B.8-1 Some Useful Constants
π≈3.1415926535 e≈2.7182818284
1
e≈0.3678794411 log102≈0.30103 log103≈0.47712
B.8-2 Complex Numbers
e±jπ/2= ±j e±jnπ=
1 neven
−1 nodd e±jθ=cosθ±jsinθ a+jb=rejθ r=√
a2+b2,θ=tan−1b
a
(rejθ)k=rkejkθ
(r1ejθ1)(r2ejθ2)=r1r2ej(θ1+θ2)
B.8-3 Sums
"n k=m
rk=rn+1−rm
r−1 r=1
"n k=0
k=n(n+1) 2
"n k=0
k2=n(n+1)(2n+1) 6
"n k=0
k rk=r+ [n(r−1)−1]rn+1
(r−1)2 r=1
"n k=0
k2rk=r[(1+r)(1−rn)−2n(1−r)rn−n2(1−r)2rn]
(1−r)3 r=1
B.8-4 Taylor and Maclaurin Series
f(x)=f(a)+(x−a)
1! ˙f(a)+(x−a)2
2! fă(a)+ ã ã ã =
"∞
k=0
(x−a)k k! f(k)(a) f(x)=f(0)+ x
1!˙f(0)+x2
2!fă(0)+ ã ã ã =
"∞
k=0
xk k!f(k)(0)
B.8-5 Power Series
ex=1+x+x2 2!+x3
3!+ ã ã ã +xn n!+ ã ã ã sinx=x−x3
3!+x5 5!−x7
7!+ ã ã ã cosx=1−x2
2!+x4 4!−x6
6!+x8 8!− ã ã ã tanx=x+x3
3 +2x5 15 +17x7
315 + ã ã ã x2< π2/4 tanhx=x−x3
3 +2x5 15 −17x7
315 + ã ã ã x2< π2/4 (1+x)n=1+nx+n(n−1)
2! x2+n(n−1)(n−2)
3! x3+ ã ã ã + n
k
xk+ ã ã ã +xn (1+x)n≈1+nx |x| 1
1
1−x=1+x+x2+x3+ ã ã ã |x|<1
B.8-6 Trigonometric Identities
e±jx=cosx±jsinx cosx=12[ejx+e−jx] sinx=2j1[ejx−e−jx] cos(x±π2)= ∓sinx sin(x±π2)= ±cosx 2 sinxcosx=sin 2x
sin2x+cos2x=1 cos2x−sin2x=cos 2x cos2x=12(1+cos 2x) sin2x=12(1−cos 2x)
cos3x=14(3 cosx+cos 3x) sin3x=14(3 sinx−sin 3x)
sin(x±y)=sinxcosy±cosxsiny cos(x±y)=cosxcosy∓sinxsiny tan(x±y)= tanx±tany
1∓tanxtany
sinxsiny=12[cos(x−y)−cos(x+y)]
cosxcosy=12[cos(x−y)+cos(x+y)]
sinxcosy=12[sin(x−y)+sin(x+y)]
acosx+bsinx=Ccos(x+θ) C=√
a2+b2,θ=tan−1−b
a
B.8-7 Common Derivative Formulas
d
dxf(u)= d duf(u)du
dx d
dx(uv)=udv dx+vdu
dx d
dx u
v
=vdudx−udvdx v2 dxn
dx =nxn−1 d
dxln(ax)=1 x d
dxlog(ax)=loge x d
dxebx=bebx d
dxabx=b(lna)abx d
dxsinax=acosax d
dxcosax= −asinax d
dxtanax= a cos2ax d
dx(sin−1ax)= a
√1−a2x2 d
dx(cos−1ax)= −a
√1−a2x2 d
dx(tan−1ax)= a 1+a2x2
B.8-8 Indefinite Integrals
#
u dv=uv−
# v du
#
f(x)˙g(x)dx=f(x)g(x)−
# ˙f(x)g(x)dx
#
sinax dx= −1 acosax
#
cosax dx=1 asinax
#
sin2ax dx=x
2−sin 2ax 4a
#
cos2ax dx=x
2+sin 2ax
# 4a
xsinax dx= 1
a2(sinax−axcosax)
#
xcosax dx= 1
a2(cosax+axsinax)
#
x2sinax dx= 1
a3(2axsinax+2 cosax−a2x2cosax)
#
x2cosax dx= 1
a3(2axcosax−2 sinax+a2x2sinax)
#
sinaxsinbx dx=sin(a−b)x
2(a−b) −sin(a+b)x
2(a+b) a2=b2
#
sinaxcosbx dx= − cos(a−b)x
2(a−b) +cos(a+b)x 2(a+b)
!
