# Fractional Frequency and negative Frequencies

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 ?

• I would rephrase your first sentence "when we sample a signal that contains frequencies higher than the Nyquist frequency". Nyquist frequency is determined by the sample rate, not by the signal. – endolith Oct 1 '12 at 17:54

$$x[n] = e^{jn\theta_0}$$
with $\theta_0$ being 0, $2\pi/8$ and $2\pi/2$. These are not cosines, but complex exponentials.