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

# T he p eriodicity property dnno d n means beyond n n

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Unformatted text preview: suppresses t he time-varying component and yields a dc component with some residual ripple. Find t he filter o utput y(t). F ind also t he dc o utput a nd t he rms value of the ripple voltage. First, we shall find t he Fourier series for the rectified signal J (t), whose period is To = 1l'. Consequently, Wo = 2, a nd where Dn ='!' 1l' 0 L I n Sec. 2.4-3, w e s howed t hat t he r esponse o f a n L TIC s ystem w ith t ransfer f unction H (s) t o a n e verlasting e xponential i nput e st is also a n e verlasting e xponential H (s ) e st . T herefore, t he s ystem r esponse t o e verlasting e xponential e jwt is a n e verlasting e xponential H ( jw )e jwt . T his i nput-output p air c an b e d isplayed as+ To compute trigonometric Fourier series coefficients, we recall t he p rogram c31.m along with c ommands t o convert D n into C n a nd en. c 31;cJg C O=Dnmag(l); C n=2*Dnmag(2:M); A mplitudes= [CO;Cn] A ngles=Dnangle(l:M); A ngles=Angles*(180/pi); d isp(' A mplitudes A ngles') [ Amplitudes A ngles] % T o P lot t he F ourier c oefficients k =O:length(Amplitudes)-l; k =k'; s Ubplot(211 ) ,stem(k,Amplitudes) s ubplot(212), s tem(k,Angles) 219 3.7 L TIC S ystem R esponse t o P eriodic I nputs &quot; 1 0 s inte- j2n 'dt= (22) 1l' 1 - 4n (3.88) +This result applies only to the asymptotically stable systems. This is because when s = j w, the integral on the right-hand side of Eq. (2.48) does not converge for unstable systems. Moreover, for marginally stable systems also, t hat integral does not converge in t he ordinary sense, and H (jw) cannot be obtamed by replacing s in H (s) with jw. 3 Signal Representation by O rthogonal Sets 220 1 50 + + Full-Wave Rectifier s in f y (t) f (t) (a) 3.7 221 L TIC S ystem Response t o P eriodic I nputs The output Fourier series coefficient corresponding to n = 0 is the dc component of the output, given by 2 /&quot;. T he remaining terms in the Fourier series constitute the unwanted component called the ripple. We can determine the rms value of the ripple voltage by finding the power of the ripple component using Eq. (3.83). The power of the ripple is the power of all the components except the dc (n = 0). Note t hat Dn, the exponential Fourier coefficient for the output y (t) is Dn= 2 1-_-47&quot;n&quot;&quot;2&quot;)-:-:j6:;-n-+:-;-;'1 ( ) - &quot;7 :( Therefore, from Eq. (3.83b), we have ( a) y (t) 00 1 . - 2 00 2_ 2 Proppl. - ~ IDnl - 2 ~ , ,(I _ 4 n2)(j6n ,--- o - 31t Prippl. = 0.0025, 31t f (t) 2 = L..,.. ,,(1 _ 4n2) e j2nt - 00 Next we find t he transfer function of the R C filter in Fig. 3.20a. This filter is identical to the circuit i n Example 1.11 (Fig. 1.32) for which the differential equation relating the output (capacitor voltage) to the input f (t) was found to be [Eq. (1.60)] RC (3D + l )y(t) = f (t) The transfer function H (s) for this system is found from Eq. (2.50) as 1 H (s)=3s+1 and 1 H (jw)=-3jw + 1 (3.89) From Eq. (3.87), t he filter output y (t) can be expressed as (with wo = 2) L 00 y (t) = DnH(jnwo)ejnwot = L D nH(j2n)ej2nt Substituting D n a nd H (j2n) f...
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