Xcell Journal Issue 78: Charge to Market with Xilinx 7 Series ...

XPLANATION: FPGA 101
Figure 2 – IDFT or impulse response of the perfect low-pass filter
band—is also apparent in this
response. Ripples in the passband will
affect the signals being filtered. The
stopband attenuation demonstrates
how much of the unwanted frequencies
remain within the filter output.
This can be critical in applications
where specific frequency rejection is
required, for example when filtering
one frequency-division multiplexed
channel from another in a communications
system.
Figure 3 – Low-pass filter impulse responses (top chart) and frequency responses
IMPULSE RESPONSE
It is from the impulse response that the
coefficients for your filter are abstracted.
However, to achieve the best performance
from your filter, the standard
practice is to use a windowing function.
Windowing is the technique of applying
an additional mathematical function to a
truncated impulse response to reduce
the undesired effects of truncation.
Figure 2 demonstrates the impulse
response extending out infinitely with
ripples which, though they diminish significantly
in amplitude, never settle at
zero. Therefore, you must truncate the
impulse response to N + 1 coefficients
chosen symmetrically around the center
main lobe, where N is the desired filter
length (please remember that N
must be an even number). This truncation
affects the filter’s performance in
the frequency domain due to the abrupt
cutoff of the new, truncated impulse
response. If you were to take a discrete
Fourier transform (DFT) of this truncated
impulse response, you would
notice ripples in both the passband and
stopband along with reduced roll-off
performance. This is why it is common
practice to apply a windowing function
to improve the performance.
STEP RESPONSE
The step response, which is obtained by
integrating the impulse response,
demonstrates the time-domain performance
of the filter and how the filter itself
modifies this performance. The three
parameters of importance when you are
observing the step response are the rise
time, overshoot and linearity.
The rise time is the number of samples
it takes to rise between 10 percent
and 90 percent of the amplitude levels,
demonstrating the speed of the filter. To
be of use within your final system, the filter
must be able to distinguish between
events in the input signal; therefore, the
step response must be shorter than the
spacing of events in the signal.
Overshoot is the distortion that the
filter adds to the signal as it is processing
it. Reducing the overshoot in the
38 Xcell Journal First Quarter 2012