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. The bandedge shaping filter,Ha ( zM ) is derived by replacing each delay element of a prototype filter Ha (z) a by M delay elements. Its complementary filter Hc( zM ) is given by
Where Na is the filter length of Ha(z). The frequency response of and are shown in Fig2.2 Two masking filters, HMa(z) and HMc(z) are used to remove the undesired frequency components from Ha ( zM ) and Hc ( zM ) respectively. The overall filter is formed according to
Ha ( zM ) HMa(z)+[ ] HMc(z)
where H(z) is z-transform transfer function of the overall filter. The length of HMa(z) and HMc(z) must be equal to ensure that the outputs from the two branches are in phase. If they are not of the same length, zero-valued coefficients must be padded to the shorter one. The frequency response of H(z) ,HMa(z) and HMc(z)
An extension of the above technique produces the FFB. The FFB has good frequency selectivity with very low computational complexity. An N-channel analysis FFB decomposes the input signal into N channels in the frequency range from dc to sampling frequency, where N is an integer power of two. Let N=2P, where P is an integer. The -channel FFB consists of P levels of filters. The structure of an example of an eight-channel analysis FFB .
Structure of an eight-channel analysis FFB.
is a delay term of and have complementary frequency responses as shown in Fig.2.4(b), whereas the frequency response of the prototype filter is shown in Fig.2.4(a). The frequency responses of the subsequent levels of prototype filters are compressed by respective factors and then shifted by an appropriate amount if necessary, to mask out the unwanted channels. For example, the cascade of , and leads to the output of channel 5. The frequency responses of , and their complements.
Frequency responses of filters for deriving channel 5 in an eight-channel FFB.
The N-channel(N=2P) FFB generates uniform filter banks with transition bands centered at(2n+1) , for n=0,1,….N-1. To have the filter banks with transition width of wt, the passband edge of the prototype half band filter is set to be . The passband edges of the subsequent levels of the prototype filters are chosen as
for level p=0,1,….P-1. The FRM and FFB provide efficient ways to design sharp transition band filters and filter banks with very low computational complexity. However, they cannot be directly extended to design variable 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 and transition width of . The maximum passband and stopband ripple magnitudes are and , respectively. is variable in a range of [ ], whereas , , , and are fixed for a given design. Thus, the frequency response of the variable filter, , satisfies the following constraints:
1- ≤ ≤1+ ,ω [0, ]
- ≤ ≤ ,ω [ ]
For ≤ ≤ , 0≤ ≤ ≤ and
The symbol of anN -channel FFB, where the input signal is decomposed into N channels. For convenience the channels in the frequency range to are considered, where, is the Nyquist frequency. The channels are re-labeled as 0,±1 ,±2 ,….. (N/2). The frequency response of the nth channel Hn , represented as for , n=0,±1 ,±2 ,….. (N/2). is shown in Fig. 3.3(a). For an input signal with spectrum X( )as shown in Fig. 3.2, the output for channel is denoted as Xn( ), for n=0,±1 ,±2 ,….. (N/2). For real input, the output and are real, whereas for n=±1 ,±2 ,….. ±( )is complex. Furthermore is the complex conjugate of for n=1 ,2 ,….. ( ). Therefore, summing and results in a real output. The same effect can be achieved by taking the real part of and then scaling it by a factor of 2.
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Discretely Tunable Filter
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 may generate lowpass filter with passband edge at for n=0,1 ,2 ,….. ( ).. 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 /8 . 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.
Continuously Tunable Filter
An example is used to illustrate the principle of the design of continuously tunable filter. Suppose that it is desired to design a lowpass filter with frequency response , as shown in Fig. 3.5(a), where is the passband edge; the transition width of the lowpass filter is . Suppose also that an eight-channel FFBis adopted, and is not located at any of the discrete values for , for N=8 and n=0, 1, 2, 3. The filter synthesis process goes as follows.
