Perhaps the simplest FIR lowpass filter is the so-called running-sum lowpass filter . 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 
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''
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