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Unformatted text preview: ential s pectra r ange from - 00 t o 0 0. By definition, the frequency of a signal is
always a positive quantity. Presence of a spectral component of a negative frequency
- nwo merely indicates t hat t he Fourier series contains terms of t he form e - jnwot .
T he s pectra o f t he t rigonometric a nd e xponential Fourier series are closely related,
a nd one can be found by t he inspection of t he o ther.
T he F ourier series coefficients C n o r D n may be computed numerically using
t he discrete Fourier transform (DFT), which can be implemented by a n efficient
F FT (fast F ourier t ransform) algorithm. This method uses No uniform samples of
f (t) over one p eriod s tarting a t t = O.
I n Sec. 3.7, w e discuss a method of finding t he response of a n LTIC s ystem t o a
periodic input signal. T he periodic input is expressed as an exponential Fourier series, which consists of everlasting exponentials o f t he form e jnwot. We also know t hat
response of a n L TIC system t o an everlasting exponential e jnwot is H (jnwo)ejnwot.
T he s ystem response is t he s um of t he s ystem's responses t o all t he e xponential
components in t he Fourier series for t he i nput. T he response is, therefore, also a n
e xponential Fourier series. Thus, the response is also a periodic signal of t he same
period as t hat o f t he i nput. P roblems 227 1
f<t) o X<I) <a) ~ (b) 1 o F ig. P 3.1-2 1 t_ 3 .1-2 ( a) F or t he s ignals f (t) a nd x (t) d epicted in Fig. P3.1-2, find t he c omponent of
t he form x ( t) c ontained i n f ( t). I n o ther w ords find t he o ptimum value o f c in t he
a pproximation f (t) ~ c x(t) so t hat t he e rror s ignal energy is m inimum.
( b) F ind t he e rror s ignal e(t) a nd i ts energy E .. Show t hat t he e rror signal is
orthogonal t o x (t), a nd t hat E f = c2Ex + Ee. C an y ou explain t his r esult in t erms
o f vectors. 3 .1-3 F or t he s ignals f (t) a nd x (t) s hown in Fig. P3.1-2, find t he c omponent o fthe form f (t)
c ontained in x (t). I n o ther words, find t he o ptimum value o f c in t he a pproximation
x (t) ~ c f(t) s o t hat t he e rror signal energy is m inimum. W hat is t he e rror signal
energy? 3 .1-4 R epeat P rob. 3.1-2 if x (t) is a sinusoid pulse shown in Fig. P3.1-4. x U) References
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Holt, R inehart a nd W inston, New York, 1989.
2. Bell, E. T ., M en o f M athematics, Simon a nd Schuster, New York, 1937.
3. D urant, W ill, a nd Ariel, T he Age o f Napoleon, History of Civilization, P art XI,
Simon a nd Schuster, New York, 1975.
4. Calinger, R ., 4 th ed., Classics o f M athematics, Moore P ublishing Co., Oak
Park, n., 1982.
5. Lanczos, C ., D iscourse o n F ourier S eries, Oliver Boyd Ltd., London, 1966.
6 . Walker, P .L., T he Theory o f F ourier S eries a nd Integrals, Wiley-Interscience,
New York, 1986.
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B. Guillemin, E . A., T heory o f L inear P hysical S ystems, Wiley, Ne...
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