In signal processing, sampling, reconstruction, and aliasing are foundational concepts that bridge the gap between continuous-time signals and discrete-time signals. A signal’s time and frequency domain representations provide complementary views of the same information. While the time domain represents how a signal evolves over time, the frequency domain represents the signal in terms of its frequency components, allowing for analysis of periodicities and signal structure. In this chapter, we will explore the relationship between the time and frequency domains with respect to sampling, reconstruction, and aliasing. We will also discuss how these concepts impact the accuracy of signal representation and processing in both domains.

A continuous-time signal x(t) is represented as a sequence of discrete-time samples x[n] when sampled at a uniform rate fs (samples per second). The discrete-time signal is given by:

x[n]=x(nT)

Where:
● T is the sampling period, T=1/fs, and fs is the sampling frequency.
● x[n] is the value of the continuous signal at time t=nT, where n is an integer.

Sampling is the process of converting a continuous-time signal into a discrete-time signal by measuring the signal at specific intervals. Mathematically, this is done by multiplying the continuous signal by a periodic impulse train (a series of Dirac delta functions spaced by T):

xs(t)=x(t)⋅∑n=−∞∞δ(t−nT)

Where:
● δ(t−nT) is the Dirac delta function, indicating the sampling instances at t=nT. This produces a discrete-time signal x[n], where each sample corresponds to the value of the signal at specific times.

The frequency domain provides a representation of a signal in terms of its frequency components, showing how much of the signal is composed of different frequencies. Fourier analysis transforms a signal from the time domain into the frequency domain, enabling us to analyze the signal’s periodic components.

Aliasing occurs when the sampling rate is too low to capture the signal's high-frequency components. In the frequency domain, aliasing is manifested as the overlapping of frequency components due to insufficient sampling. If a signal contains frequency components higher than half the sampling rate (the Nyquist frequency), these components will overlap with lower frequencies when sampled. This overlap leads to aliasing, where the high-frequency components are indistinguishable from lower-frequency components, causing distortion in the reconstructed signal.

Reconstructing a continuous-time signal from its discrete-time samples involves using an interpolation process. Ideal reconstruction is achieved by using the sinc function interpolation. The reconstructed signal is given by:

xr(t)=∑n=−∞∞x[n]⋅sinc(t−nT)

Where:
● sinc(x)=sin (πx)/πx is the interpolation function. The sinc function ensures that the reconstructed signal passes through each sample point and smoothly interpolates between them.

Choosing a Sampling Rate: The sampling rate should be high enough to capture the frequency content of the signal without aliasing. In practice, the Nyquist rate is used as a guideline, but often a higher rate is chosen to ensure that the system can handle noise and other imperfections.

Anti-Aliasing Filter: An anti-aliasing filter (typically a low-pass filter) is applied before sampling to remove high-frequency components that might cause aliasing. This ensures that the signal being sampled contains only frequencies within the Nyquist range.

Signal Reconstruction: In practical systems, ideal reconstruction is not possible, and a low-pass filter is used to approximate the ideal sinc interpolation. The quality of reconstruction depends on the filter characteristics and the sampling rate.