Continuously Variable Bandwidth Sharp FIR Filters with Low Complexity ()
1. Introduction
Real time variable frequency response characteristics are required in various telecommunication applications including the software defined radio (SDR). SDR implies that important radio characteristics can be defined by software. In every radio receiver, one of the important characteristics is the bandwidth of the filter that selects the desired channel.
Several methods exist [1-8] for realizing the discretely variable bandwidth filter. The usual method to change the frequency response of a digital filter is to increase the number of taps or the order of the finite impulse response (FIR) filter to reduce the bandwidth and to decrease the number of taps to increase the bandwidth.
This can be implemented in SDR by modifiable software or firmware operating on programmable devices having constraints in resources as well as power. In the hardware implementation of a variable bandwidth filter, we must use a variable length data array, coefficient array, adders, multipliers etc. along with the filter design algorithm operating in the background to compute the new coefficients for each desired bandwidth.
The number of taps in a FIR filter is known to be proportional to the ratio of sample rate to transition bandwidth i.e. fs/∆f. When the bandwidth of a filter is reduced or increased by a factor, the transition width is also reduced or increased respectively by the same factor i.e., the spectral shape scales with bandwidth. Hence as the number of taps varies, the transition bandwidth and hence the overall bandwidth will also vary.
Now, if the number of taps is fixed, then to obtain a continuously variable bandwidth FIR filter, a new technique was proposed by Fred Harris [9] where a filter design algorithm is not needed to operate in the background to compute new coefficients for each desired bandwidth. Here, the principle used is that the absolute bandwidth of a filter is reduced by operating it at a reduced sample rate. This is accomplished [9] by efficiently shifting the frequency spectrum of the input signal by using an arbitrary down sampler. The down-sampled signal is then processed with a fixed length, fixed bandwidth FIR filter and then shifted back to the original spectrum by using an up-sampler. The combined technique gives the effect of reducing the effective bandwidth of the filter. This method can only decrease the bandwidth. However, many applications may require a bandwidth greater than the bandwidth of the fixed length filter. But this may lead to distortion in the output signal response. The main cause for this distortion is the non-zero transition bandwidth of the fixed length filter. The length of the FIR filter is inversely proportional to the transition bandwidth.
In this paper, the technique of obtaining a continuously variable bandwidth filter using re-sampling, is modified in such a way that, we get a bandwidth increase as well as decrease. At the same time, the distortion in the output response is reduced to a minimum. Hence, the variable bandwidth filter proposed in this paper has the property of fine tuning, i.e., continuous variation of the bandwidth is possible with minimum distortion, which includes both decrease and increase of bandwidth. The fixed length FIR filter is designed by frequency-response masking (FRM) technique. This leads to a sharp transition width and hence less distortion in the output spectrum as well as reduction in the computational complexity.
The next section reviews the variable bandwidth filter. Section 3 presents the design procedure for the arbitrary sample rate converter. The principle and structure of the low complexity technique on the design of the fixed length FIR filter, more specifically, the FRM is presented in Section 4. Section 5 gives the implementation as well as the detailed performance analysis and in Section 6, the computational complexity analysis is done. Section 7 concludes the paper.
2. Review of Variable Bandwidth Filter
Figure 1 shows the concept of a variable bandwidth filter. In the case of a constant form factor FIR filter, the length of the filter is inversely proportional to the transition width, which is proportional to the bandwidth of the filter. Thus, if the bandwidth of the filter is changed by a factor, the length of the filter is also changed by the same factor.
In order to realize a variable bandwidth filter without changing the number of coefficients or coefficient values, we can utilize the relationship between the sample rate and bandwidth [9-11]. This can be further elaborated that the interval between the main lobe peak and the first zero crossing of the impulse response is the reciprocal of the filter bandwidth. Also the interval between the samples is the reciprocal of the sample rate.
The number of taps of the filter (TP) is given by 
, where “fs” is the sampling frequency and “∆f” is the transition bandwidth. If the bandwidth is changed, the transition bandwidth is also changed by the same factor. Hence if the number of taps of the filter is fixed, one method of changing the bandwidth is to change the sampling rate. The absolute bandwidth is proportional to the sampling rate [9]. So we can change the absolute bandwidth of a filter by operating it at a different sampling rate. The technique used in [9] reduces the absolute bandwidth of a filter by operating it at a reduced sample rate. In this paper, we have modified the implementation in such a way that the bandwidth of the filter can also be increased by increasing the sample rate. The use of an FRM based FIR filter, makes the transition bandwidth sharp and reduces the complexity and distortion.
The implementation of the process is represented in Figure 2. An input signal which is initially oversampled is applied to an arbitrary sample rate converter (up or down) that preserves the dynamic range and still satisfies
Figure 1. Magnitude response of variable bandwidth filter.
Figure 2. Functional block diagram of variable bandwidth filter [9].
the Nyquist criterion. The modified signal is processed by the fixed length, fixed bandwidth FRM based FIR filter. The output of the filter is then converted back to the original input sampling rate by using another arbitrary sample rate converter (down or up).
3. Arbitrary Sample Rate Converter
Figure 3 shows the interpolation indexing. This gives the method of forming a sample value at a location “y” that does not correspond to an available location of a M-path polyphase interpolator. The value of M is taken as 5 for illustration.