Test: Hilbert Transformers Design

Explanation: An ideal Hilbert transformer is a all pass filter.

Explanation: An ideal Hilbert transformer is a all pass filter that imparts a 90^o phase shift on the signal at its input.

Explanation: Hilbert transforms are frequently used in communication systems and signal processing, as, for example, in the generation of SSB modulated signals, radar signal processing and speech signal processing.

Explanation: We know that the frequency response of an ideal Hilbert transformer is given as
H(ω)= -j ;0 < ω < π
j ;-π < ω < 0
Thus the unit sample response of an ideal Hilbert transform is obtained as

Explanation: We know that the unit sample response of the Hilbert transform is given as

it sample response of an ideal Hilbert transform is infinite in duration and non-causal.

Explanation: We know that the unit sample response of the Hilbert transform is given as

Thus from the above equation, we can tell that h(n)=-h(-n). Thus the unit sample response of Hilbert transform is anti-symmetric in nature.

Explanation: In view of the fact that the ideal Hilbert transformer has an anti-symmetric unit sample response, we shall confine our attention to FIR designs in which h(n)=-h(M-1-n).

Explanation: Our choice of an anti-symmetric unit sample response is consistent with having a purely imaginary frequency response characteristic.

Explanation: We know that when h(n) is anti-symmetric, the real valued frequency response characteristic is zero at ω=0 for both M odd and even and at ω=π when M is odd. Clearly, then, it is impossible to design an all-pass digital Hilbert transformer.

Explanation: The bandwidth of Hilbert transformer need only cover the bandwidth of the signal to be phase shifted. Consequently, we specify the desired real valued frequency response of a Hilbert transformer filter is
H(ω)=1; 2π f_l < ω < 2πf_u
where fl and fu are the cutoff frequencies.