Consider the following statement regarding the Hilbert transform:

1. Hilbert transform is non causal system.

2. Hilbert transform does not change the domain of the system.

3. Hilbert transform does not change the magnitude spectrum of the signal.

Which of the above statement are correct?

Hilbert Transformer:

Hilbert Transform of a signal x(t) is defined as:

\(H\left\{ {x\left( t \right)} \right\} = x\left( t \right)*\frac{1}{{\pi t}}\)

\( = \mathop \smallint \limits_{ = \infty }^\infty x\left( \tau \right) \cdot \frac{1}{{\pi \left( {t - \tau } \right)}}d\tau \)

\( = \frac{1}{\pi }\mathop \smallint \limits_{ - \infty }^\infty \frac{{x\left( \tau \right)}}{{\left( {t - \tau } \right)}}d\tau \)

The impulse response of Hilbert Transform is given by

\(h\left( t \right) = \frac{1}{{\pi t}}\)

h(t) ≠ 0 for t < 0

Hence, the System is Non – causal.

• Hilbert transform does not change the domain of the signal.

• The magnitude of the frequency component present in x(t) remains unchanged when it is passed through the system.
• The phase of the positive frequency components is shifted by -π/2 and that of negative frequency components is shifted by +π/2.
• A signal x(t) and its Hilbert transform \({\rm{\hat x}}\left( {\rm{t}} \right)\) are orthogonal to each other.