SYSTEM ARCHITECTURE DESIGN
5.6 DIGITAL FILTER DESIGN ALGORITHMS
Commonly used techniques for the design of IIR filters are based on transformations of continuous-time IIR systems into discrete-time IIR systems. This is partly because continuous-time filter design was a highly advanced field before discrete-time filters were of interest and partly because of the difficulty of implementing a non-iterative direct design method for IIR filters [12].
5.6.1 BILINEAR TRANSFORMATION METHOD
This is by far the most important method of obtaining IIR filter coefficients. In this method, Bilinear Transformation is applied to convert an analog filter transfer function, H(s), into an equivalent digital filter transfer function, H(z) as shown below:
or T z k
kz
s 2
1 1,
1 =
+
= −
The above transformation maps the analog transfer function, H(s), from the s-plane into the discrete transfer function, H(z), in the z-plane [16].
DESIGN OF AN INTEGRATED GFSK DEMODULATOR FOR A BLUETOOTH RECEIVER 42
5.6.2 IMPULSE INVARIANT METHOD
In this method, starting with a suitable analog transfer function, H(s), the filter impulse response h(t), is obtained using the Laplace Transform. The h(t) so obtained is suitably sampled to produce h(kT), where T is the sampling interval.
5.6.3 POLE-ZERO PLACEMENT METHOD
When a zero is placed at a given point on the z-plane, the frequency response will be zero at the corresponding point. A pole, on the other hand, produces a peak at the corresponding frequency point. Poles that are close to the unit circle give rise to large peaks, whereas zeros close to or on the unit circle produce troughs or minima. Thus, by strategically placing the poles and the zeros on the z-plane, simple lowpass, or other frequency-selective filters can be obtained. For the coefficients of the filter to be real, the poles and the zeros must either be real (i.e., lie on the positive or the negative real axis) or occur in complex conjugate pairs.
In contrast to IIR filters, FIR filters are almost entirely restricted to discrete-time implementations. Consequently, the design techniques for FIR filters are based on directly approximating the specified frequency response of the discrete-time system. Furthermore, most techniques for approximating the magnitude response of an FIR system assume linear phase constraint avoiding the complications involved in the direct implementation of IIR systems [8].
An FIR filter is characterized by the following equations:
∑
where: h(k), k=0, 1, 2, . . ., N-1 are the impulse response coefficients of the FIR filter, H(z) is the transfer function of the FIR filter,
N is the FIR filter length.
The first equation is a time-domain, non-recursive difference equation of an FIR filter. The second equation is the transfer function of an FIR filter.
The sole objective of most FIR filter coefficient calculation/approximation methods is to obtain values of h(k) such that the resulting filter meets the design specifications, such as amplitude-frequency response and throughput requirements. Several methods are available for obtaining h(k). The following three methods are the most common, however, as they all can lead to linear phase FIR filters.
5.6.4 WINDOW METHOD
The simplest method of FIR filter design is called the window method. The ideal transfer function of a non-recursive digital filter is:
∑
+∞=−∞ −The corresponding ideal frequency response is:
∑
+∞=−∞ −∑
=+∞−∞ −H = and the corresponding ideal impulse response hideal(k) are related by the inverse Fourier transform:
−
∫
For an FIR filter, the ideal impulse response has to be truncated by setting hideal(k)=0 for k>N. However, this introduces undesirable ripples and overshoots called the Gibb’s phenomenon. Direct truncation of hideal(k) is equivalent to multiplying hideal(k) by a rectangular window of the form:
to overshoots and ripples in the frequency response.
A practical approach is to multiply hideal(k), by a suitable window function, w(k), whose duration is finite. This yields an impulse response that decays smoothly towards zero, however, the transition width is wider than for the rectangular window. The transition width of the filter is determined by the width of the main lobe of the window. The side lobes produce ripples in both the passband and the stopband.
Several window functions have been proposed. Some of the most common window functions are:
The Kaiser window was used to design most of the filters in the system.
5.6.5 OPTIMAL METHOD
Inherent in the process of calculating suitable filter coefficients in the window method is the problem of finding a suitable approximation to a specified or ideal frequency response. The peak ripple of filters designed by the window method occurs near the band edges, and decreases away from the band edges. If the ripples were distributed more evenly over the passband and the stopband, a better approximation of the desired frequency response can be achieved. The optimal method is based on the concept of equiripple passband and stopband.
The difference between the ideal and the practical filter response can be represented by an error function:
DESIGN OF AN INTEGRATED GFSK DEMODULATOR FOR A BLUETOOTH RECEIVER 44
[
( ) ( )]
) ( )
(ω W ω H ω H ω
E = ideal −
where: Hideal(ω) is the ideal filter response and W(ω) is a weighting function that allows the relative error of approximation between different frequency bands (passbands, stopbands, transitionbands) to be defined.
In the optimal method, the objective is to determine the filter coefficients, h(k), such that the value of the maximum weighted error, E(ω), is minimized in the passband and the stopband. Mathematically, this may be expressed as:
min[maxE(ω)]
over the passbands and the stopbands. It can be shown that when max E(ω) is minimized the resulting filter response will have equiripple passbands and stopbands, with the ripple alternating in sign between two equal-amplitude levels. For a given set of filter specifications, the location of the extremal (the maxima and the minima are known as the extrema) frequencies, apart from those at the band edges are not known a priori. Thus the main problem in the optimal method is to find the locations of the extremal frequencies. A technique employing the Remez Exchange Algorithm to find the extremal frequencies is used. Knowing the locations if the extremal frequencies, it is a simple matter to work out the actual frequency response and hence the impulse response of an FIR filter. For a given set of specifications (i.e., passband edge frequencies, N, and the ratio between the passband and the stopband ripples) the optimal method involves the following key steps:
1) use the Remez Exchange Algorithm to find the optimum set of extremal frequencies.
2) determine the frequency response using the extremal frequencies 3) obtain the impulse response coefficients
The heart of the optimal method is the first step where an iterative process is used to determine the extremal freqiuencies of the filter whose amplitude-frequency response satisfies the optimality condition. This step relies on the alternation theorem which specifies the number of extremal frequencies that can exist for a given value of N [16].
The Remez Exchange Algorithm was used to model the pre-modulation Gaussian Lowpass filter in the modulator section of the system model.
5.6.6 FREQUENCY SAMPLING METHOD
The frequency sampling method yields non-recursive FIR filters for both standard frequency-selective filters (lowpass, bandpass, highpass) as well as filters with arbitrary frequency response. A unique attraction of the frequency sampling method is that it also allows recursive implementation of FIR filters, leading to computationally efficient filters.
With some restrictions, recursive FIR filters with integer coefficients may also be designed which is convenient when implementing only primitive arithmetic operations as in systems with standard microprocessors [16].
SUMMARY
This chapter outlined the architecture-level design of the GFSK demodulator. The convolver-based architecture was selected for the demodulator where the convolvers were implemented as bandpass filters using FIR filter banks. The envelope detector was implemented with a full-wave rectifier followed by a low-pass filter.