First, a discrete tunable filter is synthesized by combining channels 0,±1 and±2 to have the discrete stopband edge smaller than Dω .The frequency response of the discrete bandedge filter, denoted as ( ) is shown in Fig. 3.5(c). Let the output of ( ) be ( ).
Second, we attempt to synthesize a bandpass filter having frequency response ( ) as shown in Fig. 3.5(d) in such a way that the combination of ( ) and ( ) produces the desired lowpass response.
Last, the output of the discretely tunable filter ( ) is then added to the output of the bandpass filter ( ) to form the desired output of the continuously tunable lowpass filter.
The bandpass filter ( ) proposed in the second step may be obtained as follows.
1) A bandpass channel is selected from the FFB. In this example, channel 3 with frequency response ( ) as shown in Fig. 3.5(e) is selected. In case the transition band of the variable filter in a given variation is located in the transition bands of the FFB channels, the two adjacent bandpass channels are selected and combined. Let the output of the selected bandpass channel(s), i.e., ( ) in this example, be ( ), as shown in Fig. 3.5(f).
2) ( ) is shifted in the frequency domain by an appropriate amount to become as shown in Fig. 3.5(g).
3) is then filtered by a lowpass filter whose frequency response, ( ) , is shown in Fig. 3.5(h), to produce the output signal Xb́( )= ( ) shown in Fig. 3.5(i). This lowpass filter ( ) is a fixed filter with the same transition width, , as that of ( ) is referred to as bandpass shaping filter. The purpose of shifting ( ) in the frequency domain and using the bandpass shaping filter is to remove the frequency component in ( ) corresponding to the stopband of . The frequency component in ( ) to be removed is shown as the shaded region in Fig. 3.5(f).
4) Xb́( ) is then shifted back to its original frequency location to form Xb́( ) as shown in Fig. 3.5(j).
5) Xb́( ) is complex. The desired output of the bandpass filter ( ) is obtained as twice the real part of Xb́( ). The spectrum of the desired output of the bandpass filter denoted as Xb́ ́( ) is shown in Fig. 3.5(k).
The block diagram of the bandpass filter ( )
Channel Selection for the Construction of Continuously Tunable Filter
For any given , the channels used to construct the discretely tunable filter ( ) and the bandpass filter ( ) must be determined.
To construct the continuously tunable filter using the technique the stopband edge of the discretely tunable filter must not be larger than , but closest to . is the stopband edge of the desired variable lowpass filter, . The discretely tunable filter consists of channel (± n) of the FFB where n satisfies the constraint
Note that channel 0 is always selected for the construction of the discretely tunable filter. As a consequence, the minimum band of the variable filter that can be synthesized is that of channel 0.
In generating the output of the discretely tunable filter, besides the output of channel 0, conjugate output of channel pairs of for n satisfying the above equation are summed together. As the same effect of such summation can be achieved by taking the real part of the output of channel , and scaling it by a factor of 2, in actual implementation, channel is n used, while channel (-n) is ignored, as shown in Fig.3.7.
In the selection of bandpass channels for the construction of the bandpass filter, there are two cases. In the first case, the transition band of is located entirely in the passband of a single channel of FFB, for example, the variation of shown in Fig. 3.6(b). The transition band of the variable filter is located inside channel 1. Channel 1 is thus selected for the construction of the bandpass filter. In the second case, the transition band of is located at the junction of two channels, as shown in Fig. 3.6(c); in this case, both channels, (channels 2 and 3 in the example of Fig. 3.6(c), are selected and their outputs are summed to construct the bandpass filter. Considering the above two cases, channel is selected for the construction of the bandpass filter if is satisfied. Note that channel (N/2) is not selected to construct the bandpass filter in any case. As a consequence, the maximum band of the variable filter that can be synthesized is not larger than the combination of channels from 0 to ±(N/2-1). Based on the above equations, the diagram of the channel selector for the construction of the discrete tunable filter and bandpass filter is illustrated in Fig. 3.7
Fig. 3.6. Illustration of the transition band location of the variable filter with respect to the FFB channels. (a) Frequency response of an eight-channel FFB. (b) Transition band of the variable filter is located within the passband of a single FFB channel, and (c) Transition band of the variable filter is located at the junction of two FFB channels.
