A real-valued signal 𝑥(𝑡) is known to be uniquely determined by its samples when the sampling frequency is 𝜔𝑠 = 1000𝜋. For what values of 𝜔 is 𝑋(𝜔) guaranteed to be zero?

ECE481 Image Processing (Spring 2023)
Homework #1
(Due date: Feb 2, 2023)
• Reading assignment: Chap. 1, 2
1. (10 pts) Consider a discrete-time system with input 𝑥[𝑛] and output 𝑦[𝑛] related by
𝑦[𝑛] = 𝑛𝑥[𝑛].
(a) (5 pts) Is this system linear? Justify your answer.
(b) (5 pts) Is this system time-invariant? Justify your answer.
2. (20 pts) The forward Fourier transform for 1-D signals is defined as follows:
𝑋(𝜔) = ∫ 𝑥(𝑡)𝑒−𝑗𝜔𝑡𝑑𝑡
∞
−∞
Calculate the Fourier transforms of the following signals.
(a) (5 pts) 𝑥(𝑡) = 𝑒𝑗𝜔0𝑡
(b) (5 pts) 𝑥(𝑡) = cos𝜔0𝑡
(a) (5 pts) 𝑥(𝑡) = 𝛿(𝑡 + 1) + 𝛿(𝑡 − 1)
(b) (5 pts) 𝑥(𝑡) = {1, |𝑡| < 𝑎 0, |𝑡| > 𝑎
3. (10 pts) The forward Fourier transform for 1-D signals is defined as follows:
𝑥(𝑡) = 1
2𝜋 ∫ 𝑋(𝜔)𝑒𝑗𝜔𝑡𝑑𝜔
∞
−∞
Calculate the inverse Fourier transforms of the following signals.
(a) (5 pts) 𝑋(𝜔) = 2𝜋𝛿(𝜔) + 𝜋𝛿(𝜔 − 4𝜋) + 𝜋𝛿(𝜔 + 4𝜋)
(b) (5 pts) 𝑋(𝜔) = {
2, 0 ≤ 𝜔 ≤ 2
−2, −2 ≤ 𝜔 ≤ 0
0, |𝜔| > 2
4. (10 pts) A real-valued signal 𝑥(𝑡) is known to be uniquely determined by its samples when the sampling frequency is 𝜔𝑠 = 1000𝜋. For what values of 𝜔 is 𝑋(𝜔) guaranteed to be zero?
5. (10 pts) A continuous-time signal 𝑥(𝑡) is obtained at the output of an ideal lowpass filter with cutoff frequency 𝜔𝑐 = 1000𝜋. If impulse-train sampling is performed on 𝑥(𝑡), which of the following sampling periods would guarantee that 𝑥(𝑡) can be recovered from its sampled version using an appropriate lowpass filter?
(a) 𝑇 = 0.5 × 10−3
(b) 𝑇 = 2 × 10−3
6. (10 pts) Consider the two image subsets, S1 and S2, shown in the figure below. For 𝑉 = {1}, determine whether these two subsets are (a) 4-adjacent, (b) 8-adjacent, or (c) 𝑚-adjacent.
7. (20 pts) Consider the image segment shown.
(a) (10 pts) Let 𝑉 = {0, 1} and compute the lengths of the shortest 4-, 8-, and m-path between 𝑝 and 𝑞. If a particular path does not exist between these two pointes, explain why.
(b) (10 pts) Repeat for 𝑉 = {1, 2}.
8. (10 pts) The forward transform kernel is said to be separable if 𝑟(𝑥, 𝑦, 𝑢, 𝑣) = 𝑟1(𝑥, 𝑢)𝑟2(𝑦, 𝑣).
In addition, the kernel is said to be symmetric if 𝑟1(𝑥, 𝑢) is functionally equal to 𝑟2(𝑦, 𝑣) so that 𝑟(𝑥, 𝑦, 𝑢, 𝑣) = 𝑟1(𝑥, 𝑢)𝑟1(𝑦, 𝑣). Prove that the discrete Fourier transform given below is separable and symmetric.
𝑇(𝑢, 𝑣) = ∑ ∑ 𝑓(𝑥, 𝑦)𝑒−𝑗2𝜋(𝑢𝑥
𝑀 +𝑣𝑦
𝑁 )
𝑁−1
𝑦=0
𝑀−1
𝑥=0

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