Question

Circle the correct words in each of the following statements.Given that x(t) is a real function with the Fourier transform X(f): a.The real part of X(f) is an even / odd function. b.The imaginary part of X(f) is an even / odd function.

Answer 1

Answer:

(a) even function

(b) odd function

Step-by-step explanation:

From the given information;

Any function x(t) can be said to be the sum of even and odd functions.

i.e.

$x(t) = x_e(t) + x_o(t)$

where;

x(t) is a real function.

For a  real and even function, the Fourier transform is also real and even.

$x_e(t) \iff x_e(f)$

Also, the Fourier Transform of a real and odd function is imaginary and odd.

$x_o(t) \iff x_o(f)$

The behavior of X(f) largely depends on the behavior of x(t).

If x(t) is real and even, then x(f) is real and even valued function

Thus, the real part of X(f) is an even function.

If x(t) is real and odd, then x(f) is imaginary and odd valued function

Thus, the imaginary part of X(f) is an odd function.