Visualising negative frequency

Negative frequency is one of those concepts that many people struggle with. Here’s a image that will hopefully help. There’s an svg version available at https://pzdsp.com/negf/negative_frequency.svg.

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Negative Frequency GUI Code

function negfgui(varargin)
% An interactive GUI which attempts to help students visualise negative frequencies
% by David Dorran, TU Dublin, March 2022
% reuses some code to track mouse movement from the zpgui function by Tom Krauss from Perdue University
% Youtube video explanation provided at https://youtu.be/Rxc_ypVdruw

global zero_locations diff_equ_text op_text_samps op_freq_pts op_points draw_plt disp_plt op_disp_plt freq_plt xyz_pts op_xyz_pts xy_pts ptr negfqui_fh freq_pts points text_samps b a y gain pole_vals zero_vals freq_resp_pts

buf_len = 50;
tail_len = 10;
max_points = 100000;
fft_len = 1024;
win_func = hamming(buf_len);

if nargin == 0
    action = 'init';
else
        action = varargin{1};
end

if(strcmp(class(action), 'matlab.ui.control.UIControl'))
    zero_vals = str2num(zero_locations.String);
    negfgui('update_freq_response');
end
switch action
    case 'init'
        zero_vals = [ -j j];
        pole_vals = [0 0];
        
        if(strcmp(class(negfqui_fh),'matlab.ui.Figure'))
            if(~isvalid(negfqui_fh))
                negfqui_fh = figure;    
            end
            if(nargin ==0)
                close(negfqui_fh);
                negfqui_fh = figure;  
            end
        else
            negfqui_fh = figure;
        end
        set(negfqui_fh,'position',[ 1          41        1366         651]);
        ptr = 1;
        points = ones(1,max_points)*NaN;
        op_points = ones(1,max_points)*NaN;
        
        freq_plt = subplot(2,3,1);
        fax = (([0:fft_len-1])/fft_len*2*pi)-pi;
        yyaxis left
        freq_pts = plot(fax,points(1:fft_len));
        hold on
        op_freq_pts = plot(fax,points(1:fft_len),'r');
        xlim([-pi pi])
        ylim([0 buf_len/2])
        ylabel('Magnitude Content')
        yyaxis right
        freq_resp_pts = plot(fax,points(1:fft_len));
        xlim([-pi pi])
        ylim([0 1])
        title('Double sided-spectrum');
        xlabel('frequency (radians per sample)')
        ylabel('Magnitude Response')
        draw_plt = subplot(2,3,2);
        xy_pts = plot([]);
        for m =1:tail_len
            hold on
            xy_pts(m) = plot(points(m)+points(m)*j,'r.','MarkerSize',tail_len-m+1);
        end
        ylabel('Y')
        xlabel('X')
        xlim([-1 1]);
        ylim([-1 1]);
        disp_plt = subplot(2,3,3);
        xyz_pts = plot3(real(points),imag(points),[0:length(points)-1]);
        hold on
        surf([0 0 ],[-1.1 1.1],[-1 length(points); -1 length(points)],'FaceAlpha',0.2)
        surf([-1.1 1.1], [ 0 0 ],[ length(points) length(points); -1 -1],'FaceAlpha',0.2)
        xlabel('X')
        ylabel('Y')
        zlabel('Samples')
        
        xlim([-1 1]);
        ylim([-1 1]);
        zlim([0 buf_len])
        set(gca,'CameraUpVector', [0 1 0 ])
        grid on
        set(draw_plt,'buttondownfcn', 'negfgui(''draw'')')
        
        op_disp_plt = subplot(2,3,6);
        op_xyz_pts = plot3(real(op_points),imag(op_points),[0:length(op_points)-1]);
        hold on
        surf([0 0 ],[-1.1 1.1],[-1 length(op_points); -1 length(op_points)],'FaceAlpha',0.2)
        surf([-1.1 1.1], [ 0 0 ],[ length(op_points) length(op_points); -1 -1],'FaceAlpha',0.2)
        xlabel('X')
        ylabel('Y')
        zlabel('Samples')
        
