Note 62. Generating Analytic Signals with Frequency-Shifted FIR Lowpass Filters
A useful technique for designing FIR analytic signal generation (ASG) filters is presented by Reilly et al. in [1]. The basic idea is to design a “standard” lowpass filter having real-valued coefficients, and then phase-shift the impulse response coefficients such that the filter’s passband is shifted to cover only positive frequencies and the stopband is shifted to cover only negative frequencies. The phase-shifted coefficients are complex-valued. The details are provided in Design Procedure 62.1.
Example 62.1 attempts to implement a broadband, or “almost-all-positive pass” ASG filter using the approach of Design Procedure 62.1. Example 62.2 implements a more pronouncedly bandpass design that is only concerned with cancelling negative frequencies in the band from –3fs/8 to –fs/8. The more generous transition bands in this second case result in an ASG filter with much higher stopband attenuation.
Design Procedure 62.1. Complex FIR ASG Filter Design
- Map out a desired ASG filter response similar to the one shown in Figure 62.1(a). The stopband must cover the entire negative-frequency Nyquist zone, and the passband must be centered on fs/4, as shown. The passband extends from f1 to f2 where
DP 62.1
Figure 62.1. Desired frequency responses for Design Procedure 62.1: (a) analytic signal generating filter response, (b) corresponding prototype lowpass response
- Using the value of fH that yields the desired passband as given by Eq. (DP 62.1), construct a response mask for a “standard” lowpass filter like the one shown in Figure 62.1(b).
- Using a standard FIR design technique like least-squares or Parks-McClellan, obtain the coefficients, h[n], for a filter that approximates the response in Figure 62.1(b).
- Compute coefficients, h[n], for the frequency-shifted filter that approximates the response in Figure 62.1(a):
Example 62.1. “Almost-All-Positive-Pass” ASG Filter
The desired response has a positive-frequency passband from 0.025fs to 0.475fs, thus, fH = 0.225fs. The coefficients for a 61-tap lowpass filter can be obtained using the following segment of MATLAB code:
b=firpm(60,[0 0.45 0.5 1],[1 1 0 0]);
The response of this filter is shown in Figure 62.2. The vector of phase-shifting factors can be generated as
bm=0:60;
bm=exp(j*pi*bm/2);
Figure 62.2. Frequency response for prototype lowpass filter from Example 62.1
Finally, the shifted filter’s coefficients are obtained as
bc=b.*bm;
The response of the ASG filter is shown in Figure 62.3.
Figure 62.3. Frequency response for analytic signal filter from Example 62.1
Example 62.2. Bandpass ASG Filter
A digital I/Q scheme described in Note 65 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 62.4(a).
Figure 62.4. Spectra for Example 62.2: (a) spectrum for typical bandpass signal, (b) spectrum for corresponding analytic signal
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 62.4(b). The coefficients for a 61-tap prototype lowpass filter can be obtained using the following segment of MATLAB code:
b=firpm(60,[0 0.25 0.5 1],[1 1 0 0]);
The response of this filter is shown in Figure 62.5. The vector of phase-shifting factors can be generated as
bm=0:60;
bm=exp(j*pi*bm/2);
Figure 62.5. Frequency response for prototype lowpass filter from Example 62.2
Finally, the shifted filter’s coefficients are obtained as
bc=b.*bm;
The response of the ASG filter is shown in Figure 62.6.
Figure 62.6. Frequency response for analytic signal filter from Example 62.2
Reference
1. A. Reilly, G. Frazer, and B. Boashash, “Analytic Signal Generation—Tips and Traps,” IEEE Trans. Signal Proc., vol. 42, n. 11, Nov. 1994, pp. 3241–3245.