### 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

 (10.5)

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 [263]

 (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 53.1.2) 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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