iVenky
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I have come up with something. I don't know if it is true or not. I just want your opinion. Nyquist said that the sampling frequency should be at least twice than that of the maximum frequency. But that is not always true if we make use of the phase of the signal. Consider the following matlab program.
For eg: take Sampling frequency fs=8000;
So t=0:1/8000:1;
Consider a signal:
y=sin(2*pi*3000*t);
Consider another signal:
y2=sin(2*pi*5000*t);
Now if we find the fft
k=fft;
k2=fft(y2);
a1=abs(k);
a2=abs(k2);
b1=angle(k);
b2=angle(k2);
figure,plot(b1),figure,plot(b2)
The magnitude spectrum of the two waves y and y2 will be same but not the angle spectrum. So take the signal together
z=y+y2;
z2=y;
k=fft(z);
k2=fft(z2);
If you find the magnitude spectrum for both they will be same which states that there is only one frequency component in both of the signals z and z2. So we say there is aliasing. But we can avoid this by taking the angle / phase spectrum into consideration which is
ang1=angle(k);
ang2=angle(k2);
If you plot both they will be different. Using this phase spectrum we can indeed differentiate two signals and say that the signal z consists of 3000Hz and 5000hz waves though the sampling frequency is less than twice that of the maximum frequency of the signal which is 5000Hz. So I believe Nyquist rate is not always true. But anyway the sampling frequency should be atleast be equal to the maximum frequency of the wave in the above case or else we will have aliasing for sure. What do you say?
For eg: take Sampling frequency fs=8000;
So t=0:1/8000:1;
Consider a signal:
y=sin(2*pi*3000*t);
Consider another signal:
y2=sin(2*pi*5000*t);
Now if we find the fft
k=fft;
k2=fft(y2);
a1=abs(k);
a2=abs(k2);
b1=angle(k);
b2=angle(k2);
figure,plot(b1),figure,plot(b2)
The magnitude spectrum of the two waves y and y2 will be same but not the angle spectrum. So take the signal together
z=y+y2;
z2=y;
k=fft(z);
k2=fft(z2);
If you find the magnitude spectrum for both they will be same which states that there is only one frequency component in both of the signals z and z2. So we say there is aliasing. But we can avoid this by taking the angle / phase spectrum into consideration which is
ang1=angle(k);
ang2=angle(k2);
If you plot both they will be different. Using this phase spectrum we can indeed differentiate two signals and say that the signal z consists of 3000Hz and 5000hz waves though the sampling frequency is less than twice that of the maximum frequency of the signal which is 5000Hz. So I believe Nyquist rate is not always true. But anyway the sampling frequency should be atleast be equal to the maximum frequency of the wave in the above case or else we will have aliasing for sure. What do you say?