1 Interpolated FIR Filters Kyle S. Marcolini, Student, University of Miami Abstract—The idea of interpolated finite impulse response III. IFIR DESIGN filters (IFIR) is a technique devised in the 1980s used for designing narrowband filters. Typically, FIR can be of high order IFIR filters were introduced to help with this problem of to achieve the same results as their infinite impulse response normal FIR filters. To achieve a narrowband filter, FIR filters (IIR) counterparts, especially considering filters that have a steep will indefinitely be very high order. IFIR filters begin with a L transition band. IFIR filters can be implemented in place of FIR model FIR filter, FM(z ), but with an introduced stretching, or filters in this case, as they will be lower order overall and interpolation factor, L, increasing the width of the passband. computational efficient. This study will give an overview of IFIR As the passband width increases, the order of the filter filters, improvements made on them, as well a advantages and disadvantages of these enhancements. decreases by L. This model filter is then run through an image filter, G(z), Index Terms—interpolated, impulse response, infinite, finite, which narrows out the passband by attenuating undesired narrowband, order redundancies, creating a lower-order, FIR narrowband filter. The use of low-order cascaded filters in this design results in lowering the order of a single FIR filter, saving computation I. INTRODUCTION because of the number of extra multipliers that are not needed. WO types of filters used in modern filtering are finite The system flow diagram is shown in Figure 1 [1]. Timpulse response (FIR) and infinite impulse response (IIR). FIR filters are attractive to use in designs for several reasons: they are easy to implement, always stable, and can be easily made linear phase. The main disadvantage of using them is their computation requirements. While IIR filters are not always stable and are cannot be made linear phase, they Figure 1. IFIR system are much more efficient and can generate the same response of FIR filters with very low order and fewer computations. L HIFIR = F(z )G(z) (3) II. FIR DESIGN where When designing FIR filters with a narrow transition band, ⎧ f (n /L) n kL, k 0, 1, 2... the order of the FIR filter must increase, which increases the = = ± ± € fL (n) = ⎨ (4) number of multipliers and delays, making it computationally ⎩ 0 otherwise costly. Using MATLAB’s fdatool [1], a narrow lowpass filter is designed with the specifications: In designing the system a stretching and interpolation factor needs to be chosen to maximize efficiency and functionality of ⎧ω p = 0.08π , δ p = 0.05 the filter. For a lowpass IFIR filter, the maximum for value for H(z) :⎨ € (1) the interpolation factor, L, is determined as a function of the ⎩ ω s = 0.09π , δs = 0.01 stopband edge frequency, ωs [1]: To determine the minimum order of the resulting filter, Kaiser’s formula is used [3]: ⎢ π ⎥ LMAX = ⎢ ⎥ (LPF), ⎣ ω SL ⎦ € 10log 13 10 ( δsδ p ) − M = −2π (2) ⎢ π ⎥ 14.6 ω −ω LMAX = ⎢ ⎥ (HPF), (5) ( s p ) π −ω ⎣ SH ⎦ The resulting order of the FIR filter must be at least 292. ⎢ 2π ⎥ This is very expensive, even for such a simple design. Thus, LMAX = ⎢ ⎥ (BPF). the use of IFIR filters is instrumental in creating a filter of a ⎣ ω S1 −ω S2 ⎦ € much lower order. The brackets denote truncation to an integer. These definitions for L guarantee a maximum factor less than π, or half the sampling frequency. € 2 IV. IFIR EXAMPLE ⎧ω p = 0.015π , δ p = 0.0005 To clarify how exactly IFIR filters function a simple G(z) :⎨ (11) ⎩ ω s = 0.1133π , δs = 0.001 example is demonstrated in this section. An FIR and IFIR will be designed using the same parameters, and their results will Using the Parks-McClellan algorithm with two separate be analyzed. The parameters are as follows: filters given these parameters yields the following specifications: ⎧ω p = 0.015π , δ p = 0.001€ H(z) :⎨ (6) ω = 0.020π , δ = 0.001 Orders: NF = 97 NG = 74 NTotal = 171 ⎩ s s Multipliers = 86 Adders = 171 The Parks-McClellan algorithm is used to determine the Pct. Reduction = 88.22% order of the filter. In this case, using Kaiser’s formula from (2), the order is determined to be 1303. The magnitude and € phase response are shown in Figure 2. Fig. 3. Interpolation model filter Below is a comprehensive table of filter orders versus Fig. 2. Magnitude and phase of FIR implementation, up to only 0.2π interpolation factors for the given specifications. In this case, for the given parameters, the optimal interpolation factor [3] is