Computing Acceptable Bandpass Sample Rates

Rick Lyons November 25, 2011 Coded in Matlab

If you need to perform analog-to-digital conversion of the analog bandpass signal whose spectrum is shown in Figure 1(a), your fs bandpass sampling rate must be within one of three frequency ranges. That is, given an analog bandpass signal centered at fc = 20 Mhz, having a bandwidth of Bw = 5 MHz, an acceptable (resulting in no aliasing errors) fs bandpass sample rate must be within one of the following three frequency ranges:

22.5 MHz -to- 35 MHz
15 MHz -to- 17.5 MHz
11.25 MHz -to- 11.6667 MHz

Figure 1(b) shows the spectrum of the sampled (discrete) bandpass signal when, for example, fs = 17.5 MHz. In that case, no spectral replications overlap in frequency and no aliasing errors occur.

Matlab Code:
Here's Matlab code to compute, display, and graphically plot the acceptable bandpass sample rate frequency ranges based on an input analog (continuous) signal's bandwidth and center frequency:

% Filename: Bandpass_Sample_Rate_Calc.m
%
%  Calculates acceptable Bandpass Sampling rate ranges 
%  based on an analog (continuous) signal's bandwidth 
%  and center freq.
%
%  Merely define bandwidth "Bw" and center frequency "Fc", in 
%  Hz, near line 22, for the analog bandpass signal and run the 
%  program.  [Example: Bw = 5, Fc = 20.]  Acceptable Fs sample  
%  rate ranges are shown in Figure 1 and displayed in Command window.
%
%  If the User defines a value for the BP sample rate Fs, near 
%  near line 28, then Figure 2 will show the locations of the 
%  positive and negative-freq spectral components after 
%  bandpass sampling.
%
%   Richard Lyons [November 2011]
%******************************************

clear, clc

Bw = 5; % Analog signal bandwidth in Hz
Fc = 20; % Analog signal center freq in Hz

% ##############################################
% Define an Fs sample rate value below

Fs = 11; % Selected Fs sample rate in Hz
% ##############################################

disp(' '), disp(['Analog Center Freq = ',num2str(Fc),' Hz.'])
disp(['Analog Bandwidth = ',num2str(Bw),' Hz.']), disp(' ')

% *****************************************************
%  Compute % display the acceptable ranges of BP sample rate 
% *****************************************************
disp('----------------------------------')
disp('Acceptable Fs ranges in Hz:')
No_aliasing = 0;  % Init a warning flag
M = 1; % Initialize a counter

while (2*Fc + Bw)/(M+1) >= 2*Bw
   FsMin = (2*Fc + Bw)/(M+1);
   FsMax = (2*Fc - Bw)/M;
   Fs_ranges(M,1) = FsMin;
   Fs_ranges(M,2) = FsMax;
   M = M + 1;
   disp([num2str(FsMin),' -to- ',num2str(FsMax)])
end
disp('----------------------------------')

% *****************************************************
%  Plot the acceptable ranges of BP sample rate 
% *****************************************************
figure(1), clf
title('Acceptable Ranges of Bandpass Sampling Rate')
xlabel('Freq (Hz)')
ylabel('This axis has no meaning')

for K = 1:M-1
   hold on
   plot([Fs_ranges(K,1),Fs_ranges(K,1)],[0.5,1.2],':g');
   plot([Fs_ranges(K,1),Fs_ranges(K,2)],[1,1],'-r','linewidth',4);
   axis([0.8*(2*Bw),1.1*max(Fs_ranges(1,2)), 0.8, 1.55])
end

plot([2*Bw,2*Bw],[0.5,1.2],'-b','linewidth',2);
text(2*Bw,1.45,'Bold red lines show acceptable Fs ranges')
text(2*Bw,1.35,'Blue line = Twice analog signal Bandwidth (2 x Bw)')
text(2*Bw,1.25,'(You can zoom in, if you wish.)')
hold off, grid on, zoom on

% **************************************************************
%  If Fs has been defined, plot spectrum of the sampled signal.  
% **************************************************************
% 
% Check if "Fs" has been defined
disp(' ')
if isempty(strmatch('Fs',who,'exact')) == 1; 
    disp('Fs sampling rate has NOT been defined.')
    % Fs does NOT exist, do nothing further.
else 
	% Fs is defined, plot the spectrum of the sampled signal.
        disp(['Fs defined as ',num2str(Fs),' Hz'])
	
    % To determine intermediate frequency (IF), check integer 
    %      part of "2Fc/Fs" for odd or even
	Temp = floor(2*Fc/Fs);
    if (Temp == 2*floor(Temp/2))
       disp(' '), disp('Pos-freq sampled spectra is not inverted.')
       Freq_IF = Fc -Fs*floor(Fc/Fs); % Computed IF frequency
	else
       disp(' '), disp('Pos-freq sampled spectra is inverted.')
       Freq_IF = Fs*(1 + floor(Fc/Fs)) -Fc; % Computed IF frequency
	end
    disp(' '), disp(['Center of pos-freq sampled spectra = ',num2str(Freq_IF),' Hz.'])

    % Prepare to plot sampled spectral range in Figure 2
    IF_MinFreq = Freq_IF-Bw/2;
    IF_MaxFreq = Freq_IF+Bw/2;
    
	figure(2), clf
	hold on
	plot([IF_MinFreq,IF_MaxFreq],[0.95, 1],'-r','linewidth',4);
	plot([Fs-IF_MaxFreq,Fs-IF_MinFreq],[1, 0.95],'-r','linewidth',4);
	plot([Fs,Fs],[0.5, 1.02],'-b','linewidth',2);
	plot([Fs/2,Fs/2],[0.5, 1.02],':b','linewidth',2);
    plot([IF_MinFreq,IF_MinFreq],[0.5, 0.95],':r');
    plot([IF_MaxFreq,IF_MaxFreq],[0.5, 1],':r');
    plot([Fs-IF_MaxFreq,Fs-IF_MaxFreq],[0.5, 1],':r');
    plot([Fs-IF_MinFreq,Fs-IF_MinFreq],[0.5, 0.95],':r');
    text(0.9*Fs,1.03,['Fs = ',num2str(Fs),' Hz'])
    text(0.8*Fs/2, 1.03,['Fs/2 = ',num2str(Fs/2),' Hz'])
    text(Fs/10,1.07,'(You can zoom in, if you wish.)')
    axis([0,1.2*Fs, 0.8, 1.1])
	
	hold off
	title('Red lines show spectral range of sampled signal')
    xlabel('Freq (Hz)')
	ylabel('This axis has no meaning')
	grid on, zoom on
    
    % ################################################################
    % Check if Fs is NOT in an acceptable freq range (aliasing occurs)
    Aliasing_Flag = 1; % Initialize a flag
    for K = 1:M-1 % Check each individual acceptable Fs range
        if Fs_ranges(K,1)<=Fs & Fs<=Fs_ranges(K,2)% & Fs<=Fs_ranges(K,2)
            % No aliasing will occur
            Aliasing_Flag = Aliasing_Flag -1;
        else, end
    end % End K loop
    if Aliasing_Flag == 1;  % Aliasing will occur
        text(Fs/10, 0.91, 'WARNING! WARNING!')
        text(Fs/10, 0.89, 'Aliasing will occur!')
    else, end
    zoom on
    % ################################################################
end

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