06-12-2012, 05:58 PM
Variable digital filters
[attachment=42860]
INTRODUCTION
Variable digital filters have wide applications in telecommunication, medical instrument, and digital radios. The variable characteristics of digital filters mainly present on the variable frequency response, such as variable cutoff/bandedge frequency and controllable fractional delay. Many researches have studied digital filters with variable cutoff/bandedge frequency, where the transition width is fixed, but the cutoff frequency, or the bandedges are variable over a range of frequencies. In this approach, each delay element of a prototype filter is replaced by a first-order all pass network to transform the frequency. The resulting filter then has an identical frequency response as that of the prototype filter, but on a distorted frequency scale. Oppenheim proposed a new class of transformation based on the above technique, so that the resulting impulse response of the filter is finite and the phase of the filter is linear.
A straightforward but practical method to implement variable bandedge FIR filter is to use a set of over-designed fixed filters, each having several times sharper transition band than that required by the variable filter. Thus, each filter is taking care of only part of the variable frequency regions. At any moment of the operation, only one of the filters is used. Due to the over-design of the filters, however, the computational complexity of the filters are high, especially when the variable filter requires sharp transition band.
It is well known that the computational complexity in terms of the number of multiplications is inversely proportional to the transition width. When the filters are made variable, the complexity is at least as high as that of their corresponding directly implemented fixed filters with the same transition widths. In contrast to the traditional variable filters that vary the bandedge of the frequency response of the filters, here we have a method to efficiently shift the input signal frequency spectrum. The frequency-shifted signal is shaped by a filter with a fixed bandedge, and then shifted back to its original frequency region. This technique achieves the same effect of varying the bandedge by shifting the signal along the frequency axis. By making use of the low-complexity techniques in fixed filter design, the overall computational complexity of the variable filter may even be lower than that of a fixed filter with the same transition width and ripple requirements implemented in its direct form. The frequency response masking technique and its extension the fast filter bank are the basics of this approach.
FREQUENCY RESPONSE MASKING AND FAST FILTER BANKS
A very efficient technique to design fixed sharp FIR filters with low complexity is frequency response masking (FRM). Just as the name implies, FRM uses masking filters to obtain the desired frequency responses. The basic idea behind this is to compose the overall filter using several subfilters, namely, the bandedge shaping filter, its complementary, and two masking filters
VARIABLE BANDEDGE FILTER
Traditionally, a design technique for lowpass filters can be transformed to design highpass and bandpass/bandstop filters. The technique proposed here can similarly be extended to the design of highpass and bandpass/bandstop filters. Here, only lowpass filters are considered.
Consider a lowpass filter with a passband edge of ω_p and transition width of satisfies the following constraints:
Discretely Tunable Filter
From Fig.3.3(a), it can been seen that FFB can serve as a variable bandedge filter by combining the proper channels; the resulting variable filter, however, has bandedges only at a set of discrete frequency values. For example, an N-channel FFB with transition width ω_t may generate lowpass filter with passband edge at (2n+1)π/N-ω_t/2 for n=0,1 ,2 ,….. (N/2-1).. by combining channels from channel 0 to channel (±n). Fig.3.3(b) shows one of the variation of a lowpass filter with transition width synthesized from an eight-channel FFB. By combining channels 0 and ±1, the resulting lowpass filter has passband edge at 3 π /8-ω_t/2. The bandedge of a filter synthesized by combining the outputs of an FFB can be adjusted only in discrete step; such a filter is called a discretely tunable filter.
[attachment=42860]
INTRODUCTION
Variable digital filters have wide applications in telecommunication, medical instrument, and digital radios. The variable characteristics of digital filters mainly present on the variable frequency response, such as variable cutoff/bandedge frequency and controllable fractional delay. Many researches have studied digital filters with variable cutoff/bandedge frequency, where the transition width is fixed, but the cutoff frequency, or the bandedges are variable over a range of frequencies. In this approach, each delay element of a prototype filter is replaced by a first-order all pass network to transform the frequency. The resulting filter then has an identical frequency response as that of the prototype filter, but on a distorted frequency scale. Oppenheim proposed a new class of transformation based on the above technique, so that the resulting impulse response of the filter is finite and the phase of the filter is linear.
A straightforward but practical method to implement variable bandedge FIR filter is to use a set of over-designed fixed filters, each having several times sharper transition band than that required by the variable filter. Thus, each filter is taking care of only part of the variable frequency regions. At any moment of the operation, only one of the filters is used. Due to the over-design of the filters, however, the computational complexity of the filters are high, especially when the variable filter requires sharp transition band.
It is well known that the computational complexity in terms of the number of multiplications is inversely proportional to the transition width. When the filters are made variable, the complexity is at least as high as that of their corresponding directly implemented fixed filters with the same transition widths. In contrast to the traditional variable filters that vary the bandedge of the frequency response of the filters, here we have a method to efficiently shift the input signal frequency spectrum. The frequency-shifted signal is shaped by a filter with a fixed bandedge, and then shifted back to its original frequency region. This technique achieves the same effect of varying the bandedge by shifting the signal along the frequency axis. By making use of the low-complexity techniques in fixed filter design, the overall computational complexity of the variable filter may even be lower than that of a fixed filter with the same transition width and ripple requirements implemented in its direct form. The frequency response masking technique and its extension the fast filter bank are the basics of this approach.
FREQUENCY RESPONSE MASKING AND FAST FILTER BANKS
A very efficient technique to design fixed sharp FIR filters with low complexity is frequency response masking (FRM). Just as the name implies, FRM uses masking filters to obtain the desired frequency responses. The basic idea behind this is to compose the overall filter using several subfilters, namely, the bandedge shaping filter, its complementary, and two masking filters
VARIABLE BANDEDGE FILTER
Traditionally, a design technique for lowpass filters can be transformed to design highpass and bandpass/bandstop filters. The technique proposed here can similarly be extended to the design of highpass and bandpass/bandstop filters. Here, only lowpass filters are considered.
Consider a lowpass filter with a passband edge of ω_p and transition width of satisfies the following constraints:
Discretely Tunable Filter
From Fig.3.3(a), it can been seen that FFB can serve as a variable bandedge filter by combining the proper channels; the resulting variable filter, however, has bandedges only at a set of discrete frequency values. For example, an N-channel FFB with transition width ω_t may generate lowpass filter with passband edge at (2n+1)π/N-ω_t/2 for n=0,1 ,2 ,….. (N/2-1).. by combining channels from channel 0 to channel (±n). Fig.3.3(b) shows one of the variation of a lowpass filter with transition width synthesized from an eight-channel FFB. By combining channels 0 and ±1, the resulting lowpass filter has passband edge at 3 π /8-ω_t/2. The bandedge of a filter synthesized by combining the outputs of an FFB can be adjusted only in discrete step; such a filter is called a discretely tunable filter.