Unformatted text preview: cos T - 1) = -~ sin ( T)
2 1 (jw + 2 }(jw + I )
Expanding the right-hand side in partial fractions (Sec. B.5) W~T sin (W;) = ~ [sin~7}r = ~ sinc (W;)
J W+ H(s}18=jw = - .- 2 yew} = H(w}F(w} (4.50b) Taking the Fourier transform of Eq. (4.49) and using the results in Eqs. (4.50), we obtain
_ w 2 F(w} = = Therefore Also, from the time-shifting property (4.37)
o(t _ to} H(w} a nd 2 Yew} (4.51) = 1
j w + 1 - jw + 2 (4.54) and
The spectrum F(w} is depicted in Fig. 4.25d. This procedure of finding the ~ourier transform can be applied to any function f (t} made up of straight-line segments. With f (t} - ; 0
as It I - ; 0 0. T he second derivative of such a signal yields a sequence of Imp~lses whose
Fourier transform can be found by inspection. This example suggest~ a n~mencal.method
of finding the Fourier transform of an arbitrary signal f (t} by approxlmatmg the signal by
l::. • E xercise E 4.10
F ind t he F ourier t ransform of rect ( -!: ), u sing t he t ime-differentiation property. yet} = ( e- t _ e - 2t )u(t} • l::. E xercise E 4.11
F or t he s ystem in E xample 4.15, show t hat t he z ero-input response t o t he i nput e tu( - t) is
y (t) = ~[etu(-t) + e - 2t u(t)].
Hint: Use p air 2 ( Table 4.1) t o find t he F ourier t ransform o f e tu( - t). \ l
t Stability implies t hat t he r egion o f convergence o f H (s) i ncludes t he j w axis. \l 4 Continuous-Time Signal Analysis: The Fourier Transform 268 4 .4-1 4.4 Signal Transmission Through LTIC Systems ................. S ignal D istortion during Transmission IH(oo)1 For a system w ith transfer function H(w), if F(w) a nd Y (w) a re t he s pectra of
the i nput a nd the o utput signals, respectively, then Y (w) = F(w)H(w) o ······..................... (4.55) 269 /H(OO)
.............. T he transmission of the input signal f (t) t hrough the system changes i t i nto t he
o utput signal y (t). E quation (4.55) shows the n ature of this change or modification.
Here F(w) a nd Y (w) are t he s pectra of t he i nput and the o utput, respectively.
Therefore, H (w) i s the spectral response of the system. The o utput s pectrum is
obtained by the i nput s pectrum multiplied by the spectral response of t he system.
Equation (4.55), which clearly brings out t he s pectral shaping (or modification) of
t he signal by t he s ystem, can be expressed in polar form as F ig. 4 .26 L TIC s ystem F requency response for distortionless transmission. T he Fourier transform of this equation yields Y (w) = k F(w)e- jwt "
B ut Y (w) = F (w)H(w) lY(w)lejLY(w) = IF(w)IIH(w)lej[LF(W)+LH(jw)] Therefore H(w) = k e- jwt " Therefore lY(w)1 = IF(w)IIH(w)1
LY(w) = LF(w) + LH(w) (4.56a)
(4.56b) During transmission, the input signal amplitude spectrum IF(w)1 is changed to
IF(w)IIH(w)l. Similarly, the input signal phase spectrum LF(w) is changed t o
LF(w) + LH(w). An input signal spectral component of frequency w is modified in
amplitude by a factor IH(w)1 a nd is shifted in phase by an angle LH(w). Clearly,
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