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

c d ijz2 zl z2 a since subtraction cannot be

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Unformatted text preview: ctly in polar form, we convert Zl and t hen (B.15a) Z2 to Cartesian form: %+ jsin %) = V 2+jV2 ,,/3 = 8 (cos t + j sin t ) = 4 + j 4v'3 Zl = 2e j ,,/4 = 2 (cos a nd (B.15b) Z2 = 8e j Therefore, Moreover, (B.15c) 2Z1 - Z2 a nd = (2V2 - 4) + j (2V2 - 4J3) (B.15d) = - 1.17 - j4.1 T his shows t hat t he o perations of multiplication, division, powers, a nd r oots c an b e carried o ut w ith r emarkable ease when t he n umbers are in polar form. • = 2(V2 + j V2) - (4 + j 4J3) ( b) E xample B .3 Determine Z lZ2 and Z l/Z2 for the numbers ( c) Zl = 3 + j 4 = 5ej53.1o Z2 = 2 + j 3 = V13ej56.3° ( d) We shall solve this problem in both polar and Cartesian forms. o M ultiplication: C artesian F orm ZlZ2 = (3 + j 4)(2 + j 3) = (6 - 12) + j (8 + 9) = - 6 + j 17 M ultiplication: P olar F orm ZlZ2 = (5ej53.10) (V13ej56.30) = 5V13e j109 .4° D ivision: C artesian F orm Zl Z2 3 + j4 2 + j3 In order to eliminate the complex number in the denominator, we multiply both the numerator a nd the denominator of the right-hand side by 2 - j3, the denominator's conjugate. This yields C omputer E xample C B.3 Determine ZlZ2 and Zl/Z2 if Zl = 3 + j 4 and Multiplication and division: Cartesian Form Z2 = 2 + j3 z 1=3+j*4; z 2=2+j*3; z 1z2=z1 * z2 z lz2--6.000+17.0000i z Lover - 2:2=z1 / z2 z l_over-z2=1.3486-0.0769i Therefore (3 + j 4)(2 + j 3) = - 6 + j 17 and (3 + j 4)/(2 + j 3) = 1.3486 - 0.0769 o 14 • B ackground E xample B .5 Consider F (w), a complex function of a real variable w: B.2 is t he p eriod. F or t he s inusoid in Eq. (B. I S), C is t he a mplitude, F a is t he f requency ( in H ertz), a nd () is t he p hase. L et u s consider two special cases of t his s inusoid when () = 0 a nd () = -11" / 2 a s follows: F(w) = 2 +jw 3 +j4w (B.16a) ( a) To o btain the real and imaginary parts of F(w), we must eliminate imaginary terms in the denominator of F(w). This is readily done by multiplying both the numerator and denominator o f F(w) by 3 - j4w, the conjugate of the denominator 3 + j4w so t hat (2 + j w)(3 - j4w) (6 + 4w 2) - j5w 6 + 4w 2 . 5w ( w)=(3+j4w)(3-j4w)= 9 + 16w 2 = 9+16w 2 - J 9+w 2 (B.16b) This is the Cartesian form of F(w). Clearly the real and imaginary parts Fr(w) and Fi(w) are given by - 5w 6 +4w 2 F;(w) = 9 + 16w2 Fr(w) = 9 + 16w2 ' ( b) «() = 0) (a) f (t) = C cos 211" Fat ( b) ( a) Express F(w) in Cartesian form, and find its real and imaginary parts. ( b) Express F(w) in polar form, and find its magnitude IF(w)1 and angle LF(w). F 15 Sinusoids f (t) = C cos (211"Fat - I) = C s in 211"Fat «() = -11" / 2) T he angle or phase c an b e e xpressed in u nits of degrees or radians. Although t he r adian is t he p roper u nit, i n t his b ook we s hall often use t he degree u nit b ecause s tudents generally have a b etter feel for t he r elative magnitudes of angles when expressed in degrees r ather t han i n radians. For example, we relate b etter t o t he angle 24° t han t o 0.419 r adians. Remember, however, when in d oubt...
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