determined in (6). ⎡ ⎤ ⎢ 2π ⎥ L (7) OPT = ⎢ ⎥ 2 ⎢ ωs +ω p + π(ωs −ω p ) ⎥ When converting the FIR specifications to IFIR, both the L model, interpolation filter (HM(z )) and the image filter (G(z)) need to be built. The new specs for the interpolation filter are € mapped as: ⎧ δ ⎪ L L p Table 1. Comparing interpolation factors to filter order [4] L ω p = Lω p, δ p = FM (z ) : ⎨ 2 , (8) ⎪ L L ⎩ω s = Lω s, δs = δs where the new image filter is defined as ⎧ δ p ⎪ω ʹp = ω p , δ ʹp = € G(z) : ⎨ 2 . (9) 2π ⎪ ω ʹ = −ω , δ ʹ = δ ⎩ s L s s s The new design specifications given the prior design are: Fig. 4. Constructed IFIR filter, magnitude and phase, L=15 L ⎧ω p = 0.225π , δ p = 0.0005 FM (z ) :⎨ (10) Comparing the IFIR filter to the FIR model, the passband € ⎩ ω s = 0.3π , δs = 0.001 and transition band appear to be identical, whereas in the IFIR filter, the stopband varies a lot more, but attenuates more. € 3 Given that it is less costly and achieves the same frequency VI. MODIFICATIONS OF IFIR FILTERS characteristics as the normal FIR filter, the IFIR filter becomes In recent years, modifications have been made to IFIR a very attractive option if practical filter design. filters in addition to decimation. These modifications seek to improve performance and explore functionality for new V. DECIMATION IN IFIR FILTERS applications. Among these are ways to design IFIR filters A fairly new technique involved in IFIR filters is a without multipliers, as well as ones with multiple stretching decimation stage in IFIR filter designs. Because of the factors. These will be discussed further in this section. possibility of aliasing in a system because of decimation, it is important to bandlimit a system before a decimation filter can A) Filter With No Multipliers be implemented. A Multiplier-free IFIR design was proposed in 2005. This Assuming the order of the filter is not a prime number, and idea seeks to optimize the IFIR filters in a three-step process efficient implementation of multiple decimation filters can be by representing coefficient values in signed canonic digits of implemented as cascaded pairs in a system. Figure 3a shows a binary digits [2]. After choosing the interpolation factor, L, as basic decimation IFIR implementation, with figure 3b being a well as the model and image filters, the process involves more efficient, cascaded implementation of the same design. choosing a small rounding factor, r. Typically, the starting point is around r = 0.01. It can be modified later. Next, each coefficient of the cascaded time-domain filters in the IFIR system needs to be rounded. The model filter, fM(n) and the image filter, g(n) become fM_r(n) and gr(n), from the Fig. 5a. IFIR design with decimation process: ⎡ h(n)⎤ h (n) = r • h (n) where h (n) = round (12) r I I ⎣⎢ r ⎦⎥ Now, the rounded model filter, fr(n), is interpolated by the factor, L. Finally, sharpening polynomials are applied, where the factors, n and m are chosen, usually equal to 1, and the € rounded filters are tested in the polynomials to see if the original parameters are met. If they are not, the values of m Fig. 5b. Optimized IFIR decimation filter and n can vary, as well as the rounding factor, r, until the The goal of this type of a design is to design a narrowband specifications are met. filter with cutoff frequency near π/M, where M is large and is the decimation factor of the system. Implementing the B) Two Stretch Factors parameters from figure 5b with the design specifications in In 2004, Mehrnia and Willson [3] devised a way to make an equation (6), the resulting response of the decimated filter is optimal IFIR filter using multiple interpolation factors. For shown below in figure 6. more difficult designs, it makes sense to make an IFIR filter out of the image filter. The new design using multiple stretch factors looks like: L L L (13) HIFIR = FM (z )[G1(z )G2(z )] With this new implementation, the stopband and passband frequencies of the model filter, F(z) are now related to the stretch factor, becoming Lωs and Lωp respectively. The first € L image filter’s (G1(z )) stopband and passband frequencies become L12π(L −ωs) and Lωp. Finally, the stop and passbands of the final image filter are 2πL1 − (2π L −ωs) and ωp. The new number of multipliers in the system is defined as: Fig. 6. Decimated IFIR filter ⎛ δ p ⎞ Similarly to the original IFIR design, the passband and MIFIR ≈ D⎜ ,δs⎟ f (L,L1) (14) transition band are nearly identical, with the stopband ⎝ C ⎠ differing more. In the decimated IFIR design, the stopband attenuation is consitent, around -80dB to -90dB.