Sign in

username:

password:



Not a member?

Search Online Books



Search tips

Free Online Books

Ads

Chapters

Spectral Audio Signal Processing
    The Filter Bank Summation (FBS) Interpretation of the Short Time Fourier Transform (STFT)
       The DFT Filter Bank
          The Running-Sum Lowpass Filter

Chapter Contents:

Search Spectral Audio Signal Processing

  

Book Index | Global Index

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

  

The Running-Sum Lowpass Filter

Perhaps the simplest FIR lowpass filter is the so-called running-sum lowpass filter [159]. The impulse response of the length $ N$ running-sum lowpass filter is given by

$\displaystyle h(n) \isdef \left\{\begin{array}{ll} 1, & n=0,1,2,...,N-1 \\ [5pt] 0, & \hbox{otherwise.} \\ \end{array} \right. \protect$ (10.3)

Figure 9.10: Black-box view of application of the running-sum lowpass filter.
\begin{figure}\input fig/blackbox.pstex_t \end{figure}

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) ... ...um_{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 (9.3) is called ``running sum'' filter. Dividing it by $ N$, it becomes a ``moving average'' filter, averaging the most recent $ N$ input samples.

The transfer function of the running-sum filter is given by [242]

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

so that its frequency response is

\begin{eqnarray*} H(e^{j\omega}) &=& \frac{1-e^{-j\omega N}}{1-e^{-j\omega }} = ... ... [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''

$\displaystyle \hbox{asinc}_N(\omega ) \isdef \frac{\sin(\omega N/2)}{N\cdot\sin(\omega /2)} $

previously in Chapter 1 (§3.1) 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$
\includegraphics[width=4in]{eps/sincabs}

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 passband 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

Order a Hardcopy of Spectral Audio Signal Processing

Previous: The DFT Filter Bank
Next: Modulation by a Complex Sinusoid
written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


Comments

No comments yet for this page


Add a Comment
You need to login before you can post a comment (best way to prevent spam). ( Not a member? )