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 $N$ running-sum lowpass filter is given by

$$h(n) \isdef \left\{\begin{array}{ll} 1, & n=0,1,2,...,N-1 \\ [5pt] 0, & \hbox{otherwise.} \\ \end{array} \right. \protect$$ (10.5)

Figure 9.10: Black-box view of application of the running-sum lowpass filter.

Figure 9.10 depicts the generic operation of filtering $x(n)$ by $h(n)$ to produce $y(n)$ , where $h(n)$ is the impulse response of the filter. The output signal is given by the convolution of $x$ and $h$ :

$$\begin{eqnarray*} y(n) &=& (h\ast x)(n) \isdef \sum_{m=-\infty}^{\infty} h(m) x(n-m) = \sum_{m=0}^{N-1} x(n-m)\\ &=& x(n) + x(n-1) + \cdots + x(n-N+1). \end{eqnarray*}$$

In this form, it is clear why the filter (9.5) is called ``running sum'' filter. Dividing it by $N$ , it becomes a ``moving average'' filter, averaging the most recent $N$ input samples. It is also called an integrated comb filter.^10.1

The transfer function of the running-sum filter is given by [263]

$$H(z) = 1 + z^{-1}+ \cdots + z^{-N+1} = \frac{1-z^{-N}}{1-z^{-1}},$$ (10.6)

so that its frequency response is

$$\begin{eqnarray*} H(e^{j\omega}) &=& \frac{1-e^{-j\omega N}}{1-e^{-j\omega }} = \frac{e^{-j\omega N/2}}{e^{-j\omega /2}} \frac{\sin(\omega N/2)}{\sin(\omega /2)}\\ [10pt] &\isdef & Ne^{-j\omega(N-1)/2} \hbox{asinc}_N(\omega ). \end{eqnarray*}$$

Recall that the term $e^{-j\omega(N-1)/2}$ is a linear phase term corresponding to a delay of $(N-1)/2$ 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''

$$\hbox{asinc}_N(\omega ) \isdef \frac{\sin(\omega N/2)}{N\cdot\sin(\omega /2)}$$ (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 $N$ running sum filter is $N$ . We could use a moving average instead of a running sum ( $h \leftarrow h/N$ ) to obtain unity dc gain.

Figure: Running-sum amplitude response for $N=5$

Figure 9.11 shows the amplitude response of the running-sum lowpass filter for length $N=5$ . The gain at dc is $N=5$ , and nulls occur at $\omega = \pm2\pi/5$ and $\pm4\pi/5$ . 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 $2\cdot 3 = 6$ samples, so $6\pi/5$ 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.^10.2