Not sure of how I can sample this Nyquist wave? Combining data problems

I was given a task of creating a continuous time signal consisting of two different sine waves added together, with this new wave then being sampled after its creation. I managed to complete the first part with ease by making two different sets of data and combining them to make a new sine. Though there came an issue with it, as I could not properl use the stem function to discretely "sample" the data.

This is what I got for my initial output:

https://gyazo.com/f15f83d7f9824ae8cfed10fac1629e93

As you can see, while the wave did end up fine in the end, discrete data was all over the place as you can see with the blue lines not fitting at all into the wave.

This is what I was supposed to get for my output:

https://gyazo.com/52423e5ae12e9b62fcbd88103c4465c4

Over in this image, you can see that the lines fit; the data is being correctly sampled discretely.

This is the code I used to get my intial output

%Time Base
t = 0:0.001:1.8;

%Nyquist Frequencies
Fn1 = 1;
Fn2 = 6;

%Nyquist Rates 
Fnr1 = 2*(Fn1);
Fnr2 = 2*(Fn2);

%Sampling Period
Sp1 = 5*(Fnr1);
Sp2 = 5*(Fnr2);

Ts1 = 1/(Sp1);
Ts2 = 1/(Sp2);
T1 = 1/(Fn1);
T2 = 1/(Fn2);

%Number of Samples
N1 = (T1/Ts1); 
n1 = 0:1:N1; 
N2 = (T2/Ts2); 
n2 = 0:1:N2; 

nTs1 = n1 * Ts1; 
nTs2 = n2 * Ts2; 
x_c = sin(2*pi*Fn1*nTs1);
x_c1 = sin(2*pi*Fn1*t);

x_c2 = sin(2*pi*Fn2*nTs2);
x_2 = sin(2*pi*Fn2*t);

signal = x_c1 + x_2;
ct = nTs1 + nTs2;
nqsignal = x_c + x_c2;

%Second Part
h = stem(ct, nqsignal, 'linewidth', 2);
hold
plot(t, signal, 'linewidth', 2)
lgd = legend('Discrete Data', 'Continuous Data');
set (lgd, "fontsize", 12)
set(gca,'XTick',[0:0.2:1.8]);
set(gca,'YTick',[-2:0.5:2]);
title('Time vs Magnitude','fontweight','bold','fontsize',16);
xlabel('Time(s)','fontweight','bold','fontsize',14)
ylabel('Magnitude','fontweight','bold','fontsize',14)
grid

[Mathieu NOE]

hello

if I understand well from the attached picture, you just need to plot the original continous sine wave and it's sampled version

your first explanation is a bit confusing, I was thinking you have 2 different signals and you need to make a combination of the two... it's not what I guess from the pictures.

[Joel Okanta]

To break it down what i'm trying to do is to get the same result as the expected output just that when I did mine, I got some messy graph with the sampling lines not fitting inside the sine waves. My question is, how do i fix my code to get a better output?

Answer 1

[Mathieu NOE]

hi

to be honest with you, I rewrote the entire code, simply because the idea is to have one signal at Fs that will be then decimated to show the sampling .

so here is it - enjoy !

%Time Base
Fs = 1000;
dt = 1/Fs;

t = 0:dt:1.8;
samples = length(t);

%Sine Frequencies
Fn1 = 1;
Fn2 = 6;
signal = sin(2*pi*Fn1*t) + 0.25*sin(2*pi*Fn2*t);

% decimated signal for tem plot
decim = 50; % decimation factor
ind = (1:decim:samples);
t2 = t(ind);
x2 = signal(ind);

%Second Part
h = stem(t2, x2, 'linewidth', 2);
hold
plot(t, signal, 'linewidth', 2)
lgd = legend('Discrete Data', 'Continuous Data');
set (lgd, "fontsize", 12)
set(gca,'XTick',[0:0.2:1.8]);
set(gca,'YTick',[-2:0.5:2]);
title('Time vs Magnitude','fontweight','bold','fontsize',16);
xlabel('Time(s)','fontweight','bold','fontsize',14)
ylabel('Magnitude','fontweight','bold','fontsize',14)
grid

[Joel Okanta]

Hey this is late but I want to ask a question regarding the code you sent me:

Firstly, is it possible to adjust the sampling frequency by 5 times the nyquist rate?

[Mathieu NOE]

hello

by definition, the nyquist frequency is Fs/2 (not another factor) , but you can set your sampling frequency Fs to higher values (not lower) , so that all signals frequencies are below the nyquist frequency.

[Joel Okanta]

Alright so how about this, say you had 4 values for the signal variable, e.g.

signal = sin(a) + sin(b)+ cos(c) and etc.

Would any extra changes be required for the program? Like in the code you gave me (For which I am very thankful for by the way) you did sin(xyz) + 0.25sin(abc)

Furthermore, what would identify as the nyquist rate on the code?

[Mathieu NOE]

hello

yiu can of course expand my code to more sine waves

%Sine Frequencies
Fn1 = 1;
Fn2 = 6;
Fn3 = 12;
Fn4 = 16; %... and so forth 
signal = sin(2*pi*Fn1*t) + 0.25*sin(2*pi*Fn2*t)+ 0.25*sin(2*pi*Fn3*t)+ 0.15*sin(2*pi*Fn4*t);

I used Fs = 1000, (so Nyquist freq = 500 Hz) to give plenty of headroom for higher sine waves frequencies , and also have a smooth "analog" sine wave plot

[Joel Okanta]

I'm kinda confused because in my case I'm doing some work that involves using this, so for me it's this:

signal = sin(2*pi*6*t) + sin(2*pi*7*t)+ cos(2*pi*8*t)- sin(2*pi*9*t);

I was given 4 different frequencies to work with and was told to sample 5 times at the nyquist rate. So i'm not sure if i still stick with your solution or do my own and make its own thing, for example:

6+7+8+9 = 30

Nyquist Rate of that is 60

Sampling Frequency is 300

Sampling Period is 1/300 which is 0.0033333

How would I include such in your code? That's what i'm confused about since you already made a base frequency of 500Hz. Is the 5x sampling necessary or just the lowest sampling you can do(so it's something like the lowest boundary you can go).

Furthermore, what i'm also confused about is why you did the 0.25 and 0.15 multiplier

[Mathieu NOE]

hello Joel

the sampling theory (Shannon theorem) states that the highest frequency of you signal must be below the Nyquist frequency but there are no fixed ratio between the two .

There is no need to add the signal frequencies to define the sampling frequency (otherwise a very complex signal like music or speech should be sampled at very high Fs !)

So you are free to choose F Nyq > 9 hz . Let's say 10 so Fs = 20 Hz min.

the multipliers (individual frequencies amplitudes) are completely free - and I don't think that it makes any difference for the purpose of this exercise.

[Joel Okanta]

I would like to know if I could gather the same results as you were getting using my previous code. I'm going to ask a new question about this and I want to know where i went wrong and how I can improve it

https://uk.mathworks.com/matlabcentral/answers/724413-nyquist-shannon-theorem-matlab-code-suffering-from-minor-issues-don-t-know-where-i-went-wrong

Answer 2

[Joel Okanta]

I would like to know if I could gather the same results as you were getting using my previous code. I'm going to ask a new question about this and I want to know where i went wrong and how I can improve it