        xlim([-1 1]);
        ylim([-1 1]);
        zlim([0 buf_len])
        set(gca,'CameraUpVector', [0 1 0 ])
        grid on
        
        text_samps = annotation('textbox','position',[.0500    0.48    0.85    0.05],'LineStyle','none','FontSize',10);
        op_text_samps = annotation('textbox','position',[.0500    0    0.85    0.05],'LineStyle','none','FontSize',10);
        diff_equ_text = annotation('textbox',...
                [0.34 0.337941628264209 0.285237188872621 0.0844854070660522],...
                'String',[''],...
                'FontSize',14,...
                'FitBoxToText','off',...
                'EdgeColor','none');
         annotation('textbox',...
                [0.34 0.2 0.285237188872621 0.0844854070660522],...
                'String',['Zero Locations:'],...
                'FontSize',14,...
                'FitBoxToText','off',...
                'EdgeColor','none');
        zero_locations =  uicontrol('style','edit','units','normalized','FontSize',14,'position',  [0.34  0.17 0.2 0.05]);
        update_button =  uicontrol('style','pushbutton', 'string','update locations','units','normalized','position',[0.34  0.12 0.2 0.05],'callback',{@negfgui});

        negfgui('update_freq_response')
    case 'update_freq_response'
        
        b = poly(zero_vals);
        a = poly(pole_vals);
        H = freqz(b,a,fft_len,'whole');
        gain = max(abs(H));
        freq_resp_pts.YData= fftshift(abs(H)/gain); %ensure max gain is 1
        b = b/gain;
        subplot(2,3,4)
        zplane(b,a);
        title('Pole-Zero Plot')
        try
            str = create_difference_equation(b,1,'tex');
            diff_equ_text.String = {['Difference Equation:'], str};
            
        catch
            diff_equ_text.String = {['b coefficients: ' num2str(b)], ['a coefficients:' num2str(a)]};
        end
        
        zero_locations.String = num2str(zero_vals);
        
        
    case 'draw'
        
        set(gcf,'userdata','')
        set(gcf,'windowbuttonmotionfcn','set(gcf,''userdata'',''motion'')')
        set(gcf,'windowbuttonupfcn','set(gcf,''userdata'',''up'')')
        
        done = 0;
       
         while ~done
            waitfor(gcf,'userdata')
            switch get(gcf,'userdata')
                case 'motion'
              
                    pt = get(draw_plt,'currentpoint');
                    if(abs(pt(1,1))>1)
                        pt(1,1) = 1*sign(pt(1,1));
                    end
                    if(abs(pt(1,2))>1)
                        pt(1,2) = 1*sign(pt(1,2));
                    end
                    points(ptr) = pt(1,1)+pt(1,2)*j;
                    xyz_pts.XData(ptr) = real(points(ptr));
                    xyz_pts.YData(ptr) = imag(points(ptr));
                    
                    %filter the signal (it's not perfect!)
                    try
                        op_points(ptr) = 0;
                        for k= 1:length(b)
                            op_points(ptr) = op_points(ptr)+ b(k)*points(ptr-k+1);
                        end
                    catch
                        
                    end
                    
                    op_xyz_pts.XData(ptr) = real(op_points(ptr));
                    op_xyz_pts.YData(ptr) = imag(op_points(ptr));
                    if(ptr>buf_len)
                         disp_plt.ZLim = [ptr-buf_len ptr];
                         op_disp_plt.ZLim = [ptr-buf_len ptr];
                         freq_pts.YData = abs(fftshift(fft(points(ptr-buf_len+1:ptr).*win_func', fft_len)));
                         op_freq_pts.YData = abs(fftshift(fft(op_points(ptr-buf_len+1:ptr).*win_func', fft_len)));
                    end
                    
                    for m =1:tail_len
                        try
                            xy_pts(m).XData(1) = real(points(ptr-m+1));
                            xy_pts(m).YData(1) = imag(points(ptr-m+1));

