Solutions Manual                    to accompany     Communication        SystemsAn Introduction to Signals and Noise in  ...
Solutions Manual to accompanyCOMMUNICATION SYSTEMS: AN INTRODUCTION TO SIGNALS AND NOISE IN ELECTRICAL COMMUNICATION,FOURT...
Chapter 22.1-1       Ae jφ                                                                Ae jφ n = m                    ...
2 A  sin (π − π n ) 2t / T0 sin (π + π n ) 2t / T0                                                                      ...
2.1-8                  ∞P = c0 + 2 ∑ cn = Af 0τ + 2 Af 0τ sinc f 0τ + 2 Af 0τ sinc2 f 0τ + 2 Af 0τ sinc3 f 0τ + L        2...
4                      4                         4b) v′(t ) =     cos ( ω0t − 90° ) +     cos ( 3ω0t − 90° ) +     cos ( 5...
2.2-2                   τ /2    2π tV ( f ) = − j 2∫          A sin cos2π ftdt                 0          τ               ...
2.2-6                                                         2πW             ( Ae − bt ) dt =               A2           ...
2.2-12                  2b                            4π a            a /πe −b t ↔                    ⇒ e−2π a t ↔        ...
2.3-5              t − 2T               t − 2T v (t ) = AΠ           + ( B − A)Π                       4T       ...
2.3-12                                 A tv (t ) = t z (t ) z (t ) =         Π                             ↔ 2 A sinc2 ...
2.4-2 y (t ) = 0                                t < 0, t > 5                           2              t       At      = ∫ ...
2.4-5 y (t ) = 0                                           t<0              t      = ∫ 2e−2λ d λ = 1 − e −2t              ...
2.4-10                     ∞Let y (t ) = ∫ v( λ ) w(t − λ )d λ where v( −t ) = v(t ), w(−t ) = w(t )                     −...
2.4-14          1 f V( f ) =    Π   W ( f ) = 4Π (2 f )          4 4Y ( f ) = V ( f )W( f ) = Π (2 f ) ↔ y (t ) = (1...
2.5-5v (t ) = Au (t ) − Au (t − 2τ )            1            1         1      1        − j 4π f τ V( f ) = A         ...
2.5-9                1/ εV( f ) =                 and V (0) = 1, so           1/ ε + j 2π f                      1/ ε     ...
Chapter 33.1-1 y (t ) = h(t ) ∗ A[δ (t + t d ) −δ (t −t d ) ] = A[ h (t + t d ) − h(t − t d ) ]                      (    ...
3.1-7 j 2π fY ( f ) + 16 Y ( f ) = j 2π fX ( f ) + 4π X ( f )                   π           Y ( f ) j 2π f + 2π 2 2 + jfH(...
3.1-11                                     2     f                1   f x (t ) = 2sinc4Wt ↔ X ( f ) =           Π    ...
3-1-15                 K                1H( f ) =                    =          1 + Kj 2π f         1                     ...
3.2-3                  n /3H ( nf 0 ) =                   arg H ( nf 0 ) = 90° − arctan( n /3)               1 + (n / 3)  ...
3.2-6               πf              − 30                                             f ≤ 15arg H ( f ) =              ...
3.2-8                                                                             α2               exp  − j (ωT − α sin...
3.2-11                                                                9 A3            3 A3y (t ) = 2 A cos ω 0t − 3 A3 cos...
3.3-3 Pout 50 ×10 −3     =          = −16 dB, mLi = 0.4 × 400 = 160 dB, gi ≤ 30 dB Pin      2m × 30 dB − 160 ≥ −16 ⇒ m ≥ 4...
3.3-7 Pout 2 ×10−6     =        = −64dB                 L = 92.4 − 14 + 20 = 98.4 Pin     5g 2 Pout     98.4 − 64   =     ...
3.4-2H ( f ) = Ke− jω td − H BP ( f ) where H BP ( f ) = Eq. (1)Thus, from Exercise 3.4-1, h( t ) = Kδ (t − t d ) − 2 BK s...
3.4-6                         Z RC                  R / jω C        LC(a) H ( f ) =                    where Z RC =       ...
3.4-9                            1             A  f x (t ) = A sinc2Wt ⇒ τ x =    , X( f ) =     Π                      ...
1                                                       1(b) td ?                                                    td = ...
3.5-4                   1           1x (t ) = cos ω 0t − cos3ω 0t + cos5ω 0t                   3           5              ...
3.5-8                           h ( t ) e− jω t dt = 2∫ h ( t ) cos ω t dt                     ∞                          ...
3.6-6          A        f  − jω tdV( f ) =          Π  4W  e          4W                                 2    ...
3.6-11x (t ) = Π (10t ) = Π                             t                       1 f                         1/10  ↔ X ...
Chapter 44.1-1vi (t ) = v1 (t ) + v2 (t) c o s α             vq (t ) = v2 (t) s i n αA(t ) = v12 ( t ) + 2 v1(t ) v2 (t) c...
( v (t) c o s ω t − v   sinω ct ) dt                   ∞(b) Ebp = ∫                                                       ...
4.1-8 f                   f0                        = (1 + δ ) ≈ 1 − δ , so                                  −1    = 1+ δ ...
4.1-11H lp ( f ) = Π   e− j(ω +ω c )td ⇒ hlp ( t ) = Be − jω ctd sinc B ( t − t d )                   f                ...
4.1-13                    f                     1                  BH lp ( f ) = e jf             X lp ( f ) = Z ( f ) =...
4.2-4               1  f sinc2 40t      ↔Λ               40  40 BT = 2W = 80 Hz4.2-5Amax = ( 2 Ac ) = 32kW ⇒ A c = 8...
4.2-8x (t ) = 2 K cos20π t + K cos12π t + K cos28π t            x max = x(0) = K (2 +1 + 1) ≤ 1 ⇒ K ≤ 1 / 4               ...
