### Uniform Running-Sum Filter Banks

Using a length running-sum filter, let's make bandpass filters tuned to center frequencies

(10.11) |

Since the bandwidths, as defined, are , the filter pass-bands overlap by 50%. A superposition of the bandpass frequency responses for is shown in Fig.9.14. Also shown is the frequency-response sum, which we will show to be exactly

*constant*and equal to . This gives our filter bank the

*perfect reconstruction property*. We can simply add the outputs of the filters in the filter bank to recreate our input signal exactly. This is the source of the name

*Filter-Bank Summation (FBS)*.

#### System Diagram of the Running-Sum Filter Bank

Figure 9.15 shows the system diagram of the complete
-channel filter bank
constructed using length
FIR running-sum lowpass filters. The
th channel computes:

#### DFT Filter Bank

Recall that the Length
*Discrete Fourier Transform* (DFT) is
defined as

(10.13) |

Comparing this to (9.12), we see that the filter-bank output , , is precisely the DFT of the input signal when ,

*i.e.*,

(10.14) |

In other words, the filter-bank output at time (the set of samples for ), equals the DFT of the first samples of ( , ). That is, taking a snapshot of all filter-bank channels at time yields the DFT of the input data from time

**0**through .

More generally, for all
, we will call Fig.9.15 the *DFT
filter bank*. The DFT filter bank is the special case of the STFT for
which a rectangular window and hop size
are used.

The *sliding DFT* is obtained by advancing successive DFTs by one
sample:

(10.15) |

When for any integer , the Sliding DFT coincides with the DFT filter bank. At other times, they differ by a linear phase term. (Exercise: find the linear phase term.) The Sliding DFT

*redefines the time origin*every sampling period (each modulation term within the DFT starts at time 0 for each transform), while the DFT Filter Bank does not redefine the time origin (modulation terms are ``free running'' as they would be in an analog filter bank). Since ``DFT time'' repeats every samples, the two treatments coincide every samples (

*i.e.*, for every integer ).

When
is a power of 2, the DFT can be implemented using a Cooley-Tukey Fast
Fourier Transform (FFT) using only
operations per
transform. By keeping track of the linear phase term (an
modification), a DFT Filter Bank can be implemented efficiently using
an FFT. Uniform FIR filter banks are very often implemented in
practice using FFT software such as `fftw`.

Note that the channel bandwidths are *narrow* compared with half
the sampling rate (especially for large
), so that the filter bank
output signals
are *oversampled*, in general. We will
later look at *downsampling* the channel signals
to
obtain a ``hopping FFT'' filter bank. ``Sliding'' and ``hopping''
FFTs are special cases of the discrete-time *Short Time Fourier
Transform* (STFT). The STFT normally also uses a *window
function* other than the rectangular window used in this development
(the running-sum lowpass filter).

#### Inverse DFT and the DFT Filter Bank Sum

The Length
*inverse* DFT is given by [264]

(10.16) |

This suggests that the DFT Filter Bank can be inverted by simply

*remodulating*the baseband filter-bank signals ,

*summing*over , and dividing by for proper normalization. That is, we are led to conjecture that

(10.17) |

This is in fact true, as we will later see. (It is straightforward to show as an exercise.)

#### Specific Windows

- Recall that the
**rectangular**window transform is , implying the rectangular window itself is , which is obvious. - The window transform for the
**Hamming family**is , implying that Hamming windows are , which we also knew. - The rectangular window transform is also
for any integer
, implying that all hop sizes given
by
for
are COLA.
- Because its side lobes are the same width as the sinc side lobes,
the Hamming window transform is also
,for any integer
, implying hop sizes
are good, for
. Thus, the available hop sizes for the Hamming
window family include
*all*of those for the rectangular window except one ( ).

#### The Nyquist Property on the Unit Circle

As a degenerate case, note that is COLA for any window, while no window transform is except the zero window. (since it would have to be zero at dc, and we do not consider such windows). Did the theory break down for ?

Intuitively, the
condition on the window transform
ensures that all nonzero multiples of the
time-domain-frame-rate
will be zeroed out over the interval
along the frequency axis. When the frame-rate equals the
sampling rate (
), there *are no* frame-rate multiples in the
range
. (The range
gives the same result.)
When
, there is exactly one frame-rate multiple at
. When
, there are two at
. When
, they are at
and
, and so on.

We can cleanly handle the special case of
by defining *all*
functions over the unit circle as being
when there are no
frame-rate multiples in the range
. Thus, a discrete-time
spectrum
is said to be
if
, for all
, where
(the ``floor function'') denotes the greatest integer less
than or equal to
.

**Next Section:**

Downsampled STFT Filter Bank

**Previous Section:**

Making a Bandpass Filter from a Lowpass Filter