Interpolated Finite Impulse Response Filters Case Study

Evan Edward Shenkman The University of Miami, Dept. of Electrical Engineering Coral Gables, USA

Abstract— The use and development of interpolated finite would be computationally impossible to achieve, especially in impulse response filters (IFIR filters) began in the 1980s. The real-time processing. concept behind IFIR filters stems from the desire to achieve filters with extremely narrow transition bands. To meet such !" !"#!" !!!! !!" specifications, an FIR filter would require a large order resulting � = −2� (2) in a very computationally costly process. On the other hand, !".!(!!!!!) using an IIR filter reduces computational costs, but it removes the ability to attain a response. This study will provide a detailed overview of IFIR filters, including the theory III. THEORY OF OPERATION - INTERPOLATION behind operation, historical and current improvements, design Interpolation or up-sampling is the process of introducing process, and the advantages and disadvantages of IFIR filtering. new data points into a discrete-time signal — assuming that

discrete-time signal was sampled at a suitable rate. In digital Keywords—finite, FIR, interpolated, narrowband, filter order , interpolation is a two stage process. The first stage involves inserting zero-samples into the digital signal I. INTRODUCTION being interpolated. For example, take x[n] shown in Figure 1. In the second graph in Figure 1 x[n] has been interpolated with All realizable filters can be broken down into two distinct an interpolation-factor (L) of 3. classes—finite impulse response (FIR) filters and infinite impulse response (IIR) filters. FIR filters are useful for many reasons. FIR filters are always stables, they are easy to design to a given specification, and are capable of having a linear phase phase-response. While there are many advantages associated with implementing FIR filters, there exists a huge limitation. Larger, and more precise filters are computationally expensive and become impractical to implement. IIR filters on the other hand can achieve the same impulse response with a much lower filter order thus reducing the number of computations. IIR filters however can become unstable and cannot be designed for linear phase. IFIR filters look to maintain the positives of FIR filters while reducing their glaring limitations [1]. Figure 1 II. THEORY OF OPERATION – FIR DESIGN The new x[n] can now be described by equation 3 where To begin an overview of how an IFIR filter functions as a v(n) below represents the second signal in Figure 1. means to improve upon narrowband FIR design, we must understand some of the theory behind its operation. First we will examine a normal FIR narrowband using the (3) below specifications. We can also observe the effects in the frequency domain. �! = 0.015� , �! = 0.001 Figure 2 below shows how the frequency mapping along – � to � � = (1) � is compressed after the zero-samples are inserted into the �! = 0.020� , �! = 0.001 digital signal. The peaks on the top graph correspond to 5kHz Using Kaiser’s formula (2), we are able to calculate and 6.5kHz with a sample-rate of 44.1kHz. In the bottom graph the minimum filter-order required to realize the described however, we can see those peaks are compressed inwards and the periodic replicas of x[n]’s frequency response have now frequency response as 1179. Obviously a filter of this order “imaged” onto xint[n]’s frequency response since – � to � now encompasses those replicas. The ratio of frequency compression in inversely proportional to the interpolation factor. This compression is the basis behind IFIR design.

Figure 3

Figure 2 The second and final stage of interpolation requires the signal enter a Low-Pass Filter to remove the induced high frequency images. The Low-Pass Filter operation will successfully complete the up-sampling process and smooth out the interpolated signal just as if it had initially been sampled at the faster rate.

IV. IFIR FILTER DESIGN IFIR Filters utilize the frequency compression inherent to interpolation to develop computationally efficient narrowband filters. To begin the design process, we start with a model FIR Filter, H (z). We then replace H (z) with H (zM). Proto Proto Proto Figure 4 Shown in eq4 and eq5, by adding the stretching factor of M we are expanding the length of the prototype filter’s impulse There exists a limitation or bounds upon which the response [7]. This new filter is called a Band-Edge Shaping interpolation factor cannot exceed. If the interpolation factor is sub-filter. It is called a sub-filter as it is a building block to the too high, there will be too much frequency compression and final IFIR filter (see Figure 3). the masking filter will have to be of a much higher order to � = � �! = !!! ℎ (�)�!!" (4) !"#$!!"#! !"#$# !!! !"#$# remove the images that are now closer to our desired filter response. The bound on the interpolation factor can be �!"#$!!"#! = � �!"#$# − 1 + 1 (5) calculated using eq6. L depends on the type of FIR filter we are implementing. This design process can be used for The “stretching” of the FIR filter’s impulse response narrowband Low-Pass, High-Pass, and Band-Pass filters [4]. should sound similar to interpolation described before – in fact it is. Our band-edge filter’s frequency response contains a � �!"# = ��� � ��� compressed version of our prototype’s frequency response as �!" well as the high-frequency images caused by the interpolation of the impulse response. The frequency response of Hband-edge ! �!"# = ��� � ��� (6) is shown in Figure 4. !!!!" The output from the band-edge filter is then fed into a Low-Pass filter known as the masking sub-filter. The masking � �!"# = ��� � ��� sub-filter G(z) is responsible for removing the high frequency �!! − �!! M images in Hproto(z )’s frequency response. The resulting frequency response is the desired baseband, narrowband FIR V. IFIR VS. FIR EXAMPLE Low-Pass filter as illustrated in Figure 4. To compare performance and observe the advantages of and disadvantages of IFIR filters let put it to the test. In this section both and FIR and IFIR filter with the same specs will be designed concurrently following the same specs laid out in ! ! (1). �! = 0.015� , �! = 0.0005 � � = ! ! (11) For the FIR filter, the Kaiser formula returned that the �! = 0.1133� , �! = 0.001 filter would require an order of 1179. We can then invoke the Remez Exchange/Parks-McClellan algorithm to find the Using the Kaiser formula (2) and the Parks-McClellan optimum impulse response to meet the specs, thus giving us algorithm we can design the two sub-filters. our FIR filter in Figure 5.