4.3-1                                                           2a           (a) vout = a1 x(t ) + a2 x 2 (t ) + a2 cos ...
4.3-4Take vin = x + cos ω 0t sovout = a1 ( x + cos ω 0t ) + a3 ( x3 + 3x 2 cos ω0t + 3x cos 3 ω0t + cos3 ω 0t )          ...
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
Bruce carlson communication systems (4th ed) solutions manual
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Bruce carlson communication systems (4th ed) solutions manual

  1. 1. Solutions Manual to accompany Communication SystemsAn Introduction to Signals and Noise in Electrical Communication Fourth Edition A. Bruce Carlson Rensselaer Polytechnic Institute Paul B. Crilly University of Tennessee Janet C. Rutledge University of Maryland at Baltimore
  2. 2. Solutions Manual to accompanyCOMMUNICATION SYSTEMS: AN INTRODUCTION TO SIGNALS AND NOISE IN ELECTRICAL COMMUNICATION,FOURTH EDITIONA. BRUCE CARLSON, PAUL B. CRILLY, AND JANET C. RUTLEDGEPublished by McGraw-Hill Higher Education, an imprint of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas,New York, NY 10020. Copyright © The McGraw-Hill Companies, Inc., 2002, 1986, 1975, 1968. All rights reserved.The contents, or parts thereof, may be reproduced in print form solely for classroom use with COMMUNICATION SYSTEMS: ANINTRODUCTION TO SIGNALS AND NOISE IN ELECTRICAL COMMUNICATION, provided such reproductions bear copyrightnotice, but may not be reproduced in any other form or for any other purpose without the prior written consent of The McGraw-HillCompanies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distancelearning.www.mhhe.com
  3. 3. Chapter 22.1-1 Ae jφ  Ae jφ n = m ∫− T0 / 2 T0 / 2cn = e j2π ( m−n )f 0t dt = Ae jφ sinc( m − n ) =  T0 0 otherwise2.1-2c0 v (t ) = 0 2 T0 / 4 2π nt T0 / 2 2π nt 2A πncn = T0 ∫ 0 A cos T0 dt + ∫ ( − A)cos T0 / 4 T0 dt = πn sin 2 n 0 1 2 3 4 5 6 7 cn 0 2A/π 0 2 A / 3π 0 2 A / 5π 0 2 A / 7πarg cn 0 ±180° 0 ±180°2.1-3c0 = v (t ) = A / 2 2 T0 /2  2 At  2π nt A Acn = T0 ∫ 0 A−  T0   cos T0 dt = πn sin π n − (π n) 2 (cos π n − 1) n 0 1 2 3 4 5 6 cn 0.5A 0.2A 0 0.02A 0 0.01A 0arg cn 0 0 0 02.1-4 2 T0 / 2 2π tc0 = T0 ∫ 0 A cos T0 =0 (cont.) 2-1
  4. 4. 2 A  sin (π − π n ) 2t / T0 sin (π + π n ) 2t / T0  T0 / 2 2 T0 / 2 2π t 2π ntcn = T0 ∫ 0 A cos T0 cos T0 dt =  T0  4(π − π n) / T0 + 4(π + π n ) / T0  0  A/2 n = ±1 [ sinc(1 − n) + sinc(1 + n )] =  A = 2  0 otherwise2.1-5c0 = v (t ) = 0 2 T0 / 2 2π nt Acn = − j T0 ∫ 0 A sin T0 dt = − j πn (1 − cos π n ) n 1 2 3 4 5 cn 2A/π 0 2 A / 3π 2 A / 5πarg cn −90° −90° −90°2.1-6c0 = v(t ) = 0 2 A  sin (π − π n ) 2t / T0 sin ( π + π n ) 2t / T0  T0 / 2 2 T0 / 2 2π t 2π ntcn = − j T0 ∫ 0 A sin T0 sin T0 dt = − j  T0  4(π − π n ) / T0 − 4(π + π n )/ T0  0  m jA / 2 n = ±1 [sinc(1 − n ) − sinc(1 + n ) ] =  A = −j 2  0 otherwise2.1-7 1  T0 / 2 ∫0 v ( t) e 0 dt + ∫T0 / 2 v(t )e 0 dt ] T0 − jnω t − jnω tcn = T0   T0 T0 / 2where ∫ T0 / 2 v(t )e − jnω0 t dt = ∫ 0 v (λ + T0 /2) e− jnω 0λ e− jnω 0T0 / 2 d λ T0 / 2 = −e jnπ ∫ v (t )e − jnω0 t dt 0since e jnπ = 1 for even n, cn = 0 for even n 2-2
  5. 5. 2.1-8 ∞P = c0 + 2 ∑ cn = Af 0τ + 2 Af 0τ sinc f 0τ + 2 Af 0τ sinc2 f 0τ + 2 Af 0τ sinc3 f 0τ + L 2 2 2 2 2 2 n =1 1where = 4 f0 τ 1 A21 + 2sinc2 1 + 2sinc2 1 + 2sinc2 3  = 0.23 A2 f > P=  τ 16  4 2 4 2 A  2 2 1 2 1 2 3 2 5 2 3 2 7 f > P= 1 + 2sinc 4 + 2sinc 2 + 2sinc 4 + 2sinc 4 + 2sinc 2 + 2sinc 4 = 0.24 A 2 τ 16   1 A 2 2 1 2 1 f > P= 1 + 2sinc 4 + 2sinc 2  = 0.21A 2 2τ 16  2.1-9  0 n even cn =  2 2  π n  n odd   2 2 1 T0 / 2  4t  2 T0 / 2  4t  1a) P = T0 ∫−T0 / 2  1 − T0  dt = T0   ∫ 0  1 −  dt =  T0  3 2 2 2  4   4   4 P′ = 2  2  + 2  2  + 2  2  = 0.332 so P′ / P = 99.6% π   9π   25π  8 8 8b) v′(t ) = 2 cos ω 0t + 2 cos3ω 0t + cos5ω 0t π 9π 25π 22.1-10  0 n even cn =  − j 2  πn n odd  1  2 2  2  2  2 2  ∫−T0 / 2 (1) dt = 1 P  π   3π   5π   T0 / 2a) P = ′ = 2   +   +    = 0.933 so P′ / P = 93.3% 2 T0   (cont.) 2-3
  6. 6. 4 4 4b) v′(t ) = cos ( ω0t − 90° ) + cos ( 3ω0t − 90° ) + cos ( 5ω 0t − 90°) π 3π 5π 4 4 4 = sin (ω 0t ) + sin ( 3ω 0t ) + sin ( 5ω0t ) π 3π 5π2.1-11 2 1 T0  t  1 1 / 2 n=0P= T0 ∫0   dt =  T0  3 cn =  1 / 2π n n ≠ 0 ∞ 4 4  2  2  1 1 1  1P =2∑   = 2    4 + 4 + 4 +L = n odd  π n  π  1 3 5  3 1 1 1 4π 2 1 1  π2Thus, + 2 + 2 +L= 3− 4= 6 12 2 3 2  2.1-12 2 2 T0 / 2  4t  1 0 n evenP= T0 ∫0  1 −  dt =  T0  3 cn =   (2/ π n) n odd 2 ∞2 2 1  1  1 2 1 1 1  1P =   + 2∑   = + 2  2 + 2 + 2 +L = 2 n =1  2π n  4 4π 1 2 3  3 1 1 1 π4 1 π 4Thus, 4 + 4 + 4 + L = = 1 3 5 2 ⋅ 24 3 962.2-1 πt τ /2V ( f ) = 2∫ cos2π ftdt A cos 0 τ τ  π (  sin τ −2π f 2 sin ) ( π 2π f ) τ2  Aτ τ+  = 2A  + = [ sinc( f τ −1/2) + sinc( f τ + 1/2) ]  2 π −2π f τ ( 2 ) (π 2π f )  2 τ+   (cont.) 2-4
  7. 7. 2.2-2 τ /2 2π tV ( f ) = − j 2∫ A sin cos2π ftdt 0 τ   ( sin 2π −2π f τ τ ) sin ( 2τπ 2π f ) τ2  = − j Aτ sinc( f τ −1) − sinc( f τ + 1) +  = − j2 A  2− 2π 2π f  [ ] 2π (  2 τ −2π f  )2 (τ )  2 + 2.2-3 τ t 2 Aτ   ωτ  V ( f ) = 2∫  A − A  cos ω tdt = 2  2sin 2   − 1 + 1 = Aτ sinc f τ 2 0  τ (ωτ )   2  2.2-4 τ t 2 AτV ( f ) = − j 2∫ A sin ω tdt = − j (sin ωτ − ωτ cos ωτ ) 0 τ (ωτ ) 2 A = −j (sinc2 f τ − cos2π f τ ) πf2.2-5 1  f v (t ) = sinc2Wt ↔ Π  2W  2W  2 ∞ ∞ 1  f  ∞ 1 1∫ sinc2Wt dt = ∫  df = ∫−∞ 4W 2 df = 2W 2 Π −∞ −∞ 2W  2W  2-5
