On a Design of Narrowband FIR Low-Pass Filters

Introduction

Stearns and David (1996) states that “for many diverse applications, information is now most conveniently recorded, transmitted, and stored in digital form, and as a result, digital signal processing (DSP) has become an exceptionally important modern tool.” Typical operation in DSP is digital filtering. Frequency selective digital filter is used to pass desired frequency components in a signal without distortion and to attenuate other frequency components (Smith, 2002; White, 2000). The pass-band is defined as the frequency range allowed to pass through the filter. The frequency band that lies within the filter stop-band is blocked by the filter and therefore eliminated from the output signal. The range of frequencies between the pass-band and the stop-band is called the transition band and for this region no filter specification is given.

Digital filters can be characterized either in terms of the frequency response or the impulse response (Diniz, da Silva & Netto, 2002). Depending on its frequency characteristic, a digital filter is either low-pass, high-pass, band-pass, or band-stop filters. A low-pass (LP) filter passes low frequency components to the output, while eliminating high-frequency components. Conversely, the high-pass (HP) filter passes all high-frequency components and rejects all low-frequency components. The band-pass (BP) filter blocks both low- and high-frequency components while passing the intermediate range. The band-stop (BS) filter eliminates the intermediate band of frequencies while passing both low- and high-frequency components.

In terms of their impulse responses digital filters are either infinite impulse response (IIR) or finite impulse response (FIR) digital filters. Each of four types of filters (LP, HP, BP, and BS) can be designed as an FIR or an IIR filter (Ifeachor & Jervis, 2001; Mitra, 2005; Oppenheim & Schafer, 1999).

The design of a digital filter is carried out in three steps (Ingle & Proakis, 1997):

• •

  Define filter specification

• •

  Approximate given specification

• •

  Implement digital filter in hardware or software.

The topic of filter design is concerned with finding a magnitude response (or, equivalently, a gain) which meets the given specifications. These specifications are usually expressed in terms of the desired pass-band and stop-band edge frequencies 𝜔_p and 𝜔_s, the permitted deviations in the pass-band (pass-band ripple) R_p, and the desired minimum stop-band attenuation A_s, (Mitra, 2005). Figure 1 illustrates a typical magnitude specification of a digital low-pass filter.

Figure 1.

Low-pass filter magnitude specification

In many applications it is often advantageous to employ FIR filters, since they can be designed with exact linear phase, and exibits no stability problems. However FIR filters have a computationally more intensive complexity compared to IIR filters. During past several years, many design methods have been proposed to reduce complexity of the FIR filters, (Chen, Chang and Vinod, 2006; Jovanovic-Dolecek & Mitra, 2007; Lian and Yang, 2001; Rodrigues & Pai, 2005; Yang and Lian, 2003;Yang and Lian, 2006; Zou & Saramaki, 2004).