a2=b2
#
cosaxcosbx dx=sin(a−b)x
2(a−b) +sin(a+b)x
2(a+b) a2=b2
#
eaxdx=1 aeax
#
xeaxdx=eax
a2(ax−1)
#
x2eaxdx=eax
a3(a2x2−2ax+2)
#
eaxsinbx dx= eax
a2+b2(asinbx−bcosbx)
#
eaxcosbx dx= eax
a2+b2(acosbx+bsinbx)
# 1
x2+a2dx=1 atan−1x
# a x
x2+a2dx=1
2ln(x2+a2)
B.8-9 L’Hôpital’s Rule
If limf(x)/g(x)results in the indeterministic form 0/0 or∞/∞, then limf(x)
g(x)=limf˙(x)
˙ g(x)
B.8-10 Solution of Quadratic and Cubic Equations
Anyquadraticequation can be reduced to the form ax2+bx+c=0 The solution of this equation is provided by
x=−b±√
b2−4ac 2a A generalcubicequation
y3+py2+qy+r=0 may be reduced to thedepressed cubicform
x3+ax+b=0 by substituting
y=x−p 3 This yields
a=13(3q−p2) b=271(2p3−9pq+27r) Now let
A= 3
−b 2+
b2 4 +a3
27 B= 3
−b 2−
b2 4 +a3
27 The solution of the depressed cubic is
x=A+B, x= −A+B
2 +A−B 2
√−3, x= −A+B
2 −A−B 2
√−3
and
y=x−p 3 R E F E R E N C E S
1. Asimov, Isaac.Asimov on Numbers.Bell Publishing, New York, 1982.
2. Calinger, R., ed.Classics of Mathematics.Moore Publishing, Oak Park, IL, 1982.
3. Hogben, Lancelot.Mathematics in the Making.Doubleday, New York, 1960.
4. Cajori, Florian.A History of Mathematics,4th ed. Chelsea, New York, 1985.
5. Encyclopaedia Britannica.MicropaediaIV, 15th ed., vol. 11, p. 1043. Chicago, 1982.
6. Singh, Jagjit.Great Ideas of Modern Mathematics.Dover, New York, 1959.
7. Dunham, William.Journey Through Genius.Wiley, New York, 1990.
P R O B L E M S
B.1-1 Given a complex numberw=x+jy, the com- plex conjugate of w is defined in rectangular coordinates asw∗=x−jy. Use this fact to derive complex conjugation in polar form.
