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.
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''
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
Modulation by a Complex Sinusoid
Computational Examples in Matlab