Lth-Band (Nyquist) filters have the property that (N-1)/2L coefficients are zero, which makes them computationally efficient. For example, the popular halfband filters are just Lth-band filters with L = 2.
ScopeFIR's Lth-Band design method really is just an interface to some of its other design methods, with the Lth-Band constraints automatically imposed.
ScopeFIR's Lth-Band (Nyquist) Editor is shown below:
To design an Lth-Band filter:
- Specify the Sampling Frequency (Fs) by hitting the Sampling Frequency button in the upper left. You will be prompted to enter the numerical frequency as well as the units of frequency/time.
- Enter the Number of Taps. This specifies the coefficients the filter will have. ScopeFIR can also Estimate the number of taps for Parks-McClellan designs, once all other specifications have been entered.
- Enter the Cutoff Frequency, which must be between 0 and Fs/2. If the Cutoff Frequency is less than Fs/2L (see below) then the filter will be a lowpass; otherwise it will be a highpass. Also, the Cutoff Frequency cannot be exactly Fs/2L, and cannot be greater than Fs/L. (Note that the nearer the Cutoff Frequency is to Fs/2L, the narrower the transition band of the filter will be, and the more taps you will need.)
- Enter L. L specifies the "pivot point" P between the passband and stopband of the filter according to the formula P = Fs/2L. L must be an integer greater than one. Use L=2 to design a "halfband" filter. The Cutoff Frequency you specify determines one band edge, and the second band edge is automatically calculated as Fs/L - Cutoff Frequency. This sets the two band edges equidistant from the Fs/2L "pivot point". For example, if Fs = 1.0, L = 2, the pivot point is Fs/2L = 0.25. Next, setting the Cutoff Frequency to 0.2 will result in a lowpass filter whose passband upper edge is 0.2, and whose stopband lower edge is Fs/L - Cutoff Frequency = 1.0 / 2 - 0.2 = 0.3. When you move between specifications or when you design a filter, the "Band Info" box will automatically show you the type of filter (highpass or lowpass), and the band edge frequencies.
- Next, select a design method from among Parks-McClellan, Windowed Sinc, and Spline Sinc. (For more information on these, see FIR Design Methods.)
For a Parks-McClellan design:
- Specify the filter in terms of either Passband Ripple or Stopband Attenuation. When you specify one of these, ScopeFIR sets the other one automatically to match (which is a design requirement of Lth-Band filters.) You can specify ripple/attenuation either in dB or in linear terms. Note, however, that for a Parks-McClellan design, the filter design depend only on the Cutoff Frequency and the Number of Taps. Therefore, the ripple/attenuation parameters are irrelevant except when you Estimate the Number of Taps, or Optimize the filter (see below).
- Set the Grid Density. The default value of 16 generally works well. However, larger values of Grid Density tend to result in less need to Zeroize (see below), at the cost of slowing the filter design process.
- Specify whether you want to Zeroize the filter. In a Parks-McClellan design, some of the coefficients which should theoretically be zero tend to converge to almost zero. The Zeroize option causes all applicable coefficients to be truncated to exactly zero. (Zeroizing results in a slightly different response than the Parks-McClellan algorithm designed, but the difference is usually negligible.) When you select zeroize after already having designed a filter, the filter will be Zeroized; if you de-select Zeroize for an existing design, the filter will be designed again.
For a Windowed Sinc design, specify the window type and the "Alpha" parameter if you have selected a "Kaiser-Bessel" window. (See The Windowed Sinc Editor for more information on Windowed Sinc filters.
For a Spline Sinc design, specify the Order of the spline, or simply select Auto to have ScopeFIR automatically select the spline Order based on the Number of Taps and the Cutoff Frequency.
Once all specifications have been entered, hit the Design button to actually design the filter. For Parks-McClellan designs, you can also hit the Optimize button to determine the minimum number of taps which will meet the specification.
After designing the filter, the Frequency Response, Impulse Response and Coefficient Editor windows will update and the Band Info box will be filled in. The Band Info box will show either Passband Upper and Stopband Lower for Lowpass designs, or will show Stopband Upper and Passband Lower for Highpass designs.