B.1-2 Express the following numbers in polar form:
(a) wa=1+j (b) wb=1+ej (c) wc= −4+j3 (d) wd=(1+j)(−4+j3) (e) we=ejπ/4+2e−jπ/4 (f) wf=12j+j
(g) wg=(1+j)/(−4+j3) (h) wh=sin1−(jj)
B.1-3 Express the following numbers in Cartesian (rectangular) form:
(a) wa=j+ej (b) wb=3ejπ/4 (c) wc=1/ej
(d) wd=(1+j)(−4+j3) (e) we=ejπ/4+2e−jπ/4 (f) wf=ej+1 (g) wg=1/2j
(h) wh=jjj (jraised to thejraised to thej) B.1-4 Showing all work and simplifying your answer,
determine the real part of the following num- bers:
(a) wa=1j(j−5e2−3j) (b) wb=(1+j)ln(1+j)
B.1-5 Showing all work and simplifying your answer, determine the imaginary part of the following numbers:
(a) wa= −jejπ/4 (b) wb=1−2je2−4j (c) wc=tan(j)
B.1-6 For complex constantw, prove:
(a) Re(w)=(w+w∗)/2 (b) Im(w)=(w−w∗)/2j
B.1-7 Givenw=x−jy, determine:
(a) Re(ew) (b) Im(ew)
B.1-8 For arbitrary complex constants w1 and w2, prove or disprove the following:
(a) Re(jw1)= −Im(w1) (b) Im(jw1)=Re(w1)
(c) Re(w1)+Re(w2)=Re(w1+w2) (d) Im(w1)+Im(w2)=Im(w1+w2) (e) Re(w1)Re(w2)=Re(w1w2) (f) Im(w1)/Im(w2)=Im(w1/w2) B.1-9 Givenw1=3+j4 andw2=2ejπ/4.
(a) Expressw1in standard polar form.
(b) Expressw2in standard rectangular form.
(c) Determine|w1|2and|w2|2.
(d) Express w1+w2 in standard rectangular form.
(e) Expressw1−w2in standard polar form.
(f) Expressw1w2in standard rectangular form.
(g) Expressw1/w2in standard polar form.
B.1-10 Repeat Prob. B.1-9 usingw1=(3+j4)2 and w2=2.5je−j40π.
B.1-11 Repeat Prob. B.1-9 using w1= j+eπ/4 and w2=cos(j).
B.1-12 Using the complex plane:
(a) Evaluate and locate the distinct solutions to (w)4= −1.
(b) Evaluate and locate the distinct solutions to (w−(1+j2))5=(32/√
2)(1+j). (c) Sketch the solution to|w−2j| =3.
(d) Graphw(t)=(1+t)ejtfor(−10≤t≤10). B.1-13 The distinct solutions to (w−w1)n =w2 lie on a circle in the complex plane, as shown in Fig. PB.1-13. One solution is located on the real axis at√
3+1=2.732, and one solution is located on the imaginary axis at√
3−1=0.732.
Determinew1,w2, andn.
0 0
Figure PB.1-13
B.1-14 Find the distinct solutions to each of the follow- ing. Use MATLAB to graph each solution set in the complex plane.
(a) w3= −278 (b) (w+1)8=1 (c) w2+j=0
(d) 16(w−1)4+81=0 (e) (w+2j)3= −8
(f) (j−w)1.5=2+j2 (g) (w−1)2.5=j4√
2 B.1-15 Ifj=√
−1, what is√ j?
B.1-16 Find all the values of ln(−e), expressing your answer in Cartesian form.
B.1-17 Determine all values of log10(−1), expressing your answer in Cartesian form. Notice that the logarithm has base 10, note.
B.1-18 Express the following in standard rectangular coordinates:
(a) wa=ln(1/(1+j)) (b) wb=cos(1+j) (c) wc=(1−j)j
B.1-19 By constrainingwto be purely imaginary, show that the equation cos(w)=2 can be represented as a standard quadratic equation. Solve this equation forw.
B.1-20 Certain integrals, although expressed in rela- tively simple form, are quite difficult to solve.
For example,$
e−x2dx cannot be evaluated in terms of elementary functions; most calculators that perform integration cannot handle this indefinite integral. Fortunately, you are smarter than most calculators.
(a) Expresse−x2 using a Taylor series expan- sion.
(b) Using your series expansion fore−x2, deter- mine$
e−x2dx.
(c) Using a suitably truncated series, evaluate the definite integral$1
0e−x2dx.
B.1-21 Repeat Prob. B.1-20 for$ e−x3dx.
B.1-22 Repeat Prob. B.1-20 for$
cosx2dx.
