### Filtering and Downsampling

Because downsampling by causes aliasing of any frequencies in the original signal above , the input signal may need to be first lowpass-filtered to prevent this aliasing, as shown in Fig.11.5.

Suppose we implement such an anti-aliasing lowpass filter as an FIR filter of length with cut-off frequency .12.1 This is drawn in direct form in Fig.11.6.

We do not need out of every filter output samples due to the : 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 times the sampling frequency of the input signal . This reduces the computation requirements by a factor of . 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 . 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

 (12.3)

is scaled by .

• Subphase signal 1

 (12.4)

is scaled by ,

• Subphase signal

is scaled by .
These scaled subphase signals are finally summed to form the output signal shown in Fig.11.7

 (12.5)

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

The summed polyphase signals of Fig.11.7 can be interpreted as serial to parallel conversion'' from an interleaved'' stream of scalar samples to a deinterleaved'' sequence of buffers (each length ) every samples, followed by an inner product of each buffer with . The same operation may be visualized as a deinterleaving through variable gains into a running sum, as shown in Fig.11.8.

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Downsampling (Decimation) Operator