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Problem 1.
In Topic 1, we have determined the Fourier Series (FS) for the function.
F(x) =



−(x − 2n) 2n − 1 ≤ x < 2n
x − 2n 2n ≤ x < 2n + 1
The FS has following sinusoidal terms:
F(x) = X∞
k=0
8 sin[πx
2
(2k + 1)]2
(2k + 1)2π
2
For example, when the first two terms are used, the expression is
F(x) ≈

8 sin[πx
2
]
2
π
2
+
8 sin[ 3πx
2
]
2
9π
2

Attempt to answer the following questions:

1. Write out the approximation expressing by using one, three, four and five terms.
2. Plotting using Mathcad 1
   . Refer to the Mathcad document for the class example,

• Plot the original F(x) against x with 0 ≤ x ≤ 10;
• The FS the first one term and the first five terms. Both plots need to be in one
  figure.
  1You complete the complete homework in Mathcad!
  2
  Problem 2
  We have an example showing how to use the basic CFT properties to determine the CFT of
  a function, f(x) = e
  −x
  2/2
  . Please review this example and repeat this process for a modified
  function f(x) = e
  −x
  2/k, where k is a constant number and k > 0.
  3
  Problem 3
  (1) Perform DFT for the following toy signal:
  {xk} = {−1, −
  1
  2
  ,
  1
  2
  , 1}
  collected at time intervals of every ∆t = 0.1 sec with
  {tk} = {0, 0.1, 0.2, 0.3}sec
  .
  Then write out all the DFT values of {Fn} as
  Fn =
  N
  X−1
  k=0
  xk exp −ı
  2π
  N
  n ∗ k
  for any n ∈ {0, 1, N − 1}.
  (2) Plot the norm of DFT against the frequency numbers.
  4
  Problem 4
  Given a time-domain function
  f(t) = 2 sin(2πν0t) + 3 sin(6πν0t)
  Answer the following questions:

1. Use the basic properties of CFT and the common CFT functions in Table 1 of Topic
   2 and determine the CFT of f(t) above.
2. Let ν0 = 10 Hz and let t changes from 0 to 1 sec. Plot f(t) against t (using Mathcad).
   In this case, please use the sampling time interval of ∆t = 0.01 · sec.
3. How many data points in your time-series that discrete f(t)? Define the corresponding
   frequency sequence vector.
4. Perform DFT using using the Mathcad function ‘dft’.
5. Plot the frequency contents by plotting the amplitudes of the DFT coefficients against
   the frequency vector.
6. Now repeat Questions 2 to 5 by considering ∆t = 0.02 · sec. What is your observation?
7. Now we will contaminate the signal by adding Whiten noises to the function y(t). The
   relevant topic is found here. Then, repeat Question 4 and 5 with ∆t = 0.01 · sec.

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