aliasing in frequency domain problem

Hello ,i am trying to show frequency domain transform of 3 sinuos signal.I samples it twice the frequency of the highest frequency in the signal.

Althogh i get 3 harmonics but i dont get them at the 10,30,70 Hz and their amplitude is not 10(As shown in the code bello)

Where did i go wrong?

Thanks.

t=0:0.001:1
f1=10;
f2=30;
f3=70;
y1=10*sin(2*pi*f1*t);
y2=10*sin(2*pi*f2*t);
y3=10*sin(2*pi*f3*t);
y4=y1+y2+y3;
% twice the sampling rate
Fs=2*70;
Ts=1/Fs;
Tn=0:Ts:1;
fft_L=length(Tn);
y4_samples=10*sin(2*pi*f1*Tn)+10*sin(2*pi*f2*Tn)+10*sin(2*pi*f3*Tn);
%stem(Tn_new,y4_samples);
ff=fft(y4,fft_L);
plot(abs(ff));

Answer 1

[Ameer Hamza]

First, you are taking fft of y4 which is sampled at 1000Hz. Check the following code to see how it can be dome properly

f1=10;
f2=30;
f3=70;
% twice the sampling rate
Fs=2.05*70; % sampling frequency is a bit above 2 times to get all the peaks.
Ts=1/Fs;
Tn=0:Ts:1;
fft_L=length(Tn);
y4_samples=10*sin(2*pi*f1*Tn)+10*sin(2*pi*f2*Tn)+10*sin(2*pi*f3*Tn);
%stem(Tn_new,y4_samples);
ff=fft(y4_samples);
ff1 = abs(ff/fft_L);
fft2 = ff1(1:floor(fft_L/2)+1);
f = Fs*(0:fft_L/2)/fft_L;
plot(f, fft2);

See this example for more detail: https://www.mathworks.com/help/matlab/ref/fft.html#buuutyt-7

[fima v]

Hello Ameer ,I am intresting in the property of aliasing, could you please show me by this simple example, by what amount our signal is copy shifted (in the frequency domain) given a certain undersampling?

Thanks.

[Ameer Hamza]

As an example check this

f1=10;
f2=30;
f3=70;
% twice the sampling rate
Fs=1.5*70; % sampling frequency is a bit above 2 times to get all the peaks.
Ts=1/Fs;
Tn=0:Ts:1;
fft_L=length(Tn);
y4_samples=10*sin(2*pi*f1*Tn)+10*sin(2*pi*f2*Tn)+10*sin(2*pi*f3*Tn);
%stem(Tn_new,y4_samples);
ff=fft(y4_samples);
ff1 = abs(ff/fft_L);
fft2 = ff1(1:floor(fft_L/2)+1);
fft2(2:end) = 2*fft2(2:end);
f = Fs*(0:fft_L/2)/fft_L;
plot(f, fft2);

The sampling frequency is 135 less than the Nyquist frequency for 70 Hz signal. So you will see that it is shifted back by the amount of this new sampling frequency (105-70=35) Hz.

[fima v]

Hello Ameer,in aliasing i was looking for a copy of our signal in a different location

our signal+our signal shifted, this just a shift.

coud you please show a shifted copy?as shown in the figure bellow.

Thanks.

Thanks.

[Ameer Hamza]

Sorry, I don't have much knowledge about this stuff. Since this question is specifically related to signal to process, I think you will have a better chance of getting an answer on an appropriate forum: https://dsp.stackexchange.com/

[fima v]

Hello Ameer, could please give me a manual to read about the theory of this?

I am struggling to understand how to contol the amplitude of the harmonics in our transform.

Thanks.