Understanding Aliasing: Mathematical Analysis and Reconstruction Conditions

1. The Phenomenon of Aliasing: A Frequency-Domain View

Let's revisit the equation for the spectrum of the impulse-sampled signal, $x_s(t)$:
$X_s(\omega )=\frac{1}{T_s}{\sum }_{k=-\infty }^{\infty }X(\omega -k{\omega }_s)$
In our previous lesson, we considered the case where the sampling frequency ${\omega }_s$ was large enough to keep the spectral replicas of $X(\omega )$ separate. Now, let's examine the case where ${\omega }_s<2{\omega }_m$.

The following video provides an excellent visualization of what happens when the Nyquist criterion is violated.

As the video illustrates, when ${\omega }_s<2{\omega }_m$, the spectral replicas overlap. This overlap is the graphical representation of aliasing. When we attempt to recover the original signal by applying an ideal low-pass filter (with a cutoff at ${\omega }_s/2$), we no longer isolate a clean copy of the original spectrum. Instead, we capture a corrupted version that includes high-frequency content from adjacent replicas, which has been folded into the baseband.

A high-frequency component at ${\omega }_{high}>{\omega }_s/2$ in the original signal can become indistinguishable from a lower frequency, its alias, at ${\omega }_{alias}={\omega }_s-{\omega }_{high}$ after sampling.

2. The Mathematical Description of Aliasing

Let's formalize this with the mathematics you prefer. How can two distinct continuous-time signals produce identical sample sequences? Consider two sinusoids:
$x_1(t)=\cos (2\pi f_{1}t)$
$x_2(t)=\cos (2\pi (f_1+k{f}_s)t)\text{for any integer }k$
where $f_s$ is the sampling frequency.

Let's sample both signals at a rate of $f_s$, meaning at time instants $t=n{T}_s=n/f_s$. The samples for $x_1(t)$ are:
$x_1[n]=x_1(n/f_s)=\cos (2\pi f_1\frac{n}{f_s})$
The samples for $x_2(t)$ are:
$x_2[n]=x_2(n/f_s)=\cos (2\pi (f_1+k{f}_s)\frac{n}{f_s})=\cos (\frac{2\pi f_{1}n}{f_s}+2\pi kn)$
Since $kn$ is always an integer, $2\pi kn$ is an integer multiple of $2\pi$. Due to the periodicity of the cosine function, $\cos (\theta +2\pi m)=\cos (\theta )$ for any integer $m$. Therefore:
$x_2[n]=\cos (\frac{2\pi f_{1}n}{f_s})=x_1[n]$
The two signals, $x_1(t)$ and $x_2(t)$, which are clearly different in continuous time, are indistinguishable from their samples. The higher frequency $f_1+k{f}_s$ is an alias of the lower frequency $f_1$.

This is why, as we saw in the previous module, the Discrete-Time Fourier Transform (DTFT) spectrum is periodic with period $2\pi$ (in normalized frequency) or $f_s$ (in Hz). All frequency information is "wrapped" into the fundamental interval $[-f_s/2,f_s/2]$.

3. Conditions for Alias-Free Reconstruction

We've established that aliasing occurs when spectral replicas overlap. The condition to prevent this is that the replicas must be separate. This leads us back to the Nyquist criterion, but now stated as a strict condition to avoid ambiguity.

The Strict Inequality and the Critical Frequency

For a signal $x(t)$ that is strictly band-limited, meaning its Fourier transform $X(f)$ is zero for all $|f|\ge B$, the condition for perfect, unambiguous reconstruction is:
$f_s>2B$
The rate $2B$ is called the Nyquist rate of the signal. The frequency $f_s/2$ is the Nyquist frequency of the sampling system. To avoid aliasing, the sampling rate must be strictly greater than the signal's Nyquist rate.

Why the strict inequality? What happens at the boundary case, $f_s=2B$?

As the reading shows, a sinusoid of frequency $B$ sampled at $f_s=2B$ can produce different sample sets depending on its phase. For example, $\cos (2\pi Bt)$ sampled at $t=n/2B$ gives $\cos (\pi n)=(-1{)}^n$. However, $\sin (2\pi Bt)$ sampled at the same instants gives $\sin (\pi n)=0$. The sine wave is completely invisible to the sampler! This ambiguity at the boundary is why the strict inequality $f_s>2B$ is required for guaranteed reconstruction of all band-limited signals.

Conclusion

In this lesson, we dissected the phenomenon of aliasing. We've seen it from multiple perspectives: as an overlap of spectra in the frequency domain, as a time-domain ambiguity where different continuous signals yield identical samples, and as an audible distortion in audio signals.

Key Takeaways:

• Aliasing is the distortion that occurs when a signal is sampled at a rate $f_s<2B$, where $B$ is the signal's highest frequency.
• Mathematically, aliasing is the consequence of the periodic nature of the sampled signal's spectrum. Frequencies outside the range $[-f_s/2,f_s/2]$ are "folded" or "wrapped" into this range.
• A high frequency $f$ becomes an alias of a lower frequency, typically $|f-k{f}_s|$ for some integer $k$.
• The strict condition for alias-free reconstruction is that the sampling rate must exceed the Nyquist rate of the signal: $f_s>2B$.
• Sampling at exactly the Nyquist rate ($f_s=2B$) can lead to ambiguity, as certain signal components (like a sine wave at frequency $B$) can become invisible to the sampler.

Preview of the Next Lesson:
We have now established a firm rule: to avoid aliasing, first ensure your signal is band-limited, then sample at more than twice its highest frequency. But what about real-world signals, which are almost never perfectly band-limited? If we sample a real-world audio or radio signal, there will always be some energy, however small, at very high frequencies. Does this mean aliasing is inevitable? The answer lies in practical filtering. In our next lesson, we will discuss the practical necessity and design considerations for anti-aliasing and reconstruction filters, the essential components that make digital signal processing possible in the real world.