**1. Introduction**

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Digital Hilbert transformers are a special class of digital filter whose characteristic is to introduce a *π*/2 radians phase shift of the input signal. In the ideal Hilbert transformer all the positive frequency components are shifted by –*π*/2 radians and all the negative frequency components are shifted by *π*/2 radians. However, these ideal systems cannot be realized since the impulse response is non-causal. Nevertheless, Hilbert transformers can be designed either as Finite Impulse Response (FIR) or as Infinite Impulse Response (IIR) digital filters [1], [2], and they are used in a wide number of Digital Signal Processing (DSP) applications, such as digital communication systems, radar systems, medical imaging and mechanical vibration analysis, among others [3]-[5].

IIR Hilbert transformers perform a phase approximation. This means that the phase response of the system is approximated to the desired values in a given range of frequencies. The magnitude response allows passing all the frequencies, with the magnitude obtained around the desired value within a given tolerance [6], [7]. On the other hand, FIR Hilbert transformers perform a magnitude approximation. In this case the system magnitude response is approximated to the desired values in a given range of frequencies. The advantage is that their phase response is always maintained in the desired value over the complete range of frequencies [8].

Whereas IIR Hilbert transformers can present instability and they are sensitive to the rounding in their coefficients, FIR filters can have exact linear phase and their stability is guaranteed. Moreover, FIR filters are less sensitive to the coefficients rounding and their phase response is not affected by this rounding. Because of this, FIR Hilbert transformers are often preferred [8]-[15]. Nevertheless, the main drawback of FIR filters is a higher complexity compared with the corresponding IIR filters. Multipliers, the most costly

© 2012 Troncoso Romero and Jovanovic Dolecek, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Troncoso Romero and Jovanovic Dolecek, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

elements in DSP implementations, are required in an amount linearly related with the length of the filter. A linear phase FIR Hilbert transformer, which has an anti-symmetrical impulse response, can be designed with either an odd length (Type III symmetry) or an even length (Type IV symmetry). The number of multipliers *m* is given in terms of the filter length *L* as *m CL* , where *C* = 0.25 for a filter with Type III symmetry or *C* = 0.5 for a filter with Type IV symmetry.

The design of optimum equiripple FIR Hilbert transformers is usually performed by Parks-McClellan algorithm. Using the MATLAB Signal Processing Toolbox, this becomes a straightforward procedure through the function firpm. However, for small transition bandwidth and small ripples the resulting filter requires a very high length. This complexity increases with more stringent specifications, i.e., narrower transition bandwidths and also smaller pass-band ripples. Therefore, different techniques have been developed in the last 2 decades for efficient design of Hilbert transformers, where the highly stringent specifications are met with an as low as possible required complexity. The most representative methods are [9]-[15], which are based in very efficient schemes to reduce complexity in FIR filters.

Methods [9] and [10] are based on the Frequency Response Masking (FRM) technique proposed in [16]. In [9], the design is based on reducing the complexity of a half-band filter. Then, the Hilbert transformer is derived from this half-band filter. In [10], a frequency response corrector subfilter is introduced, and all subfilters are designed simultaneously under the same framework. The method [11] is based on wide bandwidth and linear phase FIR filters with Piecewise Polynomial-Sinusoidal (PPS) impulse response. These methods offer a very high reduction in the required number of multiplier coefficients compared to the direct design based on Parks-McClellan algorithm. An important characteristic is that they are fully parallel approaches, which have the disadvantage of being area consuming since they do not directly take advantage of hardware multiplexing.

The Frequency Transformation (FT) method, proposed first in [17] and extended in [18], was modified to design FIR Hilbert transformers in [12] based on a tapped cascaded interconnection of repeated simple basic building blocks constituted by two identical subfilters. Taking advantage of the repetitive use of identical subfilters, the recent proposal [13] gives a simple and efficient method to design multiplierless Hilbert transformers, where a combination of the FT method with the Pipelining-Interleaving (PI) technique of [19] allows getting a time-multiplexed architecture which only requires three subfilters. In [14], an optimized design was developed to minimize the overall number of filter coefficients in a modified FT-PI-based structure derived from the one of [13], where only two subfilters are needed. Based on methods [13] and [14], a different architecture which just requires one subfilter was developed in [15].

In this chapter, fundamentals on digital FIR Hilbert transformers will be covered by reviewing the characteristics of analytic signals. The main connection existing between Hilbert transformers and half-band filters will be highlighted but, at the same time, the complete introductory explanation will be kept as simple as possible. The methods to design low-complexity FIR filters, namely FRM [16], FT [17] and PPS [11], as well as the PI architecture [19], which are the cornerstone of the efficient techniques to design Hilbert transformers presented in [9]-[15], will be introduced in a simplified and concise way. With such background we will provide an extensive revision of the methods [9]-[15] to design low-complexity efficient FIR Hilbert transformers, including MATLAB routines for these methods.
