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must be constant, a nd t he p hase response LH (w) should be a linear function of w
over a band of interest. Ideal filters, which allow distortionless transmission of a
certain band of frequencies a nd suppress all t he remaining frequencies, are physically
unrealizable (noncausal). In fact, i t is impossible t o build a physical system with
zero gain [H(w) = OJ over a finite b and of frequencies. Such systems (which include
ideal filters) can b e realized only with infinite time delay in t he response.
T he energy o f a signal f (t) is equal t o 1/271' times t he a rea u nder IF(w)2j
( Parseval's theorem). T he energy contributed by spectral components within a
b and t :.F (in Hz) is given by IF(w)1 2t:.F. Therefore, IF(w)12 is t he energy spectral
density per u nit b andwidth (in Hz). T he energy spectral density IF(w)12 of a signal
f (t) is t he F ourier transform of the autocorrelation function 1/1 f (t) of t he signal f (t).
T hus, a signal a u t ocorrelation function has a direct link t o its spectral information.
T he process of modulation shifts t he signal spectrum to different frequencies.
Modulation is u sed for many reasons: t o t ransmit several messages simultaneously
over t he s ame channel t o utilize channel's high bandwidth, t o effectively radiate
power over a r adio link, t o shift signal s pectrum a t higher frequencies t o overcome
t he difficulties associated with signal processing a t lower frequencies, t o effect the
exchange of transmission bandwidth a nd transmission power required t o t ransmit
d ata a t a c ertain r ate. Broadly speaking there are two types of modulation; amplitude a nd a ngle modulation. Each of these two classes has several subclasses.
Amplitude m odulation b andwidth is generally fixed. T he b andwidth in angle modulation, however, is controllable. T he higher t he b andwidth, t he more immune is
t he scheme t o noise.
In practice, we often need t o t runcate d ata. Truncating d ata is like viewing it
t hrough a window, which permits a view of only certain portions of t he d ata a nd
hides (suppresses) t he remainder. A brupt t runcation of d ata a mounts t o a rectangular window, which assigns a u nit weight t o d ata seen from t he window a nd assigns
zero weight t o t he remaining d ata. T apered windows, on t he o ther hand, reduce
t he weight gradually from 1 t o O. D ata t runcation can cause some unsuspected
problems. For example, in computation of t he Fourier transform, windowing ( data
t runcation) causes spectral spreading (spectral smearing) t hat is characteristic of
t he window function used. A rectangular window results in t he least spreading, b ut
it does so a t t he cost of a high a nd oscillatory spectral leakage outside t he signal
b and which decays slowly as l /w. C ompared to a rectangular window, t apered windows in general have larger spectral spreading (smearing), b ut t he s pectral leakage
is smaller a nd d ecays faster with frequency. I f we t ry to reduce spectral leakage
by using a s moother window, t he s pectral spreading increases. Fortunately, the Problems 30...
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