Multistage Design Of Decimators/Interpolators
This example shows how to design multistage decimators and interpolators. The example Efficient Narrow Transition-Band FIR Filter Design showed how to apply the IFIR and the MULTISTAGE approaches to single-rate designs of lowpass filters. The techniques can be extended to the design of multistage decimators and/or interpolators. The IFIR approach results in a 2-stage decimator/interpolator. For the MULTISTAGE approach, the number of stages can be either automatically optimized or manually controlled.
Reducing the Sampling-Rate of a Signal
Decimators are used to reduce the sampling-rate of a signal while simultaneously reducing the bandwidth proportionally. For example, if the rate is to be reduced by a factor of 8, the following are typical specifications for a lowpass filter that will reduce the bandwidth accordingly.
Fpass = 0.11; Fstop = 0.12; Apass = 0.02; % 0.02 dB peak-to-peak ripple Astop = 60; % 60 dB minimum attenuation M = 8; % Decimation factor of 8 Hfd = fdesign.decimator(M,'lowpass',Fpass,Fstop,Apass,Astop);
A single-stage equiripple design for these specifications requires on average 649/8 = 81.125 multiplications per input sample (MPIS):
Hm = design(Hfd,'equiripple','SystemObject',true); cost(Hm)
ans = struct with fields: NumCoefficients: 650 NumStates: 648 MultiplicationsPerInputSample: 81.2500 AdditionsPerInputSample: 81.1250
As we mentioned, an IFIR design results in a two stage implementation. Notice that in this case only 27/4+172/8 = 28.25 MPIS are required.
Hm_ifir = design(Hfd,'ifir','SystemObject',true); cost(Hm_ifir)
ans = struct with fields: NumCoefficients: 199 NumStates: 194 MultiplicationsPerInputSample: 28.2500 AdditionsPerInputSample: 27.8750
A multistage design results in approximately the same cost when implementing the design with 3 stages. The number of MPIS required for this design is 9/2+14/4+172/8 = 29.5
Hm_multi = design(Hfd,'multistage','SystemObject',true); cost(Hm_multi)
ans = struct with fields: NumCoefficients: 195 NumStates: 190 MultiplicationsPerInputSample: 29.5000 AdditionsPerInputSample: 28.6250
Tweaking Designs for Improved Efficiency
The IFIR design has a joint optimization option that can improve the efficiency of the design even further (although it may not converge in some cases). Note that this design may take a long time. The number of MPIS is reduced to 13/4+163/8 = 23.625
Hm_ifir_jo = design(Hfd,'ifir','JointOptimization',true,... 'SystemObject',true); cost(Hm_ifir_jo)
ans = struct with fields: NumCoefficients: 176 NumStates: 174 MultiplicationsPerInputSample: 23.6250 AdditionsPerInputSample: 23.2500
The multistage design can also be improved by noticing that the decimators used involve a decimation factor of 2. Two of three stages can be replaced with halfband filters, for which only about half of the coefficients are non-zero. The number of MPIS is reduced to 27.25.
Hm_multi_hb = design(Hfd,'multistage','UseHalfbands',true,... 'SystemObject',true); cost(Hm_multi_hb)
ans = struct with fields: NumCoefficients: 188 NumStates: 194 MultiplicationsPerInputSample: 27.2500 AdditionsPerInputSample: 26.3750
Using Nyquist Filters for Improved Efficiency
When decimating by a factor M, it is often computationally advantageous to use Nyquist designs (Where the band is also M). In this example, we have M = 8.
TW = 0.01; % Transition width is the same as in previous design Astop = 60; % 60 dB minimum attenuation M = 8; % Decimation factor of 8 Hfd_nyq = fdesign.decimator(M,'nyquist',M,TW,Astop);
A single-stage design using a Kaiser window results in a filter length of 727. However, given that it is a Nyquist filter, about one out of every M coefficients is zero, so the actual number of multipliers is 637 so the number of MPIS is 79.625 which is less than the single-stage equiripple design for the same transition width and stopband attenuation. Moreover, the passband ripple is smaller.
Hm_nyq = design(Hfd_nyq,'kaiserwin','SystemObject',true); measure(Hm) measure(Hm_nyq)
ans = Sample Rate : N/A (normalized frequency) Passband Edge : 0.11 3-dB Point : 0.11379 6-dB Point : 0.11496 Stopband Edge : 0.12 Passband Ripple : 0.019632 dB Stopband Atten. : 60.0576 dB Transition Width : 0.01 ans = Sample Rate : N/A (normalized frequency) Passband Edge : 0.12 3-dB Point : 0.12384 6-dB Point : 0.125 Stopband Edge : 0.13 Passband Ripple : 0.01648 dB Stopband Atten. : 60.1578 dB Transition Width : 0.01
ans = struct with fields: NumCoefficients: 637 NumStates: 720 MultiplicationsPerInputSample: 79.6250 AdditionsPerInputSample: 79.5000
Halfband filters are a special case of Nyquist filters when the band is equal to two. One way to obtain an 8th-band Nyquist filter that decimates by 8 is to cascade three halfbands, each of which decimates by two. Multistage designs will do this automatically when the cost is found to be minimal. Given that only about half the coefficients are non-zero the number of MPIS required in this case is only 15.375, the lowest by far of all designs tried.
Hm_multi_nyq = design(Hfd_nyq,'multistage','SystemObject',true); cost(Hm_multi_nyq)
ans = struct with fields: NumCoefficients: 99 NumStates: 186 MultiplicationsPerInputSample: 15.3750 AdditionsPerInputSample: 14.5000
hfvt = fvtool(Hm_nyq,Hm_multi_nyq, 'Color', 'white'); legend(hfvt,'Single-stage Nyquist design','Multistage Nyquist design',... 'Location','NorthEast')
The cases shown above apply equally to the design ofinterpolators. For example:
Fpass = 0.11; Fstop = 0.12; Apass = 0.02; % 0.02 dB peak-to-peak ripple Astop = 60; % 60 dB minimum attenuation L = 8; % Interpolation factor of 8 Hfi = fdesign.interpolator(L,'lowpass',Fpass,Fstop,Apass,Astop);
A multistage design that uses halfbands can be computed with identical syntax as before:
Hmi = design(Hfi,'multistage','UseHalfbands',true,'SystemObject',true); measure(Hmi) hfvt = fvtool(Hmi); hfvt.Legend = 'off';
ans = Sample Rate : N/A (normalized frequency) Passband Edge : 0.11 3-dB Point : 0.114 6-dB Point : 0.11515 Stopband Edge : 0.12 Passband Ripple : 0.012937 dB Stopband Atten. : 60.053 dB Transition Width : 0.01
Notice that as with all interpolators, the passband gain (in linear units) is equal to the interpolation factor. The minimum distance (in dB) between the passband and the stopband is about 60 dB as specified by Astop.
The use of multistage techniques can provide significant computational savings when implementing decimators/interpolators. In particular, the use of multistage Nyquist filters can be extremely efficient.