Filter Design Toolbox 4.6

Complex Bandpass Filter Design

Complex bandpass filters are used in many applications from IF subsampling digital down converters to vestigial sideband modulation schemes for analog and digital television broadcast. One easy way to design a complex bandpass filter is to start with a lowpass prototype and apply a complex shift frequency transformation. In this demo, we review several cases of lowpass prototypes from single-stage single-rate FIR filters to multistage multirate FIR filters to IIR filters.

Contents

Single-Stage Single-Rate FIR Design

In the case of a single-rate FIR design, we simply multiply each set of coefficients by (aka 'heterodyne with') a complex exponential. In the next example, we rotate the zeros of the lowpass Nyquist filter prototype by a normalized frequency of .6.

Hlp = design(fdesign.nyquist(8));     % Lowpass prototype
N = length(Hlp.Numerator)-1;
Fc = .6;                              % Desired frequency shift
j = complex(0,1);
Hbp = copy(Hlp);
Hbp.Numerator = Hbp.Numerator.*exp(j*Fc*pi*(0:N));
hfvt = fvtool(Hlp,Hbp,'Color','white');
legend(hfvt,'Lowpass Prototype','Complex Bandpass','Location','NorthWest')

The same technique also applies to single-stage multirate filters.

Multirate Multistage FIR Design

In the case of multirate multistage FIR filters, we need to account for the different relative frequencies each filter operates on. In the case of a multistage decimator, the desired frequency shift applies only to the first stage. Subsequent stages must also scale the desired frequency shift by their respective cumulative decimation factor.

f = fdesign.decimator(16,'nyquist',16,'TW,Ast',.01,75);
Hd = design(f,'multistage');
N1 = length(Hd.Stage(1).Numerator)-1;
N2 = length(Hd.Stage(2).Numerator)-1;
N3 = length(Hd.Stage(3).Numerator)-1;
M12 = Hd.Stage(1).DecimationFactor; % Decimation factor between stage 1 and 
2
M23 = Hd.Stage(2).DecimationFactor; % Decimation factor between stage 2 and 
3
Fc  = -.2;                          % Desired frequency shift
Fc1 = Fc;                           % Frequency shift applied to the first s
tage
Fc2 = Fc*M12;                       % Frequency shift applied to the second 
stage
Fc3 = Fc*M12*M23;                   % Frequency shift applied to the third s
tage
Hdbp = copy(Hd);
Hdbp.Stage(1).Numerator = Hdbp.Stage(1).Numerator.*exp(j*Fc1*pi*(0:N1));
Hdbp.Stage(2).Numerator = Hdbp.Stage(2).Numerator.*exp(j*Fc2*pi*(0:N2));
Hdbp.Stage(3).Numerator = Hdbp.Stage(3).Numerator.*exp(j*Fc3*pi*(0:N3));
set(hfvt,'Filters',[Hd,Hdbp])
legend(hfvt,'Lowpass Prototype','Complex Bandpass','Location','NorthWest')

Similarly, in the case of a multistage interpolator, the desired frequency shift applies only to the last stage. Previous stages must also scale the desired frequency shift by their respective cumulative interpolation factor.

f = fdesign.interpolator(16,'nyquist',16,'TW,Ast',.01,75);
Hi = design(f,'multistage');
N1 = length(Hi.Stage(1).Numerator)-1;
N2 = length(Hi.Stage(2).Numerator)-1;
N3 = length(Hi.Stage(3).Numerator)-1;
L12 = Hi.Stage(2).InterpolationFactor; % Interpolation factor between stage 
1 and 2
L23 = Hi.Stage(3).InterpolationFactor; % Interpolation factor between stage 
2 and 3
Fc = .4;                               % Desired frequency shift
Fc3 = Fc;                              % Frequency shift applied to the thir
d stage
Fc2 = Fc*L23;                          % Frequency shift applied to the seco
nd stage
Fc1 = Fc*L12*L23;                      % Frequency shift applied to the firs
t stage
Hibp = copy(Hi);
Hibp.Stage(1).Numerator = Hibp.Stage(1).Numerator.*exp(j*Fc1*pi*(0:N1));
Hibp.Stage(2).Numerator = Hibp.Stage(2).Numerator.*exp(j*Fc2*pi*(0:N2));
Hibp.Stage(3).Numerator = Hibp.Stage(3).Numerator.*exp(j*Fc3*pi*(0:N3));
set(hfvt,'Filters',[Hi,Hibp])
legend(hfvt,'Lowpass Prototype','Complex Bandpass','Location','NorthWest')

Single-Rate IIR Design

Finally in case of single-rate IIR designs, we can either use a complex shift frequency transformation or a lowpass to complex bandpass IIR transformation. In the latter case, the bandwidth of the bandpass filter may also be modified.

Fp = .2;
Hiirlp = design(fdesign.lowpass(Fp,.25,.5,80),'ellip');
Fc  = .6;                         % Desired frequency shift
Hiircbp = ciirxform(Hiirlp, ...   % Shift frequency transformation
    'zpkshiftc', 0, Fc);          % DC shifted to Fc
Hiircbp2 = iirlp2bpc(Hiirlp, ...  % Lowpass to complex bandpass transformati
on
    Fp, [Fc-Fp, Fc+Fp]);          % Lowpass passband frequency mapped to ban
dpass passband frequencies
set(hfvt,'Filters',[Hiirlp,Hiircbp,Hiircbp2])
legend(hfvt,'Lowpass Prototype','Complex Bandpass #1',...
    'Complex Bandpass #2','Location','NorthWest')