Allpass filter for lowpass to complex bandpass transformation
returns the numerator,
AllpassDen] = allpasslp2bpc(
AllpassNum, and the denominator,
AllpassDen, of the first-order allpass mapping filter for performing
a real lowpass to complex bandpass frequency transformation. For more information, see Real Lowpass to Complex Bandpass Frequency Transformation.
Design the allpass mapping filter changing the real lowpass filter with the cutoff frequency
0.5 into a complex bandpass filter with band edges
Wt2 precisely defined at
0.4, respectively. Calculate the frequency response of the mapping filter in the full range.
Wo = 0.5; Wt = [0.2 0.4]; [AllpassNum,AllpassDen] = allpasslp2bpc(Wo,Wt); [h,f] = freqz(AllpassNum,AllpassDen,'whole'); plot(f/pi,abs(angle(h))/pi,Wt,Wo,'ro'); title('Mapping Function Wo(Wt)'); xlabel('New Frequency, Wt'); ylabel('Old Frequency, Wo');
Wo— Frequency value to be transformed from prototype filter
Frequency value to be transformed from the prototype filter, specified as a real
scalar in the range
(0,1) with 1 corresponding to half the sample
Wt— Desired frequency locations in target filter
Desired frequency locations in the transformed target filter, specified as a real
vector with values in the range
(-1,1) with 1 corresponding to half
the sample rate.
AllpassNum— Numerator of mapping filter
Numerator of the mapping filter, returned as a complex-valued vector.
Complex Number Support: Yes
AllpassDen— Denominator of mapping filter
Denominator of the mapping filter, returned as a complex-valued vector.
Complex Number Support: Yes
Real lowpass to complex bandpass frequency transformation effectively places one feature of the original filter, located at frequency -Wo, at the required target frequency location, Wt1, and the second feature, originally at +Wo, at the new location, Wt2. It is assumed that Wt2 is greater than Wt1.
Relative positions of other features of the original filter do not change in the target filter. This means that it is possible to select two features of an original filter, F1 and F2, with F1 preceding F2. Feature F1 will still precede F2 after the transformation. However, the distance between F1 and F2 will not be the same before and after the transformation.
Choice of the feature subject to the lowpass to bandpass transformation is not restricted only to the cutoff frequency of an original lowpass filter. In general it is possible to select any feature for example, the stopband edge, the DC, the deep minimum in the stopband, or other ones.
Lowpass to bandpass transformation can also be used for transforming other types of filters, for example, real notch filters or resonators can be doubled and positioned at two distinct desired frequencies at any place around the unit circle forming a pair of complex notches/resonators. This transformation can be used for designing bandpass filters for radio receivers from the high-quality prototype lowpass filter.