Note 64. Generating Analytic Signals with Complex Equiripple FIR Filters
This note describes the design and use of complex equiripple FIR filters to generate discrete-time analytic signals of the sort discussed in Note 60.
Such filters are designed using a complex-valued extension of the Parks-McClellan algorithm and offer more flexibility than other analytic signal generation (ASG) techniques when it comes to specifying the passband, stopband, and transition bands of the filter. However, this flexibility has a price—extra constraints placed on the filter’s configuration usually demand an increased number of taps to meet the specified performance.
The complex filter coefficients for an FIR analytic signal generator can be determined using the MATLAB cfirpm (formerly cremez) function. This function requires that the user provide a piecewise specification of the desired response. For generating analytic signals, the ideal desired response has the characteristics listed in List 64.1.
Example 64.1 deals with the bandpass ASG case that is featured in the digital I/Q scheme discussed in Note 65.
Example 64.3. Equiripple Bandpass ASG Filter
A digital I/Q scheme, described in Note 60, involves a real-valued bandpass signal that has been frequency-shifted and sampled such that the passband extends from fs/8 to 3fs/8, as shown in Figure 64.1(a). Before additional processing can be performed, the negative-frequency passband must be eliminated to produce an analytic signal having the positive-only spectrum shown in Figure 64.1(b).
• A desired response that meets the criteria in List 64.1 is shown in Figure 64.2(a).
Figure 64.1. Spectra for Example 64.1: (a) spectrum for typical bandpass signal, (b) spectrum for corresponding analytic signal
Figure 64.2. Desired response for the complex filter in Example 64.1: (a) band edges specified in terms of the sampling rate, (b) band edges specified in units acceptable to MATLAB cfirpm function
• The cfirpm function requires the band-edge frequencies to be specified in terms of a normalized radian frequency, ω/π, so all specified values will range from –1.0 to +1.0. Figure 64.2(b) shows the desired response recast in frequency units acceptable to cfirpm. A value of 0.05 has been assumed for fL.
• The following segment of MATLAB code designs the filter:
• This specification yields a 91-tap filter with the response shown in Figure 64.3.
Figure 64.3. Frequency response for 91-tap filter designed in Example 64.1
Reference
1. M. Z. Komodromos, S. F. Russell, and P. T. P. Tang, “Design of FIR Filters with Complex Desired Frequency Response Using a Generalized Remez Algorithm,” IEEE Trans. Circuits and Systems, vol. 42, no. 4, April 1995.