Si l d S tSignals and SystemsEE 112Lecture 13: Fourier TransformLecture 13: Fourier TransformKhosrow GhadiriElectrical Engineering Department Electrical Engineering Department San Jose State University© Khosrow GhadiriJean Baptiste Charles FourierJean Baptiste Charles Fourier (3/21/17685/16/1830)© Khosrow GhadiriJean Baptiste Charles Fourier (3/21/1768-5/16/1830)2Signal and System EE Dept. SJSUOutlineContinuous-time Fourier transform definitionContinuoustime Fourier transform definition Different Fourier transform forms Localization properties FT examples IFT example© Khosrow Ghadiri 3Signal and System EE Dept. SJSUFourier transform definitionThe continuous Fourier transform is one of the specific forms of The continuous Fourier transform is one of the specific forms of Fourier analysis which maps a time-domain function into its frequency-domain representation of the original function. In this specific case, both domains are continuous and p,unbounded. There are several conventions for defining the Fourier transform of complex-valued Lebesgue integrable function .ft For example, The Fourier transform decomposes a function into a continuous spectrum of its frequency components, and the inverse transform synthesizes a function from its spectrum of frequency componentsfrequency components. An analogy is the relationship between set of notes in musical notation (The frequency components) and the sound of the musical chord represented by these notes ( the function/ © Khosrow GhadiriSignal itself).4Signal and System EE Dept. SJSUFourier transform in communication and signal processingThe Fourier transform is a linear operator that maps a function The Fourier transform is a linear operator that maps a function into another function. For example, The Fourier transform decomposes a function into a continuous spectrum of its frequency components, and the pqyp,inverse transform synthesizes a function from its spectrum of frequency components. An analogy is the relationship between set of notes in musical t ti (Th f t ) d th d f th notation (The frequency components) and the sound of the musical chord represented by these notes ( the function/ Signal itself).Physically the Fourier transform of signal can be thought ftPhysically, the Fourier transform of signal can be thought of as a representation of a signal in the “frequency domain”. Physically, the Fourier transform of signal can be thought of as a representation of a signal in the “frequency domain”.ftft© Khosrow Ghadiri © 5Signal and System EE Dept. SJSUFourier transform formsThe popular forms of the Fourier transform:The popular forms of the Fourier transform: The Fourier transform in mathematics. Angular frequency ω (rad./sec.) Unitary angular frequency 1231112222jtFftedtFF Non-unitary angular frequency 1222212jtft F e d 21322212jtjtFftedtFFft F e d The Fourier transform in communication and signal processing. Ordinary frequency f (Hertz) 2312222 2jftjftFf fte dt FfFfffdf© Khosrow Ghadiri 6Signal and System EE Dept. SJSU23jftftFfedfDifferent Fourier presentation111Angular Frequency Unitary 1231112222jtFftedtFF 112jtftFedqy(rad/s)Non-unitary2 21322defjtFftedtFFNonunitaryOrdinary 212jtftFedOrdinaryFrequency(hertz)unitaryf 231222 2defjftFf fte dt F f F f 23jftftFfedf© Khosrow Ghadiri 7Signal and System EE Dept. SJSUThe generalized Fourier transform definitionThe general definition of forward 1-D Fourier transform is given by: The general definition of forward 1D Fourier transform is given by: 12jb tabFfte dt And the inverse 1-D Fourier transform is given by:12jb tabft F e d Where a and b are arbitrary constant, and the transform definitions are symmetric; they can be reversed by changing the signs of a and b. In the ordinary frequency convention a=0 and b= , in this case the 22yq y,variable is changed to . If and carry units, their product must be dimensionless. (t in second f in Hertz=1/second) In the unitary, angular frequency convention a=0 and b=1In the nonunitaryangular frequencyconvention a=1 and b=1fft© Khosrow GhadiriIn the non-unitary,angular frequencyconvention a=1 and b=18Signal and System EE Dept. SJSUThe Fourier uncertainty prrincipleThe uncertainty principle shows that one can not jointly localize a signal The uncertainty principle shows that one can not jointly localize a signal in time and frequency arbitrarily well; either one has poor frequency localization or poor time localization. In signal processing we are faced with the problem of taking a finite block of signal and using it to represent the whole signal This is known block of signal and using it to represent the whole signal. This is known as the window problem. The sharp corners of the block, in particular, produce lobes of spurious frequencies that severely contaminate the signal. For this reason we use what are called window functions to smooth out the edges Thus in estimating the spectrum of a signal two smooth out the edges. Thus, in estimating the spectrum of a signal, two workers might get different results depending on their choice of window function. This is also a manifestation of the Uncertainty Principle and we cannot accurately estimate the spectrum of a signal from only part of its time record though the accuracy improves as the length of our record time record, though the accuracy improves as the length of our record increases. The uncertainty principle for
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