B.1-23 For each function, determine a suitable series expansion.
(a) fa(x)=(2−x2)−1 (b) fb(x)=(0.5)x
B.1-24 Consider the functionf(x)=1+x+x2+x3. (a) Express f(x) using a Taylor series with
expansion point ofa=1. Explicitly write out every term. [Hint:See Sec. B.8-4.]
(b) Describe a good reason why you might want to express a function that is already a simple polynomial using such a series.
B.1-25 Determine the Maclaurin series expansion of each of the following. [Hint: See Sec. B.8-4.]
(a) fa(x)=2x (b) fb(x)=1
3
x
B.2-1 Determine the fundamental period T0, fre- quency f0, and radian frequency ω0 for the following sinusoids:
(a) cos(5πt+3) (b) 7 sin2t−π
3
B.2-2 Determine an expression for a sinusoid that oscillates 15 times per second, that has a value of -1 att=0, and whose peak amplitude is 3.
Use MATLAB to plot the signal over 0≤t≤1.
B.2-3 Let x1(t)=2 cos(3t+1) and x2(t)= −3 cos (3t−2).
(a) Determine a1 and b1 so that x1(t) = a1cos(3t)+b1sin(3t).
(b) Determine a2 and b2 so that x2(t) = a2cos(3t)+b2sin(3t).
(c) DetermineCandθ so thatx1(t)+x2(t)= Ccos(3t+θ).
B.2-4 In addition to the traditional sine and cosine functions, there are the hyperbolic sine and cosine functions, which are defined by sinh(w) = (ew−e−w)/2 and cosh(w) = (ew+e−w)/2. In general, the argument is a complex constantw=x+jy.
(a) Show that cosh(w) = cosh(x)cos(y)+ jsinh(x)sin(y).
(b) Determine a similar expression for sinh(w) in rectangular form that only uses functions of real arguments, such as sin(x), cosh(y), and so on.
B.2-5 Use Euler’s identity to solve or prove the following:
(a) Find real, positive constantscandφfor all real t such that 2.5 cos(3t)−1.5 sin(3t+ π/3)=ccos(3t+φ). Sketch the resulting sinusoid.
(b) Prove that cos(θ±φ)=cos(θ)cos(φ)∓ sin(θ)sin(φ).
(c) Given real constantsa,b, andα, complex constantw, and the fact that
# b
a
ewxdx= 1
w(ewb−ewa) evaluate the integral
# b
a
ewxsin(αx)dx
B.2-6 A particularly boring stretch of interstate high- way has a posted speed limit of 70 mph.
A highway engineer wants to install “rumble bars” (raised ridges on the side of the road) so that cars traveling the speed limit will produce quarter-second bursts of 1 kHz sound every second, a strategy that is particularly effective at startling sleepy drivers awake. Provide design specifications for the engineer.
B.3-1 By hand, accurately sketch the following signals over(0≤t≤1):
(a) xa(t)=e−t (b) xb(t)=sin(2π5t) (c) xc(t)=e−tsin(2π5t)
B.3-2 In 1950, the human population was approxi- mately 2.5 billion people. Assuming a doubling time of 40 years, formulate an exponential model for human population in the formp(t)= aebt, wheretis measured in years. Sketchp(t) over the interval 1950≤t≤2100. According to this model, in what year can we expect the population to reach the estimated 15 billion carrying capacity of the earth?
B.3-3 Determine an expression for an exponen- tially decaying sinusoid that oscillates three
times per second and whose amplitude enve- lope decreases by 50% every 2 seconds. Use MATLAB to plot the signal over−2≤t≤2.
B.3-4 By hand, sketch the following against indepen- dent variablet:
(a) xa(t)=Re
2e(−1+j2π)t (b) xb(t)=Im
3−e(1−j2π)t (c) xc(t)=3−Im
e(1−j2π)t
B.4-1 Consider the following system of equations:
−1 2
3 −4
! x1 x2
!
= 3
−1
!
