Multirate digital filters and filter banks find applications in communications, speech processing, image compression, antenna systems, analog voice privacy systems, and in the digital audio industry. There has been substantial progress in multirate system research. This includes design of decimation and interpolation filters, analysis/synthesis filter banks, and the development of new sampling theorems [17].
The parallel filter bank system in [16] has an 18-band filter bank implemented by using infinite impulse response (IIR) filters. Figure 3-1(a) shows the parallel filter bank block diagram of [16]’s auditory compensation system. The input signal X(n) passes the parallel 18-band filters called F22 ~ F39 that decompose the input into different frequency components and then enters a “Gain & Compression” system to get different gains and compression ratios. Then the six octave bands’ outputs are added. For multirate systems, the aliasing that produced by down-sampler can be avoided if the input signal X(n) is a low pass signal bandlimited to the region |ω| <
π/M, where M is the down-sample factor. The frequency ranges of second octave filters are all smaller than π/2 so that they can be down-sampled by two with no aliasing. Also, the frequency ranges of the third octave to six octave bands are four to thirty two times smaller than first octave so that they can be down-sampled by four to thirty two. Figure 3-1(b) shows the multirate filter bank with basic down-sampler on analysis bank. In the synthesis bank, the down-sampled data should up-sample to recover origin data rate. However, the up-sample operation causes the image components in synthesis bank. So we should further design the synthesis filters, F’22 ~ F’36, to filter image components such that each 1/3-octave band filter can meet the ANSI S1.11 specification.
Figure 3-1 Various filter bank structures: (a) parallel, (b) reduced data rate, (c) reduced data rate with combined interpolation filters
From Figure 3-1(b), F’22 ~ F’36 in synthesis bank are used to eliminate the alias and image components. Since the images occur on the high frequencies, we can use low pass filters to suppress them. The image components of two to thirty two times up-sampler generate on frequency ranges that bigger than π/2 to π/32 so we can use low pass filters that to eliminate image components. Figure 3-1(c) shows that the image terms of F’22 ~ F’36 that generated by two to thirty two up-sampler are suppressed by using low-pass filters called interpolation filters, I2, I4, I8, I16, and I32,
where I2 keeps the frequency range from zero to π/2 and I4 keeps the frequency range from zero to π/4 and so on. From this architecture, we do not need to design F’22 ~ F’36. Instead, we only need to design five low-pass interpolation filters instead of designing F’22 ~ F’36 to suppressthe image components.
Before we down-sample the input signal, we should remove the high frequency components to avoid the alias term. For example, if we want to down-sample the input signal by two, we should remove the frequency range that bigger than π/2 to avoid the aliasing first. Therefore, we need a low pass decimation filter (D) that keeps the frequency component smaller than π/2 and remove the frequency component that bigger than π/2. Similarly, if we want to down-sample the input by thirty two, we should apply another low-pass decimation filter that removes the signals bigger than π/32 to avoid the aliasing. However, it is not efficient to design different decimation filters. We can use the characteristic of 1/3-octave bands and apply a multirate architecture to reuse the decimation filter that shown in Figure 3-2.
In Figure 3-2, we just use three band-pass filters (F37 ~ F39), a decimation filter (D), and an interpolation filter (I) to cover F22 ~ F39 to implement the whole system. D and I filters are low-pass filters with cut-off around π/2. In analysis bank, the decimation filter is used to avoid aliasing. For second octave bands, the decimation
filter removes the frequency components bigger than π/2 and the two times down-sample operation stretch the frequency response of other components by two times. After down-sampling, the frequency ranges of F34 ~ F36 become equal to F37 ~ F39 because the characteristic of the 1/3-octave band filters. For third octave bands, the first decimation filter and down-sampler remove the frequency components that bigger than π/2 and stretch the frequency components smaller than π/2 by two times.
The second decimation filter and down-sampler remove the frequency components >
π/4 and stretch the frequency components bigger than π/4 by two times again since the input frequency range from π/4 to π/2 has been stretched to π/2 ~ π by passing first down-sample operation and so the other octaves. Finally, the sixth octave passes the decimation filter and two times down-sampler by five times. Input frequency components that bigger than π/32 are removed and the frequency range of F22 ~ F24
has become F37 ~ F39. The interpolation filter in synthesis bank is like decimation filter. The interpolation filter removes the image components that bigger than π/2 each time that causes from two times up-sampler. The two times up-sampler reduce the frequency components by half and induce the image term on high frequencies. When the specifications of F37 ~ F39 pass the two times up-sampler, they will become F34 ~ F36 when neglecting the image component. Similarly, the responses of F22 ~ F33 can be generated by F37 ~ F39 and four to thirty two times up-sampler without taking images into considerations. The interpolation filter will remove the image components of up-samplers, just like the decimation filter. Finally, the sixth octave passes the interpolation filter and two times up-sampler by five times. Image components bigger than π/32 are removed.
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Figure 3-2 Proposed multirate filter bank system
For the final structure, we use five filters to implement an 18-band filter bank F22
~ F39. The frequency response of nth band is shown in equation (3-1).
From ANSI S1.11 specification, we know that the specification on each band’s passband ripple is 1dB and the stopband attenuation is 60dB. Furthermore, the frequency response of each band-pass filter’s should under M(ω) and above the m(ω), where M(ω) and m(ω) are the limitations on the maximum and minimum attenuation of the nth band from frequency range 0 ~ π that defined in Section 2-3. The design constrain donates as equation (3-2)
39
~ 22 ), ( )
( )
( ≤ H e ≤M n=
mn ω n jw n ω , (3-2)
whereHn(ejw)is the frequency response of nth band.
For first octave design, the F37 ~ F39 should meet the ANSI S1.11 37th ~ 39th bands specification. Besides, the passband ripple of the three bands should be small enough because the ripple has effects on H22 ~ H36. For decimation filter and interpolation filters, the alias and image components should be suppressed and the passband ripple should be small enough because it also has effects on H22 ~ H36.