  8. 8. 2.2-6 2πW ( Ae − bt ) dt = A2 A2 A2 ∞E=∫ E′ = 2∫ 2 2W df = arctan 0 2b 0 b + (2π f )2 πb bE′ 2 2πW 50% W = b / 2π = arctan =E π b 84% W = 2b / π2.2-7 ∞ ∞ ∞ ∫ v (t )w( t) dt =∫−∞ v( t )  ∫−∞ W ( f ) e jω t df dt −∞     ∞ ∞ ∞= ∫ W ( f )  ∫ v (t) e− j (− ω ) t dt  df = ∫ W ( f )V (− f ) df −∞  −∞    −∞ ∞ ∞ ∞ ∫ v 2 ( t ) dt =∫ V ( f )V * ( f )df = ∫ V ( f ) df 2V ( − f ) = V * ( f ) when v (t) is real, so −∞ −∞ −∞2.2-8 ∗ ∗∫−∞ w∗ (t)e− j 2π ft dt =  ∫−∞ w(t )e j 2π ft dt  =  ∫−∞ w( t)e− j 2π ( − f )t dt  = W ∗ ( f ) ∞ ∞ ∞         ∗ ∗ ∗Let z ( t ) = w ( t ) so Z ( f ) = W ( − f ) and W ( f ) = Z ( − f ) ∞ ∞Hence ∫−∞ v( t )z( t) dt = ∫ V ( f ) Z ( − f ) df −∞2.2-9 t  1  f Π   ↔ A sinc Af so sinc At ↔ Π    A A  A 2t τ  fτ  2v (t ) = sinc ↔ V ( f ) = Π   for A = τ 2  2  τ2.2-10 πt  t  BτB cos Π   ↔ [ sinc( f τ − 1/2) + sinc( f τ + 1/2) ] τ τ  2 Bτ π (− f )  − f  πf  f so [sinc(tτ − 1/2) + sinc(tτ + 1/2) ] ↔ B cos Π  = B cos Π  2 τ  τ  τ τ Let B = A and τ = 2W ⇒ z (t ) = AW [sinc(2Wt − 1/2) + sinc(2Wt + 1/2) ]2.2-11 2π t  t  BτB sin Π  ↔− j [sinc( f τ − 1) + sinc( f τ + 1) ] τ τ  2 Bτ 2π (− f )  − f  2π f  f so − j [ sinc(tτ − 1) + sinc( tτ + 1) ] ↔ B sin Π  = − B sin Π  2 τ  τ  τ τ Let B = − jA and τ = 2W ⇒ z ( t ) = AW [ sinc(2Wt − 1) + sinc(2Wt + 1) ] 2-6
  9. 9. 2.2-12 2b 4π a a /πe −b t ↔ ⇒ e−2π a t ↔ = 2 b + (2π f ) 2 2 (2π a ) + (2π f ) 2 2 a + f2 (e ) 2 ∞ 2 1 ∞ a /π a ∞ df∫ =∫ ∫ −2 π a t dt = df = 2   2π a −∞ a + f π  (a + f2) −∞ 2 2 0 2 2 2 ∞ dx 1 π  1 πThus, ∫ =   = 3 (a + x2 ) 2  a  2π a 4 a 0 2 22.3-1z (t ) = v (t − T ) + v( t + T ) where v( t ) = AΠ (t /τ ) ↔ Aτ sinc f τso Z( f ) = V ( f ) e− jω T + V ( f )e jωT = 2 Aτ sinc f τ cos2π fT2.3-2z (t ) = v (t − 2T ) + 2v( t ) + v( t + 2T ) where v(t ) = aΠ (t /τ ) ↔ Aτ sinc f τZ ( f ) = V ( f )e − j 2ωT + V ( f ) + V ( f )e j2ωT = 2 Aτ (sinc f τ )(1 + cos4π fT )2.3-3z (t ) = v (t − 2T ) − 2v( t ) + v( t + 2T ) where v (t ) = aΠ (t /τ ) ↔ Aτ sinc f τZ ( f ) = V ( f )e − j 2ωT − 2V ( f ) + V ( f )e j2ωT = 2 Aτ (sinc f τ )(cos4π fT − 1)2.3-4  t −T   t −T / 2 v (t ) = AΠ   + ( B − A)Π    2T   T  − j ωTV ( f ) = 2 AT sinc2 fTe + ( B − A)T sinc fTe− jω T / 2 2-7
  10. 10. 2.3-5  t − 2T   t − 2T v (t ) = AΠ   + ( B − A)Π    4T   2T V ( f ) = 4 AT sinc4 fTe − j 2ωT + 2( B − A)T sinc2 fTe− j 2ωT2.3-6 1Let w(t ) = v( at ) ↔ W ( f ) = V ( f / a) a 1Then z (t ) = v[a(t − td / a)] = w(t − td / a) so Z ( f ) = W ( f ) e− jω td / a = V ( f / a) e− jω td / a a2.3-7 ∞ ∞F  v (t) e jω ct  = ∫ v(t )e jω ct e− jω t dt =∫ v (t) e− j 2π ( f − fc ) t dt =V ( f − f c )   −∞ −∞2.3-8v (t ) = AΠ (t /τ )cos ωc t with ω c = 2π f c = π / τ Aτ Aτ AτV( f ) = sinc( f − f c )τ + sinc( f + f c )τ = [ sinc( fτ − 1/2) + sinc( f τ + 1/2) ] 2 2 22.3-9 v( t ) = AΠ (t /τ )cos(ωc t − π /2) with ω c = 2π f c = 2π / τ e− jπ / 2 e jπ / 2V( f ) = Aτ sinc( f − f c )τ + Aτ sinc( f + f c )τ 2 2 Aτ = −j [sinc( f τ − 1) − sinc( f τ + 1) ] 22.3-10 2Az (t ) = v (t) c o s ω ct v(t ) = Ae − t ↔ 1 + (2π f ) 2 1 1 A AZ ( f ) = V ( f − fc ) + V ( f + fc ) = + 2 2 1 + 4π ( f − f c ) 1 + 4π ( f + f c )2 2 2 22.3-11 Az (t ) = v ()cos(ωc t − π /2) t v (t ) = Ae −t for t ≥ 0 ↔ 1 + j 2π f e − jπ / 2 e jπ / 2 − jA / 2 jA / 2Z( f ) = V ( f − fc) + V ( f + fc ) = + 2 2 1 + j 2π ( f − f c ) 1 + j 2π ( f + f c ) A/2 A/2 = − j − 2π ( f − f c ) j − 2π ( f + f c ) 2-8
  11. 11. 2.3-12  A tv (t ) = t z (t ) z (t ) = Π  ↔ 2 A sinc2 f τ  τ τ d d  sin2π f τ  2A Z ( f ) = 2A   = (2π f τ )2  (2πτ ) f cos2π f τ − 2πτ sin2π f τ  2df df  2π f τ    1 d − jAV( f ) = Z( f ) = ( sinc2 f τ − cos2π f τ ) − j 2π df πf2.3-13 −b t 2 Abz (t ) = tv (t ) v( t ) = Ae ↔ b + (2π f ) 2 2 1 d  2 Ab  j 2 AbfZ( f ) =  2 2 = − j 2π df  b + (2π f )   b + (2π f ) 2  2 2  2.3-14 Az (t ) = t 2v( t ) v (t ) = Ae −t for t ≥ 0 ↔ b + j2π f 1 d  A  2AZ( f ) =  b + j 2π f  = ( − j 2π f )  [ b + j 2π f ] 2 3 df 2.3-15 1v (t ) = e−π (bt) ↔ V ( f ) = e −π ( f / b) 2 2 b d j 2π f −π ( f / b)2( a) v (t ) = −2π b 2te −π ( bt ) ↔ 2 e dt b 1 d f −π ( f / b )2(b ) te− π (bt) ↔ V( f ) = 2 e − j 2π df jbBoth results are equivalent to bte −π ( bt ) ↔ − jf e−π ( f / b) 2 22.4-1 y (t ) = 0 t<0 2 t At = ∫ Aλ d λ = 0<t <2 0 2 2 = ∫0 Aλ d λ = 2A t >2 2-9
  12. 12. 2.4-2 y (t ) = 0 t < 0, t > 5 2 t At = ∫ Aλ d λ = 0 <t< 2 0 2 = ∫0 Aλ d λ = 2 A 2 2<t <3 2 A = ∫ Aλ d λ =  4 − ( t − 3) 2  3 < t < 5 t− 3 2 2.4-3 y (t ) = 0 t < 0, t > 3 t At 2 = ∫0 Aλ d λ = 0 < t <1 2 t A = ∫ Aλ d λ = (2t − 1) 1< t < 2 t −1 2 2 A = ∫ Aλ d λ =  4 − (t − 1) 2  2 < t < 3 t −1 2 2.4-4 y (t ) = 0 t <4 t = ∫ 2 Ad λ = 2 At − 8 A 4 ≤ t ≤ 6 4 8 = ∫ 2A dλ = 4 A t >6 6 2-10
  13. 13. 2.4-5 y (t ) = 0 t<0 t = ∫ 2e−2λ d λ = 1 − e −2t 0 ≤t≤ 2 0 t = ∫ 2e −2λ d λ = e −2t  e4 − 1   t>2 t− 22.4-6 y (t ) = 0 t < −1, t ≥ 3 t = ∫ 2(1 + λ ) d λ = t2 + 2t + 1 −1 ≤ t < 0 −1 0 t = ∫ 2(1 + λ ) d λ + ∫ 2(1 − λ )d λ = −t 2 + 2t + 1 0 ≤ t < 1 −1 0 0 1 = ∫ 2(1 + λ )d λ + ∫ 2(1 − λ ) d λ = −t 2 + 2t + 1 1 ≤ t < 2 t− 2 0 1 = ∫ 2(1 − λ ) d λ = t2 − 6t + 9 2≤t <3 t− 22.4-7 y (t ) = 0 t≤0 = ∫ Ae− aλ Be− b( t− λ )d λ =[ AB /( a − b)][ e− bt −e − at ] t > 0 t 02.4-8 j jπ t j − jπ tv (t ) = Ae −at w(t ) = sin π t = e − e = B1e −b1t + B2e− b2 t 2 2y (t ) = v ∗ w1 (t ) + v ∗ w2 (t ) = [ AB1 /(a − b1 )][ e −b1t − e − at ] + [ AB2 /( a −b2 )][ e− b2 t − e − at ]Let B1 = j /2, b1 = − jπ , B2 = − j /2, b2 = jπ and simplify2.4-9 ∞v ∗ w(t ) = ∫ v (λ )w(t − λ ) d λ let µ = t − λ −∞ −∞ ∞ = − ∫ v (t − µ ) w( µ ) d µ = ∫ w( µ) v (t − µ )d µ = w ∗ v (t ) ∞ −∞ 2-11