Expressing all answers in rational form (ratio of integers), use Cramer’s rule to determinex1and x2. Perform all calculations by hand, including matrix determinants.
B.4-2 Consider the following system of equations:
⎡
⎣ 1 2 0
0 3 4
5 0 6
⎤
⎦
⎡
⎣ x1
x2 x3
⎤
⎦=
⎡
⎣ 7 8 9
⎤
⎦
Expressing all answers in rational form (ratio of integers), use Cramer’s rule to determinex1, x2, and x3. Perform all calculations by hand, including matrix determinants.
B.4-3 Consider the following system of equations.
x1+x2+x3=1 x1+2x2+3x3=3 x1−x2= −3
Use Cramer’s rule to determinex1,x2, andx3. Matrix determinants can be computed by using MATLAB’sdetcommand.
B.5-1 Determine the constantsa0, a1, and a2 of the partial fraction expansion
F(s)= s (s+1)3
= a0
(s+1)3+ a1
(s+1)2+ a2
(s+1) B.5-2 Compute by hand the partial fraction expan-
sions of the following rational functions:
(a) Ha(s)= s3s+s2+25s+s+1+6 , which has denominator poles ats= ±jands= −1
(b) Hb(s)=H1
1(s)=s3s+s2+25s+s+1+6
(c) Hc(s)=(s+1)21(s2+1)
(d) Hd(s)=3ss22++2s+15s+6
B.5-3 Compute by hand the partial fraction expan- sions of the following rational functions:
(a) Fa(x)=(x−(x1−)(3x)−22)
(b) Fb(x)=(3x−1)(2x−1)(x−1)2
(c) Fc(x)=(3x−1)(x−1)2(2x−1)2
(d) Fd(x)=2xx22−+8x+65x+6
(e) Fe(x)=2xx22−3x−11−x−2
(f) Ff(x)=−33+2x+2x+2x2 (g) Fg(x)=x3+2xx22+1+3x+4
(h) Fh(x)=1+2x+3xx2+5x+62
(i) Fi(x)=3x3−xx22+14x+4+4
(j) Fj(x)=2xx−5+6x−1−1+−12x (k) Fk(x)=3−5x2−9x+23
x2+x−2
B.6-1 A system of equations in terms of unknownsx1
andx2and arbitrary constantsa,b,c,d,e, andf is given by
ax1+bx2=c dx1+ex2=f
(a) Represent this system of equations in matrix form.
(b) Identify specific constantsa,b,c,d,e, and f such that x1=3 andx2= −2. Are the constants you selected unique?
(c) Identify nonzero constantsa,b,c,d,e, and fsuch that no solutionsx1andx2exist.
(d) Identify nonzero constantsa,b,c,d,e, and f such that an infinite number of solutions x1andx2exist.
B.6-2 Using a matrix approach, solve the following system of equations:
x1+x2+x3+x4=4 x1+x2+x3−x4=2 x1+x2−x3−x4=0 x1−x2−x3−x4= −2
B.6-3 Using a matrix approach, solve the following system of equations:
x1+x2+x3+x4=1 x1−2x2+3x3=2 x1−x3+7x4=3
−2x2+3x3−4x4=4
B.6-4 A signal f(t)=acos(3t)+bsin(3t)reaches a peak amplitude of 5 at t=1.8799 and has a zero crossing att=0.3091. Use a matrix-based approach to determine the constantsaandb.
B.6-5 Define
x= 1 3
−2 4
!
, y= −5 2
! ,
and z= 0 1
−1 0
!
By hand, calculate the following:
(a) fa=yTy (b) fb=yyT (c) fc=xy (d) fd=xTy (e) fe=yTx (f) ff=xz (g) fg=zxz (h) fh=xT−z
B.7-1 Use MATLAB to produce the plots requested in Prob. B.3-4.
B.7-2 Use MATLAB to plot the function x(t) = tsin(2πt) over 0≤t≤10 using 501 equally spaced points. What is the maximum value of x(t)over this range oft?
