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

F(ω) 
Real 
→ 
Conjugate symmetric 
Conjugate symmetric 
→ 
Real 
Imaginary 
→ 
Conjugate antisymmetric 
Conjugate Anti symmetric 
→ 
Imaginary 
Real + Even 
→ 
Real + Even 
Imaginary + Even 
→ 
Imaginary + Even 
Real + odd 
→ 
Imaginary + odd 
Imaginary + odd 
→ 
Real + odd 
Discrete 
→ 
Periodic 
Periodic 
→ 
Discrete 
Continuous 
→ 
Aperiodic 
Aperiodic 
→ 
Continuous 
Continuous + periodic 
→ 
Discrete + Aperiodic 
Continuous + Aperiodic 
→ 
Continuous + Aperiodic 
Discrete + Periodic 
→ 
Discrete + Periodic 
Discrete + Aperiodic 
→ 
Continuous + Periodic 
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.