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So I just read something that two signals are aliasing to each other. I know what a single signal is being aliased - that is when sampling rate is not higher than or equal to the Nyquist rate. But I am just confused of what two signals aliasing to each other mean.

Let's say that each signal only carries single frequency. Let $f_s$ be sampling frequency and $f_1$ be the first signal and $f_2$. $f_s$ is assumed to be lower than Nyquist rate for both signals. In such case, in what condition of $f_1$ and $f_2$ would aliasing occur, given $f_s$?

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  • $\begingroup$ It's a particular case of two inputs being mapped to the same output. $\endgroup$ – Emre May 12 '14 at 18:26
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Aliasing would occur if $f_2 = f_1 + kf_s$, where $k$ is an integer. Consider the following example:

$$ x_1(t) = \cos(j2\pi f_1 t + \phi_1) \\ x_2(t) = \cos(j2\pi f_2 t + \phi_2) $$

An ideal sampler samples the above at a sample rate of $f_s$ to yield discrete-time signals $x_1[n]$ and $x_2[n]$:

$$ x_1[n] = \cos\left(2\pi f_1 \frac{n}{f_s} + \phi_1\right) \\ x_2[n] = \cos\left(2\pi f_2 \frac{n}{f_s} + \phi_2\right) $$

We got the above by noting that the sampler takes samples at times $\left[0, \frac{1}{f_s}, \frac{2}{f_s}, \ldots\right]$. To see the aliasing effect, substitute back in the relationship between $f_1$ and $f_2$ that we started with:.

$$ x_2[n] = \cos\left(2\pi (f_1 + kf_s) \frac{n}{f_s}+ \phi_2\right) \\ x_2[n] = \cos\left(2\pi f_1 \frac{n}{f_s} + 2\pi kf_s \frac{n}{f_s}+ \phi_2\right) \\ x_2[n] = \cos\left(2\pi f_1 \frac{n}{f_s} + 2\pi kn + \phi_2\right) $$

Now we recall a property of sinusoids:

$$ \cos(x + 2\pi m) = \cos(x)\ \forall m \in \mathbb{Z} $$

That is, if we add any integer multiple of $2\pi$ in the argument of a sinusoid, we get the same value out of the function (stated differently, sinusoids are $2\pi$ periodic). That means that we can drop the $2\pi kn$ out of the above expression for $x_2[n]$:

$$ x_2[n] = \cos\left(2\pi f_1 \frac{n}{f_s} + \phi_2\right) $$

Now compare the two discrete-time signals again:

$$ x_1[n] = \cos\left(2\pi f_1 \frac{n}{f_s} + \phi_1\right) \\ x_2[n] = \cos\left(2\pi f_1 \frac{n}{f_s} + \phi_2\right) $$

When we look at the sampled versions of $x_1(t)$ and $x_2(t)$, they appear to be sinusoids at the exact same frequency. In the discrete-time domain, since one or both of them were not sampled at at least the Nyquist rate, and the two frequencies have the relationship that was stated above, we are unable to discriminate between the two frequency components. This is what is called aliasing.

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