This example shows how to efficiently convert sample rates between arbitrary factors.
The need for sample rate conversion by an arbitrary factor arises in many applications (e.g. symbol synchronization in digital receivers, speech coding and synthesis, computer simulation of continuous-time systems, etc...). In this example, we will examine an example where cascades of polynomial-based and polyphase filters form an efficient solution when it is desired to convert the sampling rate of a signal from 8 kHz to 44.1 kHz.
Polyphase structures are generally considered efficient implementations of multirate filters. However in the case of fractional sample rate conversion, the number of phases, and therefore the filter order, can quickly become excessively high. To resample a signal from 8 kHz to 44.1 kHz, we interpolate by 441 and decimate by 80 (8*441/80=44.1).
L = 441; % Interpolation factor M = 80; % Decimation factor TW = 1/(L*2); % Transition Width Astop = 50; % Minimum stopband attenuation f = fdesign.rsrc(L,M,'Nyquist',L,'TW,Ast',TW,Astop); h1 = design(f);
Let's examine the implementation cost of this filter:
ans = Number of Multipliers : 5178 Number of Adders : 4738 Number of States : 11 Multiplications per Input Sample : 64.725 Additions per Input Sample : 59.225
Although the number of operations (65 multiplications and 59 additions) per input sample is reasonable, 5178 coefficients would have to be stored in memory in this case.
Polynomial-based filters overcome this problem by computing on the fly the coefficients that are applied to each input sample. Farrow structures are efficient implementations for such filters. To convert a polyphase filter to a polynomial-based (a.k.a. Farrow) filter, we simply approximate each column of the polyphase decomposition by a low-order polynomial.
p=polyphase(h1); % Polyphase decomposition Np = size(p,2); % Length of each polyphase filter N = 4; % Polynomial order X = 0:1/L:(L-1)/L; for i=1:Np, fp(Np-i+1,:)=polyfit(X,p(:,i).',N); end h1far = mfilt.farrowsrc(L,M,fp);
With 3rd-order polynomials we get only 49 dB a attenuation in the stopband. We verify that 4th-order polynomials form a good approximation of the original polyphase filter, meeting the specified transition width and minimum stopband attenuation. For the analysis of multirate filters, we ignore the polyphase implementations and use an "upsample-filter-downsample" equivalent model that allow us to attach a sampling frequency value for each filter.
W = linspace(0,44.1e3,2048); % Define the frequency range analysis Fs1 = 8e3*441; % The equivalent single stage filter is clocked at 3.528 MHz hfvt = fvtool(h1,h1far,'FrequencyRange','Specify freq. vector', ... 'FrequencyVector',W,'Fs',[Fs1 Fs1], ... 'NormalizeMagnitudeto1','on', 'Color', 'white'); legend(hfvt,'Polyphase Interpolator','Farrow Interpolator',... 'Location','NorthEast')
Let's examine the implementation cost of this filter:
ans = Number of Multipliers : 64 Number of Adders : 59 Number of States : 11 Multiplications per Input Sample : 352.8 Additions per Input Sample : 325.2375
When compared to the cost of the polyphase filter, the number of coefficients to store in memory is drastically reduced from 5178 to 64. However the number of operations per input sample has increased significantly (from 65 to 353 multiplications per sample and from 59 to 325 additions per sample). Depending on the application and the hardware target, this loss of performance may or may not be acceptable.
We now try to design a hybrid solution that would take advantage of the two types of filters that we have previously seen. Polyphase filters are particularly well adapted for interpolation or decimation by an integer factor and for fractional rate conversions when the interpolation and the decimation factors are low. Farrow filters can efficiently implement arbitrary (including irrational) rate change factors. First, we interpolate the original 8 kHz signal by 4 using a cascade of FIR halfband filters.
TW = .125; % Transition Width f2 = fdesign.interpolator(4,'Nyquist',4,'TW,Ast',TW,Astop); hfir = design(f2,'multistage','HalfbandDesignMethod','equiripple');
Then, we interpolate the intermediate 8x4=32 kHz signal by 44.1/32 = 1.378125 to get the desired 44.1 kHz final sampling frequency. We use a cubic Lagrange polynomial-based filter for this purpose.
Np = 3; % Polynomial Order [L,M]=rat(1.378125); % Interpolation and decimation factors ffar = fdesign.polysrc(L,M,'Fractional Delay','Np',Np); hfar = design(ffar,'lagrange');
The overall filter is simply obtained by cascading the two filters.
h2 = cascade(hfir,hfar); cost(h2)
ans = Number of Multipliers : 32 Number of Adders : 29 Number of States : 21 Multiplications per Input Sample : 94.15 Additions per Input Sample : 85.6375
The number of coefficients of this hybrid design is the lowest so far (32 compared to 64 for the single stage Farrow design and 5178 for the single stage polyphase design). Moreover, the number of operations per input sample has increased only a little (from 65 to 94 multiplications per sample and from 59 to 86 additions per sample) when compared to the single stage polyphase design. These numbers make the hybrid design the most efficient solution so far. We now turn to the analysis of the frequency response of these different filters.
Fsfir = 32e3; % The equivalent filter is clocked at 32 kHz Fsfar = 32e3*441; % The equivalent filter is clocked at 14.112 MHz set(hfvt, 'Filters', [hfir,hfar],'Fs',[Fsfir Fsfar]); legend(hfvt,'Polyphase Interpolator','Farrow Interpolator',... 'Location','NorthEast')
On this magnitude response plot, we can see that the Farrow filter suppresses the replicas of the polyphase interpolator. We then overlay the frequency responses of the single-stage and the multistage designs.
Fs2 = 32e3*441; % The equivalent 2-stage filter highest clock is 14.112 MHz set(hfvt,'Filters',[h1,h2],'Fs',[Fs1 Fs2]); legend(hfvt,'Single Stage Polyphase Interpolator', ... 'Cascade Farrow and FIR Polyphase Interpolator')
The two designs have almost identical transition widths and they both meet the specification of a 50 dB minimum attenuation in the stopband.
If the exact phase linearity constraint can be relaxed, we can use IIR halfband design techniques in lieu of the 4th band FIR Nyquist filter to get an approximately linear phase filter in the passband.
hiir = design(f2,'multistage','HalfbandDesignMethod','iirlinphase'); h3 = cascade(hiir,hfar); cost(h3)
ans = Number of Multipliers : 19 Number of Adders : 25 Number of States : 21 Multiplications per Input Sample : 76.15 Additions per Input Sample : 80.6375
IIR polyphase filters require even less coefficients than their FIR polyphase counterparts. The number of coefficients is further reduced from 32 to 19. Furthermore, the number of multiplications (76 compared to 94) and additions (81 compared to 86) per input sample is also reduced.
set(hfvt,'Filters',[h2,h3],'Fs',[Fs2 Fs2]); legend(hfvt,'Cascade Farrow and FIR Polyphase Interpolator', ... 'Cascade Farrow and IIR Polyphase Interpolator')
The IIR polyphase design has a slightly narrower transition width when compared with the FIR polyphase design. It also has less ripples in the passband.
axis([0 5, -.1, 0.1])
Multirate Farrow filters along with FIR and IIR polyphase filters constitute essential building blocks of modern multirate multistage systems. Polyphase filters are particularly well adapted for interpolation or decimation by an integer factor and for fractional rate conversions when the interpolation and the decimation factors are low. Farrow filters can efficiently implement arbitrary (including irrational) rate change factors.