Hilbert Transformers (Digital Filter Design Toolkit)
June 2008
NI Part Number:
371988B-01
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The ideal frequency response of a Hilbert transformer is as follows:
H(f) = −jsgn(f)
where f is the normalized frequency with the range [0, 0.5]. The frequency response has −90° phase shift for positive frequencies and 90° phase shift for negative frequencies.
If you compare this equation with Equation B in the Linear Phase Filters topic, you can see that an ideal Hilbert transformer has a linear phase and is a Type III or Type IV linear phase FIR filter.
You can use the DFD Remez Design VI to design a Hilbert transformer by setting the filter type input to Hilbert. The following sections explain how to create Type IV and Type III Hilbert transformers.
Designing Type IV Hilbert Transformers
Type IV (odd order, antisymmetric) filters make ideal Hilbert transformers. When you design a Type IV Hilbert transformer with the DFD Remez Design VI, you can specify a single band that contains two points with equal weights. For example, consider a frequency range of [0.1, 0.5] with an amplitude of [−1, −1]. To design a Type IV Hilbert transformer with an order of 11 using this band, enter the specifications shown in the following figure into the DFD Remez Design VI.
The following figure shows the magnitude response and impulse response of the designed Type IV Hilbert transformer.
In the previous figure, you can see that magnitude response of the designed filter is fairly constant in the range [0.1, 0.5]. The magnitude response of an ideal Hilbert transformer, which has an infinitely large order, has a constant value in the specified frequency range. The impulse response of the designed filter shows that this filter is antisymmetric.
Designing Type III Hilbert Transformers
Type III (even order, antisymmetric) Hilbert transformers are useful if you want to filter out high frequencies. Type III filters constrain the amplitude to zero at the Nyquist frequency of 0.5 Hz. By changing the frequency range of the previous example to [0.05, 0.45] and maintaining the corresponding amplitude at [−1, −1], you can design a bandpass-like Hilbert transformer. For this example, you design a Type III Hilbert transformer with an order of 12 by entering the specifications shown in the following figure into the DFD Remez Design VI.
The following figure shows the magnitude response of the resulting Type III Hilbert transformer.
In the first figure, you can see that the magnitude response of the designed filter is fairly constant in the range [0.05, 0.45] and is zero at 0.5 Hz. The figure also shows that the magnitude response contains ripples. To minimize the ripples, you can specify a larger value for the order input.