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Unformatted text preview: compared t o t he triangular case (411"/T r ad/s).
2. Although H (w) is bandlimited, t he windowed filters are not. B ut t he s topband
behavior o f t he triangular case is superior t o t hat of the rectangular case.
For t he r ectangular window, the leakage in the stopband decreases slowly (as
l /w) c ompared t o t hat o f t he t riangular window (as 1/w 2 ). Moreover, the
rectangular case has a higher peak sidelobe amplitude compared t o t hat of the
triangular window. 4 .10 307 * Summary I n C hapter 3 we represented periodic signals as a sum of (everlasting) sinusoids
or exponentials (Fourier series). In this chapter we extended this result t o aperiodic
signals, which a re represented by the Fourier integral (instead of t he Fourier series).
An aperiodic signal j et) may be regarded as a periodic signal with period To - >
0 0, so t hat t he Fourier integral is basically a Fourier series with a fundamental
frequency approaching zero. Therefore, for aperiodic signals, t he Fourier spectra
are continuous. This continuity means t hat a signal is r epresented as a s um of
sinusoids (or exponentials) of all frequencies over a continuous frequency interval.
T he Fourier transform F(w), therefore, is the spectral density (per unit bandwidth
An ever-present aspect of t he Fourier transform is t he duality between time
a nd frequency, which also implies duality between the signal I (t) a nd its transform
F(w). T his duality arises because of near-symmetrical equations for direct and
inverse Fourier transforms. T he duality principle has far-reaching consequences
and yields m any valuable insights into signal analysis.
T he scaling property of the Fourier transform leads to t he conclusion t hat t he
signal b andwidth is inversely proportional t o signal duration (signal width). Time t t In a ddition t o t runcation, we also n eed t o delay t he t runcated function b y
i n o rder t o r ender
i t c ausal. However, t he t ime d elay only a dds a l inear p hase t o t he s pectrum w ithout c hanging t he
a mplitude s pectrum. For t his reason, we shall ignore t he d elay in order t o simplify o ur discussion. II o e: e: 308 4 Continuous-Time Signal Analysis: T he Fourier Transform shifting of a signal does not change its amplitude spectrum, b ut adds a linear phase
spectrum. Multiplication of a signal by a n e xponential e jwot results in shifting t he
s pectrum t o t he r ight by woo In practice, spectral shifting is achieved by multiplying a signal w ith a sinusoid such as cos wot ( rather t han t he e xponential e jwot ).
T his process is k nown as amplitude modulation. Multiplication of two signals results in convolution of their spectra, whereas convolution of two signals results in
mUltiplication of t heir s pectra.
For an LTIC s ystem with t he t ransfer function H (w), t he i nput a nd o utput
s pectra F (w) a nd Y (w) are related by t he e quation Y (w) = F (w)H(w). T his is
valid only for asymptotically stable systems. For distortionless transmission of
a signal through a n LTIC system, t he a mplitude response IH(w)1 of t he s ys...
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