Consider the 2 Statement: 1. An odd and imaginary signal always has an odd and imaginary Fourier transform. 2. The convolution of an odd Fourier transform with an even Fourier transform is always even. Which of the above statements is/are true: 

Consider the 2 Statement: 1. An odd and imaginary signal always has an odd and imaginary Fourier transform. 2. The convolution of an odd Fourier transform with an even Fourier transform is always even. Which of the above statements is/are true:  Correct Answer None

Explanation:

Let F(ω) is the Fourier transform of f(t).

Hence an odd and imaginary signal always has an odd and real Fourier transform

Hence statement (1) is wrong.

Convolution:

Convolution is a mathematical operation used to express the relation between input and output of an LTI system. It relates the input, output, and impulse response of an LTI system as

y(t) = x(t) * h(t)

Where y (t) = output of LTI

x (t) = input of LTI

h (t) = impulse response of LTI

1. Convolution of two even signals or two odd signals always results in an even signal.

2. Convolution of odd signal and even signal always results in the odd signal.

Hence statement (2) is also false.

So option (1) is the correct answer.