ECEN314 - Section 502
Assignment 1
Chapter 0 (Complex Numbers)
Solution Release Date: Sunday, January 23, 2022 (12:00 PM)
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Only the solution to the following problems will be released:
• Problem 1: a, e, i
• Problem 2: c, f, i
• Problem 3: a, b, c
• Problem 4: b, f
• Problem 5: c, e
• Problem 6: b, d
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Problems:
1. Express each of the following in Cartesian form:
(a)
(b)
(c)
1 jπ
3e
1 −jπ
3e
1 jπ/2
2e
(e)
1 −jπ/2
2e
e−j3π/2
(h)
(f)
2ejπ/4
(i)
(d)
√
(g)
√
√
√
2e−jπ/4
2ej7π/4
2e−j7π/4
2. Express each of the following in polar form:
1
2
√
3
2
(a) 6
(d)
(b) −3
(e) 1 − j
(h)
(c) −4j
(f) (1 − j)2
(i)
+j
1
(g) j(1 + j)
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1−j
1+j
√
√
2+j√ 2
1+j 3
3. Express each of the following in both Cartesian (real part and imaginary part) and polar
(magnitude and angle) forms:
√
(a) ( 3 + j 3 )(1 − j)
(b)
√
2−j6/ 3
√
2+j6/ 3
√
√
(c) ( 3 + j)2 2e−jπ/4
4. Using Euler’s formula, derive the following relationships:
(a) cos2 θ = 12 (1 + cos 2θ)
(b) sin2 θ = 12 (1 − cos 2θ)
1
2 sin 2θ
(sin θ)(sin φ) = 12 (cos (θ − φ) − cos (θ + φ))
(cos θ)(cos φ) = 12 (cos (θ − φ) + cos (θ + φ))
(sin θ)(cos φ) = 12 (sin (θ − φ) + sin (θ + φ))
(c) (sin θ)(cos θ) =
(d)
(e)
(f)
5. Evaluate each of the following sums, and write the final result in Cartesian form:
(a)
(d)
!10
πk
k=0 cos ( 2 )
!N −1
πk
k=0 cos ( N )
(b)
(e)
(fixed N > 0)
!10
j2πk/12
k=0 e
!N −1 jπk/N
k=0 e
(c)
(f)
(fixed N > 0)
!∞
1 k
πk
k=0 ( 3 ) cos ( 2 )
πk
k
k=0 α cos ( N )
!∞
(fixed 0 < α < 1)
6. Evaluate each of the following integrals, and write the final result in Cartesian form:
(a)
(d)
"3
jπt/2 dt
0 e
" T jat
0 e dt
(fixed T > 0, fixed a)
(b)
(e)
"∞
−2t sin (3t)dt
0 e
" ∞ −at
sin (ωt)dt
0 e
(c)
(f)
(fixed a > 0, fixed ω)
2
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"1
−jπt dt
0 te
" T −jωt
dt
0 te
(fixed T > 0, fixed ω)