                            text_samps.String = num2str(fliplr(round(points(ptr-m+1:ptr)*100)/100));
                            op_text_samps.String = num2str(fliplr(round(op_points(ptr-m+1:ptr)*100)/100));
                        catch
                            
                            text_samps.String = num2str(fliplr(round(points(1:ptr)/100)*100));
                        end
                    end
                    
                    ptr=ptr+1;
                    if(ptr> max_points)
                       negfgui('init') 
                    end
                case 'up'
                    done = 1;
            end
            set(gcf,'userdata','')
        end
        set(gcf,'windowbuttonmotionfcn','')
        set(gcf,'windowbuttonupfcn','')

end

        

Digital Signal Processing Notes

I’ve made some note relating to Digital Signal Processing and the Discrete Fourier Transform available at http://pzdsp.com/docs.

I intend to add more documents to this over time.

Code to extract music notation from Flowkey app

% Matlab code to extract music notation from flowkey screen grab video . See https://youtu.be/4oOMWkMRCCg

v = VideoReader('takemetochurch.mp4');
ind = 5*25;
frame =v.read(ind);
im = frame(430:660 ,: ,:);
new_im = rgb2gray(im);
prev_x = sum(new_im); % used to find bes overlap of frames

while 1
    ind = ind+5*25;
    if(ind > v.NumFrames)
        break
    end
    frame =v.read(ind);
    im = frame(430:660 ,: ,:);
    img = rgb2gray(im); %conver to greyscale
    x = sum(img);
    xc =[]; %stores cross correlation result
    seg2 = prev_x(end-600+1:end); %end of previous frame
    for w = 1: 300
        seg1 = x(1+w:600+w);
        xc(w) = sum(seg1.*seg2)./(sqrt(sum(seg1.^2)*sum(seg2.^2))); %cross correlation
    end
    [val loc] = max(xc); % loc is where the max correlation occurs
    
    [ r c d] = size(im);
    new_seg =img(:,201:end);
    [ r1 c1 d1] = size(new_seg);
    new_im(:,end-600+1-loc+200:end-600-loc+200+c1) = new_seg;
    prev_x = x;
    
end
img = new_im;
% find where the bars occur in the image
for k =1: length(img)
    x(k) = sum(img(95:145,k)); %the bar lines cross are clear from rows 75 to 125
end
bars = find(x<5000); %obtained value of 5000 by visually inspecting plot(x)


op_image_max_width = 2000;
op_image_max_height = 1200;

overlap = 10;
start_pt = overlap+1;
prev_end_pt = bars(1);
op_img = img(1,1);
k = 1;
img_num = 1;
while(k<length(bars))
    end_pt = bars(k);
    if(end_pt-start_pt>op_image_max_width)
        [rows cols ] = size(img(:,start_pt-overlap:prev_end_pt+overlap));
        [rows1 cols1] = size(op_img);
        if(rows+rows1 > op_image_max_height)
            imwrite(op_img, ['notation_' num2str(img_num) '.jpg']);
            op_img = img(1,1);
            
            img_num = img_num + 1;
        end
        op_img(end+1:end+rows,1:cols,:) = img(:,start_pt-10:prev_end_pt+10);
        start_pt = prev_end_pt;
    end
    prev_end_pt = end_pt;
    k = k + 1;
end

imwrite(op_img, ['notation_' num2str(img_num) '.jpg']);

Arduino sinuoidal waveform generator and signal capture

I have put together an example on how to use the Arduino nano as a signal generator and digital capture device. It runs a pretty low sampling rate 2 kHz but I thought it might be of some use. Check out a video explanation at  https://youtu.be/_31sQn1Cg9g.  Files used are available at http://pzdsp.com/shared/arduino_sin_gen.zip

Virtual Oscilloscope

I’ve created a virtual oscilloscope that hopefully will help students learn how to work with real ones! It doesn’t have too many features so students can focus on the fundamentals that appear on all scopes: scaling, positioning and triggering.