  14. 14. 2.4-10 ∞Let y (t ) = ∫ v( λ ) w(t − λ )d λ where v( −t ) = v(t ), w(−t ) = w(t ) −∞ ∞ ∞y (−t ) = ∫ v (λ )w( −t − λ ) d λ = ∫ v (λ )w(t + λ ) d λ −∞ −∞ ∞ ∞ = − ∫ v( −µ ) w( t − µ ) d µ = ∫ v( µ ) w(t − µ )d µ = y (t ) −∞ −∞2.4-11 ∞Let y (t ) = ∫ v (λ )w(t − λ ) d λ where v ( −t ) = −v (t ), w(−t ) = −w(t) −∞ ∞ ∞y (−t ) = ∫ v (λ )w(−t − λ ) d λ = − ∫ v (λ )w(t + λ ) d λ −∞ −∞ ∞ ∞ = ∫ v (− µ ) w(t − µ ) d µ = ∫ v (µ ) w( t − µ )d µ = y (t ) −∞ −∞2.4-12Let w(t ) = v ∗ v (t ) = τΛ (t /τ ) 0 t +τ / 2 3 v ∗ w(t ) = ∫ (τ + λ) d λ + ∫ (τ − λ) d λ = τ 2 − t 2 0 ≤ t < τ / 2 t −τ / 2 0 4 2 τ 1 3  = ∫t −τ / 2 (τ − λ) d λ =  t − τ  τ / 2 ≤ t < 3τ / 2 2 2   3 2 2  4τ −t t <τ / 2  1  2 3 Thus v ∗ v ∗ v (t ) =   t − τ  τ / 2 ≤ t < 3τ / 2 2 2   0 t ≥ 3τ / 2  2.4-13F {v (t ) ∗ [ w(t ) ∗ z (t )]} = V ( f ) [W (f )Z ( f )] = [V ( f )W( f ) ] Z ( f ) so v (t ) ∗[w(t ) ∗ z (t) ] = F −1 {[V ( f )W( f )]Z ( f )} = [v( t ) ∗ w(t) ] ∗ z (t ) 2-12
  15. 15. 2.4-14 1 f V( f ) = Π   W ( f ) = 4Π (2 f ) 4 4Y ( f ) = V ( f )W( f ) = Π (2 f ) ↔ y (t ) = (1/2)sinc( t /2)2.5-1 t Aτ Aτz (t ) = AΠ   cos ωc t Z ( f ) = sinc( f − f c )τ + sinc( f + f c )τ τ  2 2As τ → 0 the cosine pulse z (t) gets narrower and narrower while maintaining height A.This is not the same as an impulse since the area under the curve is also getting smaller.As τ → 0 the main lobe and side lobes of the spectrum Z ( f ) get wider and wider, howeverthe height gets smaller and smaller. Eventually the spectrum will cover all frequenci es withalmost zero energy at each frequency. Again this is different from what happens in the caseof an impulse.2.5-2  W ( f ) = v ( f )e − j 2π ftd =  ∑ cv ( nf 0 )δ ( f − nf 0 )  e− j 2π ftd  n  = ∑ cv ( nf 0 ) e  − j 2π n f t d 0 δ ( f − nf 0 ) ⇒ cw ( nf 0 ) = cv (nf 0 )e − j2π n f0 td  n2.5-3  W ( f ) = j 2π fV ( f ) = j 2π f  ∑ cv (nf 0 )δ ( f − nf 0 )  = ∑ [ j 2π nf 0 cv ( nf 0 ) ]δ ( f − nf 0 )  n  n⇒ cw ( nf 0 ) = j 2π nf 0 cv ( nf 0 )2.5-4 1 1  W( f ) = [V ( f − mf0 ) ] = ∑ cv ( nf0 )δ ( f − kf0 − mf0 ) + ∑ cv ( nf0 )δ ( f − kf0 + mf0 )  2 2 n n  1  = ∑ cv [(k − m) f 0 ]δ ( f − kf 0 ) + ∑ cv [(k + m) f 0 ]δ ( f − kf 0 )  2 k k  1 = ∑ {cv [(n − m) f 0 ] + cv [(n + m) f 0 ]}δ ( f − nf 0 ) n 2 1so cw ( nf 0 ) = {cv [(n − m) f 0 ] + cv [(n + m) f 0 ]} 2 2-13
  16. 16. 2.5-5v (t ) = Au (t ) − Au (t − 2τ )  1 1  1 1  − j 4π f τ V( f ) = A + δ ( f ) − + δ ( f ) e   j 2π f 2  j 2π f 2   − j 4π ft − j0But δ ( f ) e = e δ ( f ), soV( f ) = A j 2π f (1 − e − j 4π fτ ) = 2 Aτ sinc2 f τ e− j 2π f τ  t −τ  − j 2π f τAgrees with v( t ) = Π   ↔ 2 Aτ sinc2 f τ e  2τ 2.5-6v (t ) = A − Au (t + τ ) + Au (t − τ )   1 1   1 1  V ( f ) = A δ ( f ) −  + δ ( f )  e j 2π f τ −  + δ ( f )  e− j 2π fτ    j 2π f 2   j 2π f 2   j 2π f τ − j 2π f τBut δ ( f ) e = δ ( f )e = e δ ( f ), so j0  V ( f ) = A δ ( f ) − 1 j 2π f (e j 2π f τ − e− j2π f τ ) = Aδ ( f ) − 2 Aτ sinc2 f τ  Agrees with v( t ) = A − AΠ (t / 2τ ) ↔ Aδ ( f ) − 2 Aτ sinc2 f τ2.5-7v (t ) = A − Au (t + T ) − Au( t − T )   1 1  j 2π fT  1 1  − j 2π fT V ( f ) = A δ ( f ) −  + δ ( f ) e − + δ ( f ) e    j 2π f 2   j 2π f 2  But δ ( f ) e j 2π fT = δ ( f ) e− j2π fT = e j 0δ ( f ) = δ ( f ), so −AV( f ) = j 2π f (e j 2π fT + e − j2π fT ) = j− Af cos2π fT π −AIf T → 0, v(t ) = − A sgn t ↔ V ( f ) = , which agrees with Eq. (17) jπ f2.5-8V ( f ) = sinc f ε and V (0) = 1, so sinc f ε 1W(f ) = + δ(f) j 2π f 2 1 1If ε → 0, w( t ) = u (t ) and W ( f ) = + δ ( f ), which agrees with Eq. (18) j 2π f 2 2-14
  17. 17. 2.5-9 1/ εV( f ) = and V (0) = 1, so 1/ ε + j 2π f 1/ ε 1W(f ) = + δ(f ) ( j 2π f )(1/ ε + j 2π f ) 2 1 1If ε → 0, w( t ) = u (t ) and W ( f ) = + δ ( f ), which agrees with Eq. (18) j 2π f 22.5-10z (t ) = AΠ ( t / τ ) ∗ [δ (t − T ) +δ (t + T ) ]so Z( f ) = ( Aτ sinc f τ ) ( e− jω T + e jωT ) = 2 Aτ sinc f τ cos2π fT2.5-11z (t ) = AΠ ( t / τ ) ∗ [δ (t − 2T ) + 2δ (t ) + δ (t + 2T ) ]so Z ( f ) = ( Aτ sinc f τ ) ( e − jω 2T + 2 + e jω 2 T ) = 2 Aτ sinc f τ (1 + cos4π fT )2.5-12z (t ) = AΠ ( t / τ ) ∗ [δ (t − 2T ) − 2 δ ( t ) + δ (t + 2T ) ]so Z ( f ) = ( Aτ sinc f τ ) ( e − jω 2T − 2 + e jω 2T ) = 2 Aτ sinc fτ (cos4π fT − 1)2.5-13 n 0 1 2 3 4 5 6 7 8sin(π t) δ (t − 0.5n) 0 1 0 1 0 1 0 1 0 v (t ) 0 1 1 2 2 3 3 4 42.5-14 n -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0cos(2π t )δ (t − 0.1n) 1 0.81 0.31 -0.31 -0.81 -1 -0.81 -0.31 0.31 0.81 1 v (t ) 1 1.81 2.12 1.81 1 0 -0.81 -1.12 -0.81 0 1 v () for n = 1,10 t 1.81 2.12 1.81 1 0 -0.81 -1.12 -0.81 0 1 2-15
  18. 18. Chapter 33.1-1 y (t ) = h(t ) ∗ A[δ (t + t d ) −δ (t −t d ) ] = A[ h (t + t d ) − h(t − t d ) ] ( )Y ( f ) = H ( f )A e jω td − e − jω td = j 2 AH ( f )sin2π ft d3.1-2 y (t ) = h( t ) ∗ A δ ( t + t d ) + δ ( t )  = A h ( t + td ) + h ( t )     ( )Y ( f ) = H ( f )A e jω td + 1 = 2AH ( f )cos π ft d e jπ ftd3.1-3 y (t ) = h( t ) ∗ Ah (t − t d ) = Ah(t ) ∗ h(t − t d )Y ( f ) = H ( f ) AH ( f )e − jω td = AH 2 ( f ) e− jω ftd3.1-4 t −t dy (t ) = h( t ) ∗ Au (t − t d ) = A ∫ h (λ ) d λ ∞  1 1  A AY ( f ) = H( f )A  + δ ( f )  e − jω td = H ( f )e − j2 π ftd + H (0)δ ( f )  j 2π f 2  j 2π f 23.1-5  1 1  1 1F [ g (t )] = H ( f )  + δ ( f ) = H ( f ) + H(0)δ ( f )  j 2π f 2  j 2π f 2F [ dg (t) / dt ] = j 2π f F [ g (t ) ] = H ( f ) = F [ h( t )] Thus h( t ) = dg (t) / dt3.1-6 j 2π fY ( f ) + 4πY ( f ) = j 2π fX ( f ) + 16π X ( f ) Y( f ) j 2π f + 2π 8 8 + jfH(f ) = = = X ( f ) j 2π f + 2π 2 2 + jf 64 + f 2 f f H( f ) = arg H ( f ) = arctan − arctan 4+ f2 8 2 3-1
