EngMathG
Worksheet1
March, 2021
[1] Find all the (complex) roots of the following equations:
(a) 𝑥𝑥 2 + 4𝑥𝑥 + 5 = 0
(b) 𝑥𝑥 2 + 4𝑥𝑥 − 5 = 0
(c) 𝑥𝑥 2 − 4𝑥𝑥 + 5 = 0
(d) 𝑥𝑥 2 − 4𝑥𝑥 − 5 = 0
(e) 𝑥𝑥 2 + 4𝑥𝑥 + 4 = 0
(f) 𝑥𝑥 2 + 3𝑥𝑥 + 3 = 0
(g) 𝑥𝑥 2 + 3𝑥𝑥 − 3 = 0
(h) 𝑥𝑥 2 + 3𝑥𝑥 + 2 = 0
[2] Given four complex numbers 𝑧𝑧1 = 2 + 𝑗𝑗 , 𝑧𝑧2 = −2 + 𝑗𝑗, 𝑧𝑧3 = 1 + 2𝑗𝑗, 𝑧𝑧4 = −4 − 2𝑗𝑗 , evaluate
the following quantities using the rectangular-form operations, and represent them in the rectangular
form.
(a) 𝑧𝑧2 ∙ 𝑧𝑧3
(b) 𝑧𝑧3 ∙ 𝑧𝑧4 ∙ 𝑧𝑧1
(c) 𝑧𝑧3 ∙ 𝑧𝑧4 ∙ 𝑧𝑧�1 ∙ 𝑧𝑧2
(d) 𝑧𝑧1 /𝑧𝑧2
(e) 𝑧𝑧4 /𝑧𝑧3
(f)
(𝑧𝑧1 ∙ 𝑧𝑧3 )/(𝑧𝑧2 ∙ 𝑧𝑧4 )
(g) (𝑧𝑧1 ∙ 𝑧𝑧2 )/(𝑧𝑧�3 ∙ 𝑧𝑧4 )
[3] Given four complex numbers 𝑧𝑧1 = 2 + 𝑗𝑗, 𝑧𝑧2 = −2 + 𝑗𝑗, 𝑧𝑧3 = 1 + 2𝑗𝑗, 𝑧𝑧4 = −4 − 2𝑗𝑗, represent
the four complex numbers in the Euler’s form. For the Euler’s form, no a numerical value, but an
𝑦𝑦
expression of tan−1 �𝑥𝑥 �-form may be allowable.
[4] Given four complex numbers 𝑧𝑧1 = 2 + 𝑗𝑗 , 𝑧𝑧2 = −2 + 𝑗𝑗, 𝑧𝑧3 = 1 + 2𝑗𝑗, 𝑧𝑧4 = −4 − 2𝑗𝑗 , evaluate
the following quantities using the Euler-form operations, and represent them in the Euler’s form.
(a) 𝑧𝑧2 ∙ 𝑧𝑧3
(b) 𝑧𝑧3 ∙ 𝑧𝑧4 ∙ 𝑧𝑧1
(c) 𝑧𝑧3 ∙ 𝑧𝑧4 ∙ 𝑧𝑧�1 ∙ 𝑧𝑧2
(d) 𝑧𝑧1 /𝑧𝑧2
(e) 𝑧𝑧4 /𝑧𝑧3
(f) (𝑧𝑧1 ∙ 𝑧𝑧3 )/(𝑧𝑧2 ∙ 𝑧𝑧4 )
(g) (𝑧𝑧1 ∙ 𝑧𝑧2 )/(𝑧𝑧�3 ∙ 𝑧𝑧4 )
[5] Find all the (complex) roots of the following equations:
(a) 𝑥𝑥 8 − 1 = 0
(b) (𝑗𝑗𝑗𝑗)8 + 𝑗𝑗 = 0
(c) 𝑥𝑥 −8 + 𝑗𝑗 = 0
(d) 𝑥𝑥 8 − 64 = 0
(e) (2𝑥𝑥)8 + 64 = 0
(f) 𝑥𝑥 4 − 16 = 0
𝑥𝑥 4
(g) �2� + 16 = 0
[6] Find the 𝑛𝑛-th power of the complex number 𝑧𝑧 = 𝑒𝑒 𝑗𝑗(𝜋𝜋/2) .
(a) 𝑛𝑛 = 2
(b) 𝑛𝑛 = 3
(c) 𝑛𝑛 = 4
(d) 𝑛𝑛 = 5
(e) 𝑛𝑛 = 6
[7] Find the 𝑛𝑛-th power of the complex number 𝑧𝑧 = 2𝑒𝑒 𝑗𝑗(𝜋𝜋/2) .
