Aliasing

Given a sampling rate $\Delta t$, we want to show that a harmonic function (here, a cosine) with a frequency higher than the the Nyquist frequency $\omega_\text{Ny} = \frac{\pi}{\Delta t}$ cannot be distinguished by a lower frequency harmonic, sampled with the same time step.

Definitions

First, we import a Matlab-like set of commands,



In [1]:

    
%pylab inline









    



Populating the interactive namespace from numpy and matplotlib

To be concrete, we'll use $\Delta t = 0.4$ s and a fundamental period $T_n=20$ s, hence a number of samples per period $N=50$, or $2.5$ samples per second.



In [2]:

    
Tp   = 20.0
N    = 50
step = Tp/N

To the values above, we associate the fundamental frequency of the DFT and the corresponding Nyquist frequency.



In [3]:

    
dw = 2*pi/Tp
wny  = dw*N/2
print("omega_1 =", dw)
print("Nyquist freq. =",wny,"rad/s =", wny/dw, '* omega_1')









    



omega_1 = 0.3141592653589793
Nyquist freq. = 7.853981633974483 rad/s = 25.0 * omega_1

For comparison, we want to plot our functions also with a high sampling rate, so that we create the illusion of plotting a continuous function, so we say



In [4]:

    
M = 1000

The function linspace generates a vector with a start and a stop value, with that many points in it (remember that the number of intervals is the number of points minus one),



In [5]:

    
t_n=linspace(0.0,Tp,N+1)
t_m=linspace(0.0,Tp,M+1)

The Nyquist circular frequency is $25\Delta\omega$.

The functions that we want to sample and plot are

$$\cos(h\Delta\omega t)\quad\text{and}\quad\cos((h-N)\Delta\omega t),$$

in this example it is $h=47$ but it works with different values of $h$ as well...

In the following, hs and ls mean high and low sampling frequency, while hf and lf mean high and low cosine frequency. Note that t_m and t_n are vectors, and also c_hs_hf etc are vectors too.



In [6]:

    
hf = 47
lf = hf - N

c_hs_hf = cos(hf*dw*t_m)
c_hs_lf = cos(lf*dw*t_m)

c_ls_hf = cos(hf*dw*t_n)
c_ls_lf = cos(lf*dw*t_n)

First, we plot the harmonics with a high frequency sampling (visually continuous, that is).



In [7]:

    
figsize(12,2.4)
figure(1);plot(t_m,c_hs_hf,'-r')
ylim((-1.05,+1.05))
grid()
title(r'$\cos(%+3d\omega_1t)$, continuous in red, 50 samples in blue'%(hf,))
figure(2);plot(t_m,c_hs_lf,'-r')
ylim((-1.05,+1.05))
grid()
title(r'$\cos(%+3d\omega_1 t)$, continuous in red, 50 samples in blue'%(lf,))









    Out[7]:





Text(0.5,1,'$\\cos( -3\\omega_1 t)$, continuous in red, 50 samples in blue')

Not surprisingly, the two plots are really different.

In the next plots, we are going to plot the continuous functions in red, and to place a blue dot in every (t,f) point that was chosen for a low sampling rate.



In [8]:

    
figure(1) ; plot(t_m,c_hs_hf,'-r',t_n,c_ls_hf,'ob')
ylim((-1.05,+1.05));grid();
figure(2) ; plot(t_m,c_hs_lf,'-r',t_n,c_ls_lf,'ob')
ylim((-1.05,+1.05));grid();

If you look at the patterns of the dots they seem, at least, very similar. What happens is aliasing!

It's time to plot only the functions samplead at a low rate:

the high frequency cosine, sampled at 2.5 samples per second, blue line,
the low frequency cosine, sampled at 2.5 samples per second, red crosses only.



In [9]:

    
figure(3) ; grid()
title('The two cosines, sampled at 2.5 points per second')
figure(3)
plot(t_n,c_ls_hf,'-b', linewidth=.33)
plot(t_n,c_ls_lf,'xr', markersize=8)
xticks((2,4,6,8,10,12,14,16,18,20))
ylim((-1.05,+1.05));