Check it out at pzdsp.com/elab

FFT based FIR filter design

%% This script deals issues related with 'fft' based FIR filter design
% See youtube video at https://youtu.be/R4RpNG_Botk


% GO TO lines 115 to 133 if you want to skip the preamble

%% First show fft of finite number of impulse response samples  lie on the continuous spectrum
%Design a band reject filter using buit-in fir1 function
b = fir1(22, [0.3913 0.6522],'stop'); %cutoff frequency is (0.3, 0.6 x pi) radians/sample = 0.3,0.6 x fs/2;
h = b; % The non-zero values of the impulse response of any fir filter 
       %   is equal to the b coefficients
H = fft(h); % Frequency response bin values
bin_freqs = [0:length(H)-1]/length(H)*2*pi; 

%% Create the continuous spectrum
% Going to do this three ways - just for the purpose of demonstration. 
% The mathematical formula for the frequency response of an mth order FIR is:
%   H(w) = b0 + b1*e^-jw + b2*e^-2jw + b3*e^-2*jw + ... ... + bm*e^(-m*jw)
%
%   This equation comes from the fact that H(w) = H(z) when z= e^jw i.e. when H(z) is
%   evaluated around the unit circle (see https://youtu.be/esZ_6n-qHuU )
%  The transfer function H(z) = b0*Z^0+b1*z^-1+b2*z^-2+b3*z-3+ ... +bmz^-m

% We'll create a 'continuous' spectrum by evaluating this equation for a
% "large" number of frequencies, w.

% Two alternatives to creating a 'continuous' spectrum are to zero pad the
% b coefficients by a "large" amount prior to taking the fft; and to use the built in freqz
% function. I'll use all three approaches here for comparison purposes.

% METHOD 1 - zero-pad method
H_cont_1 = abs(fft([h zeros(1,10000)])); % the value of 10000 could be any relatively large number i.e. large enough to capture the spectral detail of the frequency response
w_cont_1 = [0:length(H_cont_1)-1]/length(H_cont_1)*2*pi;

% METHOD 2 - freqz method
[H_cont_2_complex w_cont_2] = freqz(b,1,length(H_cont_1), 'whole'); % the second parameter of freqz is the a coefficients
H_cont_2 = abs(H_cont_2_complex);

% METHOD 3 -evaluate H(z) around the unit circle method (see
%https://youtu.be/esZ_6n-qHuU)
H_cont_3_complex = [];
k = 0;
w_cont_3 = [0:length(H_cont_1)-1]/length(H_cont_1)*2*pi;
for w = w_cont_3
    k = k + 1;
    H_cont_3_complex(k) = b(1);
    for ind = 2:length(b)
        H_cont_3_complex(k) = H_cont_3_complex(k) + b(ind)*exp(-1i*w*(ind-1));
    end
end

H_cont_3= abs(H_cont_3_complex);

%% plot 'continuous' frequency response and overlay 'sampled' bin values
figure
plot(w_cont_1, H_cont_1,'LineWidth', 6)
hold on
plot(w_cont_2, H_cont_2,'LineWidth', 3)
plot(w_cont_3, H_cont_3)
plot(bin_freqs, abs(H),'g.','MarkerSize', 16) 
xlabel('Frequency (radians per sample)')
ylabel('Magnitude')
legend('Cont. Freq Resp Method 1', 'Cont. Freq Resp Method 2', 'Cont. Freq Resp Method 3', 'Bin Values')
title('Frequency response of filter using a built-in filter design technique')
set(gcf,'Position',[22 416 613 420]);

%% Now let's try and design a filter by starting with the desired frequency response
% First we'll look at the band pass filter
H_desired = [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0];
freq_bins_desired = [0:length(H_desired)-1]/length(H_desired)*2*pi;
b_ifft = ifft(H_desired); %b coefficients of newly designed filter
h_ifft = b_ifft; 

