In 1743 a famous Swiss mathematician named Leonard Euler (1707-1783) derived the formula ejz = cos(z) + j sin(z): 150 years later the physicist Arthur E: Kennelly and the scientist Charles P: Steinmetz used this formula to introduce the complex notation of harmonic wave forms in electrical engineering, that is ej! t = cos(! t) + j sin(! t): Later on, in the beginning of the 20th century, the German scientist David Hilbert (1862-1943) ? nally showed that the function sin(! t) is the Hilbert transform of cos(! t).
This gave us the §? 2 phase-shift operator which is a basic property of the Hilbert transform. b A real function f(t) and its Hilbert transform f(t) are related to each other in such a way that they together create a so called strong analytic signal. The strong analytic signal can be written with an amplitude and a phase where the derivative of the phase can be identi? ed as the instantaneous frequency. The Fourier transform of the strong analytic signal gives us a one-sided spectrum in the frequency domain. It is not hard to see that a function and its Hilbert transform also are orthogonal.
This orthogonality is not always realized in applications because of truncations in numerical calculations. However, a function and its Hilbert transform has the same energy and therefore the energy can be used to measure the calculation accuracy of the approximated Hilbert transform. The Hilbert transform de? ned in the time domain is a convolution between the Hilbert transformer 1=(? t) and a function f(t) [7]. This is motivated later on. b De? nition 1. 1. The Hilbert transform f(t) of a function f (t) is de? ned for all t by 1 Z 1 f(? ) b f(t) = P d? ; ? ?1 t ? ? when the integral exists.
The Term Paper on Digital Tv Analog Signal Television
Digital Television Television is a medium for recording and transferring images and sound from one point to another. The Image portion of the system is a light-To-light system, gathering variations in light at the source via pickup device such as a camera and recreating those variations in light as a visual image via a display device such as cathode ray tube (CRT). Television was initially ...
When a function f(t) is real, we only have to look on the positive frequency axis because it contains the complete information about the waveform in the time domain. Therefore, we do not need the negative frequency axis and the Hilbert transform can be used to remove it. This is explained below. Let us de? ne the Fourier transform F (! ) of a signal f(t) by F (! ) = Z 1 ? 1 f(t)e? i! t dt: R (2. 5) 1 This de? nition makes sense if f 2 L1( f”w”,”F(w)”g]; Plot[ff,Hfg],fn,-2*M/Pi,2*M/Pig,AxesLabel -> f”t”,”f(t),Hf(t)”g]; 32 References [1] Aniansson J. et al, Fouriermetoder, KTH, Stockholm, 1989. 2] Goldberg R. R. , Fourier transforms, Cambrige university press, Cambridge. [3] Hahn Stefan L. , Hilbert transforms in signal processing, Artech House, Inc. , Boston, 1996. [4] Lennart HellstrAm, LinjAr analys, HAgskolan i VAxjA, 1995. o a o a o [5] Henrici Peter, Applied and computational complex analysis, Volume 1, John Wiley & Sons, Inc. , New York, 1988. [6] Proakis John G. , Salehi Masoud, Communication systems engineering, Prentice-Hall, Inc. , New Jersey, 1994. [7] Sa® E. B. , Snider A. D. , Complex analysis for mathematics, sience and enginering, Prentice-Hall, Inc. , New York, 1976. 33
