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    Purdue Basics SIGNALS AND SYSTEMS

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    mdasifhassanapspdcl
    This document provides 12 questions about computing integrals and sums of various functions. Students enrolled in ECE 301-003 and OL1 are assigned this homework due on January 29, 2021. The questions cover reviewing calculus and arithmetic skills like computing integrals, finding derivatives, performing partial fraction decompositions, and evaluating sums of discrete sequences.

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    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd

    Purdue Basics SIGNALS AND SYSTEMS

    Uploaded by

    mdasifhassanapspdcl
    154 views4 pages
    This document provides 12 questions about computing integrals and sums of various functions. Students enrolled in ECE 301-003 and OL1 are assigned this homework due on January 29, 2021. The questions cover reviewing calculus and arithmetic skills like computing integrals, finding derivatives, performing partial fraction decompositions, and evaluating sums of discrete sequences.

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    purdue basics SIGNALS AND SYSTEMS

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    This document provides 12 questions about computing integrals and sums of various functions. Students enrolled in ECE 301-003 and OL1 are assigned this homework due on January 29, 2021. The questions cover reviewing calculus and arithmetic skills like computing integrals, finding derivatives, performing partial fraction decompositions, and evaluating sums of discrete sequences.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    154 views4 pages

    Purdue Basics SIGNALS AND SYSTEMS

    Uploaded by

    mdasifhassanapspdcl
    This document provides 12 questions about computing integrals and sums of various functions. Students enrolled in ECE 301-003 and OL1 are assigned this homework due on January 29, 2021. The questions cover reviewing calculus and arithmetic skills like computing integrals, finding derivatives, performing partial fraction decompositions, and evaluating sums of discrete sequences.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 4

    ECE 301-003 and OL1, Homework #1 (CRN: 11474 and 26295)

    Due date: Friday 1/29/2021

    https://engineering.purdue.edu/~chihw/21ECE301S/21ECE301S.html

    Review of calculus and arithmetics:

    Question 1: Compute the values of the following integrals. You can leave your answers
    in the form of trigonometric function values such as “cos(1.35).” There is no need to use
    calculator to find the numerical value of the final answer.
    ∫ 3π
    πω
    sin(2ω + 0.5π + )dω
    2π 2
    ∫ 3π
    cos(π/3 − u)du
    0
    ∫ 2 ∫ 2.5
    3xπ yπ
    cos( + )dxdy
    y=1 x=0 4 3

    Question 2: Compute the values of the following integrals.


    ∫ 4π
    2−|t−π| e−j 2 t dt
    π

    −2π
    ∫ 3
    π
    ω cos( ω)dω
    −2 3
    ∫ 3
    (|z − 2|)ej2πz dz
    −3

    Question 3: Compute the values of the following integrals.


    ∫ 4
    | cos(ω) − j(2ω + 1)|2 dω
    1
    ∫ 5∫ 4 2
    1

    (2 − t) + j2√2t dsdt
    3 3
    Question 4: Compute the expressions of the following integrals.
    ∫ 2
    ω
    (t + 2s)e−jωπ(2t+s) ds
    0
    (∫ ∫ 2T )
    T /2
    1 T 2kπ T 2kπ
    and ( − t)ej T t dt + (t − )ej T t dt ,
    T −T 2 T /2 2

    where k is an integer.

    Question 5: Suppose f (s) = s2 + 2s − 3. Define g(t) = f (1 − t). Compute the following


    values.

    g(3)
    ∫ 5
    g(1 + 2s)ds
    2
    ∫ 3
    and g ′ (s)ejπs ds,
    0

    where g ′ (t) is the first order derivative of g(t).

    Question 6: Suppose f (t) = |1 − t|. Define


    ∫ t+2
    g(t) = f (s)ds
    t−1

    Compute the following values.

    g(3)
    ∫ 4
    g(s)ds
    1
    ∫ 1
    and f (s)g(1 − s)ds.
    −1

    Question 7: Consider a discrete series f [n] = 2n + 1. Define

    ∑
    1
    g[n] = kf [k − n]
    k=−2
    Compute the following values.

    g[3]
    ∑
    1
    4k g[k]
    k=−2
    ∑∞
    3−k−1 g[k + 1]
    k=3
    ∑
    −2
    and 2k g[k]
    k=−∞

    Hint: If |r| < 1, then we have the following formulas for computing the sum of a geometric
    sequence.
    ∑
    ∞
    a
    ark−1 =
    k=1
    1−r
    ∑
    ∞
    a
    kark−1 = .
    k=1
    (1 − r)2

    Question 8: Consider a discrete series f [n] = 3n − 1. Define


    1
    g[n] = (f [n] − f [−n]) . (1)
    2
    Compute the following values.

    g[3]
    ∑
    1
    and f [k]g[2 − k]
    k=−1

    Question 9: Consider a discrete series such that f [n] = 1 − j · |n| if −2 ≤ n ≤ 1 and


    f [n] = 0 otherwise. Define

    ∑
    ∞
    g[n] = f [n − k]2|k| .
    k=−∞

    Compute g[−3]. Find the expression of g[n].


    Define
    ∑
    ∞
    h[n] = f [k]2|n−k| .
    k=−∞

    Compute h[−3]. Find the expression of h[n].

    Question 10: Suppose f (t) = cos(0.5πt − π). A discrete series g[n] is defined as

    g[n] = f (6n).

    Find the values of g[0], g[1], g[2], g[3]. [Optional] Find the expression of g[n] for general
    n values.

    Question 11: Suppose f [n] = (2n − 1)−2 . A function g(t) is defined as

    g(t) = e−j2t (cos(t)f [3] + j sin(t)f [−1]).

    Compute the following value


    ∫ π
    |g(s)|2 ds.
    −π/2

    Question 12: Compute the following partial fractions:


    1 a d
    = + .
    5− ω2− 6jω b + c · jω e + f · jω
    Namely, find the real-valued coefficients a, b, c, d, e, and f .
    Compute the following partial fractions:
    1 3 − 2jω a d g
    × = + + .
    5− ω2− 6jω 1 − jω b + c · jω e + f · jω (b + c · jω)2

    Namely, find the real-valued coefficients a, b, c, d, e, f , and g.

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