% Now compute the continuous spectrum using any of the methods above. (Zero
% padding b_ifft prior to taking the fft is the most straightforward
% appraoch - I think!)
H_cont_desired = abs(fft([b_ifft zeros(1,1000)]));
w_cont_desired = [0:length(H_cont_desired)-1]/length(H_cont_desired)*2*pi;

figure
plot(w_cont_desired, H_cont_desired);
hold on
plot(freq_bins_desired, H_desired,'r.','MarkerSize', 12) 

xlabel('Frequency (radians per sample)')
ylabel('Magnitude')
legend('Continuous Frequency Response','DFT bin values of desired frequency response')
title('Frequency response of band pass filter using ''inverse fft'' filter design technique')
set(gcf,'Position',[649   190   610   420]);

%% 
% We'll now mimic the 'desired' filter shown in the video i.e. a band reject filter
H_desired = [1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1];
freq_bins_desired = [0:length(H_desired)-1]/length(H_desired)*2*pi;
b_ifft = ifft(H_desired);


% Now compute the continuous spectrum using any of the methods above. (Zero
% padding b_ifft prior to taking the fft is the most straighforward
% appraoch I think)
H_cont_desired = abs(fft([b_ifft zeros(1,1000)]));
w_cont_desired = [0:length(H_cont_desired)-1]/length(H_cont_desired)*2*pi;

figure
plot(w_cont_desired, H_cont_desired);
hold on
plot(freq_bins_desired, H_desired,'r.','MarkerSize', 12) 

xlabel('Frequency (radians per sample)')
ylabel('Magnitude')
legend('Continuous Frequency Response','DFT bin values of desired frequency response')
title('Frequency response of band reject filter using ''inverse fft'' filter design technique')
set(gcf,'Position',[649   190   610   420]);

%%
% Now try using a 'practical' linear phase response rather than phase values of zero

H_desired = [1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1];
% An alternative way to create H_desired to ensure first half is a mirror
% image of the second half (for odd lengths only!)
% H_desired = [1 1 1 1 1 0 0 0 1 1 1 1];
% H_desired = [H_desired fliplr(H_desired(2:end))]

phase_diff = pi/length(H_desired)-pi;
phase_desired = [0:floor(length(H_desired)/2)]*phase_diff;
phase_desired = [phase_desired fliplr(phase_desired(2:end))*-1];

freq_bins_desired = [0:length(H_desired)-1]/length(H_desired)*2*pi;
h_ifft = ifft(H_desired.*exp(j*phase_desired));


b_ifft = h_ifft; %b coefficients of newly designed filter
%b_ifft = h_ifft.*hanning(length(h_ifft))'; %windowed b coefficients of newly designed filter 

% Now compute the continuous spectrum using any of the methods above. (Zero
% padding b_ifft prior to taking the fft is the most straighforward
% approach - I think!)
H_cont_desired = abs(fft([b_ifft zeros(1,1000)]));
w_cont_desired = [0:length(H_cont_desired)-1]/length(H_cont_desired)*2*pi; %frequency axis

plot(w_cont_desired, H_cont_desired, 'Linewidth', 1);
hold on
plot(freq_bins_desired, H_desired,'r.','MarkerSize', 12) 

xlabel('Frequency (radians per sample)')
ylabel('Magnitude')
legend('Continuous Frequency Response','DFT bin values of desired frequency response')
title('Frequency response of band reject filter using ''inverse fft'' filter design technique')
set(gcf,'Position',[649   190   610   420]);

%% Demonstration of filters being a sum of prototype bandpass filters
% H_acc is the accumulation of individual prototype bandpass filter

H_prot = zeros(size(H_desired)); %initialise
%start with the DC prototype filter
H_prot(1) = H_desired(1).*exp(j*phase_desired(1));
H_acc = fft(ifft(H_prot), length(H_cont_desired)); 

% loop through 11 prototype band pass filters
show_plots = 0;
if(show_plots)
    figure
end
for k = 1: 11 
    prot_bins = [k+1 24-k] %pair of bins associated with positive and negative frequencies
    H_prot = zeros(size(H_desired)); %initialise
    H_prot(prot_bins) = H_desired(prot_bins).*exp(j*phase_desired(prot_bins));
    
