fourier series and transforms ppt

Ngày tải lên: 12/06/2014, 16:17

... and odd. Let and refer to real and imaginary, and to even and odd, and lower-case and upper-case letters to time and frequency functions. A summary of the symmetries of Fourier transform is shown ... direction (counterclockwise), and to rota- tion in the other direction. To see why, suppose and . Then . The Fourier transform is (1.34) The integrand oscillates and averages out to zero, except ... real and imaginary parts are plotted on the - and -axes, and time runs down the axis of the screw. The complex conjugate of this signal reverses the -axis and gives the screw an opposite handedness....

Ngày tải lên: 17/12/2013, 06:15

... tationhasbothamagnitudeandaphasespectrum.Forexample,themagnitudeofthea n isplotted inFig.1.4forthesawtoothwaveformofFig.1.3.Thefactthatthea n constituteadiscretesetis consistentwiththefactthataperiodicsignalhasa“linespectrum,”i.e.,thespectrumcontainsonly integermultiplesofthefundamentalfrequencyω 0 .Therefore,theequationpairgivenby(1.5a) and( 1.5b)canbeinterpretedasatransformpairthatissimilartotheCTFouriertransformfor periodicsignals.ThisleadstotheobservationthattheclassicalFourierseriescanbeinterpreted asaspecialtransformthatprovidesaone-to-oneinvertiblemappingbetweenthediscrete-spectral domainandtheCTdomain.Thenextsectionshowshowtheperiodicityconstraintcanberemoved toproducethemoregeneralclassicalCTFouriertransform,whichappliesequallywelltoperiodic andaperiodictimedomainwaveforms. FIGURE1.3:PeriodicCTsignalusedinFourierseriesexample. FIGURE1.4:MagnitudeoftheFouriercoefficientsforexampleofFigure1.3. 1.2.2 TheTrigonometricFourierSeries AlthoughFourierseriesexpansionsexistforcomplexperiodicsignals,andFouriertheorycanbe generalizedtothecaseofcomplexsignals,thetheoryandresultsaremoreeasilyexpressedforreal- valuedsignals.Thefollowingdiscussionassumesthatthesignals(t)isreal-valuedforthesakeof simplifyingthediscussion.However,allresultsarevalidforcomplexsignals,althoughthedetailsof thetheorywillbecomesomewhatmorecomplicated. Forreal-valuedsignalss(t),itispossibletomanipulatethecomplexexponentialformoftheFourier seriesintoatrigonometricformthatcontainssin(ω 0 t)andcos(ω 0 t)termswithcorrespondingreal- c  1999byCRCPressLLC ... TheDiscreteTimeFourierTransform ThediscretetimeFouriertransform(DTFT)canbeobtainedbyusingtheDTsamplingmodeland consideringtherelationshipobtainedin(1.12)tobethedefinitionoftheDTFT.LettingT=1so thatthesamplingperiodisremovedfromtheequationsandthefrequencyvariableisreplacedwith c  1999byCRCPressLLC Jenkins, W.K. Fourier Series, Fourier Transforms, and the DFT” Digital Signal Processing Handbook Ed. Vijay K. Madisetti and Douglas B. Williams Boca Raton: CRC Press ... ConvergenceoftheFourierSeries TheFourierseriesrepresentationofaperiodicsignalisanapproximationthatexhibitsmeansquared convergencetothetruesignal.Ifs(t)isaperiodicsignalofperiodT,ands  (t)denotestheFourier seriesapproximationofs(t),thens(t)ands  (t)areequalinthemeansquaresenseif MSE=  T/2 −T/2 |s(t)−s(t)  | 2 dt=0...

Ngày tải lên: 27/01/2014, 03:20

Ngày tải lên: 24/12/2013, 12:16

... Participation in Central and Eastern Europe 148 5.1 Constitutional and Legal Provisions for Decentralization and Participation in Bangladesh, India, Indonesia, the Philippines, and Thailand 160 6.1 Budget ... arguments must understand the perspective of government and how it influences whether they are supportive of participation and willing to listen to public feedback. Understanding the administrative ... between councillors and other officials, and a lack of resources. Sixth,Charas Suwanmala reviews Thailand’s experience with participatory budgeting. Local government in Thailand comprises five units:...

Ngày tải lên: 06/03/2014, 10:20

... Trigonometric Series (v. Arts. 7 and 8). DEVELOPMENT IN TRIGONOMETRIC SERIES. 38 25. We see, then, that any function of x which is single-valued, finite, and continuous between x = 0 and x = π, ... = 2 π π  0 f(α) sin mα.dα; (2) and the series and the function will be identical for all values of x between x = 0 and x = π, not including the values x = 0 and x = π unless the given function is ... that when m = 0, J −m (x) and J m (x) become identical and (5) reduces to z = (A + B)J 0 (x) and contains but a single arbitrary constant and is not the general solution of Fourier s Equation (8)...

Ngày tải lên: 15/03/2014, 00:20

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Ngày tải lên: 09/08/2014, 13:21

... applications issues and trends. ✔ It serves as a resource to customers and prospects of wireless local area products and wireless personal area products and to industry press and analysts. February ... the IEEE Standards Review Committee. ✔ It is published as a standard ● The standard may then be forwarded to the International Organization for Standardization (ISO) for approval and publication ... Standard. ✔ The 802.11 is the base standard (ratified July 1997, updated in 1999 and reaffirmed in 2003). ✔ All other 802.11 standards simply amend the base standard and are identified by lower case...

Ngày tải lên: 13/09/2012, 11:16

... and other characteristics can be measured, improved, and, to some extent, guaranteed in advance. ✔ QoS provides subscribers better reception for full-motion video, high- fidelity audio, and ... modifies the rules associated with DCF and PCF to create, respectively: ✔ an Enhanced Distributed Channel Access Function (EDCAF) for DCF or DCF plus QoS ✔ And a Hybrid Coordination Function ... the Virtual Carrier- Sensing indicates the medium is idle and the station can transmit. ● For the station to transmit both the physical and virtual carrier sense must report an idle condition...

Ngày tải lên: 13/09/2012, 11:16

Ngày tải lên: 08/07/2013, 01:25

... listen to the tape to get information in order to know the journey to Ba’s village 3 Nought and crosses: Where What How many How far Who is does village bus Answer the qusetions: a, Where ... ? B : It takes an hour . A : What do people do for a living in your village ? B : They plant rice and raise cattle . A : Does your village have a river ? B : There aren’t any rivers, but there ia...

Ngày tải lên: 08/07/2013, 01:25

... for your attendance ! Good bye and see you later! c. What time do they start and finish work? b. When do they plant and water trees? On Feb 2 They start at 7a.m and finish at 10 a.m -Possible ... streets -Help elderly people -Take part in sports -Help handicapped/ street children -Collect and empty garbage -Plant trees and flowers -Water trees and flowers 2. Checking Vocabulary: A rest home An ... dialogue Lan: we collect and empty garbage? Mai: At Dong Xuan Market Lan: we collect Mai: On Janaury 9th (1) (3) Mai: 8 a.m and 5 p.m (2) (6) (7) Lan: we start and (4) (5) Where do When do and empty garbage ...

Ngày tải lên: 22/07/2013, 01:28

... 3 Chip: Can I use your _______? I forgot mine. Sandy: Sure. Go ahead. Chip: ________ camera Thanks Điền từ thích hợp vào chỗ trống __________, ________ and wombats are marsupials Kangaroos koalas Number...

Ngày tải lên: 08/09/2013, 22:10