### The Running-Sum Lowpass Filter

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 :

(10.6) |

so that its frequency response is

*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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