DFT Filter Banks

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3.4.2 DFT Filter Banks

Basic Forms

The DFT is a well-known approach for splitting a signal into several bands. Figure 3.47

shows a basic N-channel DFT filter bank analyzer. The input signal, x(n), is first multiplied

by the complex sinusoids e^−jωkn, k = 0, 1, 2, ...,N − 1, resulting in N complex

signals. These modulated signals are then lowpass filtered by h(n) and downsampled by

a factor of N (critically sampled), and thereby effectively converting them to baseband

signals. The channel outputs, X_k(m), can be expressed as

[+] Enlarge Image

Figure 3.47: Basic Form of a DFT Filter Bank Analyzer (when D = N system is critcally

sampled).

and when critically sampled (D = N), X_k(m) is

Recalling that the N-point DFT of a signal x(n) has the form

for practical purposes, we can assume that the filter h(n) has a finite impulse response of

lengthN. With this, we can conclude that Equation 3.67 expresses the N-pointDFT of the

sequence h(mN − l)x(l) for each value of m.

In the DFT filter bank synthesizer, as shown in Figure 3.48, all the N channel signals

are interpolated (upsampled and lowpass filtered) first by a factor of N that restores them

to their original sampling rate. Finally, all the channel signals are upconverted and summed

to get the synthesizer output.

Figure 3.48: Basic Form of a DFT Filter Bank Synthesizer.

Mathematically, each of the channel signals can be expressed as

The operation [Y_k(m)]N↑ represents placing N − 1 zeros between samples in Y_k(m).

From the definition of the IDFT,

The output of the synthesizer is the sum of all the channel signals as given below.

Equation 3.72 says that synthesis is achievable by taking the IDFT across channels and

interpolating each resulting time sample with the LPF g0(n).

Alternative Architectures

An alternative way of implementing the DFT filter bank can be developed as follows.⁸

From Section 3.4.2, the basic DFT filter bank analyzer output is given by

This time it is assumed that the signal is not critically sampled, i.e., . Equation 3.73

can then be written as

where

is a complex BPF centered at the frequency,  and, as

noted before, [ ]D↓ is the operation of taking 1 of every D samples. Figure 3.49 is the

alternate structure for the analyzer. In each channel, the signal is bandpass filtered by

, downsampled by a factor of D, and then downconverted by the complex sinusoid

 In the case of critical sampling, i.e., D = N, Equation

3.74 becomes

Similarly, an alternative structure can be created for the synthesizer. In the basic DFT

filter bank synthesizer, each channel signal as given by Equation 3.70 is (assuming no

critical sampling)

[+] Enlarge Image

Figure 3.49: Alternate Implementation of a DFT Filter Bank Analyzer.

Another way of expressing Equation 3.78 is

Again, let

Then,

Note that this equation resembles that of the interpolation process shown in Figure 3.8

where effectively only a subset of coefficients engage the data, and thus, it can be written

as

 

Figure 3.50: Alternate Implementation of a DFT Filter Bank Synthesizer.

 

Note that  is the complex BPF representation of the LPF g₀(n) with

the center frequency at ω_k. This alternative implementation of the basic synthesizer is

shown in Figure 3.50.

In this alternative implementation, each of the channel input signals is separately multiplied

by the complex sinusoid, translating the frequency bands to their original spectral

location. The upconverted signals are then interpolated by a factor of D, converting the

sample rate to the original value. The interpolation filter is the complex BPF, the impulse

response of which is given by  which selects the desired

bandpass image. For the case of critical sampling, i.e., D = N, Equation 3.79 reduces to

Polyphase Implementation of the Digital Filter Bank

Polyphase filters⁹ take advantage of the similarities in the digital filter banks with the DFT

and IDFT implementation structures. The polyphase structures for the decimator and interpolator,

as discussed in Section 3.3, can be applied to the DFT filter banks, resulting in a

highly efficient filter bank realization.

The basic DFT filter bank analyzer, discussed earlier, implements Equation 3.67:

With the assumption of critical sampling, the impulse response h0(n) of the LPF in each

channel can be expressed in polyphase form as

where i refers to the ith branch in the N-branch polyphase decomposition of h₀(n) in any

of the k channels, k = 0, 1, 2, ...,N − 1. From Equation 3.82, replacing l by rN − i and

using the definition in Equation 3.83 yields

where the input signal to each branch of the polyphase network for a single channel, i, in

the DFT filter bank structure is expressed as

Equation 3.85 can be interpreted as advancing the original input x(N) by i samples and

then downsampling it by a factor of N. A closer inspection of the result in Equation 3.84

reveals that it can be expressed as the DFT of a convolution operation, i.e.,

The polyphase implementation of the DFT filter bank analyzer is shown in Figure 3.51

for a single filter bank channel along with the equivalent clockwise commutator model (see

Figure 3.41).

In a similar fashion, the output of the DFT filter bank synthesizer as expressed by

Equation 3.72 with the assumption of critical sampling is

Figure 3.51: Polyphase/Convolution Implementation of a DFT Filter Bank Analyzer.

Let y(n) be divided into N sets of polyphase branch output signals of the form

This actually implies that the output y(n) is computed by sequentially taking the polyphase

branch outputs, y_i(n) , i = 0, 1, 2, ...,N − 1. Using Equation 3.88, we can reformulate

Equation 3.89 as

By polyphase decomposition of the LPF g0(n),

Applying this definition, Equation 3.90 takes the form

According to this equation, first the IDFT is calculated using the N-channel DFT filter

bank synthesizer input signals, Y_k(m) , k = 0, 1, 2, ...,N − 1 for each sample time, m.

The ith IDFT output V_i(m) is the input of the ith branch of the polyphase decomposition

of the LPF g0(n). Then for that instant of time, m, the output of the ith branch filter,

y_i(m), is the convolution of the input, V_i(m) and filter coefficients for that branch, q_i(m).

The final output y(n) is the interleaved version of these polyphase branch outputs. The

implementations of Equations 3.89 and 3.92 are shown in Figure 3.52.

Figure 3.52: Polyphase/Convolution Implementation of a DFT Filter Bank Synthesizer.

Relevant NI products

Customers interested in this topic were also interested in the following NI products:

• RF and Communication Hardware and Software
  
• Other Modular Instruments (digital multimeters, digitizers, switching, etc...)
• LabVIEW Graphical Programming Environment

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