## Polyphase Decomposition

The previous section derived an efficient polyphase implementation of
an FIR filter
whose output was downsampled by the factor
. The
derivation was based on *commuting* the downsampler with the FIR
summer. We now derive the polyphase representation of a filter of any
length *algebraically* by splitting the impulse response
into
*polyphase components*.

### Two-Channel Case

The simplest nontrivial case is channels. Starting with a general linear time-invariant filter

(12.6) |

we may separate the even- and odd-indexed terms to get

(12.7) |

We define the

*polyphase component filters*as follows:

and
are the *polyphase components*
of the *polyphase decomposition* of
for
.

Now write in terms of its polyphase components:

(12.8) |

As a simple example, consider

(12.9) |

Then the polyphase component filters are

and

(12.10) |

### N-Channel Polyphase Decomposition

For the general case of arbitrary , the basic idea is to decompose into its periodically interleaved subsequences, as indicated schematically in Fig.11.9. The polyphase decomposition into channels is given by

where the subphase filters are defined by

(12.12) |

with

(12.13) |

The signal can be obtained by passing through an advance of samples, followed by downsampling by the factor . as shown in Fig.11.10.

### Type II Polyphase Decomposition

The polyphase decomposition of
into
channels in
(11.11) may be termed a ``type I'' polyphase decomposition. In
the ``type II'', or *reverse* polyphase decomposition, the powers
of
progress in the opposite direction:

(12.14) |

We will see that we need type I for

*analysis*filter banks and type II for

*synthesis*filter banks in a general ``perfect reconstruction filter bank'' analysis/synthesis system.

### Filtering and Downsampling, Revisited

Let's return to the example of §11.1.3, but
this time have the FIR lowpass filter h(n) be length
,
. In this case, the
polyphase filters,
, are
each length
.^{12.2} Recall that

(12.15) |

leading to the result shown in Fig.11.11.

Next, we commute the
:
downsampler through the adders and
upsampled (stretched) polyphase filters
to obtain
Fig.11.12. Commuting the downsampler through the
subphase filters
to obtain
is an example of a
*multirate noble identity*.

### Multirate Noble Identities

Figure 11.13 illustrates the so-called *noble identities* for
commuting downsamplers/upsamplers with ``sparse transfer functions''
that can be expressed a function of
. Note that downsamplers
and upsamplers are linear, time-varying operators. Therefore,
operation order is important. Also note that adders and multipliers
(any memoryless operators) may be commuted across downsamplers and
upsamplers, as shown in Fig.11.14.

**Next Section:**

Critically Sampled Perfect Reconstruction Filter Banks

**Previous Section:**

Upsampling and Downsampling