The following problems are related to the entries in Table 4.3.(a) Prove the validity of t...

The following problems are related to the entries in Table 4.3.

(a) Prove the validity of the discrete convolution theorem (entry 6) for the 1-D case.

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(b) Repeat (a) for 2-D.

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(c) Prove the validity of entry 7.

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(d) Prove the validity of entry 12.

(Note: Problems 4.18,4.19, and 4.31 are related to Table 4.3 also.)

TABLE 4.3

Summary of DFT pairs.The closedform expressions in 12 and 13 are valid only for continuous variables.They can be used with discrete variables by sampling the closed-form, continuous expressions.

4.18 Show that the DFT of the discrete function f(x, y) = 1 is

4.19 Show that the DFT of the discrete function f(x, y) = sin(2πu0x + 2πu0y) is

4.31 A continuous Gaussian lowpass filter in the continuous frequency domain has the transfer function

Show that the corresponding filter in the spatial domain is