  19. 19. 3.1-7 j 2π fY ( f ) + 16 Y ( f ) = j 2π fX ( f ) + 4π X ( f ) π Y ( f ) j 2π f + 2π 2 2 + jfH(f ) = = = X ( f ) j 2π f + 2π 8 8 + jf 4+ f2 f f H( f ) = arg H ( f ) = arctan − arctan 64 + f 2 2 83.1-8 j 2π fY ( f ) − 4π Y ( f ) = − j2 π fX ( f ) + 4π X ( f ) Y ( f ) − j2 π f + 2π 2 2 − jfH(f ) = = = X( f ) j 2π f − 2π 2 2 + jf 2+ f2 f H( f ) = = 1 for all f arg H ( f ) = −2arctan 2+ f2 23.1-9 BH(f ) ≈ for f ≥W? B jf B 1Thus Y ( f ) ≈ X ( f ) = 2π B X ( f ) for f ≥W jf j 2π f tand y (t ) ≈ 2π B∫−∞ x (λ ) d λ since X (0) ≈ 03.1-10 jfH(f ) ≈ for f ≤ W = B B jf 1Thus Y ( f ) ≈ X(f)= j 2π fX ( f ) for f ≤W B 2π B 1 dx (t )so y(t) ≈ 2π B dt 3-2
  20. 20. 3.1-11 2  f  1  f x (t ) = 2sinc4Wt ↔ X ( f ) = Π = Π  4W  4W  2W  4W  ∞ 2W 1 1Ex = ∫ X ( f ) df = ∫ 2 df = −∞ −2W 4W 2 W Π 1 1 f Y( f ) =  4W  1 + j ( f / B ) 2W   2W 1 / 4W 2 B 2WE y = 2∫0 df = arctan 1+ ( f / B ) 2 2 2W BEy B 2W = arctanE x 2W B3.1-12h (t ) = F −1 [ H1 ( f ) H 2 ( f ) ] = h1 (t ) ∗ h2 (t )where h1( t ) = u (t ) − u (t − T1 ) h2 ( t ) = u (t ) − u (t − T2 )3.1-13h (t ) = F [ H 1 ( f ) H 2 ( f ) ] = h1 (t ) ∗ h2 (t ) −1where h1( t ) = 2π Be−2π Bt u(t ) h2 ( t ) = u (t ) − u (t − T )3.1-14 j 2π f 1 1H(f ) = = j 2π f 1 + jK 2π f K 1/ K + j 2π f 1 d −t / K 1 1h (t ) = e u (t )  = δ (t ) − 2 e −t / K u (t )   K K dt K  1/ K  1 −t / Kg ( t ) = F −1   = e u (t ) 1/ K + j2π f  K 3-3
  21. 21. 3-1-15 K 1H( f ) = = 1 + Kj 2π f 1 + j 2π f Kso h (t ) = e− t / K u (t )g ( t ) = ∫ h( λ ) dλ = K(1 − e−t / K ) u (t ) t −∞3.1-16Since h( t ) is real, H r ( f ) = H e ( f ) and Hi ( f ) = H o ( f ), so ∞ ∞ ∞h (t ) = ∫ −∞ [ He ( f ) + jHo ( f ) ] e jω t df = 2∫ H r ( f )cos ω t df + j 2 ∫ jH i ( f )sin ωt df 0 0 ∞ ∞ = 2  ∫ H r ( f )cos ωt df − ∫ Hi ( f )sin ω t df   0  0   ∞ ∞h (t ) = 0 for t < 0 ⇒ ∫ H i ( f )sin ωt df = ∫ Hr ( f )cos ω t df 0 0 ∞ ∞Hence, for t > 0, − ∫ Hi ( f )sin ω t df = ∫ H r ( f )cos ω t df 0 0 ∞ ∞so h( t ) = ∫ Hi ( f )sin ω t df = ∫ H r ( f )cos ω t df 0 03.2-1 1 −1 / 2 H ( f ) = 1 + ( f / B )  ( f / B )2 + L ≈ 1 =1 − 2  2 3 f f 1 f  farg H ( f ) = − arctan = − +   + L ≈ − B B 3 B  Bfor f ≤W= B3.2-2 −1 / 2H ( nf 0 ) = 1 + ( n /3 )  arg H ( nf 0 ) = − arctan(n /3) 2  y (t ) = (0.95)(4)cos (ω 0t − 18° ) + (0.71)(4/9)cos ( 3ω 0t − 45° ) + (0.5)(4/25)cos ( 5ω 0t − 59°) = 3.79cos (ω 0t −18° ) + 0.31cos ( 3 ω 0t − 45° ) + 0.08cos ( 5ω0t − 59°) 3-4
  22. 22. 3.2-3 n /3H ( nf 0 ) = arg H ( nf 0 ) = 90° − arctan( n /3) 1 + (n / 3) 2y (t ) = (0.32)(4)cos (ω 0t − 72° ) + (0.71)(4/9)cos ( 3ω 0t − 45° ) + (0.86)(4/25)cos ( 5ω0t − 31°) = 1.28cos ( ω0t − 72° ) + 0.31cos ( 3ω0t − 45° ) + 0.14cos ( 5ω 0t − 31°)3.2-4 2  f  1  f X( f ) = Π = Π 40  40  20  40      20  f  − jω 5Y( f ) = Π 40  40  e   1  f  − jω 5 Π Y ( f ) 2  40  eH(f ) = =   = 10e − jω 5 X( f ) 1  f  Π 20  40   Note that 300π = 2π and the phase actually wrapped around several times. Undernormal plotting conventions we would go from −π to π and repeat this pattern 300times.3.2-5 − arctan( f / B ) 1td ( f ) = B = 2kHz lim t d ( f ) = 2π f f→0 2π Bf kHz t d ( f ), ms 0 -0.08 0.5 -0.078 1 -0.074 2 -0.062 3-5
  23. 23. 3.2-6  πf − 30  f ≤ 15arg H ( f ) =   −π f > 15  2   1  π  1 1 d − −  = f ≤ 15tg ( f ) = − arg H ( f ) =  2π  30  60 2π df   0 f > 15  π f /30 1  = f ≤ 15 − arg H ( f )  2π f 60td ( f ) = = 2π f  π /2 = 1 f > 15  2π f 4 f so t d ( f ) = t g ( f ) for f ≤ 153.2-7  (a) H c ( f ) = 1 + 2α ( e jωT + e − jωT )  e− jω T = α + e − jω T + α e − jω 2T 1  2  Thus, y (t ) = α x(t ) + x (t − T ) + α x (t − 2T ) 2T(b) τ = 3 4T τ= 3 3-6
  24. 24. 3.2-8  α2 exp  − j (ωT − α sin ωT )  = e− jω T e jα sinωT = e − jωT  1 + jα sin ωT −   sin2 ωT + L   2  αIf α = π /2, H c ( f ) ≈ e − jωT + jα sin ωT e − jωT = e − jωT + ( e jω T − e − jωT ) e − jω T 2 α α ≈ + e − jωT − e − jω 2 T 2 2 α αThus, y (t ) ≈ x( t ) + x (t − T ) − x (t − 2T ) 2 { 2 14243 leading echo inverted trailing echo3.2-9 − j ω ( t d −T ) j 0.4sin ω TH eq ( f ) = Ke ee j0.4sin ωT = 1 + j 0.4sin ωT − 0.8sin 2 ωT + L ≈ 1 + 0.2 ( e jω T − e − jω T )so H eq ( f ) ≈ Ke − jω ( td −T ) (1 + 0.2e jω T − 0.2e − jω T )Take K = 1 and t d = 2T , so H eq ( f ) ≈ ( 0.2e jω T + 1 − 0.2e − jω T ) e − jω THence, ∆ = T , M = 1, c−1 = 0.02, c0 = 1, c1 == −0.23.2-10H eq ( f ) = Ke − jω ( td − T) (1 + 0.8cos ωT ) −1Expanding using the first 3 terms(1 + 0.8cos ω T ) −1 = 1 − 0.8cos ωT + 0.64cos 2 ωT − 0.51cos3 ωTwhere cos ωT = 2 ( e + e − jωT ) , cos2 ωT = 1 + 1 cos2ωT = 1 + 1 ( e j 2ωT + e − j 2ω T ) 1 jω T 2 2 2 4cos 3 ωT = ( 3cos ω T + cos3ωT ) = ( e jω T + e − jωT ) + ( e j 3ω T + e− j3ωT ) 1 3 1 4 8 8Take K = 1 and t d = 4T , so  0.13 j3ω T 0.64 j2ω T  0.8 0.38  jωT 0.64  0.8 0.38  − jω TH eq ( f ) =  − e + e − +  e + 1+ 2 −  2 + 2  e  2 4  2 2    0.64 − j 2ωT 0.13 − j3ωT  − j3ω T + e − e e 4 2 Hence, ∆ = T , M = 3, c−3 = c3 = −0.065, c−2 = c2 = 0.16, c−1 = c1 = −0.59, c0 = 1.32 3-7
