Digital Signal Processing Reference
In-Depth Information
As the excess bandwidth is reduced to approach the ideal low-
pass filter frequency response, the lobes in the impulse response
become higher, approaching the sinc impulse response. The
signal with the smaller amplitude lobes has a larger excess
bandwidth, or wider spectrum.
The pulse shaping filter should have a zero response at
intervals of T in time so that a given symbol's pulse response will
not have a contribution to the signal at the sampling times of the
neighboring symbols. It should also minimize the height of the
lobes of the impulse (time) response and have it decay quickly,
to reduce the sensitivity to ISI if the receiver doesn't sample
precisely at the correct time for each symbol. As the roll off
factor increases, this is exactly what happens in the figures (D).
The impulse response goes to zero very quickly, and the lobes of
the filter impulse response are very small. However, they have
a frequency spectrum that is excessively wide. A better
compromise would be a roll off factor somewhere between 0.25
and 0.5 (B and C). Here the impulse response decays relatively
quickly with small lobes, requiring a pulse shaping filter with
a small number of taps, while still keeping the required band-
width reasonable.
The roll off factor controls the compromise between:
• spectral bandwidth requirement.
• length or number of taps of pulse shaping filter.
• receiver sensitivity to ISI.
Another significant aspect of the pulse shaping filter is that it is
always an interpolating filter. In the figures, this is shown as a four
times interpolation filter. Looking carefully at the impulse
response in Figure 17.10, the zero crossing occurs every four
samples. This corresponds to t
¼
N
$
T in the time domain, due to
the 4
interpolation.
The pulse shaping filter must an interpolating filter, as the I
and Q baseband signals must meet the Nyquist criterion. In this
example, the symbol rate is 1 MSPS. Using a high roll off factor,
the baseband spectrum of the I and Q signals can be as high as
1 MHz. So a minimum sampling rate of 2 MHz, or twice the
symbol rate, is required. Therefore, the pulse shaping filter will
need to interpolate by at least a factor of two, and is often
interpolated significantly higher than this.
Using pulse shaped and interpolated I and Q baseband digital
signals, digital-to-analog converters create the analog I and Q
baseband signals. These signals can be used to drive an analog
mixer which can create a passband signal. A passband signal is
a baseband signal upconverted or mixed with a carrier
frequency. This process can also be done digitally, with a DAC
