I'm trying to plot the Gauss sums according to the equation shown in the image s(t), but I keep receiving errors.
Can you please show me what am I doing wrong ?
%%
Fs = 1000; % Sampling frequency
T = 1/Fs; % Sampling period
L = 1024; % Length of signal
t = 2*(0:L-1)*T; % Time vector
x = 0;
k = 0;
s = 0;
p = primes(L);
% s(t) = cumsum((k/p)(1:length(p)-1)).*exp(1i*k*t);
for k=1:p-1
s(t) = s(t) + (k/p).*exp(1i*k*t);
end
figure
subplot(2,2,1)
plot(t,s)
title('signal')
You're treating the Legendre symbol as fraction - which it is not despite the deceivingly similar appearance.
Furthermore the index for your summation doesn't make a whole lot of sense, you probably just wan to use s as summing variable. So you just have to replace k/p in your summation expression with the Legendre symbol.
Related
I'm trying to plot the equation shown in the image, but I'm not sure how to apply the Galois Seq. "gf()" here.
Can you please help me with this?
enter image description here
clear all
close all
clc
%%
Fs = 1000; % Sampling frequency
T = 1/Fs; % Sampling period
L = 1024; % Length of signal
t = 2*(0:L-1)*T; % Time vector
signal = 0;
m = 5;
for k=1:((2^m)-1)
g = gf(2^m);
signal = signal + g(k).*exp(1i*2*pi*k*t)/((2^m)-1);
end
figure
subplot(2,2,1)
plot(t,signal)
title('signal')
The formula for the discrete double Fourier series that I'm attempting to code in MATLAB is:
The coefficient in front of the trigonometric sum (Fourier amplitude) is what I'm trying to extract from the fitting of the data through the double Fourier series seen above. Using my current code, the original function is not reconstructed, therefore my coefficients cannot be correct. I'm not certain if this is of any significance or insight, but the second term for the A coefficients (Akn(1))) is 13 orders of magnitude larger than any other coefficient.
Any suggestions, modifications, or comments about my program would be greatly appreciated.
%data = csvread('digitized_plot_data.csv',1);
%xdata = data(:,1);
%ydata = data(:,2);
%x0 = xdata(1);
lambda = 20; %km
tau = 20; %s
vs = 7.6; %k/s (velocity of CHAMP satellite)
L = 4; %S
% Number of terms to use:
N = 100;
% set up matrices:
M = zeros(length(xdata),1+2*N);
M(:,1) = 1;
for k=1:N
for n=1:N %error using *, inner matrix dimensions must agree...
M(:,2*n) = cos(2*pi/lambda*k*vs*xdata).*cos(2*pi/tau*n*xdata);
M(:,2*n+1) = sin(2*pi/lambda*k*vs*xdata).*sin(2*pi/tau*n*xdata);
end
end
C = M\ydata;
%least squares coefficients:
A0 = C(1);
Akn = C(2:2:end);
Bkn = C(3:2:end);
% reconstruct original function values (verification check):
y = A0;
for k=1:length(Akn)
y = y + Akn(k)*cos(2*pi/lambda*k*vs*xdata).*cos(2*pi/tau*n*xdata) + Bkn(k)*sin(2*pi/lambda*k*vs*xdata).*sin(2*pi/tau*n*xdata);
end
% plotting
hold on
plot(xdata,ydata,'ko')
plot(xdata,yk,'b--')
legend('Data','Least Squares','location','northeast')
xlabel('Centered Time Event [s]'); ylabel('J[\muA/m^2]'); title('Single FAC Event (50 Hz)')
I have the following code:
Fs = 1000;
T = 1/Fs;
L = 1000;
t = (0:L-1)*T;
k = 25:1:50;
m = 1:1:25;
where k and m are corresponding. I want to plot the 25 sine waves resulting from:
x = m*sin(2*pi*k*t);
I thought about doing it using a for loop that takes one value from m and k each time, but I'm unsure how to proceed.
Below is a very basic plotting solution. You will notice that it's very difficult to see what's going on in the plot, so you might want to consider other ways to present this data.
function q45532082
Fs = 1000;
T = 1/Fs;
L = 1000;
t = (0:L-1)*T;
k = 26:1:50;
m = 1:1:25;
%% Plotting
assert(numel(m) == numel(k)); % We make sure that the number of elements is the same.
figure(); hold on; % "hold" is needed if you want to see all curves at the same time.
for ind1 = 1:numel(m)
plot(t,m(ind1)*sin(2*pi*k(ind1)*t));
end
This is the result:
Note that the number of elements in k and m in your code is different, so I had to change it.
