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Spectral Audio Signal Processing
    Multirate Polyphase Filter Banks
       Upsampling and Downsampling
          Filtering and Downsampling

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Filtering and Downsampling

Figure 11.5: Lowpass filtering followed by downsampling.
\includegraphics[scale=0.8]{eps/downsampledfilter}
Because downsampling by $ N$ will cause aliasing for any frequencies in the original signal above $ \vert\omega\vert > \pi/N$, the input signal may need to be first lowpass filtered to prevent aliasing, as shown in Fig.11.5. Suppose we implement such an anti-aliasing lowpass filter $ h(n)$ as an FIR filter of length $ M$ with a cutoff frequency $ \pi/N$. This is drawn in direct form in Fig.11.6.

Figure 11.6: Direct-form implementation of an FIR anti-aliasing lowpass filter followed by a downsampler.
\includegraphics[scale=0.8]{eps/down_FIR}

We do not need $ N-1$ out of every $ N$ filter output samples due to the $ N:1$ downsampler. To realize this savings, we can commute the downsampler through the adders inside the FIR filter to obtain the result shown in Fig.11.7. The multipliers are now running at $ 1/N$ times the sampling frequency of the input signal, $ x(n)$. This reduces the computation requirements by a factor of $ 1/N$. The downsampler outputs are called polyphase signals. This is a summed polyphase filter bank in which each ``subphase filter'' is a constant scale factor $ h(m)$.

Figure 11.7: FIR lowpass filter with downsampler commuted inside the direct-form filter.
\includegraphics[scale=0.8]{eps/down_FIR_com}

The summed polyphase signals of Fig.11.7 can be interpreted in the following ways:

  1. A ``serial to parallel conversion'' from a stream of scalar samples $ x(n)$ to a sequence of length $ M$ buffers every $ N$ samples, followed by a dot product of each buffer with $ h(0:M-1)$.

  2. The overall system is equivalent to a round-robin demultiplexor, with a different gain $ h(m)$ for each output, followed by an $ M$-sample summer which adds the ``de-interleaved'' signals together:

Figure: Demultiplex-and-sum interpretation of the polyphase signal sum of Fig.11.7.
\begin{figure}\input fig/periodicGain.pstex_t \end{figure}

The polyphase processing in the anti-aliasing filter of Fig.11.7 is as follows:

  • The 0th subphase signal,

    $\displaystyle x(nN)\left\vert _{n=0}^{\infty}\right. = [x_0,x_N,x_{2N},\ldots], $

    is scaled by $ h(0)$.

  • Subphase signal 1,

    $\displaystyle x(nN-1)\left\vert _{n=0}^{\infty}\right.=[x_1,x_{N-1},x_{2N-1},\ldots] $

    is scaled by $ h(1)$,

  • $ \cdots$

  • Subphase signal $ m$,

    $\displaystyle x(nN-m)\left\vert _{n=0}^{\infty}\right.=[x_m,x_{N-m},x_{2N-m},\ldots] $

    is scaled by $ h(m)$
These scaled subphase signals are summed together to form the output signal

$\displaystyle y(n) = \sum_{m=0}^{N-1} h(m)x(nN-m). $

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Previous: Example: Upsampling by 2
Next: Polyphase Filtering
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.


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