  25. 25. 3.2-11 9 A3 3 A3y (t ) = 2 A cos ω 0t − 3 A3 cos3 ω 0t 3 A3 cos 3 ω0t = cos ω 0t + cos3ω 0t 4 4  9 A3  3 A3so y (t ) =  2 A − cos ω 0t − cos3ω 0t  4  42 nd harmonic distortion = 0 3 A3 4 300% A = 13rd harmonic distortion = × 100 =  2A − 9 A3  42% A = 2 43.2-12 y (t ) = 5 A cos ω 0t − 2 A2 cos 2 ω0 t + 4 A3 cos 3 ω0 t2 A2 cos 2 ω 0t = A2 + A2 cos2ω0t 4 A3 cos 3 ω 0t = 3 A3 cosω 0 t + A3 cos3 0 t ωso y (t ) = − A2 + ( 5 A + 3 A3 ) cos ω 0t − A 2 cos2ω0 t + A3 cos3ω0 t A2 12.5% A =12 nd harmonic distortion = ×100 =  5 A + 3A3 11.8% A=2 A3 12.5% A =1 rd3 harmonic distortion = × 100 =  5 A + 3 A3 23.5% A=23.3-1Pin = 0.5W = 27dBm l = 50km α = 2dB/kmPout = 50mW = 17dBm 20µW = −17dBm27dBm − 2l1 = −17dBm ⇒ l1 = 22km ⇒ l 3 = 50 − 22 = 28km−17dBm + g2 − 2 × 28 = −17dBm ⇒ g2 = 56dB−17dBm + g4 = 17dBm ⇒ g 4 = 34dB3.3-2Pin = 100mW = 20dBm l = 50km α = 2dB/kmPout = 0.1W = 20dBm 20µW = −17dBm20dBm − 2l1 = −17dBm ⇒ l 1 = 18.5km ⇒ l 3 = 40 − 18.5 = 21.5km−17dBm + g2 − 2 × 21.5 = −17dBm ⇒ g2 = 43dB−17dBm + g4 = 20dBm ⇒ g 4 = 37dB 3-8
  26. 26. 3.3-3 Pout 50 ×10 −3 = = −16 dB, mLi = 0.4 × 400 = 160 dB, gi ≤ 30 dB Pin 2m × 30 dB − 160 ≥ −16 ⇒ m ≥ 4.8 so m = 5g = (160 − 16)/5 = 28.8 dB3.3-4Li = 0.5 × 3000/ m = 1500/ m dB Pin = 5mW = 7dBm Pin ≥ 67µW = −11.75dBm L1 15007dBm − ≥ −11.75 ⇒ m ≥ 80 m Pout mgi 1500 = = 1 ⇒ g i = Li = = 18.75dB Pin mLi 803.3-5Li = 2.5 × 3000/ m = 7500/ m dB Pin = 5mW = 7dBm Pin ≥ 67µW = −11.75dBm L1 75007dBm − ≥ −11.75 ⇒ m ≥ 400 m Pout mgi 7500 = = 1 ⇒ gi = Li = = 18.75dB Pin mLi 4003.3-6 Pout = 2 ×10 −6 /5 = −64 dB, L = 92.4-6 + 26 = 112.4 dB Ping 2 Pout = ⇒ g = (112.4 − 64)/2 = 24.2 dB = 263L Pin 4π (π r 2 )( 0.5 ×109 ) 2so = 263 ⇒ r = 1.55 × 10−3 km = 1.55 m (3 × 10 ) 5 2 3-9
  27. 27. 3.3-7 Pout 2 ×10−6 = = −64dB L = 92.4 − 14 + 20 = 98.4 Pin 5g 2 Pout 98.4 − 64 = ⇒g= = 17.2dB = 52.5 L Pin 2 4π (π r 2 )( 0.2 ×109 ) 2so = 52.5 ⇒ r = 1.7 ×10−3 km = 1.7m (3 × 10 ) 5 23.3-8 gT gR grpt gT g RWith repeater Pout = Pin Without repeater Pout = P L1L2 L ing T g R g rpt g g LL = 1.2 T R ⇒ grpt = 1.2 1 2 L1L2 L LL1d B = 92.4 − 20log f GHz + 20log25km = 120 + 20log fL2 = L1L = 92.4 − 20log f + 20log50 = 126 + 2 − log fLet f = 1GHzL1 = L2 = 120dB ⇒ 1012 L = 126dB ⇒ 3.98 ×1012 1012 ×1012g rpt = 1.2 = 0.3 ×1012 = 115dB 3.98 × 10123.3-9Lu = 92.4 + 20log17 + 20log3.6 × 104 = 208Ld = 92.4 + 20log12 + 20log3.6 ×10 4 = 205Pin = 30dBW so Psatin = 30 + 55 − 208 + 20 = −103dBWbased on parameters from Example 3.3-1 g amp = 18 + 144 = 162dBPsatout = −103 +162 = 59dBW so Pout = 59 + 16 − 205 + 51 = −79dBW ⇒ 1.26 ×10 -8W3.4-1  f  − jω tdH ( f ) = Ke− jω td − K Π  e h (t ) = K δ (t − t d ) − 2 Kf l sinc2 f l ( t − td )  2 fl  3-10
  28. 28. 3.4-2H ( f ) = Ke− jω td − H BP ( f ) where H BP ( f ) = Eq. (1)Thus, from Exercise 3.4-1, h( t ) = Kδ (t − t d ) − 2 BK sinc B( t − td ) cos ω c (t − t d )3.4-3 −1 H (0.7 B) = 1 + (0.7)2 n  ≥ 10−1/10 = 1/1.259 2  so 1 + (0.7) ≤ 1.259 or (0.7) 2n ≤ 0.259 2nn (0.7) 2n1 0.49 ⇒ select n = 22 0.24 −1/2 −1 / 2 H (3B) = 1 + 32n    = 1 + 34    = 0.11 = −19dB3.4-4 −1 H (0.9B ) = 1 + (0.9) 2n  ≥ 10 −1/10 = 1/1.259 2  so 1 + (0.9) ≤ 1.259 or (0.9) 2n ≤ 0.259 2nn (0.9) 2n6 0.282 ⇒ select n = 77 0.229 −1/2 −1 / 2 H (3B) = 1 + 32n    = 1 + 314    = 4.6 ×10− 4 = −66.8dB3.4-5 −1  f  f   2H ( f ) = 1 + j 2 −    from Table 3.4-1   B  B   −1  s  s   2H ( s) = H ( f ) | f =s / j 2π = 1 + j 2 −    2π B  2π B     2b2 = b = 2π B / 2 ( s + b ) + b2 2so h( t ) = 2be − bt sin b t u (t ) 3-11
  29. 29. 3.4-6 Z RC R / jω C LC(a) H ( f ) = where Z RC = = Z RC + jω L R + 1/ jω C 1 + jω LC 1Thus, H ( f ) = 1 + jω LC − ω LC 2 −1so H ( f ) = (1 − ω 2 LC ) + (ω 2 LC )  = 1 − ( f / f 0 ) + ( f / f 0) with f 0 = 2 2 2 2 4 1     2π LC(b) H ( B) = 1 / 2 ⇒ 1 − ( B / f 0 ) + ( B / f0 ) =2 2 2 4 2 B 1so   = 1 + 5  f0  2 ( ) B= 1 2 ( ) 1 + 5 f0 = 1.27 f 03.4-7 −2π Bt11− e = 0.1 ⇒ t1 = 0.11/2π B  2.30 − 0.11 1  tr = t 2 − t1 = =1− e −2π Bt2 = 0.9 ⇒ t 2 = 2.30/2π B   2π B 2.87 B3.4-8g ( t ) = ∫ h( λ )d λ = 2b ∫ e − bλ sin bλ d λ = 1 − e − bt ( sin bt + cos bt ) for t ≥ 0 t t −∞ 0 0.5π 0.5π 2 1bt1 / π ≈ 0.1, bt2 / π ≈ 0.6 tr ≈ = = b 2π B 2.8 B 3-12
  30. 30. 3.4-9 1 A  f x (t ) = A sinc2Wt ⇒ τ x = , X( f ) = Π W 2W  2W     A  f   2W Π  2W  for B > W  f    Y( f ) = Π X(f )=  2B   A Π  f  for B < W  2W  2 B      A sinc2Wt ⇒ τ y = 1/ W for B > W y (t ) =  B W A sinc2 Bt ⇒ τ y = 1/ B for B < W 3.4-10 ∞ ∞H (0) = ∫ h(t )e − jω t dt = ∫ h ( t) dt −∞ f =0 −∞ ∞ ∞ ∞h( t ) = ∫ −∞ H ( f ) e jω t df ≤ ∫ −∞ H ( f )e jω t df = ∫ −∞ H ( f ) df H (0) H (0) 1Thus τ eff = ≥ ∞ = h( t ) max ∫ −∞ H ( f ) df 2Beff3.4-11 sinc2 B ( t − td ) e − jω t dt = 2 KBe − jωtd ∫ sinc2Bλ e− jωλ d λ 2 td td(a) H ( f ) = 2 KB∫ 0 −t d 1 where sin2π Bλ cos2π f λ = sin2π ( f + B ) λ − sin2π ( f − B ) λ  2  t d sin2 ( f ± B ) λ π 1 2π ( f ± B)td sin α 1 and ∫ dλ = ∫0 dα = Si  2π ( f ± B ) td  0 2π Bλ 2π B α 2π B   K π  { Thus H ( f ) = e− jω td Si  2π ( f + B ) td  − Si 2π ( f − B ) td     } (cont.) 3-13
  31. 31. 1 1(b) td ? td = B 2B3.5-1 1 ∞ δ (λ ) 1 1 1(a) δ ( t ) = ∫ ˆ dλ = = π −∞ t −λ π t − λ λ=0 π t F δˆ (t )  = ( − j sgn f ) F [δ (t ) ] = − j sgn f   Thus, F - 1 [ − j sgn f ] = δˆ (t ) = 1 πt 1  −1  1 1 · 1 (b) δ ( t ) ∗ = δ( t) and δ ( t ) ∗   = − ∗ = −  ˆ ˆ πt πt  πt πt  πt  ·  1  Thus,   = −δ ( t ) πt 3.5-2 t AΠ   = x ( t + τ / 2 ) where x (t ) = A [u ( t ) − u ( − τ )] t τ  ˆ  t  ˆ τ  Aso AΠ   = x  t +  = ln t +τ / 2 = ln A 2t + τ τ   2 π t +τ / 2 −τ π 2t − τ t A 2t + τ ANow let v(t ) = lim AΠ   so v (t ) = lim ln ˆ = ln1 = 0 τ →∞ τ  τ →∞ π 2t − τ π3.5-3 1  f  F [ x (t )] = ( − j sgn f ) Π 2W  2W  ˆ  j  f +W / 2  j  f −W / 2  = Π  − 2W Π  W  2W  W   Thus, x (t ) = sinc Wt ( e − jπWt − e jπ Wt ) = sincWt sinc π Wt = π Wt sinc2 Wt j ˆ 2 3-14