Using the functionality of plot you can also plot all sine waves without a loop:
Fs = 1000;
T = 1/Fs;
L = 1000;
t = (0:L-1)*T;
k = 26:1:50;
m = 1:1:25;
x = m.*sin(2.*pi.*bsxfun(#times,t.',k)); %this results in an L*25 matrix, each column is data of one wave
% or, if you have version 2016b or newer:
% x = m.*sin(2.*pi.*t.'*k);
plot(t,x) % plot all sines at ones
and as #Dev-iL noted, I also had to change k.
The result with L = 1000 is too crowded, so I plot it here with L = 50:
I have written a matlab function (Version 7.10.0.499 (R2010a)) to evaluate incoming FT signal and calculate the morlet wavelet for the signal. I have a similar program, but I needed to make it more readable and closer to mathematical lingo. The output plot is supposed to be a 2D plot with colour showing the intensity of the frequencies. My plot seems to have all frequencies the same per time. The program does make an fft per row of time for each frequency, so I suppose another way to look at it is that the same line repeats itself per step in my for loop. The issue is I have checked with the original program, which does return the correct plot, and I cannot locate any difference beyond what I named the values and how I organized the code.
function[msg] = mile01_wlt(FT_y, f_mn, f_mx, K, N, F_s)
%{
Fucntion to perform a full wlt of a morlet wavelett.
optimization of the number of frequencies to be included.
FT_y satisfies the FT(x) of 1 envelope and is our ft signal.
f min and max enter into the analysis and are decided from
the f-image for optimal values.
While performing the transformation there are different scalings
on the resulting "intensity".
Plot is made with a 2D array and a colour code for intensity.
version 05.05.2016
%}
%--------------------------------------------------------------%
%{
tableofcontents:
1: determining nr. of analysis f, prints and readies f's to be used.
2: ensuring correct orientation of FT_y
3:defining arrays
4: declaring waveletdiagram and storage of frequencies
5: for-loop over all frequencies:
6: reducing file to manageable size by truncating time.
7: marking plot to highlight ("randproblemer")
8: plotting waveletdiagram
%}
%--------------------------------------------------------------%
%1: determining nr. of analysis f, prints and readies f's to be used.
DF = floor( log(f_mx/f_mn) / log(1+( 1/(8*K) ) ) ) + 1;% f-spectre analysed
nr_f_analysed = DF %output to commandline
f_step = (f_mx/f_mn)^(1/(DF-1)); % multiplicative step for new f_a
f_a = f_mn; %[Hz] frequency of analysis
T = N/F_s; %[s] total time sampled
C = 2.0; % factor to scale Psi
%--------------------------------------------------------------%
%2: ensuring correct orientation of FT_y
siz = size(FT_y);
if (siz(2)>siz(1))
FT_y = transpose(FT_y);
end;
%--------------------------------------------------------------%
%3:defining arrays
t = linspace(0, T*(N-1)/N, N); %[s] timespan
f = linspace(0, F_s*(N-1)/N, N); %[Hz] f-specter
%--------------------------------------------------------------%
%4: declaring waveletdiagram and storage of frequencies
WLd = zeros(DF,N); % matrix of DF rows and N columns for storing our wlt
f_store = zeros(1,DF); % horizontal array for storing DF frequencies
%--------------------------------------------------------------%
%5: for-loop over all frequencies:
for jj = 1:DF
o = (K/f_a)*(K/f_a); %factor sigma
Psi = exp(- 0*(f-f_a).*(f-f_a)); % FT(\psi) for 1 envelope
Psi = Psi - exp(-K*K)*exp(- o*(f.*f)); % correctional element
Psi = C*Psi; %factor. not set in stone
%next step fits 1 row in the WLd (3 alternatives)
%WLd(jj,:) = abs(ifft(Psi.*transpose(FT_y)));
WLd(jj,:) = sqrt(abs(ifft(Psi.*transpose(FT_y))));
%WLd(jj,:) = sqrt(abs(ifft(Psi.*FT_y))); %for different array sizes
%and emphasizes weaker parts.
