GC2011A Datasheet, PDF (23/50 Pages) Texas Instruments – 3.3V DIGITAL FILTER CHIP

GC2011A 3.3V DIGITAL FILTER CHIP
SLWS129A
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 e^jwt and the negative frequency component e^-jwt. The Hilbert transform of the sinewave will be
just the positive component e^jwt.
The coefficients for a Hilbert transform can be generated by designing a linear phase low pass filter with a
passband from 0 to F_S/4 and a stopband from F_S/4 to F_S/2, where F_S is the signal’s sample rate. The low pass filter’s
impulse response is then mixed up to be centered on F_S/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, 0)
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.
Table 8: Hilbert Transform Mode Control Register Settings
Dual Path or
Cascaded
Dual
# of Taps
(N)
63
A-PATH
REG0 REG1
60C8 2E84
B-PATH Cascade
REG0 REG1
20C8 2E78
REG
2000
Latency
45
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.
Texas Instruments Incorporated
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