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:
![\begin{eqnarray*}
E_0(z)&=&\sum_{n=-\infty}^{\infty}h(2n)z^{-n}\\ [5pt]
E_1(z)&=&\sum_{n=-\infty}^{\infty}h(2n+1)z^{-n}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img1958.png)
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
![\begin{eqnarray*}
E_0(z) &=& 1 + 3z^{-1}\\ [5pt]
E_1(z) &=& 2 + 4z^{-1}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img1963.png)
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




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