COMPLEXITY ANALYSIS
Here, the complexity of the proposed technique, in terms of computational complexity and implementation complexity is analyzed and compared with other variable digital filters.
Computational Complexity
In the operation of the proposed variable filters, multiplications dominate the computational complexity. In this analysis, the computational complexity is measured in the number of multiplications required to generate one output sample.
It is obvious that the overall computational complexity Cv, for a variable filter with normalized transition width and ripple magnitudes and , is given by
for P=4, where . is the computational complexity of FFB to separate the signals into 16 channels, is that of bandpass shaping filter, and is that of modulation and demodulation of . For a given ripple specification, is independent of the transition width of the variable filter. Therefore, when the transition width of the variable filter is small, the computational complexity for the FFB remains unchanged.
Consider a fixed filter with the normalized transition width , and ripple in both passband and stopband. The passband and stopband ripple magnitudes are assumed to be the same to simplify the discussion. The computational complexity of the fixed filter, denoted as Cf, is given by
=
From the above equations it can be seen that the increase of due to the decrease of is slower than that of , since is proportional to , whereas is proportional to . The plots of the curves of and versus for typical values of 0.1, 0.01 and 0.001 are shown in Fig. 13. It can be seen from Fig. 13 that when is smaller than 0.0057, 0.0083, and 0.0093, for 0.1, 0.01 and 0.001, respectively, the computational complexity of the variable filter is even lower than its corresponding fixed filters with the same. .
Implementation Complexity
The implementation complexity of the computational elements is consistent with the computational complexity, i.e., the number of multipliers to be implemented is the same as the number of multiplications to generate one output sample. Besides the computational elements, the non-computational components used in the proposed technique includes digital comparators, delay elements and memory block. In this subsection, the implementation complexity in terms of the number of digital comparators and delay elements is discussed.
1) Digital Comparator: Digital comparators are used to control the switches in the channel selector, as shown in Fig.3.7. In anN -channel FFB synthesis, the required numbers of switches and comparators are both N-2. While switches could be synthesized using transmission gates, the implementation complexity of a comparator is comparable to (and generally is less than) that of an adder with the same bit width. An adder could directly serve as a digital comparator, since the sign of the sum of one operand and the negative value of the other operand is the result of the comparison. When the difference of the two operands are not required, as in the current case, simpler components could be used. Compared with the implementation complexity of the computational elements, the implementation overhead due to the digital comparators is negligible.
2) Delay Element: To synthesize the FFB, each delay element in the prototype filter in level p is replaced by elements. Meanwhile, in the bandpass shaping filter, the half band filter is realized in FRM technique, where each delay in the bandedge shaping filter is replaced by LS delays. LS is given in above equation. Total number of delay elements in the proposed technique is approximately
when . This value generally is higher than the number of delay elements required in a fixed filter with the same transition width and ripples. when P=4. This value generally is higher than the number of delay elements required in a fixed filter with the same transition width and ripples.
CONCLUSION
An efficient approach to design variable band edge FIR filters with sharp transition band is discussed in this seminar. The variable filter is constructed from a fixed FFB and a fixed half band filter, whereas the variation of the filter is realized by shifting the signals in the frequency domain. Since fixed filters plus FRM technique are used, the proposed technique achieves extremely low computational complexity when the transition band is sharp. In contrast to the traditional variable filters that vary the band edge of the frequency response of the filters, here we introduced a method to efficiently shift the input signal frequency spectrum. The frequency-shifted signal is shaped by a filter with a fixed band edge, and then shifted back to its original frequency region. The discussed method achieves the same effect of varying the band edge 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.
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