%prep for next round
f_store (jj) = f_a; % storing used frequencies
f_a = f_a*f_step; % determines the next step
end;
%--------------------------------------------------------------%
%6: reducing file to manageable size by truncating time.
P = floor( (K*F_s) / (24*f_mx) );%24 not set in stone
using_every_P_point = P %printout to cmdline for monitoring
N_P = floor(N/P);
points_in_time = N_P %printout to cmdline for monitoring
% truncating WLd and time
WLd2 = zeros(DF,N_P);
for jj = 1:DF
for ii = 1:N_P
WLd2(jj,ii) = WLd(jj,ii*P);
end
end
t_P = zeros(1,N_P);
for ii = 1:N_P % set outside the initial loop to reduce redundancy
t_P(ii) = t(ii*P);
end
%--------------------------------------------------------------%
%7: marking plot to highlight boundary value problems
maxval = max(WLd2);%setting an intensity
mxv = max(maxval);
% marks in wl matrix
for jj= 1:DF
m = floor( K*F_s / (P*pi*f_store(jj)) ); %finding edges of envelope
WLd2(jj,m) = mxv/2; % lower limit
WLd2(jj,N_P-m) = mxv/2;% upper limit
end
%--------------------------------------------------------------%
%8: plotting waveletdiagram
figure;
imagesc(t_P, log10(f_store), WLd2, 'Ydata', [1 size(WLd2,1)]);
set(gca, 'Ydir', 'normal');
xlabel('Time [s]');
ylabel('log10(frequency [Hz])');
%title('wavelet power spectrum'); % for non-sqrt inensities
title('sqrt(wavelet power spectrum)'); %when calculating using sqrt
colorbar('location', 'southoutside');
msg = 'done.';
There are no error message, so I am uncertain what exactly I am doing wrong.
Hope I followed all the guidelines. Otherwise, I apologize.
edit:
my calling program:
% establishing parameters
N = 2^(16); % | number of points to sample
F_s = 3.2e6; % Hz | samplings frequency
T_t = N/F_s; % s | length in seconds of sample time
f_c = 2.0e5; % Hz | carrying wave frequency
f_m = 8./T_t; % Hz | modulating wave frequency
w_c = 2pif_c; % Hz | angular frequency("omega") of carrying wave
w_m = 2pif_m; % Hz | angular frequency("omega") of modulating wave
% establishing parameter arrays
t = linspace(0, T_t, N);
% function variables
T_h = 2*f_m.*t; % dimless | 1/2 of the period for square signal
% combined carry and modulated wave
% y(t) eq. 1):
y_t = 0.5.*cos(w_c.*t).*(1+cos(w_m.*t));
% y(t) eq. 2):
% y_t = 0.5.*cos(w_c.*t)+0.25*cos((w_c+w_m).*t)+0.25*cos((w_c-w_m).*t);
%square wave
sq_t = cos(w_c.*t).*(1 - mod(floor(t./T_h), 2)); % sq(t)
% the following can be exchanged between sq(t) and y(t)
plot(t, y_t)
% plot(t, sq_t)
xlabel('time [s]');
ylabel('signal amplitude');
title('plot of harmonically modulated signal with carrying wave');
% title('plot of square modulated signal with carrying wave');
figure()
hold on
% Fourier transform and plot of freq-image
FT_y = mile01_fftplot(y_t, N, F_s);
% FT_sq = mile01_fftplot(sq_t, N, F_s);
% Morlet wavelet transform and plot of WLdiagram
%determining K, check t-image
K_h = 57*4; % approximation based on 1/4 of an envelope, harmonious
%determining f min and max, from f-image
f_m = 1.995e5; % minimum frequency. chosen to showcase all relevant f
f_M = 2.005e5; % maximum frequency. chosen to showcase all relevant f
%calling wlt function.
name = 'mile'
msg = mile01_wlt(FT_y, f_m, f_M, K_h, N, F_s)
siz = size(FT_y);
if (siz(2)>siz(1))
FT_y = transpose(FT_y);
end;
name = 'arnt'
msg = arnt_wltransf(FT_y, f_m, f_M, K_h, N, F_s)
The time image has a constant frequency, but the amplitude oscillates resempling a gaussian curve. My code returns a sharply segmented image over time, where each point in time holds only 1 frequency. It should reflect a change in intensity across the spectra over time.
hope that helps and thanks!
