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9.3 1D FIR Filter Design Methods

The design of FIR filters usually starts with a specification of the desired frequency behavior. For linear-phase FIR filters, it is sufficient to specify the shape of the amplitude response to uniquely define the filter coefficients. The amplitude characteristic may either be given as a set of amplitude samples on a uniform frequency domain grid, as in the frequency sampling method, or as piecewise constant amplitude values over disjoint regions of the fundamental interval, as in window-based and equiripple design methods. In the latter case, the desired passband and stopband regions are uniquely identified by a list of pairs of band-edge frequencies, while the amplitude specification is a list of amplitude values, one for each band. Referring to the example lowpass filter in Figure 9.1.1, we get the following simple list of four frequencies (i.e., two bands): {0., 0.2, 0.25, 0.5}. The corresponding amplitude values in each band form a numeric list: {1,0}. Additionally, the filter specification usually includes the maximum tolerable passband ripple, p, and the maximum tolerable stopband ripple, s, the latter also known as the minimum stopband attenuation, s. These quantities are graphically illustrated by the heights of the passband and stopband rectangles in Figure 9.1.1, where 2p0.1 and s0.05.
EquirippleFilter[N, {f1, f2, ...}, {v1, v2, ...}, {w1, w2, ...}, type]
returns an optimum linear-phase FIR filter of length N and type given by"type",with band edge frequencies given byf1, f2, ...,desired magnitude values given byv1, v2, ..., and relative weights in each band given byw1, w2,... that exhibits equiripple behavior in the frequency range of interest
FrequencySamplingFilter[N, {a1, a2, ... }]
returns a linear-phase FIR filter given a length specification N and list of desired real amplitude values in the normalized frequency range 0 ≤ f≤ 0.5
WindowFilter[N, {f1,f2, ...},{v1,v2, ...}]
returns a linear-phase FIR filter given a specification which consists of filter length N, cutoff frequencies f over the symmetric, normalized frequency range 0, and values of the frequency response vi

1D filter design functions.

EquirippleFilter returns linear-phase FIR filters with a frequency response that is optimal in the sense of minimizing the maximum approximation error [McC73b]. The approximation error is the difference between the desired and the actual frequency responses. The approximation error is spread evenly across the passbands and stopbands of the filter, resulting in a magnitude response that exhibits equal-sized ripples across the frequency domain (see Figure 9.1.1). The EquirippleFilter design is a Chebyshev approximation problem and uses the Remez exchange algorithm to find a solution.
Here is an example of the design of a 2-band lowpass filter with passband edge at f=0.2 Hz and stopband edge at f=0.25 Hz. The length of the filter is typically obtained from the specification for the desired minimum stopband attenuation or peak ripple. Here we assume a length N=21.
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This returns the desired optimum equiripple filter according to the specifications given in the example.
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It is easy to verify the frequency response characteristics of the filter. This is accomplished by computing the discrete-time Fourier transform (DTFT) of filter h [Opp89]. The DTFT and the better-known DFT are related; the DFT is a sampled representation of the DTFT of a sequence. Here we evaluate the DTFT of h.
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Here we demonstrate two common methods of plotting the magnitude spectrum.
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Two options control the accuracy of the approximation and the maximum number of iterations, respectively.

Options of EquirippleFilter.

FrequencySamplingFilter is a simple linear-phase FIR filter design method that returns a filter with an optimum mean squared error fitted to a set of amplitude values spaced uniformly in the normalized frequency range . There are two possible sequences of sample positions:
Given a frequency domain specification of a filter's amplitude response on a uniformly spaced frequency grid, the impulse response coefficients may be obtained from a set of design equations. Here are the design equations for a Type 1 (symmetric, odd length) filter.
where and M is the filter length. The design equations guarantee a linear phase impulse response. Thus the user need only specify the length of the filter, the desired real-valued amplitudes, the desired symmetry, and sample positions by specifying the value of (see Equation (9.3.1)).

Options for FrequencySamplingFilter.

Given the design equations, the design process reduces to specifying a desired amplitude response. Here a length M=21, symmetric lowpass FIR filter with a bandwidth of approximately 0.2 Hz will be obtained.
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The actual edge of the passband for this filter is located at Hz. This is the impulse response.
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The magnitude spectrum of the filter may be obtained by calculating the DTFT of the impulse response sequence h. Note the rather poor attenuation in the stopband.
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The poor performance of the filter is due to the sharp transition from passband to stopband implied by the specification. By definition, the transition band width is Hz. To increase the stopband attenuation (decrease the passband and stopband ripple), the transition band must be widened by the introduction of one or more transition samples that take values in the range 01. A significant improvement results with a selection of one transition element with a value of . Here is a comparison.
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Tables have been published that, for a given filter specification, give the value of the best transition coefficient(s) [Rab70]. These coefficients minimize the maximum deviation of the frequency response of the synthesized filter from the frequency response of the desired filter. This is known as minimax optimization.
The window method is another common method of designing FIR digital filters. It is well suited for designing filters with simple frequency responses, such as ideal lowpass, bandpass, bandstop and highpass filters, differentiators, and Hilbert transformers. Typically, the filter is designed in two steps. First, a double-sided infinite impulse response is obtained from a desired ideal frequency response by the inverse discrete-time Fourier transform (IDTFT). The window method then calls for the truncation and smoothing of the impulse response coefficients using an appropriate smoothing sequence to achieve the desired passband and stopband attenuation. The window design method returns an FIR filter ht[n] that minimizes the integral squared error between the desired frequency response given by Hd() and the DTFT of ht[n] given by Ht().
The desired frequency response Hd() is typically specified as a piecewise constant with sharp transitions between passbands and stopbands. The corresponding impulse response hd[n] is then infinite in extent and non-realizable. It has been shown that the filter ht[n] that minimizes the integral squared error is a truncated copy of hd[n].
Truncation may be implemented as a multiplication operation by a finite-length sequence called the window
where w[n] is the so-called rectangular window sequence defined as
Typically, the rectangular window is replaced by windows with smoother frequency response characteristics. Commonly used windows, such as Hamming and Blackman-Harris, are variations on the basic idea of smoothing with a raised cosine sequence. They are limited to a fixed stopband ripple that depends on the window shape. Alternatively, adjustable windows, such as the Kaiser window, may be used to satisfy any design requirements. The Kaiser window has near optimal performance (in the sense of minimizing the sidelobe energy of the window), and a simple implementation.

1D window sequences.

Following the outlined steps, we first obtain the truncated impulse response corresponding to an ideal, piecewise-constant approximation of the filter shown in Figure 9.1.1. The ideal filter specification presupposes that passband and stopband regions are contiguous with no transition region (i.e., edge frequencies of two neighboring regions are same). It is common to chose this breakpoint frequency (called the cutoff frequency) in the middle of the transition region. Here we choose fc=0.225 Hz. This gives the impulse response.
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The stopband may be omitted to yield a shorter specification.
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In step 2, we use Equation (9.3.5) to smooth the filter coefficients with a Kaiser window of the same length.
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Finally, we verify the magnitude spectrum of our filter.
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Typically, the length of the filter is not known and must be estimated from the filter specifications. Formulas have been developed to assist the designer [ Her73, Kai74 ] in calculating the length given the width of the transition region and maximum tolerable peak ripple or stopband attenuation.

Filter design utility functions.

Here we obtain the length of a filter meeting the frequency domain characteristics illustrated in Figure 9.1.1.
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When using the Kaiser window, it is also necessary to estimate the shape coefficient for a given level of stopband attenuation. Here we calculate its value for a peak ripple of 0.05.
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