  32. 32. 3.5-4 1 1x (t ) = cos ω 0t − cos3ω 0t + cos5ω 0t 3 5 1 1x (t ) = sin ω0t − sin3ω0t + sin5ω0tˆ 3 53.5-5 4 4x (t ) = 4cos ω 0 t + cos3ω 0t + cos5ω 0t 9 25 4 4x (t ) = 4sin ω0 t + sin3ω0t + sin5 0tˆ ω 9 253.5-6 1  f  1  f x (t ) = sinc2Wt ↔ X ( f ) = Π  X(f) = Π  2W  2W  2W  2W  1 f − W  − jπ / 2 1  f +W  jπ / 2x (t ) = πWt sinc Wt = sinc Wt sin π Wt ↔ X ( f ) =ˆ 2 ˆ Π 2 e + Π 2 e 2W  W    2W  W    1  f − W  − jπ /2 1  f + W  + jπ / 2X(f) = ˆ Π + Π  W   W  2 2 e e 2W   2W    Note that the cross term is zero since there is no overlap. From the graph we see that thetwo rectangle functions form one larger function so 1  f  X(f) = ˆ Π  = X(f ) 2W  2W 3.5-7x (t ) = A cos ω0t x(t ) = A sin ω0 t ˆ ∞ ∞∫−∞ x ( t ) x( t) dt = A2 ∫ cos ω 0t sinω 0t dt ˆ −∞ T  A2 A2   A2 1  sin ( ω0 − ω0 ) t dt + sin (ω 0 + ω 0 ) t dt  = lim 0 + T T= lim  T →∞  2 ∫−T 2 ∫− T  T →∞  2 2ωo cos2ω 0t  −T  A2 = lim  T →∞ 4ω ( cos2ω0T − cos ( −2ω 0T ) )  = 0  0  3-15
  33. 33. 3.5-8 h ( t ) e− jω t dt = 2∫ h ( t ) cos ω t dt ∞ ∞1F [ he ( t ) ] = ∫ 1 −∞ 2 0 2 ∞ ∞ = ∫0 h (t) c o s ω t dt = ∫−∞ h (t) c o s ωt dt = H e ( f ) 1  1  )H ( f ) = F (1 + sgn t ) he (t )  = H e ( f ) +   ∗ H e ( f ) = He ( f ) − j H e ( f ) ∗  = He ( f ) − jH e ( f ) j 2π f  πf )Thus, H o ( f ) = − He ( f )3.6-1 ∗Rwv (τ ) = w(t) v∗ ( t − τ ) = w∗ (t )v(t −τ ) ∗ = v [ t + ( −τ ) ] w∗ ( t ) = Rvw ( −τ ) *3.6-2Rv (τ ± mT0 ) = v ( t + τ ± mT0 ) v ∗ (t )but v ( t + τ ± mT0 ) = v ( t + τ ) so Rv (τ ± mT0 ) = v ( t + τ ) v∗ ( t ) = Rv (τ )3.6-3 2 2Pw = v (t +τ ) = v (t ) = Pv 2 Rv (τ ) = v( t) w∗ (t ) 2 ≤ Pv Pv = Rv2 (0) so Rv (τ ) ≤ Rv (0)3.6-4 1x (t ) = cos2ω 0t From Eq. (12) Rx (τ ) = cos2ω 0τ 2 1y (t ) = sin2ω 0t = cos ( 2ω 0t − 90° ) ⇒ R y (τ ) = cos2ω0τ 2(Note that the phase delay does not appear in the autocorrelation)Since Ry (τ ) = Rx (τ ) we conclude that y (t ) is similar to x (t ). This is the expected conclusionsince y( t ) is just a phase shifted version of x (t) .3.6-5V ( f ) = AD sinc f D e− jωtdGv ( f ) = ( AD ) 2 sinc2 fD ⇒ Rv (τ ) = A2 D Λ (τ / D ) , Ev = Rv (0) = A2 D 3-16
  34. 34. 3.6-6  A   f  − jω tdV( f ) =   Π  4W  e  4W    2  A   f  A2 A2Gv ( f ) =   Π  ⇒ Rv (τ ) = sinc4W τ , Ev = Rv (0) =  4W   4W  4W 4W3.6-7 AV( f ) = b + j 2π f A2 A2 −b τ A2Gv ( f ) = ⇒ Rv (τ ) = e , Ev = Rv (0) = b 2 + (2π f ) 2 2b 2b3.6-8 A1 jφ jω 0 t A1 − jφ − jω 0tv (t ) = A0 + e e + e e 2 2 2 AGv ( f ) = A02δ ( f ) + 1  δ ( f − f 0 ) + δ ( f + f 0 ) 4   A2 A2Rv (τ ) = A0 + 1 cos ω 0τ , Pv = Rv (0) = A0 + 1 2 2 2 23.6-9v (t ) = 2 ( A1 jφ1 jω0 t A ) ( ) e e + e − jφ1 e − jω0 t + 2 e− jπ / 2e j 2ω 0 t e jφ1 + e jπ / 2e − j 2ω0 t e− jφ1 2 2 A A2Gv ( f ) = 1 δ ( f − f 0 ) + δ ( f + f 0 ) + 2 δ ( f − 2 f 0 ) + δ ( f + 2 f 0 )  4   4   A2 A2 A2 A2Rv (τ ) = 1 cos ω oτ + 2 cos2ω0τ Pv = Rv (0) = 1 + 2 2 2 2 23.6-10 1 T /2 0 t < τRv (τ ) = lim T →∞ T ∫ −T / 2 A2 u(t )u (t − τ ) dt where u(t )u (t − τ ) =  1 t > τ T/2 T /2 TTake T / 2 > τ > 0, so ∫ u (t )u (t − τ ) dt = ∫ dt = − τ -T/2 τ 2 A T  A 2 2Thus Rv (τ ) = lim  2 −τ  = 2 for all τ T →∞ T   A2 A2Pv = Rv (0) = Gv ( f ) = δ(f) 2 2 3-17
  35. 35. 3.6-11x (t ) = Π (10t ) = Π  t  1 f  1/10  ↔ X ( f ) = 10 sinc 10    f   f H ( f ) = Ke − jω td Π   = 3e − jω 0.05 Π   2B   40  2 2 2Gy ( f ) = H ( f ) Gx ( f ) = H ( f ) X ( f ) since x( t ) is an energy signal 2 2 − jω 0.05 f  1 f   f   1 f  = 3e Π  sinc =  9Π    sinc 2   40  10 10   40   100 10  20 9 fso Ry (τ ) = ∫ sinc 2 e j2π f τ df −20 100 10 3-18
  36. 36. Chapter 44.1-1vi (t ) = v1 (t ) + v2 (t) c o s α vq (t ) = v2 (t) s i n αA(t ) = v12 ( t ) + 2 v1(t ) v2 (t) c o s α + v2 (t ) ≈ v1 (t ) + v2 ()cos α 2 t v2 (t) s i n α v (t) s i n αφ (t ) = arctan ≈ 2 v1(t ) + v2 (t) c o s α v1 (t )4.1-2vi (t ) = [ v1( t ) + v2 (t )] cos ω 0t vq (t ) = [ v2 ( t ) − v1 (t )] sinω 0tA(t ) = v12 ( t ) + 2 v1 (t ) v2 ( t) c o s 2ω0t + v 2 (t ) ≈ v1 (t ) + v2 (t) c o s 2ω 0t 2 [ v2 (t) − v1(t )] sinω 0tφ (t ) = arctan ≈ −ω 0t [ v1 (t) + v2 (t )] cos ω0t4.1-3 ∞ ∞ ∞(a) ∫−∞ vbp (t )dt = ∫ vi (t) c o s ω ct dt − ∫ vq ()sin ωc t dt −∞ t −∞ ∞ ∞ vi (t) c o s ω ct dt = ∫ Vi ( f ) δ ( f − f c ) + δ ( f + f c )  df = [Vi ( f c ) +Vi ( − fc) ] = 0 1 ∗ 1∫ −∞ −∞ 2   2since f c > W and Vi ( f ) = 0 for f > W ∞ ∞ 1 ∗∫ vq (t) s i n ω ct dt = ∫ Vq ( f ) e − jπ / 2δ ( f − fc ) + e jπ / 2δ ( f + f c )  df   −∞ −∞ 2 1 = Vq ( f c ) e− jπ / 2 + Vq (− fc )e jπ / 2  = 0 2  ∞Thus, ∫ −∞ bp v (t ) dt = 0 (cont). 4-1
