Notes › ENGR 2341: Signals and Systems Lecture 14

Aliasing and Reconstruction

calendar_today March 12, 2025 article 767 words access_time 5 min replay Modified April 1, 2025

Table of Contents

Aliases

Discrete-time frequencies differing by $2\pi k$ have equal amplitudes and are considered aliases
Frequencies inside the Nyquist interval are called principal aliases
There are an infinite number of aliases outside the Nyquist interval
Sinusoids with different frequencies can produce the identical discrete sequences
- This occurs when $|f_1 - f_2| = k \cdot f_s$, which implies their principle aliases coincide
- This ambiguity can be avoided by sampling at the Nyquist rate of the sinusoid with the greatest $f_\text{max}$

Sampling in the Time Domain

Impulse trains are used to retrieve continuous values at intervals of $T_s$
- This is possible due to the sampling property of impulse

$$ \begin{align*} x[n] &= x(t) \cdot \sum_{n=-\infty}^{\infty} \delta(t - nT_s) \\ &= \sum_{n=-\infty}^{\infty} x(nT_{s})\delta(t - nT_s) \end{align*} $$

Therefore, $x[n] = x(nT_s)$, since the impulse train samples $x(t)$ every $T_s$

Sampling in the Frequency Domain

Multiplying $x(t)$ by an impulse train in the time domain corresponds to convolution in the frequency domain
- The spectrum of a sampled signal, $X_s(j\omega)$, is obtained by convolving $X(j\omega)$ with an impulse train
- The convolution causes periodic replications of $X(j\omega)$, spaced apart by $\omega_s$
- The amplitude of each spectra is also scaled down by $\frac{1}{T_s}$

$$X_{s}(j\omega) = \frac{1}{T_{s}} \sum_{k=-\infty}^{\infty} X(j(\omega - k \omega_{s}))$$

If the sampling frequency is less than the Nyquist rate, the copies of the spectra will overlap

Reconstruction

The process of creating analog signals from samples using a reconstruction filter that extracts only the principal aliases (only one copy of the repeated spectrum)

Time Domain

In the time domain, this corresponds to replacing each sample with a weighted sinc pulse. When those pulses are summed together, it interpolates between the samples to reconstruct the analog signal
- This is because the sinc function has a peak at $t = 0$ and decays as $t$ moves away from zero. When these sinc pulses centered at each sample point are summed together, a smooth, continuous function is obtained
- In more practical applications, you use rectangular pulses instead of sinc

$$ h_r(t) = \frac{\sin\left(\frac{\omega_s}{2} t\right)}{\frac{\omega_s}{2} t} $$

Frequency Domain

In the frequency domain, this corresponds to using a low-pass filter to extract the spectra inside the Nyquist interval
- Passes frequencies in the range $|\omega| \leq \frac{\omega_s}{2}$
- Removes frequencies outside this range to eliminate aliasing

$$ H_r(j\omega) = T_s \left[ u\left(\omega + \frac{\omega_s}{2}\right) - u\left(\omega - \frac{\omega_s}{2}\right) \right] $$

In piecewise form:

$$ H_r(j\omega) = \begin{cases} T_s, & |\omega| \leq \frac{\omega_s}{2} \\ 0, & |\omega| > \frac{\omega_s}{2} \end{cases} $$

Reconstruction Formulas

The general form of reconstruction formula in the time domain is

$$y_r(t) = \sum_{n=-\infty}^{\infty} y[n] p(t - n T_s)$$

where $y[n]$ are the sampled values of the signal and $p(t)$ is a pulse function

Sinc Pulse

The reconstruction formula using the sinc pulse is:

$$ y_r(t) = \sum_{n=-\infty}^{\infty} y[n] \cdot \text{sinc}\left( \frac{t - n T_s}{T_s} \right) $$

This sums over all the samples $y[n]$ and shifts each sinc pulse to the corresponding sample location $n T_s$
- The resulting signal is a continuous-time signal that approximates the original analog signal

Rectangular Pulse

In practical applications, using an ideal sinc pulse is not feasible due to the infinite duration of the sinc function. Instead, a more commonly used approximation is a rectangular pulse, which is computationally simpler
A rectangular pulse is defined as:

$$ p_{\text{rect}}(t) = \begin{cases} 1, & |t| \leq \frac{T_s}{2} \\ 0, & |t| > \frac{T_s}{2} \end{cases} $$

This pulse has a finite duration of $T_s$ and is centered at each sample point. When we use this rectangular pulse to reconstruct the signal, we sum the shifted rectangular pulses weighted by the samples $y[n]$:

$$ y_{r}(t) = \sum_{n=-\infty}^{\infty} y[n] \cdot p_{\text{rect}}\left( \frac{t - n T_s}{T_s} \right) $$

This results in a piecewise constant approximation of the continuous signal, which is often used in digital-to-analog converters

Summary of both Methods

Sinc Pulse: Ideal interpolation with infinite duration. This gives the best reconstruction, as it perfectly interpolates between samples and avoids any distortion
Rectangular Pulse: Practical approximation with finite duration. This results in a piecewise constant reconstruction, which is easier to implement but introduces some distortion due to the abrupt transitions between samples, referred to at the Gibbs phenomenon

Sampling and Reconstruction Process Overview

Sampling maps analog frequencies to discrete frequencies and replicates the discrete spectrum
Signal bandwidth must be less than half of the Nyquist rate to avoid aliasing while sampling
Finite duration signals are not bandlimited, so practical signals must be filtered to remove frequency components greater than $\frac{f_s}{2}$ via bandlimiting
Reconstruction filter extracts the principal aliases to reconstruct the smoothest signal corresponding to the samples, denoted $y(t)$

References

Aliasing and reconstruction (course handout)
Whittaker–Shannon interpolation formula - Wikipedia]

Sources

Aliasing and reconstruction

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