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CAP 5415: Lecture 3 Worksheet
Marshall Tappen
September 3, 2009
Basic Equations
Assumptions: 1D image with N pixels. Use circular boundary handling. Convolution of image f [x] with kernel k[x]:
f [x] ∗ k[x] =
N∑ − 1
n=
f [n]k[n − x] (1)
Discrete Fourier Transform:
F [u] =
N∑ − 1
x=
f [x]e−^2 πj^
ux N (^) (2)
Inverse Transform:
f [x] =
N
N∑ − 1
u=
F [u]e^2 πj^
ux N (3)
Euler’s Equation: ejθ^ = cos θ + j sin θ (4)
Trig:
cos(−θ) = cos(θ) (5) sin(−θ) = − sin(θ) (6) (7)
Problem 1:
Calculate the DFT of a signal of length,f , N , with f [0] = 1 and the f [1... N − 1] = 0. Sketch the magnitude on the axes. The magnitude will be a discrete function, but you can sketch it as a continuous function if you would like.
− N 2
N (^02)
Problem 2:
Calculate the DFT of a signal, f , of length N , with f [0... N − 1] = 1. Sketch the magnitude on the axes. The magnitude will be a discrete function, but you can sketch it as a continuous function if you would like.
− N 2 0 N 2
Problem 3:
Calculate the DFT of a signal of length N , with f [0] = 1, f [1] = 1, and f [N − 1] = 1. Sketch the magnitude on the axes. The magnitude will be a discrete function, but you can sketch it as a continuous function if you would like.
− N 2 0 N 2
Problem 4:
Calculate the DFT of a signal of length N , with f [0] = − 1 and f [1] = 1. Sketch the magnitude on the axes. The magnitude will be a discrete function, but you can sketch it as a continuous function.
− N 2
N (^02)