B.7-3 Use MATLAB to plot x(t)= cos(t)sin(20t) over a suitable range oft.
B.7-4 Use MATLAB to plot x(t)=%10
k=1cos(2πkt) over a suitable range of t. The MATLAB commandsummay prove useful.
B.7-5 When a bell is struck with a mallet, it pro- duces a ringing sound. Write an equation that approximates the sound produced by a small, light bell. Carefully identify your assumptions.
How does your equation change if the bell is large and heavy? You can assess the quality of your models by using the MATLABsound command to listen to your “bell.”
B.7-6 You are working on a digital quadrature amplitude modulation (QAM) communication receiver. The QAM receiver requires a pair of quadrature signals: cosn and sinn. These can be simultaneously generated by following a simple procedure: (1) choose a pointwon the unit circle, (2) multiplywby itself and store the result, (3) multiplywby the last result and store, and (4) repeat step 3.
(a) Show that this method can generate the desired pair of quadrature sinusoids.
(b) Determine a suitable value of w so that good-quality, periodic, 2π×100,000 rad/s signals can be generated. How much time is available for the processing unit to compute each sample?
(c) Simulate this procedure by using MATLAB and report your results.
(d) Identify as many assumptions and limi- tations to this technique as possible. For example, can your system operate correctly for an indefinite period of time?
B.7-7 Using MATLAB’sresiduecommand, (a) Verify the results of Prob. B.5-2a.
(b) Verify the results of Prob. B.5-2b.
(c) Verify the results of Prob. B.5-2c.
(d) Verify the results of Prob. B.5-2d.
B.7-8 Using MATLAB’sresiduecommand, (a) Verify the results of Prob. B.5-3a.
(b) Verify the results of Prob. B.5-3b.
(c) Verify the results of Prob. B.5-3c.
(d) Verify the results of Prob. B.5-3d.
(e) Verify the results of Prob. B.5-3e.
(f) Verify the results of Prob. B.5-3f.
(g) Verify the results of Prob. B.5-3g.
(h) Verify the results of Prob. B.5-3h.
(i) Verify the results of Prob. B.5-3i.
(j) Verify the results of Prob. B.5-3j.
(k) Verify the results of Prob. B.5-3k.
B.7-9 Determine the original length-3 vectorsaandb need to produce the MATLAB output:
>> [r,p,k] = residue(b,a) r = 0 + 2.0000i
0 - 2.0000i
p = 3 -3
k = 0 + 1.0000i
B.7-10 Let N = [n7,n6,n5,...,n2,n1] represent the seven digits of your phone number. Construct a rational function according to
HN(s)=n7s2+n6s+n5+n4s−1 n3s2+n2s+n1
Use MATLAB’s residue command to com- pute the partial fraction expansion ofHN(s).
B.7-11 When plotted in the complex plane for
−π ≤ ω ≤ π, the function f(ω) = cos(ω)+ j0.1 sin(2ω) results in a so-called Lissajous figure that resembles a two-bladed propeller.
(a) In MATLAB, create two row vectorsfrand ficorresponding to the real and imaginary portions off(ω), respectively, over a suit- able numberNsamples ofω. Plot the real portion against the imaginary portion and verify the figure resembles a propeller.
(b) Let complex constantw=x+jybe repre- sented in vector form
w= x y
!
Consider the 2×2 rotational matrixR:
R= cosθ −sinθ sinθ cosθ
!
Show thatRwrotates vectorwbyθradians.
(c) Create a rotational matrixRcorresponding to 10◦and multiply it by the 2×Nmatrixf
= [fr;fi];. Plot the result to verify that the “propeller” has indeed rotated counter- clockwise.
(d) Given the matrixRdetermined in part (c), what is the effect of performingRRf? How aboutRRRf? Generalize the result.
(e) Investigate the behavior of multiplyingf(ω) by the functionejθ.