(a) 𝑛𝑛 = 2
(b) 𝑛𝑛 = 3
(c) 𝑛𝑛 = 4
(d) 𝑛𝑛 = 5
(e) 𝑛𝑛 = 6
[8] Given a transfer function 𝐻𝐻(𝑠𝑠) =
1
, answer the following questions:
2𝜋𝜋
2𝜋𝜋
𝑗𝑗
−𝑗𝑗
�𝑠𝑠−𝑒𝑒 3 ��𝑠𝑠−𝑒𝑒 3 �
(a) Convert the transfer function to a rational function (e.g., 𝐻𝐻(𝑠𝑠) =
𝑁𝑁(𝑠𝑠)
𝐷𝐷(𝑠𝑠)
, with 𝑁𝑁(𝑠𝑠) and 𝐷𝐷(𝑠𝑠)
are the enumerated numerator polynomial and enumerated denominator polynomial,
respectively.)
(b) Find the frequency response expression 𝐻𝐻(𝑗𝑗𝑗𝑗) as a function of 𝜔𝜔, in the rectangular form.
(c) Find the frequency response expression 𝐻𝐻(𝑗𝑗𝑗𝑗) as a function of 𝜔𝜔, in the Euler’s form.
[9] Given a transfer function 𝐻𝐻(𝑠𝑠) =
𝑠𝑠
, answer the following questions:
2𝜋𝜋
2𝜋𝜋
𝑗𝑗
−𝑗𝑗
�𝑠𝑠−𝑒𝑒 3 ��𝑠𝑠−𝑒𝑒 3 �
(a) Convert the transfer function to a rational function (e.g., 𝐻𝐻(𝑠𝑠) =
𝑁𝑁(𝑠𝑠)
𝐷𝐷(𝑠𝑠)
, with 𝑁𝑁(𝑠𝑠) and 𝐷𝐷(𝑠𝑠)
are the enumerated numerator polynomial and enumerated denominator polynomial,
respectively.)
(b) Find the frequency response expression 𝐻𝐻(𝑗𝑗𝑗𝑗) as a function of 𝜔𝜔, in the rectangular form.
(c) Find the frequency response expression 𝐻𝐻(𝑗𝑗𝑗𝑗) as a function of 𝜔𝜔, in the Euler’s form.
[10] Find the phasor of the following sinusoids:
(a) 𝑥𝑥(𝑡𝑡) = 10 cos(20𝜋𝜋𝜋𝜋 + π/4)
(b) 𝑥𝑥(𝑡𝑡) = 10 cos(20𝑡𝑡 + π/4)
(c) 𝑥𝑥(𝑡𝑡) = 5 sin(20𝑡𝑡 + π/4)
(d) 𝑥𝑥(𝑡𝑡) = −5 cos(20𝑡𝑡 + π/4)
(e) 𝑥𝑥(𝑡𝑡) = −5 sin(50𝑡𝑡 + π/4)
[11] Find the corresponding sinusoids of the following phasors:
(a) 𝐗𝐗 = 10𝑒𝑒 𝑗𝑗45° (𝜔𝜔 = 10 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟/𝑠𝑠)
(b) 𝐗𝐗 = 10𝑒𝑒 𝑗𝑗45° (𝜔𝜔 = 100 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟/𝑠𝑠)
(c) 𝐗𝐗 = 10𝑒𝑒 𝑗𝑗45° − 10 (𝜔𝜔 = 100 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟/𝑠𝑠)
(d) 𝐗𝐗 = 8 + 𝑗𝑗6 (𝜔𝜔 = 100 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟/𝑠𝑠)
π
π
[12] Given 𝑥𝑥1 (𝑡𝑡) = 10 cos �𝑡𝑡 + 4 � and 𝑥𝑥2 (𝑡𝑡) = 10 �2𝑡𝑡 + 4 �, answer the following questions:
(a) Find the corresponding phasors of the sinusoids 𝑥𝑥1 (𝑡𝑡) and 𝑥𝑥2 (𝑡𝑡).
𝑑𝑑 2 𝑥𝑥 (𝑡𝑡)
𝑑𝑑𝑥𝑥 (𝑡𝑡)
1
(b) Find the phasors of 𝑧𝑧1 (𝑡𝑡) = 𝑑𝑑𝑡𝑡12 + 3 𝑑𝑑𝑑𝑑
+ 4𝑥𝑥1 (𝑡𝑡) and 𝑧𝑧2 (𝑡𝑡) =
(c) Find the corresponding sinusoids of 𝑧𝑧1 (𝑡𝑡) and 𝑧𝑧2 (𝑡𝑡).
