Fractional Frequency and negative Frequencies

Question

So, when we Sampled any signal at the rate lower than Nyquist Frequency we get aliasing, for instance,

If the frequency of Cosine wave is 5Hertz and I had to make 5cycles in a second and if the sampling frequency is 8Hz,

Frequency = 5Hz

t = 0.2 <- 1 complete oscillation takes 0.2seconds

Sampling Frequency = 8Hz then Sampling rate would be, 0.125seconds

which means every sample will be generated on each instance of 0.125s

the 8 samples would then be,

  0.1250    0.2500    0.3750    0.5000    0.6250    0.7500    0.8750    1.0000

signal can't be completely recovered at this sampling rate, and only first few cycles will be generated.

the Fractional frequency will then would be, 5/8 (8 samples per second and 5 cycles per second).

According to the above example the Fractional frequency is produced due to the aliasing of the signal.

but as i studied, the Fractional frequency is the reason which cause the imaginary part of the frequencies to be generated.

here is what I've found, Here the cosine (real, labelled "R") and sine (imaginary, labelled "I") parts:

in figure #4 , both the waves are sine wave, so the Fractional frequency is -3/8 So if there is no Cosine does that mean that this point has no Real part ? Then how it is generated while calculating the DFT ?

Answer 1

On figures 1, 2 and 3, you are plotting 2 curves which you call real and imaginary parts. These can be interpreted as the complex exponential

$$ x[n] = e^{jn\theta_0} $$

with $\theta_0$ being 0, $2\pi/8$ and $2\pi/2$. These are not cosines, but complex exponentials.

Now, on figure 4, you only draw the sine part (what you call Imaginary), and not the cosine part. So this is why you don't find a cosine part!! You forgot to draw it. It has nothing to do with aliasing (which you interpret correctly).