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Texas Instruments Incorporated

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GC2011A 3.3V DIGITAL FILTER CHIP

SLWS129A

This document contains information which may be changed at any time without notice

3.7

DHILBERT TRANSFORM FILTERS

A Hilbert transform filter converts real signals to complex signals by passing the signal’s positive spectral

frequencies and rejecting its negative frequencies. For example, a sinewave of frequency “w” has both the positive

frequency component ejwt and the negative frequency component e-jwt. The Hilbert transform of the sinewave will be

just the positive component ejwt.

The coefficients for a Hilbert transform can be generated by designing a linear phase low pass filter with a

passband from 0 to FS/4 and a stopband from FS/4 to FS/2, where FS is the signal’s sample rate. The low pass filter’s

impulse response is then mixed up to be centered on FS/4 by multiplying the coefficients by the sequence: (j, -1, -j, 1, j,

-1, -j,

…).

For example, the coefficients:

(h0,h1,h2,h3,h4,h5,h6,h7,h6,h5,h4,h3,h2,h1,h0)

would become:

(jh0,-h1,-jh2,h3,jh4,-h5,-jh6,h7,jh6,-h5,-jh4,h3,jh2,-h1,-jh0).

These coefficients then split into the real coefficients:

( 0, -h1,

0, h3, 0, -h5,

0, h7,

0, -h5,

0, h3,

0, -h1,

and the imaginary coefficients:

(h0,

0, -h2,

0, h4,

0, -h6,

0, h6,

0, -h4,

0, h2,

0, -h0).

As seen in this example, the real coefficients of a Hilbert transform filter have odd symmetry with the center

tap non-zero and every other tap equal to zero. The imaginary coefficients have negative odd symmetry.

A special, but important, version of the Hilbert transform exists when the filter has half-band symmetry.

Half-band symmetry forces all of the real coefficients except the center tap to be zero. The real half filter, for the

half-band Hilbert Transform, is, therefore, just a delay line.

The following table shows how to configure the GC2011A chip for the Hilbert Transform. The A-path is used

for the real part and the B-path for the imaginary part.

Since the coefficients are symmetric, only 32 of the 63 low pass filter coefficients are stored in the chip. If the

low-pass filter coefficients are h(k), for k=0 to 31, where h(31) is the center tap, then coefficient register 0 of each filter

cell is loaded as:

Store -h(4k)

in memory address 192+8*k for k=0 to 7

Store -h(4k+1) in memory address 128+8*k for k=0 to 7

Store +h(4k+2) in memory address 196+8*k for k=0 to 7

Store +h(4k+3) in memory address 132+8*k for k=0 to 7

Note that the odd coefficients are stored in the A-path, and that the even coefficients are stored in the B-path. Also note

that every other odd and every other even coefficient are negated. In the half-band Hilbert transform only h(31) will be

non-zero in the A-path.

Table 8: Hilbert Transform Mode Control Register Settings

Dual Path or

Cascaded

# of Taps

(N)

A-PATH

B-PATH

Cascade

Latency

REG0

REG1

REG0

REG1

REG

Dual

60C8

2E84

20C8

2E78

2000

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