Figure 5

While converting the FIR design to an IFIR design we need to conscious of the fact that an IFIR filter has two internal sub-filters (the band-edge filter and the masking filter). To begin we must decide the optimal interpolation factor. The optimal interpolation factor will give our IFIR the maximum possible percentage of computational reduction. To compute Lopt we can use this equation outlined by Mehrina [3].

!! �!"# = (7) !!!!!! !!(!!!!!)

Based on the specifications, Lopt is calculated to be 15. Now that we have found our interpolation value we can begin to adjust the overall specifications to build the band-edge and masking filters. Figure 7 ! �! = �� , �! = ! ! ! ! ! ! �!" � = (8) ! ! The table below shows IFIR filter architecture and �! = ��! , �! = �! performance for different interpolation factors.

! �! = � , �! = ! L-factor Order H Order G IFIR Order %reduction ! ! ! ! be � � = !! (9) 1 726 8 734 49.38 �! = − � , �! = � ! ! ! ! ! 5 291 20 311 78.54 10 146 44 190 85.90 After eq8 and eq9 are calculated, we can spec the two sub- 15 97 74 171 88.32 filters.

! ! ! �! = 0.225� , �! = 0.0005 �!" � = ! ! (10) �! = 0.300� , �! = 0.001 VI. IMPROVEMENTS AND DEVELOPMENTS to design and realize, FIR filters are unique in that they can be There are a few experiments concerning the development designed for linear phase. For a system to be linear phase is and advancement of IFIR filters in attempts to improve their very important and often it is a must. Since most applications performance. While the IFIR filter greatly improves upon a in digital signal processing require constant group delay, IIR standard FIR filter there are still many trade-offs associated filters, while computationally are much less expensive, aren’t with IFIR filters. A drawback to up-sampling is the as often implemented [6]. IFIR filters take advantage of the requirement of more memory space to store a digital signal. principles behind digital signals and manipulate them in a There is new work involving a decimation stage after the clever way to improve upon the more basic FIR filter design. masking sub-filter so as to remove redundant samples post- The sheer amount of work saved when implementing IFIR filtering. filters is extremely important as it allows very particular, narrowband, linear phase filters to exist within large and Another improvement proposed in 2005 was an IFIR filter complex digital systems without drawing too much power, without any multipliers [2]. Without a single multiply stage, resources or time. Due to the provided overhead, IFIR filters an IFIR filter would become extremely computationally are still very hot research topics to this day. efficient and the processing time during a single filtering operation would be almost insignificant. This is accomplished REFERENCES through an algorithm similar to the Perceptron Learning [1] Neuvo, Y.; Doug Cheng-Yu; Mitra, S.; “Interpolated finite impulse algorithm in the field of machine learning. A small round response filters,” Acoustics, Speech and Signal Processing, IEEE value is selected from the start and the filter coefficients are Transactions, vol.32, no.3, pp. 563-570, Jun 1984 weighted prior to any input signal [5]. The system is then hit [2] Jovanovic-Dolecek, G.; Mitra, S.K.; “Multiplier-free FIR filter design with a parameterized test signal and the system checks and based on IFIR structure and rounding,” Circuits and Systems, 2005. 48th tunes itself based on the output. Midwest Symposium, pp.559-562 Vol. 1, 7-10 Aug. 2005. [3] Mehrina, A; Wilson, A.N., Jr.; “On optimal IFIR filter design,” Circuits and Systems, 2004. ISCAS ’04. Further improvements include the use of two or more [4] T. Saramaki, Y. Neuvo, and S. K. Mitra, “Design of computationally interpolation factors. Since the imaging filter is just a Low- efficient interpolated FIR filters,” IEEE Trans. Circuits Syst., vol. CAS- Pass filter by adding more interpolation factors you change 35, pp. 70-88, Jan. 1988. your imaging filter to an IFIR filter itself. By having these [5] A. Bartolo, B. D. Clymer, R. C. Burges, and J. P. Turnbull, “An efficient recursive IFIR filters within each other the computational method for FIR filtering based on impulse response rounding,” IEEE Trans. On Signal Processing, vol.46, No. 8, August 1998, pp. 2243- requirements start to vanish much quicker than if there was 2248. only a single interpolation factor implemented. [6] Jovanovic-Dolecek, G.; Javier Diaz Carmona, J. “Lowpass minimum phase filter design using IFIR filters,” Electronics Letters, vol.33, no.23, pp.1933-1935, 6 Nov 1997. VII. CONCLUSION [7] R. Lyons, “Interpolated narrowband lowpass FIR filters,” IEEE Signal Despite their cost effectiveness and computational burden, Process. Mag., pp.50-57, Jan. 2003. FIR filters are an extremely important aspect of modern digital signal processing. In addition to being reliable and very simple