𝑑𝑑 2 𝑥𝑥2 (𝑡𝑡)
𝑑𝑑𝑡𝑡 2
+3
𝑑𝑑𝑥𝑥2 (𝑡𝑡)
𝑑𝑑𝑑𝑑
[13] Find the impedances 𝑍𝑍𝑎𝑎𝑎𝑎 of the following circuit parts in terms of 𝜔𝜔, 𝑅𝑅, 𝐿𝐿, and 𝐶𝐶:
(a)
(b)
(c)
(d)
(e)
[14] Answer the following questions for the circuit below:
(a) Find the impedance 𝑍𝑍𝑎𝑎𝑎𝑎 as a function of 𝜔𝜔.
(b) Find the value of 𝜔𝜔 such that 𝑍𝑍𝑎𝑎𝑎𝑎 = 𝑅𝑅 = 10𝑘𝑘Ω.
+ 4𝑥𝑥2 (𝑡𝑡).
(c) Given 𝑣𝑣𝑖𝑖 (𝑡𝑡) = 10 cos(106 𝑡𝑡 + π/4), find 𝑣𝑣𝐿𝐿 (𝑡𝑡) using the phasor approach.
[15] A 2nd-order moving average filter is described by the following difference equation.
𝑦𝑦[𝑛𝑛] = 𝑥𝑥[𝑛𝑛] + 𝑥𝑥[𝑛𝑛 − 1] + 𝑥𝑥[𝑛𝑛 − 2]
(a) Find the transfer function 𝐻𝐻(𝑧𝑧).
(b) Find the poles and zeros of 𝐻𝐻(𝑧𝑧)
(c) Plot the corresponding pole-zero diagram.
(d) What is the analog frequency 𝑓𝑓 that this filter can remove, when the sampling rate 𝑓𝑓𝑠𝑠 =
24,000 Hz.
[16] The difference equation of a digital filter is given by
1
1
3
1
1
𝑦𝑦[𝑛𝑛] = 4 𝑦𝑦[𝑛𝑛 − 1] + 4 𝑦𝑦[𝑛𝑛 − 2] − 8 𝑦𝑦[𝑛𝑛 − 3] + 𝑥𝑥[𝑛𝑛] − 2 𝑥𝑥[𝑛𝑛 − 1] − 2 𝑥𝑥[𝑛𝑛 − 2]
(a) Find the transfer function 𝐻𝐻(𝑧𝑧).
(b) Find the poles and zeros of 𝐻𝐻(𝑧𝑧).
(c) Plot the pole-zero diagram of 𝐻𝐻(𝑧𝑧).
[17] The poles and zeros of a digital filter are as follows:
𝜋𝜋
𝜋𝜋
𝜋𝜋
𝜋𝜋
1
1
1
1
Poles: 𝑝𝑝1 = 𝑒𝑒 𝑗𝑗 4 , 𝑝𝑝2 = 𝑒𝑒 −𝑗𝑗 4 , 𝑝𝑝3 = 2 𝑒𝑒 𝑗𝑗 2 , 𝑝𝑝4 = 2 𝑒𝑒 −𝑗𝑗 2
√2
√2
Zeros: 𝑧𝑧1 = 0, 𝑧𝑧2 = 0, 𝑧𝑧3 = 1, 𝑧𝑧4 = −1
(a) Plot the pole-zero diagram of the digital filter.
(b) Find the transfer function 𝐻𝐻(𝑧𝑧).
(c) Find the difference equation [The current output should be expressed as a function of previous
outputs and current and previous inputs].
[18] You are asked to design a digital filter that will remove the two frequencies 𝑓𝑓𝑟𝑟1 = 5,500 Hz and
𝑓𝑓𝑟𝑟2 = 11,000 Hz in the signal, whose spectrum is shown below. Note that the sampling frequency
𝑓𝑓𝑠𝑠 = 55,000 Hz.
(a) Find the zeros and poles by setting the corresponding poles to 𝑝𝑝i = 0.99𝑧𝑧𝑖𝑖 .
(b) Plot the pole-zero diagram.
(c) Obtain the transfer function 𝐻𝐻(𝑧𝑧).
(d) Obtain the difference equation [The current output should be expressed as a function of
previous outputs and current and previous inputs].
Hint: Fill the table.
-----------------------------------------------------------------------------------------------------------------------------------Sampling frequency
Target frequency
Zeros
Poles (with 𝑟𝑟 = 0.99)
-----------------------------------------------------------------------------------------------------------------------------------𝑓𝑓𝑟𝑟1 = 5,500 Hz
𝑧𝑧1 =
𝑝𝑝1 =
𝑓𝑓𝑠𝑠 = 55,000 Hz
𝑧𝑧2 =
𝑝𝑝2 =
𝑓𝑓𝑟𝑟2 = 11,000 Hz
𝑧𝑧3 =
𝑝𝑝3 =
𝑧𝑧4 =
𝑝𝑝4 =