Perhaps the simplest FIR lowpass filter is the so-called
*running-sum lowpass filter* [175]. The impulse
response of the length
running-sum lowpass filter is given by

Figure 9.10 depicts the generic operation of filtering
by
to produce
, where
is the impulse response of the
filter. The output signal is given by the *convolution* of
and
:

In this form, it is clear why the filter (9.5) is called ``running sum'' filter. Dividing it by , it becomes a ``moving average'' filter, averaging the most recent input samples.

The transfer function of the running-sum filter is given by [262]

(10.6) |

so that its frequency response is

Recall that the term
is a *linear phase
term* corresponding to a delay of
samples (half of the FIR
filter order). This arises because we defined the running-sum lowpass
filter as a *causal, linear phase filter*.

We encountered the ``aliased sinc function''

(10.7) |

previously in Chapter 5 (§3.1.2) and elsewhere as the Fourier transform (DTFT) of a sampled rectangular pulse (or rectangular window).

Note that the dc gain of the length
running sum filter is
. We
could use a *moving average* instead of a running sum (
) to obtain unity dc gain.

Figure 9.11 shows the amplitude response of the running-sum
lowpass filter for length
. The gain at dc is
, and nulls
occur at
and
. These nulls occur
at the sinusoidal frequencies having respectively one and two periods
under the 5-sample ``rectangular window''. (Three periods would need
at least
samples, so
doesn't ``fit''.) Since
the pass-band about dc is not flat, it is better to call this a
``dc-pass filter'' rather than a ``lowpass filter.'' We could also
call it a *dc sampling filter*.^{10.1}

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