    H_cont_prot = fft(ifft(H_prot), length(H_cont_desired));
    H_acc = H_acc + H_cont_prot; %accumulate each individual band pass prototype
    if(show_plots)
        plot(abs(H_acc))
        hold on
        plot(abs(H_cont_prot))
        hold off
        pause(1)
    end
end

sum(abs(H_acc) - abs(H_cont_desired))

DSP Foundations Notes

I’ve put together a set of notes introducing Digital Signal Processing. There’s only 25 pages but I hope that its enough to get someone started.

Download DSP Foundations

interpLTspice

% This function interpolates data returned from the runLTspice function so
% as to make the time vector evenly spaced.
%
% example usage:
%   asc_file = 'filename.asc';
%   volt_curr_list =  runLTspice(asc_file);
%   results = runLTspice(asc_file, volt_curr_list);
%   fs = 10000;
%   interp_results = interpLTspice(results, fs); 
%
% By David Dorran (david.dorran@dit.ie)
function data_struct = interpLTspice(data_struct, fs)
%check if the data is structured as expected
data_ok = 0;
if(isstruct(data_struct))
    data_ok = 1;
    if ~(isfield(data_struct,'data') && isfield(data_struct,'data_name') && isfield(data_struct,'time_vec'))
        data_ok = 0;
    end
end
if(~data_ok)
    error('The variable passed to this function is not a data structure returned from runLTspice and cannot be processed')
end

if~(isnumeric(fs) && length(fs) ==  1 && fs > 0)
    error('The second parameter used in this function should be a positive number representing the sampling frequency')
end



try t = data_struct.time_vec(1):1/fs:data_struct.time_vec(end);
catch
    error(sprintf('A sampling rate of %f resulted in too much data (memory limits exceeded).\n\nTry a different sampling frequency using interpLTspcice(data_struct,fs);', fs));
end
for k = 1:length(data_struct.data)
    data_struct.data{k} =  interp1(data_struct.time_vec, data_struct.data{k}, t);
end
data_struct.time_vec = t;

runLTspice demo

%% Demonstration of the runLTspice function
%
% Demo video at http://youtu.be/ax9H4eKuZv4
%
% Download runLTspice from:
%         https://dadorran.wordpress.com/2014/10/29/runltspice-m/
% 
% Download this code from https://dadorran.wordpress.com
%
% LTspice schematic available from:
%       https://www.dropbox.com/s/y7j1vmyscvppsn4/RC.asc?dl=0
%

%% read the voltage and currents from the circuit
result = runLTspice('C:\Temp\RC.asc', 'V(n002)');

plot(result.time_vec, result.data{1})
xlabel('Time(seconds)')
ylabel('Voltage')
title(result.data_name{1})

%% apply a step input and plot the capacitor current and voltage wavforms
fs = 200;
step = [zeros(1, fs/2) ones(1, fs/2)];

res = runLTspice('C:\Temp\RC.asc', 'V(n002), I(C1)' , 'V1', step, fs);

plot(res.time_vec, res.data{1})
hold on
plot(res.time_vec, res.data{2}*1000,'r')
xlabel('Time(seconds)')
ylabel('Amplitude')
hold off
legend([ res.data_name{1} ' (v)'],  [res.data_name{2} ' (mA)'])

%% apply a noisy square wave and compare the supply voltage with capacitor voltage
sq_wav = [step step step step];
high_f_noise = filter(1, [1 0.9], randn(1, length(sq_wav))*0.1);
ns_sq_wav = sq_wav + high_f_noise;

r = runLTspice('C:\Temp\RC.asc', 'V(n001), V(n002)' , 'V1', ns_sq_wav, fs);

plot(r.time_vec, r.data{1})
hold on
plot(r.time_vec, r.data{2},'r')
legend('supply voltage','smoothed voltage')
hold off