Datasheet

AD9786
Rev. B | Page 40 of 56
Figure 72 shows this effect at the DAC output for a signal
mirrored asymmetrically about dc that is produced by complex
modulation without a Hilbert transform. The signal bandwidth
was narrowed to show the aliased negative frequency
interpolation images.
In contrast, Figure 73 shows the same waveform with the
Hilbert transform applied. Clearly, the aliased interpolation
images are not present.
03152-073
0.5–0.5 –0.4 –0.3 –0.2 –0.1 0 0.1 0.2 0.3 0.4
dBFS
–150
0
–50
–100
Figure 73. Effects of Hilbert Transform
If the output of the AD9786 is used with a quadrature modulator,
negative frequency images are cancelled without the need for
a Hilbert transform.
HILBERT TRANSFORM IMPLEMENTATION
The Hilbert transform on the AD9786 is implemented as a
19-coefficient FIR. The coefficients are given in Table 36.
Table 36.
Coefficient Integer Value
H(1) –6
H(2) 0
H(3) –17
H(4) 0
H(5) –40
H(6) 0
H(7) –91
H(8) 0
H(9) –318
H(10) 0
H(11) +318
H(12) 0
H(13) +91
H(14) 0
H(15) +40
H(16) 0
H(17) +17
H(18) 0
H(19) +6
The transfer function of an ideal Hilbert transform has a +90°
phase shift for negative frequencies, and a –90° phase shift for
positive frequencies. Because of the discontinuities that occur at
0 Hz and at 0.5 × the sample rate, any real implementation of
the Hilbert transform trades off bandwidth vs. ripple.
Figure 74 and Figure 75 show the gain of the Hilbert transform
vs. frequency. Gain is essentially flat, with a pass-band ripple of
0.1 dB over the frequency range of 0.07 × the sample rate to
0.43 × the sample rate.
Figure 76 shows the phase response of the Hilbert transform
implemented in the AD9786. The phase response for positive
frequencies begins at –90° at 0 Hz, followed by a linear phase
response (pure time delay) equal to nine filter taps (nine
DACCLK cycles). For negative frequencies, the phase response
at 0 Hz is +90°, followed by a linear phase delay of nine filter
taps. To compensate for the unwanted 9-cycle delay, an equal
delay of nine taps is used in the AD9786 digital signal path
opposite the Hilbert transform. This delay block is shown as Δt
in the Functional Block Diagram (Figure 1).
03152-074
1000100 200 300 400 500 600 700 800 900
–100
10
–20
–60
0
–40
–30
–70
–80
–90
–10
–50
Figure 74. Hilbert Transform Gain
03152-075
1000100 200 300 400 500 600 700 800 900
–
1.0
1.0
0.4
–
0.4
0.8
0
0.2
–
0.6
–
0.8
0.6
–
0.2
Figure 75. Hilbert Transform Ripple