I found the error. There is a 0 rather than an o in the first instance of Psi. Thinking I'll maybe rename the value as sig or something. besides this the code works. sorry for the trouble there
I need help in plotting the Bit error curve or the symbol error curve for BPSK modulation scheme for varying Signal to Noise ratios or Eb/N0. The plot should show the simulated versus the theoretical curve, but I cannot figure out how to mitigate the problems when using the Constant Modulus Algorithm as an Equalizer which are:
(1)
Error using *
Inner matrix dimensions must agree.
Error in BER_BPSK_CMA (line 50)
yy = w'*x;
(2) I want to use the filter function instead of conv in order to model a moving average channel model, chanOut = filter(ht,1,s). But, when I use filter, I am getting an error. How can I use filter function here?
(3) Bit error rate calculation
UPDATED Code with the Problem 1 solved. However, I am still unable to use filter and unsure if BER curve is proper or not.
Below is the code I wrote:
% Script for computing the BER for BPSK modulation in 3 tap ISI
% channel
clear
N = 10^2; % number of bits or symbols
Eb_N0_dB = [0:15]; % multiple Eb/N0 values
K = 3; %number of users
nTap = 3;
mu = 0.001;
ht = [0.2 0.9 0.3];
L = length(ht);
for ii = 1:length(Eb_N0_dB)
% Transmitter
ip = rand(1,N)>0.5; % generating 0,1 with equal probability
s = 2*ip-1; % BPSK modulation 0 -> -1; 1 -> 0
% Channel model, multipath channel
chanOut = conv(s,ht);
% chanOut = filter(ht,1,s); %MA
n = 1/sqrt(2)*[randn(1,N+length(ht)-1) + j*randn(1,N+length(ht)-1)]; % white gaussian noise, 0dB variance
% Noise addition
y = chanOut + 10^(-Eb_N0_dB(ii)/20)*n; % additive white gaussian noise
%CMA
Le =20; %Equalizer length
e = zeros(N,1); % error
w = zeros(Le,1); % equalizer coefficients
w(Le)=1; % actual filter taps are flipud(w)!
yd = zeros(N,1);
r = y';
% while(1)
for i = 1:N-Le,
x = r(i:Le+i-1);
%x = r(i:(Le+i-1));
yy = w'*x;
yd(i)= yy;
e(i) = yy^2 - 1;
mse_signal(ii,i) = mean(e.*e);
w = w - mu * e(i) * yy * x;
end
sb=w'*x; % estimate symbols (perform equalization)
% receiver - hard decision decoding
ipHat = real(sb)>0;
% counting the errors
nErr_CMA(ii) = size(find([ip- ipHat]),2);
% calculate SER
end
simBer_CMA = nErr_CMA/N;
theoryBer = 0.5*erfc(sqrt(10.^(Eb_N0_dB/10))); % theoretical ber
for i=1:length(Eb_N0_dB),
tmp=10.^(i/10);
tmp=sqrt(tmp);
theoryBer(i)=0.5*erfc(tmp);
end
figure
semilogy(theoryBer,'b'),grid;
hold on;
semilogy(Eb_N0_dB,simBer_CMA,'r-','Linewidth',2);
%axis([0 14 10^-5 0.5])
grid on
legend('sim-CMA');
xlabel('Eb/No, dB');
ylabel('Bit Error Rate');
title('Bit error probability curve for BPSK in ISI with CMA equalizer');
There's an error in these three lines:
sb=w'*x; % estimate symbols (perform equalization)
% receiver - hard decision decoding
ipHat = real(sb)>0;
they worked inside the while loop but you are now performing a post-estimation, so the correct lines are:
sb=conv(w,y); % estimate symbols (perform equalization)
% receiver - hard decision decoding
ipHat = real(sb(Le+1:end-1))>0; % account for the filter delay
There is still some issue with the output... but I can't go further in the analisys.
Your first problem is easily solved: change the line to
x = y(i:(Le+i-1));
Your call of filter looks OK. Which error do you get?
Maybe this is a place to start looking.
Or here (would Fig. 4 be the type of plot you're after?)