The following problems are related to the properties in Table 4.1.
(a) Prove the validity of property 2.
(b) Prove the validity of property 4.
(c) Prove the validity of property 5.
(d) Prove the validity of property 7.
(e) Prove the validity of property 9.
(f) Prove the validity of property 10.
(g) Prove the validity of property 11.
(h) Prove the validity of property 12.
(i) Prove the validity of property 13.
TABLE 4.1 Some symmetry properties of the 2-D DFT and its inverse. R(u, v) and I(u, v) are the real and imaginary parts of F(u, v), respectively. The term complex indicates that a function has nonzero real and imaginary parts.
| Spatial Domain † |
| Frequency Domain † |
1) | f(x, y) real | ⇔ | F*(u,v) = F(−u, −v) |
2) | f(x, y) imaginary | ⇔ | F*(−u, −v) = −F(u, v) |
3) | f(x, y) real | ⇔ | R(u, v) even; I(u, v) odd |
4) | f(x, y) imaginary | ⇔ | R(u, v) odd;I(u, v) even |
5) | f(-x, -y) real | ⇔ | F*(u, v) complex |
6) | f(−x, −y) complex | ⇔ | F(−u, −v) complex |
7) | f(x, y) complex | ⇔ | F*(−u −v) complex |
8) | f(x, y) real and even | ⇔ | F(u, v) real and even |
9) | f(x, y) real and odd | ⇔ | F(u, v) imaginary and odd |
10) | f(x, y) imaginary and even | ⇔ | F(u, v) imaginary and even |
11) | f(x, y) imaginary and odd | ⇔ | F(u, v) real and odd |
12) | f(x, y) complex and even | ⇔ | F(u, v) complex and even |
13) | f(x, y) complex and odd | ⇔ | F(u, v) complex and odd |
†Recall that x, y, u, and v are discrete (integer) variables, with x and u in the range [0, M − 1], and y, and v in the range [0, N − 1]. To say that a complex function is even means that its real and imaginary parts are even, and similarly for an odd complex function.
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