Signal Processing and Linear Systems-B.P.Lathi copy

444b ht hw lb understand finer points of eq 444b

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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;) 2 1 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 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 straight-line segments. 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, IH(w...
View Full Document

This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

Ask a homework question - tutors are online