  37. 37. ( v (t) c o s ω t − v sinω ct ) dt ∞(b) Ebp = ∫ 2 i c q −∞ 1 ∞ 2 ∞ ∞ ∞ ∞ =∫−∞ vi dt + ∫−∞ vq dt + ∫−∞ vi cos2ωc t dt + ∫ −∞ vq sin2ω ct dt + ∫−∞ vivq sin2ωc t dt  2 2 2 2   but vi , vq , and vi vq are bandlimited in 2W < 2 fc so, from the analysis in part (a) 2 2 ∞ ∞ ∞∫ vi2 cos2ω ct dt = ∫ vq sin2ω ct dt = ∫ vivq sin2ωc t dt = 0 2 −∞ −∞ −∞ ∫−∞ vi dt + ∫−∞ vq dt  = 2 ( Ei + Eq ) 1 ∞ 2 ∞ 1Hence, Ebp = 2 2   4.1-4 f + 100 Vlp ( f ) = Π   400   vlp (t ) = 400sinc400t e − j 2π 100t = 400sinc400t ( cos2π 100t + j sin2π 100t )vi (t ) = 800sinc400t cos2π 100t vq (t ) = −800sinc400t sin2π 100t4.1-5 1  f − 75   f + 50 Vlp ( f ) = Π  +Π   2  100   150  150 j 2 π 75 t − j 2π 50 tvlp (t ) = sinc150t e + 100sinc100t e 2vi (t ) = 2Re  vlp (t )  = 150sinc150t cos2π 75t + 200sinc100t cos2π 50t  vq (t ) = 2Im  vl p (t )  = 150sinc150t sin2π 75t − 200sinc100t sin2π 50t  4.1-6vbp (t ) = 2 z( t ) cos ( ±ω 0t + α ) cos ω ct − sin ( ±ω 0t + α ) sinω c t   so vi (t ) = 2 z( t) cos ( ±ω0t + α ) vq (t ) = 2 z( t )sin ( ±ω 0t + α ) 1vlp (t ) = 2 z ( t )  cos ( ±ω 0t + α ) + j sin ( ±ω 0t + α )  = z (t) e j( ±ω 0 t+α )   24.1-7 −1  f0   2 2 f 1  f f  H ( f ) = 1 + Q  −   = 2 ⇒ Q  − 0  = ±1    f0 f    2  f0 f so Q 2 f0 f f ± f = Qf 0 = 0 ⇒ f l , f u = 0 1 + 4Q 2 ± 1 2Q ( ) f ( f )B = f l − f u = 0 1 + 4Q2 + 1 − 0 1 + 4Q2 − 1 = 0 2Q 2Q f Q ( ) 4-2
  38. 38. 4.1-8 f f0 = (1 + δ ) ≈ 1 − δ , so −1 = 1+ δ , f0 f { } −1 1H ( f ) ≈ 1 + jQ  1+ δ − ( 1− δ )    = 1 + j 2Qδ f f − f0But δ = −1 = so f0 f0 1 f = f 0 (1 + δ ) > 0H(f ) ≈ for 1 + j 2 Q( f − f 0 ) / f 0 f − f0 = δ f0 = f 04.1-9 1H(f ) ≈ stagger-tuned 1 + ( f − fc + b) / b 2 1 + ( f − fc − b) / b2 2 2 1 ≈ single tuned 1 + ( f − f c ) / 2b 2 24.1-10 1 πBH lp ( f ) = = ⇒ hlp (t ) = π Be −π Bt u(t ) 1 + j 2 f / B π B + j2π f A jω t  Axbp (t ) = 2Re  u (t )e c  ⇒ xlp (t ) = u (t ) 2  2 π BA t −π B(t −λ ) d λ = (1 − e− π Bt ) u (t ) Aylp (t ) = hlp ∗ xlp (t ) = 2 ∫0 e 2ybp (t ) = 2Re  ylp (t ) e  = A (1 − e ) cos ωc t u(t ) j ωc t −π Bt   4-3
  39. 39. 4.1-11H lp ( f ) = Π   e− j(ω +ω c )td ⇒ hlp ( t ) = Be − jω ctd sinc B ( t − t d ) f B   A  Axbp (t ) = 2Re  u (t )e jω ct  ⇒ xlp (t ) = u (t ) 2  2 BA − jωc td tylp (t ) = hlp ∗ xlp (t ) = 2 e ∫−∞ sinc B ( λ − td ) d λ B ( t− td ) = e − jω ctd  ∫ sinc µ d µ + ∫ sinc µ d µ  A 0 2  −∞  0   = e − jω ctd  + Siπ B ( t − t d )  A 1 1 2 π  2   1 1 ybp (t ) = 2Re  ylp (t )e jωc t  = A  + Si π B ( t − t d )  cos ω c ( t − t d )   2 π 4.1-12  1  + δ ( f m f0 )  jα ± jω 0t jαxlp (t ) = 2e u( t) e ⇒ Xlp ( f ) = e   jπ ( f m f 0 )   f  BH lp ( f ) = Π   with = f 0 so δ ( f m f 0 ) falls outside passband. B 2 e jα f  eα f  BThus, Ylp ( f ) = Π ≈ Π   since f 0 ? f for f < jπ ( f m f 0 )  B  m jπ f 0  B  2  e jα ylp (t ) ≈ ± j   B sinc Bt  π f0  2B 2Bybp (t ) ≈ sinc Bt Re  ± je jα e jω ct  = m   sinc Bt sin (ω ct + α ) π f0 π f0 4-4
  40. 40. 4.1-13 f  1 BH lp ( f ) = e jf X lp ( f ) = Z ( f ) = 0 f ≤ W ≤ 2 /b B   2 2 2 1 1 f 2 f2Ylp ( f ) = e jf /b Z ( f ) ≈ 1 + j  Z ( f ) since = 1 for f ≤ W 2 2 b  b 1 j  1 j d2  ≈  Z ( f ) − 2 ( j 2π f ) Z ( f )  ⇒ ylp (t ) ≈  z ( t ) − 2 2 z (t) 2 4π b  2 4π b dt 2  1  d2 Thus, ybp (t ) ≈ z ()cos ω ct − t 2  z (t )  sin ω ct 4π b  dt 2 4.2-14.2-24.2-3 Ac (1 + µ 2 Sx ) = (1 + 0.6 2 ) = 68W 1 2 100AM: BT = 400Hz ST = 2 2 1 100DSB: BT = 400Hz ST = Ac2S x = = 50W 2 2 4-5
  41. 41. 4.2-4 1  f sinc2 40t ↔Λ  40  40 BT = 2W = 80 Hz4.2-5Amax = ( 2 Ac ) = 32kW ⇒ A c = 8kW 2 2 2 A (1+ µ 2 S x ) = 6kW 1 1 2µ = 1, S x = ⇒ ST = 2 2 c4.2-6 1 1  µ2  4Sx = , ST = Ac2 1 +  = 1kW ⇒ Ac = 2 kW 2 2  2  2+ µ2 (1 + µ )2 = (1 + µ ) A = 4 2A 2 2 kW ≤ 4kW 2 + µ2 max cso 1 + 2 µ + µ 2 ≤ 2 + µ 2 ⇒ µ ≤ 0.54.2-7 x max = x(0) =3 K (1 + 2) ≤1 ⇒ K ≤1 / 9 2 13  1 45 2 2 45 2Psb =  KAc  + ( 3KAc ) = K Ac = 2 K Pc 2 2  2 8 4 45 2 K Pc2Psb 2 45K 2 45 = = ≤ ≈ 22% ST 45 2 2 + 45 K 2 207 Pc + K Pc 2 4-6
  42. 42. 4.2-8x (t ) = 2 K cos20π t + K cos12π t + K cos28π t x max = x(0) = K (2 +1 + 1) ≤ 1 ⇒ K ≤ 1 / 4 2 1 11  3 3Psb = ( KAc ) + 2 ×  KAc  = K 2 Ac2 = K 2 Pc 2 2 22  4 22Psb 3K 2 Pc 3 = =≤ ≈ 16% ST Pc + 3K Pc 2 194.2-9 π 1 1x (t ) = 4sin t = 4sin2π t B T = 2W = kHz 2 4 2 BT0.01 < < 0.1 ⇒ 10BT < f c < 100 BT fc5 kHz < f c < 50 kHz4.2-10xc (t ) = Ac [1 + x (t )] cos ω ct ⇒ A(t ) = Ac [1 + x (t )] ≥ 0 for no phase reversals to occurSince x(t ) min = −4 there is no value of Ac that can keep A( t) from going negative.Therefore phase reversals will occur whenever x (t ) goes negative.4.2-11  1 xc (t ) = Ac  cos2π 40t + cos2π 90t  cos2π f ct  2  4-7
  43. 43. 4.3-1  2a  (a) vout = a1 x(t ) + a2 x 2 (t ) + a2 cos 2 ω c t + a1 1 + 2 x (t ) cos ω c t  1444 a1 424444  3 desired term Select a filter centered at f c =10 kHz with a bandwidth of 2W = 2 ×120 = 240 Hz.  2a   1  (b) a1 1 + 2 x(t ) cos ωc t = Ac [1 + µ x( t )] cos ω ct = 10 1 + x(t ) cos ωc t  a1   2  2a2 1 5 ⇒ a1 = 10 = ⇒ a2 = a1 2 24.3-2xc (t ) = aK 2 ( x + A cos ω c t ) − b ( x − A cos ω ct ) 2 2 = ( aK 2 − b )( x2 + A2 cos 2 ω c t ) + 2 A ( aK 2 + b ) x cos ω c t b = 4 Abx(t) c o s ωc t if K = a4.3-3xc (t ) = aK 2 ( v + A cos ω ct ) − b ( v − A cos ω c t ) 2 2 = ( aK 2 − b )( v 2 + A2 cos 2 ω ct ) + 2 A ( aK 2 + b ) v cos ω ct = 4 Ab [1 + µ x (t )] cos ω ct b if K = and v(t ) = 1 + µ x( t ) a 4-8
  44. 44. 4.3-4Take vin = x + cos ω 0t sovout = a1 ( x + cos ω 0t ) + a3 ( x3 + 3x 2 cos ω0t + 3x cos 3 ω0t + cos3 ω 0t )  3   3  3 1 =  a1 + a3  x + a 3x3 +  a1 + a 3 + 3a3 x 2  cos ω 0t + a3 x cos2ω 0t + a3 cos3ω0t  2   4  2 4Take f c = 2 f 0 where f 0 + 2W < 2 f 0 − W so f c > 6W4.3-5 Take vin = y + cos ω 0t , where y = Kx (t) , so vout = a1 ( y + cos ω 0t ) + a3 ( y 3 + 3 y 2 cos ω0t + 3 y cos 2 ω 0t + cos 3 ω0t )  3   3  3 1 =  a1 + a3  y + a 3 y 3 +  a1 + a 3 + 3a3 y 2  cos ω 0t + a3 y cos2ω 0t + a 3 cos3ω 0t  2   4  2 4Take f c = 2 f 0 where f 0 + 2W < 2 f 0 − W so f c > 6W 3   3a K xc (t ) =  a3Kx( t ) + Ac  cos ω ct = Ac 1 + 3 x (t )  cos ω ct 2   2 Ac  4-9

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