Filtering and Downsampling

Because downsampling by $ N$ causes aliasing of any frequencies in the original signal above $ \vert\omega\vert > \pi/N$ , the input signal may need to be first lowpass-filtered to prevent this aliasing, as shown in Fig.11.5.

Figure 11.5: Lowpass filtering followed by downsampling.
\includegraphics[width=2in]{eps/downsampledfilter}

Suppose we implement such an anti-aliasing lowpass filter $ h(n)$ as an FIR filter of length $ M\le N$ with cut-off frequency $ \pi/N$ .12.1 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[width=0.6\twidth]{eps/down_FIR}

Figure 11.7: FIR lowpass filter with downsampler commuted inside the direct-form filter.
\includegraphics[width=0.6\twidth]{eps/down_FIR_com}

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 in Fig.11.7 are called polyphase signals. The overall system is a summed polyphase filter bank in which each ``subphase filter'' is a constant scale factor $ h(m)$ . As we will see, more general subphase filters can be used to implement time-domain aliasing as needed for Portnoff windows (§9.7).

We may describe the polyphase processing in the anti-aliasing filter of Fig.11.7 as follows:

  • Subphase signal 0

    $\displaystyle x(nN)\left\vert _{n=0}^{\infty}\right. \eqsp [x_0,x_N,x_{2N},\ldots]$ (12.3)

    is scaled by $ h(0)$ .

  • Subphase signal 1

    $\displaystyle x(nN-1)\left\vert _{n=0}^{\infty}\right.\eqsp [x_{-1},x_{N-1},x_{2N-1},\ldots]$ (12.4)

    is scaled by $ h(1)$ ,

  • $ \cdots$

  • Subphase signal $ M-1$

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

    is scaled by $ h(M-1)$ .
These scaled subphase signals are finally summed to form the output signal shown in Fig.11.7

$\displaystyle y(n) \eqsp \sum_{m=0}^{M-1} h(m)\, x(nN-m),$ (12.5)

which we recognize as a direct-form-convolution implementation of a length $ M$ FIR filter $ h$ , with its output downsampled by the factor $ N$ .

The summed polyphase signals of Fig.11.7 can be interpreted as ``serial to parallel conversion'' from an ``interleaved'' stream of scalar samples $ x(n)$ to a ``deinterleaved'' sequence of buffers (each length $ M$ ) every $ N$ samples, followed by an inner product of each buffer with $ h = [h(0),\ldots,h(M-1)]$ . The same operation may be visualized as a deinterleaving through variable gains into a running sum, as shown in Fig.11.8.

Figure: Demultiplex-and-sum interpretation of the polyphase signal sum of Fig.11.7.
\includegraphics[width=0.7\twidth]{eps/periodicGain}


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Downsampling (Decimation) Operator