9 April, 2024
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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:
- Write out the approximation expressing by using one, three, four and five terms.
- 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:
- 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. - 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. - How many data points in your time-series that discrete f(t)? Define the corresponding
frequency sequence vector. - Perform DFT using using the Mathcad function ‘dft’.
- Plot the frequency contents by plotting the amplitudes of the DFT coefficients against
the frequency vector. - Now repeat Questions 2 to 5 by considering ∆t = 0.02 · sec. What is your observation?
- 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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